Statistical Physics
763620S
Jani Tuorila
Department of Physics
University of Oulu
December 6, 2012
• P. Pietiläinen, Statistical Physics. Previous lecture
notes of the course that lay the foundation for these
notes.
General
The website of the course can be found at:
https://wiki.oulu.fi/display/763620S/Etusivu
It includes the links to this material, exercises, and to their
solutions. Also, the possible changes on the schedule below
can be found on the web page.
• J. Arponen, Statistical Physics. Old notes were based
on this.
Contents
Schedule and Practicalities
The course will cover the following topics:
Lectures and exercise sessions will follow the schedule
below
• Background
• Lectures:
Monday 12-14 room TE320
Tuesday 10-12 room FY254-2
• Exercises:
Friday
– Probability
– Random walk
12-14 room FY254-2
– Central limit theorem
• Statistical approach to many-body problem
By solving the exercises you can earn extra points for the
final exam. The more problems you solve, the higher is
the raise. Maximum raise is one grade point. It is worthwhile to remember that the biggest gain in trying to solve
the exercise is, however, in the deeper and more efficient
learning of the course!
– Statistical formulation
– Equal a priori probabilities
– H-theorem
• Statistical Thermodynamics
Literature
– Phase space and Liouville’s theorem
This course is not based on a single book, but the main
parts of these lecture notes can be found in any of the
following books
– Ergodicity theorem
– Ensemble theory
– Thermodynamic limit
• R. Pathria and P. Beale, Statistical Mechanics, 3rd ed..
This is used widely as a textbook. I have been mainly
reading this.
• Quantum Statistics
– Density matrix
• F. Reif, Fundamentals of Statistical and Thermal
Physics. A classic but missing some modern examples.
– Bose and Fermi statistics
• Ideal Quantum Gases
– Bose-Einstein condensation
• J. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity. A book with modern flavor, available free online (http://pages.physics.
cornell.edu/~sethna/StatMech/).
– Magnetism
• Interacting Systems
– Cluster expansion
I strongly encourage you to read at least one of the books
above. It will give you more deeper understanding on the
subject than mere lectures. In addition, you can read this
course material.
– Second quantization
• Phase Transitions
– Landau’s mean field theory
Other literature:
– Critical exponents
• R. Fitzpatrick, Thermodynamics & Statistical Mechanics: An Intermediate Level Course. Lecture
notes (http://farside.ph.utexas.edu/teaching/
sm1/sm1.html), elegantly written but not advanced
enough.
– Scaling
• Non-equilibrium Statistics
– Langevin equation
– Fluctuation-Dissipation theorem
• L. Reichl, A Modern Course in Statistical Physics.
• D. Chandler, Introduction to Modern Statistical Mechanics.
1
1. Introduction
scope of the theory and leaves many interesting properties
All matter we see around us consists of vast number of of nature unexplained.
microscopic particles that are in constant motion, and in
The idea that the macroscopic properties of matter are
interaction with one another. At any given instant of time, caused by the microscopic constituents was first introduced
it is possible in principle to find the state of this kind of in the second half of the 19th century by James Clerk
many-body system. The laws of physics are such that the Maxwell, Ludwig Boltzmann, Max Planck, Rudolf Claufuture evolution of the state can then be solved from the sius, and J. Willard Gibbs. The theory was completed in
governing equations of motion. However in practise, the the first half of 20th century by the advent of quantum
number of the constituent particles is typically of the order mechanics that gives the correct description of atomic size
of the Avogadro’s number
particles. By applying the probabilistic ideas to systems
in equilibrium, one can obtain all results of the thermodyNA = 6 · 1023 ,
namics together with the ability to relate the macroscopic
parameters of the system to its microscopic constituents.
leading to stupendously large set of coupled equations.
This powerful method will be subject in the first half of
Even the storing of the initial state of these equations
this course. The description of systems not in equilibrium
would exceed in multitude the combined memory of all
is much more difficult task. One can still make some genof the computers in the world. This is the so-called manyeral statements of such systems which leads to statistical
body problem of physics.
physics of irreversible processes. This will be discussed
When the number of the constituent particles is im- briefly in the latter part of the course.
mense, it has turned out useful to use the tools of probabilStatistical physics can, in principle, be applied to
ity calculus to describe their emergent behaviour. In this
any state of matter: solids, liquids, gases, matter comcourse, we will not be interested in the details of individposed of several phases/or several components, matter
ual particles, but in the relations between the macroscopic
under extreme conditions of temperature and pressure,
properties that result. In fact, we lack most of the informamatter in equilibrium with radiation, and so on. In
tion required to describe the internal state of the system,
addition, the concepts and methods of statistical physics
and therefore our approach has to be probabilistic in nahave turned out useful in many other fields of science:
ture. This is the starting point of statistical physics.
chemistry, study of dynamical systems, communications,
The equations governing the macroscopic properties (e.g. bioinformatics, complexity, and even stock markets.
energy, pressure, temperature) of a many-body system are
statistical averages of those of the constituent particles.
Therefore, it is implicitly assumed that the averaged quantities experience statistical deviations.
For example, the familiar equation of state of the ideal
gas
P V = nRT
“If a million monkeys typed ten hours a day, it is
extremely unlikely that their output would exactly equal
all the books of the richest libraries of the world; and yet,
relates the average pressure P and average volume V to in comparison, it is even more unlikely that the laws of
the average temperature T . Quite generally, the deviations statistical mechanics would ever be violated, even briefly.”
from the√average in a many-body system with N particles
go as 1/ N . The astounding number of particles, which - Emile Borel, 1913
makes the calculations of the macroscopic properties from
first principles practically impossible, leads to such an accuracy of the statistical results that they can be taken as
exact physical laws.
Motivation
The systematic studies of macroscopic systems started
from the phenomenological studies in the 18th and 19th
centuries. These were done by the likes of James Watt,
William Rankine, Rudolf Clausius, Sadi Carnot, and
William Thomson. The resulted theory of thermodynamics is concerned on heat and its relation to other forms of
energy and work. The strength of thermodynamics is in
its generality, emerging from the fact that the theory does
not depend on any detailed assumptions on the microscopic
properties of the system. This is also the main weakness
of thermodynamics, since only relatively few statements
can be made on such general grounds. This restricts the
2
2. Background
performs successive steps, each of which has a length l and
is taken on a random direction. The successive steps are
statistically independent, so we can denote
2.1 Ensemble and Probability
The laws of Physics are deterministic in a sense that
given the state of a system at one time one can calculate
the state at all later times, by using the time-dependent
Schrödinger equation or classical mechanics. Then, the
outcome of an experiment on the system should be determined completely by this final state. Even though this can
be done in principle, it is practically impossible to determine an initial state of a system with NA particles, let
alone to calculate its time evolution.
p = probability of a right step
q = 1 − p = probability of a left step
One obtains a great insight by considering the number p as
the fraction of right steps in the ensemble of mental copies
of single steps.
The random walk of N individual steps is formed by taking a subset, i.e. a sample, from the ensemble and calculating the sum of its members. After N steps, the particle
Statistical physics circumvents this problem by considlies at
ering a (infinitely) large set of similar experiments, instead
x = ml,
of a single one. This kind of set of mental copies of similar
1
systems is called an ensemble .
where
m = nr − nl
(2)
Let then S denote a general system and consider an experiment on the system with a general outcome X. We
denote with Σ the ensemble of mental copies of S. Now,
the probability of observing the outcome X is given by
P (X) =
lim
Ω(Σ)→∞
Ω(X)
,
Ω(Σ)
and nr (nl ) denotes the number of steps to right (left).
Naturally,
−N ≤ m ≤ N
and
(1)
N = nr + nl .
(3)
where Ω(Σ) is the number of systems in the ensemble and
By taking into account the number of different ways takΩ(X) is the number of systems exhibiting the outcome X.
ing
nr steps out of N , we arrive at the probability
This is the scientific definition of probability.
WN (nr ) =
Example: Throwing a dice
One could in principle determine the initial position and
orientation of a dice. Also, one could figure out exactly how
the dice is thrown out of hand. By using these as initial
conditions, one could use the classical equations of motion
to determine exactly which of the six numbers turns up.
Practically, this is impossible.
N ! nr nl
p q
nr !nl !
(4)
of taking nr steps to the right. Probability function (4) is
referred to as the binomial distribution since it resembles
the terms in the binomial expansion.
By using Eqs. (2) and (3), one can show (exercise) that
the probability of finding a particle at position m after N
steps is
Instead, we usually consider a large group of (nearly)
similarly thrown dices. The result of a single throw is
PN (m) = WN (nr )
not anymore deterministic, since we do not have the ex(5)
N!
act knowledge of the initial state. Nevertheless, we can
p(N +m)/2 q (N −m)/2 .
=
[(N
+
m)/2]![(N
−
m)/2]!
find out the probability of a particular outcome if we can
assume that each outcome is equally likely2 . In such a case,
we obtain
1
Moments of a discrete distribution
P (X) = .
6
Before continuing with the random walk example, let us
This can, of course, be compared with the actual dice
recall a couple of concepts that characterize a given discrete
throws. Indeed, if we notice a deviation from this assumpdistribution (later, we will generalize these for continuous
tion of equally probable outcomes, we immediately suspect
distributions).
that the dice is crooked!
The mean of an arbitrary (normalized) distribution P (u)
is
given by
2.2 Random Walk
Next, we will consider one-dimensional random walk as a
simple example, grasping the relevant concepts in probability calculus needed in this course. Assume, that a particle
µ ≡ hui =
M
X
i=1
1 Ensemble is the French word for group. We will discuss in a later
chapter the reasons why this kind of treatment can be done.
2 This is in fact a postulate of so-called a priori probabilities that
lies in the foundation of statistical physics, and will be discussed more
thoroughly in the following chapter!
ui P (ui ) =
lim
Ω(Σ)→∞
M
X
ui Ω(ui )
i=1
Ω(Σ)
,
(6)
where ui are the M possible values the variable u can have.
The latter equality allows the interpretation of the mean
as the ensemble average. The mean has the familiar interpretation due to the law of large numbers:
3
If one repeats endlessly the same random exMean values of the random walk
periment, the average of the results approaches
First, let us calculate the mean and variance of the whole
the mean of the ensemble formed by mental ensemble
copies of the single experiment.
2
2
2
This law is, of course, a straightforward consequence of µ = p − q , σ = p(1 − (p − q)) + q(−1 − (p − q)) = 4pq.
our definition of probability.
The random walk with length N is formed by taking a
Let us consider a series of random steps (l = 1) and, sample from the ensemble. It can be characterized either
after each step, calculate the average displacement of a with distribution (4) or (5). According to Eq. (4), we can
show that the binomial distribution is normalized, i.e.
single random walk
x̄ = l
nr − nl
→ l(p − q) = hxi.
N
N
X
WN (nr ) = 1.
nr =0
net displacement
1
Thus, we can calculate the mean number of steps to the
right as
hnr i = N p,
where we have used the result
500
1000
steps
nr pnr = p
∂ nr
(p ).
∂p
The mean displacement is then given by
hmi = N (p − q) = N µ,
-1
We have assumed p = 0.5 and l = 1 in the figure. As which is (naturally) zero when p = q. We see that the
one increases the number of steps, the size of the sample average displacement is formed by N independent single
grows and the average of the steps approaches that of the displacements with the mean µ = p − q.
ensemble.
Correspondingly, the variance of the right steps in the
random
walk is
Assume that f (u) and g(u) are any functions of the discrete variable u and c is a constant. Then,
hf (u)i =
M
X
σr2 = h(nr − hnr i)2 i = N pq.
Now, the variance of the net displacement is
f (ui )P (ui )
2
σm
= h(m − hmi)2 i = 4σr2 = N σ 2 ,
i=1
hf (u) + g(u)i = hf (u)i + hg(u)i
where we have used the relations (2) and (3). For the
number of right steps, we obtain the standard deviation
r
σr
q 1
√ .
=
(9)
hnr i
p N
√
We see that the relative width decreases as 1/ N with
increasing sample size N .3
hcf (u)i = chf (u)i.
The mean is called the first moment of the distribution.
In general, the nth moment is defined as
huk i =
M
X
uki P (ui ).
(7)
WN H nr L
i=1
In principle, the whole distribution can be resolved by calculating all of its moments.
æ
æ
A useful mean value is called the variance, or the second
moment about the mean
σ 2 ≡ h(u − hui)2 i =
M
X
æ
0.15
æ
æ
æ
æ
æ
0.10
(ui − hui)2 P (ui ).
æ
(8)
æ
i=1
æ
æ
æ
æ
æ
æ
0.05
æ
ææ
æ
The variance has a great importance since its square root
æ
æ
æ
æ
σ, the standard deviation, can be used to characterize the
æ
æ
æ
ææ
æ
æ
ææææ
æææææææææææ
ææææææææææææ
nr
width of the range over which the random variable is dis10
20
30
40
tributed around its mean. The following relation is often
√
3 This kind of 1/ N -law is common in statistics, and is the reason
useful
2
2
2
why statistical mechanics gives (nearly) exact results.
σ = hu i − hui .
4
Binomial distributions of the number of right steps with where W̃N = WN (hnir ) can be determined from the norN = 20 (red) and N = 40 (blue). The arrows indicate the malization condition5
standard deviations.
r
Z ∞
N
X
|B2 |
WN (nr ) ≈
WN hnir +η dη = 1 ⇒ W̃N =
.
2π
−∞
nr =0
Large N limit
Let us then assume that N is large, meaning that we are
in the limit of a long random walk, or, of a large sample.
Due to simpler calculations, we do the calculation for the
number of right steps nr , but the same reasoning applies
also for the net displacement m. First, we note that since
p
hnr i = N p and σr = N pq 1.
Thus, we have obtained that, in the limit of large N and
nr , the binomial distribution converges to the Gaussian
distribution
2
WN (nr ) = p
WN H nr L
we have that near the mean hnr i and in the limit of large
N
|WN (nr + 1) − WN (nr )| WN (nr ).
(n −hn i)
1
− r 2σ2r
r
e
.
2πσr2
(11)
æ
æ
æ
0.15
This allows us to consider WN as a continuous function of
continuous nr .
æ
æ
æ
æ
0.10
Now, the location of the maximum nr = ñr can be found
approximately from the equation
æ
æ
æ
æ
æ
æ
æ
æ
æ
0.05
d ln WN
= 0.
dnr
æ
ææ
æ
æ
æ
æ
æ
æ
ææææææææææææ
We make a Taylor expansion of ln WN in the vicinity of ñr ,
and obtain4
10
æ
æ
æ
æ
ææææ
20
ææ
æææææææææææ
30
40
Binomial distribution (dots) with p = 0.5, N = 20 and
1
1
ln WN (nr ) = ln WN (ñr ) + B1 η + B2 η 2 + B3 η 3 + · · · , N = 40, together with the corresponding Gaussian distri2
6
(10) butions (solid lines).
where η = nr − ñr and
Bk =
2.3 Central Limit Theorem
dk ln WN .
dnkr
nr =ñr
The
preceding
discussion
is
one
special
case of the so-called central limit theorem 6 :
By using Stirling’s formula (exercise)
If we sum N (N → ∞) systems that are statistically independent and identical, the resultant
distribution is under very general conditions a
Gaussian.
ln n! ≈ n ln n − n,
we obtain
In terms of the random walk, we started with a single
step with a probability distribution
p X = right
P (X) =
q X = left.
B1 = − ln nr + ln(N − nr ) + ln p − ln q.
At the maximum (nr = ñr ), B1 = 0 which implies
ñr = N p = hnir .
Then, we combine a large number N of such random
steps and calculate the net displacement to the right m =
nr − nl , where nr (nl ) is the number of steps to the right
(left). In the limit of N → ∞, we obtain
Similarly,
B2 = −
1
1
=− 2 <0
N pq
σr
as it should for a maximum. The higher order derivatives
Bk ∝ N −(k−1) , and can be neglected in the limit of large
N in the expansion (10).
nr
N
nl
N
m
N
We have thus obtained
1
→
p
→
q
→ p − q = µ.
5 We have seen that the distribution has a pronounced maximum,
so the continuation of the integral from −∞ to ∞ is an excellent
approximation.
6 Central limit theorem is notoriously difficult to prove, which is
therefore omitted here.
2
WN (nr ) = W̃N e− 2 |B2 |η ,
4 We use ln W
N instead of WN since it’s expansion converges more
rapidly.
5
nr
This is the law of large numbers. Additionally for a fixed the probability that u has the value ui and v the value vj .
N , the central limit theorem tells us that the measured We require the normalization
values for individual random walks are spread around the
N X
M
X
mean. This spread is characterized by the relevant stanP (ui , vj ) = 1.
7
dard deviation .
i=1 j=1
We showed that for a random walk, the number of right
steps obeys a Gaussian distribution (11) when N is large.
We can use the relations (2) and (3) to show that also other
resultant random variables are distributed normally. For
example, let us study the average displacement m̄ = m/N .
Now,
m = 2nr − N , hm̄i =
Naturally,
P (ui ) =
M
X
P (ui , vj ) and P (vj ) =
N
X
j=1
σ2
hmi
2
= µ , σm̄
.
=
N
N
P (ui , vj ).
i=1
When the outcome of one variable is independent of the
outcome of the other, we say that the two variables are
statistically independent, or uncorrelated. In this case,
Because the change of variables is linear, we have that
dn (m̄−hm̄i)2
−
1
r
2
2σm̄
e
PN (m̄) = WN (nr )
.
= p
2
dm
2πσm̄
P (ui , vj ) = P (ui )P (vj ).
Let F (u, v), G(u, v), f (u) and g(v) be arbitrary functions. Now,
distribution
8
hF i =
6
N X
M
X
F (ui , vj )P (ui , vj )
i=1 j=1
hF + Gi = hF i + hGi
4
hf i =
2
M
N X
X
f (ui )P (ui , vj ) =
i=1 j=1
- 0.15
- 0.10
- 0.05
0.00
0.05
0.10
0.15
average displacement
N
X
f (ui )P (ui )
i=1
hf gi = hf ihgi,
where the last equality holds only for uncorrelated u and
v. The proofs are left as an exercise.
In the figure, N = 400 and p = 0.5. We have calculated
m̄ for 3000 random walks. The height of the bars denote
the number of random walks with m̄ within the interval
m + δm. The interval δm is the width of the bars. The
height of the bars is normalized so, that the total area
under them is 1. The red curve is the Gaussian distribution
with
1
hm̄i = 0 and σm̄ = √ .
N
Continuous distributions
We have already had one example in the previous sections on the probability distribution of a continuous random variable in the form of the Gaussian. Let us consider
such cases more generally.
Assume that u can obtain any value in the continuous
range a1 < u < a2 . The probability of finding the variable
between u and u + du is
2.4 Several Variables and Continuous Distributions
P (u) = P(u)du,
(12)
Several variables
In statistical physics, the typical problems deal with sys- where P is the probability density. The normalization
tems with 1024 , or so, particles. Therefore, it is evident condition and the definition of the expectation value are
that we have to deal with probability distributions with straightforwardly generalized from the discrete case. Thus,
Z a2
more than one variable. We consider here the case of two
P(u)du = 1
(13)
variables, but the results are straightforwardly generaliza1
able to cases with any number of variables.
Z a2
Assume that we have two random variables u and v with and
possible values ui and vj , where i = 1, . . . , N and j =
hf (u)i =
f (u)P(u)du,
(14)
a1
1, . . . , M . We denote with
where f is an arbitrary function of u. These results are
P (ui , vj )
easily generalized to the cases where one has more than
7 The mean is only the most probable outcome. For example, if
one continuous random variable (exercise).
you toss a coin four times, the mean is two heads and two tails. But,
sometimes you might get all four tails!
6
3. Statistical Formulation
Assume that we have tossed two heads. This is the
”macrostate” of the three coin system. Nevertheless from
In this chapter, we will present the formal description of
the statistical approach to the analysis of the internal mo- the Table, we see that we can obtain two heads in three diftions of a many-body system. This approach holds for sys- ferent ways. If we do not posses any additional knowledge
tems in equilibrium 8 . The non-equilibrium statistics will on the system (e.g. labels on the coins), we can only say
that the system is after the toss in one of the corresponding
be studied in the latter part of the course.
three ”microstates” 2,3 or 4.
Assume then that the number of coins is of the order of
NA . One can easily imagine that there exists, in general,
We want to describe a system that is macroscopic, by an extremely large number of possible states for a given
which we mean that the number N of its constituent par- value of heads.
ticles obeys
We can generalize the above coin flip example to the
1
√ 1.
(15) system consisting of macroscopic number of microscopic
N
particles. The extensive macroscopic parameters, that deCharacterization of the state
Typically, N ≈ NA ≈ 1024 . This means that the statistical arguments discussed in the previous chapter can be
applied.
termine the macroscopic state, give us only partial information about the system. Nevertheless, they impose constraints to the possible states the system is allowed to occupy. The macroscopic state, i.e macrostate, of the system
Macroscopic state can be characterized with the so-called
is a function of a set of f coordinates, where f denotes the
state variables. Extensive variables are proportional to the
number of the degrees of freedom in the system (i.e. the posize of the system, i.e. to the particle number N or to
sition coordinates and the possible spins of the constituent
the volume V . For example, the total energy E of N nonparticles) and is proportional to the number N of the coninteracting particles is
stituent particles. Any given macrostate can be achieved
X
by several different sets of values for the coordinates. Each
E=
ni εi ,
(16)
such set is called a microstate.
i
where ni denotes the number of particles with energy εi .
Naturally,
X
N=
ni .
(17)
Postulate of equal a priori probabilities
From the simple coin flip example, we can see that for
a given macrostate the number of possible microstates can
i
increase very rapidly as the number of random variables
Intensive variables are independent on the size, and can increases. What can we then say about the probability of
be determined from any sufficiently large volume element finding the system in any of its allowed microstates? As
of the system. Examples include temperature T , pressure an example, consider an isolated system. By definition,
p, chemical potential µ, ratios of extensive variables, such the energy is conserved which imposes a constraint on the
possible microstates the system can occupy. Usually, there
as density ρ = N/V .
are however many such states.
The macroscopic state, or macrostate, is determined by
General statements can be made when the system is in
the values of the extensive variables, e.g. N , V and E.
equilibrium. The equilibrium is characterized by the fact
that the probability of finding the system in any of its
Statistical ensemble
accessible microstates is independent on time. There is
nothing in the laws of physics that would make one assume
that some of these microstates would be occupied more
Example: Coin flips
frequently than the others. So, in the case of any other
Assume that we flip three coins. There are altogether prior knowledge, we assume that:
four different possible outcomes: 0,1,2 or 3 heads.
Isolated system in equilibrium is equally likely
State index Outcome Number of heads
to be found in any of its accessible states.
1
hhh
3
In statistical physics, the above assumption is called the
2
hht
2
postulate
of equal a priori probabilities. Therefore, it can3
hth
2
not
be
proven
but one can only derive its consequences and
4
thh
2
compare
them
with measurements. This kind of measure5
htt
1
ments
have
been
done in multitude, and the agreement has
6
tht
1
9
so
far
been
very
good
.
7
tth
1
9
8
ttt
0
We naturally assume this postulate when we play a game of
chance, such as coin toss, dice toss, or cards. In the absence of any
additional information, we always assume that the different outcomes
occur with equal probabilities, as we discussed in the first dice toss
example!
8 We
will define the equilibrium in the course of the discussion. At
the moment, it is sufficient to consider it as the state of the system
which does not show any temporal macroscopic changes.
7
3. τ ≈ texp : Distribution over accessible states is not
uniform and changes in the experimental time scales.
Problem cannot be reduced to a discussion of equilibrium situations.
Approach to equilibrium
The postulate of a priori probabilities indicates that the
equilibrium of the isolated system is characterized by a
uniform distribution of its accessible microstates. The accessible states are those that do not violate the constraints
According to the H-theorem, after the relaxation the
set by the known values of systems macroscopic parameters
quantity H is a static constant, and can be written for
y1 , y2 , . . ., yn (such as E, V and N ). Thus, the number of
an isolated system as
the accessible states can be written in functional form
X
Heq =
Pr ln Pr = − ln Ωf .
Ω = Ω(y1 , y2 , . . . , yn ),
r
meaning that the parameter α lies in the range yα and Thus, we see that
yα + δyα .
If some constraints of an isolated system are
Assume that initially the isolated system is in equilib- removed then the parameters of the system tend
rium, meaning that it can be found equally likely in any to spontaneously readjust themselves in such a
of its Ωi states. A removal of a constraints leads to an way that
increase of accessible states
Ω(y1 , . . . , yn ) → maximum.
Ωf ≥ Ω i .
We call process the transition from the initial to the final
macrostate. When Ωf = Ωi , the process is called reversible.
The initial macrostate can be recovered by reimposing the
constraint since the system is throughout the process in
equilibrium. If Ωf > Ωi the corresponding process is irreversible. The imposition or removal of additional conConsider a system in such a state of non-equilibrium. Let
straint cannot spontaneously recover the initial macrostate
Pr be the probability of the system to be in the microstate
because of the uniform distribution over Ωf states.
r. The so-called H-theorem 10 says that (exercise)
We see that the quantity ln Ω measures the degree of
dH
irreversibility of the macrostate of an isolated system in
≤ 0,
(18)
dt
equilibrium. Especially for any physical process operating
on an isolated system,
where H = hln Pr i and the equality holds when Pr = Ps
for all microstates r and s. This means that the nonln Ωf − ln Ωi ≥ 0.
equilibrium state changes so, that the quantity H decreases until the uniform equilibrium distribution of the For historical reasons, this measure of irreversibility is written in the units joules/kelvin. Thus, it is multiplied with
microstates is reached.
the Boltzmann constant (kB = 1.381 · 10−23 J/K)
The H-theorem guarantees that isolated systems approach the equilibrium, eventually. The time scale that
S = kB ln Ω,
(19)
characterizes the process is called the relaxation time τ .
The relaxation time depends on the particular system, es- and termed the (equilibrium) entropy.
pecially on the inter-particle interactions. The postulate of
equal a priori probabilities holds for the isolated systems
Implications of the statistical definition of enin equilibrium. In order for the postulate to hold, it is tropy
therefore important to wait a number of relaxation times
The H-theorem gives straightforwardly the second law of
after the decoupling of the system from the environment.
thermodynamics 11 :
By comparing with the experimental time scale texp we can
The entropy of an isolated system tends to inseparate three distinctive cases:
crease with time and can never decrease.
1. τ texp : System reaches equilibrium very quickly
As one goes to lower energies, one has to turn into quanand equilibrium statistics are applicable.
tum mechanical treatment12 . This is because in classical
physics there is no lower bound on the energy of the system.
2. τ texp : Equilibrium is reached very slowly. In fact, Accordingly, we can write the third law of thermodynamics:
it is so slow in the experimental time scale that it can
11 In thermodynamics, the second law results after rather cumberbe considered to be in equilibrium (with additional
some considerations of Carnot cycles. This has been extensively studconstraints that prevent the reaching to the equilib- ied in in the Thermodynamics course and will be omitted here.
12 In order to have unique value for the entropy, one actually has
rium). Hence, the statistical arguments apply.
When Ωf > Ωi , this means that immediately after the
removal of the constraint the system is not distributed
equally between its allowed microstates, it is not in equilibrium, and its macroscopic parameters change in time.
10 The
to treat it quantum mechanically. This is because the volume elements in classical phase-space (discussed later) are not bounded by
the uncertainty principle.
theorem was first proposed and proven by L. Boltzmann in
1872.
8
Entropy of a system approaches to a constant 4. Statistical Thermodynamics
value as the temperature approaches zero.
The previous chapter indicates that the statistiThe constant is determined solely by the degeneracy of cal physics of a macrostate follows from the number
the quantum mechanical ground state of the system. In Ω(E, V, N ) of allowed microstates. In general, this number
the case of non-degenerate ground state (one accessible mi- is a function of the macroscopic energy of the system, in
crostate, Ω = 1), the entropy is zero.
addition to the external variables such as the particle number and volume. We leave the detailed computations of Ω
Probability calculations
to the next chapter, and concentrate, in this chapter, in its
Consider an isolated system in equilibrium whose (con- relation with thermodynamic quantities. It is worthwhile
stant) energy is known to be somewhere between E and to remember that thermodynamics is expressed in terms
of macroscopic parameters, not relying on any microscopic
E + δE. We denote with Ω(E) the total number of states
with energy in this range. Let us assume that we measure knowledge on the matter under study.
the property y, and let yk be the possible outcomes of that
measurement and Ω(E; yk ) the number of the state with
such outcome. Based on our discussions in the previous
chapter, we form an ensemble of the Ω(E) states. Then,
our fundamental postulate of a priori probabilities tells us
that each of the Ω(E) states in the ensemble is equally
likely to occur. Thus, the probability of obtaining yk from
the measurement is
P (yk ) =
We have learned that the macroscopic quantities are statistical in nature. This means that in equilibrium their
values are statistical averages of NA , or so, particles. The
largeness of NA means that the distributions of the averages are Gaussians whose deviations are small compared
with the most probable value, e.g.
1
σE
∼√
.
hEi
NA
Ω(E; yk )
.
Ω(E)
Thus, we can replace macroscopic quantities, such as energy, temperature and pressure, by their mean values. This
We assume in the above that δE is sufficiently large, so is the essence of classical thermodynamics.
that we can take Ω(E) → ∞. This is generally true in
macroscopic systems.
4.1 Contact Between Statistics and Thermodynamics
We see that the statistical physics calculations reduce
simply to counting states, subject to relevant constraints
(such as for E in the above discussion).
Thermal interaction
Consider two systems A1 and A2 separately in equilibrium. The number of microstates in system A1 in a
macrostate with energy in range E1 to E1 + δE1 is denoted
with Ω1 (E1 ). Correspondingly, we denote the number of
microstates of the system A2 with Ω2 (E2 ). Assume that
the systems are brought in a thermal contact, meaning that
they can exchange energy, but that their particle numbers
and volumes stay fixed. We thus assume that the energies
E1 and E2 can vary, but the variations are restricted by
the energy conservation condition
E (0) = E1 + E2 = constant.
Each microstate in A1 can be combined with a state in
A2 , so the total number of microstates in the composite
system is
Ω(0) = Ω1 (E1 )Ω2 (E2 ) = Ω1 (E1 )Ω2 (E (0) − E1 ).
Based on the discussion in the previous chapter, after we
have brought the systems together and after waiting several relaxation times, the combined system can be assumed
to have reached the equilibrium. We have shown with the
H-theorem that the equilibrium is such that Ω(0) is maximized.
Thus, (we use partial derivative since we assume purely
thermal interaction in which the other parameters are kept
9
fixed)
∂Ω (E ) 1
1
Ω2 hE2 i
∂E1
E1 =hE1 i
∂Ω2 (E2 ) ∂E2
+ Ω1 hE1 i
= 0.
∂E2
E2 =hE2 i ∂E1
This implies (∂E2 /∂E1 = −1) that the equilibrium condition of the composite system can be written as
!
!
∂ ln Ω2 (E2 ) ∂ ln Ω1 (E1 ) =
or
∂E1
∂E2
E1 =hE1 i
β=
∂ ln Ω(N, V, E)
∂E
σX (E) = −
E2 =hE2 i
β1 = β2 ,
where
We want to calculate how the total number of accessible
states Ω(E, x) changes as we increase x with dx. First, we
define the generalized force conjugate to the parameter x
as
∂Er
.
(22)
X=−
∂x
For a fixed value of X the number of such states that are
located within energy −Xdx below E, and change to a
value greater than E is
!
ΩX (E, x)
Xdx,
δE
where ΩX (E, x) is the number of such microstates that
have their energy Er within the interval above, and the
generalized force within X and X + δX.
.
(20)
E=hEi
The parameter β has the following properties:
1. If two systems in equilibrium with the same value of
β are brought into thermal contact, they will remain
in equilibrium.
Thus, we obtain the total number of states σ(E) that
change from an energy value lower than E to one that is
larger
X
Ω(E, x)
Xdx,
σ(E) =
σX (E) = −
δE
X
where
X=
X
1
ΩX (E, x)X
Ω(E, x)
X
2. If two systems in equilibrium with different values of
β are brought into thermal contact, the system with is the average of the generalized forces over all accessible
higher value of β will absorb heat from the other until states obeying uniform distribution. We have also denoted
the total number of states with energy within E to E + δE
the two β values are the same (exercise).
as
X
Ω(E, x) =
ΩX (E, x).
In addition, we notice that β has the units of inverse energy.
X
We define the thermodynamic or absolute temperature with
!
The total number of states within E and E +δE changes
1
∂S(E, V, N ) = kB β =
,
(21) as
T
∂E
V,N,E=hEi
∂Ω(E, x)
∂σ
dx = σ(E) − σ(E + δE) = −
δE.
where the subscripts contain the assumption of constant
∂x
∂E
volume and particle number. We thus see that the thermodynamic temperature of a macroscopic system depends We obtain
∂
∂Ω
only of the rate of changes of the number of accessible mi=
(ΩX).
∂x
∂E
crostates with the total energy.
This implies that
According to the previous discussion, it is easy to arrive
at the zeroth law of thermodynamics:
∂ ln Ω
∂ ln Ω
∂X
=
X+
.
∂x
∂E
∂E
If two systems are separately in thermal equilibrium with a third system then they must also Since X is intensive but Ω and E are extensive variables,
be in thermal equilibrium with one another.
one sees that the second term is negligible compared with
the first in the macroscopic limit.
General interaction
In the most general case, the external parameters do not
remain fixed and the system is not thermally insulated.
Additionally, the (essentially quantum mechanical) energies Er of the accessible microstates are dependent on the
external parameters Er = Er (x1 , . . . , xn ), where xi are the
relevant external macroscopic parameters. For simplicity,
we consider here the case where only one external parameter x is allowed to vary. In addition to the constraint due
to x, we assume that the energy of the system is found
from the interval E to E + δE. When we change x by dx
the energy Er of a microstate r changes by (∂Er /∂x)dx.
10
Thus, we have obtained that a change in an external
parameter results in a mean generalized force given by the
relation
∂S
1
= X.
(23)
∂x
T
In general, if we calculate
entropy S = S(E, x1 , . . . , xn ),
!
n
X
∂S
dS =
dE +
∂E
i=1
x1 ,...,xn
the infinitesimal change in
we obtain
!
∂S
dxi ,
∂xi
E,x1 ,...,xi−1 ,xi+1 ,...,xn
(24)
where the subscripts denote the variables that are kept is an exact differential. For example, the integrating factor
constant during partial derivation.
of heat is 1/T 13
dQ
In order to apply the results (20) and (23), one should
dS =
.
T
take care that the process leading to the change in S is
reversible. This guarantees that the average values have
throughout the process well defined values. The reversibility is achieved by requiring that the changes in E and the
external parameters xi are carried out so slowly that the
system remains arbitrarily close to equilibrium at all stages
of the process. This is called a quasi-static process. The
slowness is, naturally, a characteristic of the system and
determined by the length of the relaxation time.
For a quasi-static process, one can write
dS =
n
1X
1
dE +
Xi dxi .
T
T i=1
(25)
We see that the latter term is a sum of components of
the form generalized force times the a change in the corresponding coordinate. Thus, it can be related to the decrease in the internal energy dE during the process. Accordingly, we call it the work done by the system and denote
it with
n
X
Xi dxi .
dW =
Now, the total change in the mean energy changes during
the process is
∆E = Ef − Ei = Q − W,
(28)
where W is the macroscopic work done by the system as a
result of the change in external parameters. The quantity
Q is the heat absorbed by the system, i.e. the change in the
macroscopic energy due to the thermal interaction. For a
quasi-static process, this is a restatement of the first law
of thermodynamics, which in its most general form it says:
Energy of an isolated system is conserved.
Equilibrium conditions
Let us consider the case with an exchange of the external
parameters V and N between the subsystems A1 and A2 ,
in addition to energy. Again, we consider a process leading
from initial to final equilibrium. The change in entropy
S1 = S1 (E1 , V1 , N1 ) due to infinitesimal changes in the
arguments can be written as
i=1
∂S1 ∂S1 ∂S1 dE1 +
dV1 +
dN1 .
dS1 =
The change in the internal energy caused by the thermal
∂E1 ∂V1 ∂N1 V
,N
E
,N
E
,V
interaction is called heat
1
1
1
1
1
1
(26) One can write a similar formula for the change in entropy
of the system A2 by replacing 1 → 2 in the above formula.
One should pay attention to the notation. Special sym- Due to the fact that the system is isolated, we have that
bols dQ and dW denote the fact that the heat and work dE2 = −dE1 , dV2 = −dV1 and dN2 = −dN1 . Thus, the
are infinitesimal quantities that do not correspond to any change in entropy of the total system can be written as
!
differences between two heats or two works of the initial
∂S2 ∂S1 −
dE1
dS =dS1 + dS2 =
and final states of the process. Such infinitesimal quanti∂E1 ∂E2 V1 ,N1
V2 ,N2
ties are called inexact differentials. In contrast, the state
!
variables (such as E, p, V , T , . . .) that have well defined
∂S1 ∂S2 +
dV1
(29)
−
values for the initial and final states are referred to as exact
∂V1 ∂V2 E1 ,N1
E2 ,N2
differentials.
!
∂S1 ∂S2 Generally speaking, consider a function F (x, y) of two
+
−
dN1
∂N1 ∂N2 independent variables x and y. The differential
dQ = dE + dW,
E1 ,V1
∂F
∂F
dF =
dx +
dy
∂x
∂y
is exact if
∂2F
∂2F
=
.
∂x∂y
∂y∂x
Thus, dF = dF is integrable and
Z 2
dF = F (2) − F (1),
1
E2 ,V2
In the equilibrium, small deviations in the total entropy
have to vanish, leading to new equilibrium conditions
∂S1 ∂S2 =
(30)
∂V1 ∂V2 E1 ,N1
E2 ,N2
∂S2 ∂S1 =
.
(31)
∂N1 ∂N2 (27)
E1 ,V1
E2 ,V2
By recalling from thermodynamics, the definitions of pressure and chemical potential are respectively
∂E ∂E In the case of inexact differentials, the integrals depend
= −p ,
= µ.
on the taken path, but there always exists an integrating
∂V ∂N S,N
S,V
factor λ(x, y) so that
meaning that the integral of an exact differential does not
depend on the path of integration.
13 Note that dS ≥ dQ/T for an arbitrary process, the equality holds
only for reversible ones.
λdF = df
11
ρ = N/V → 0. For such a system, one can readily write
the dependence of Ω(E, V, N ) on V . We will settle for now
in order-of-magnitude calculation, and improve it once we
start to discuss the calculation of Ω more generally.
Using the mathematical identities, valid for any function
f of variables x and y,
!−1
∂f ∂x
∂f ∂x ∂y = −1 ,
=
∂x ∂y ∂f ∂x ∂f
y
f
x
y
y
one arrives at statistical definition for pressure
p
∂S =
T
∂V E,N
and that for chemical potential
µ
∂S − =
T
∂N .
Let us consider a single particle in the gas. Since the system has a fixed number of particles, the number of its accessible microstates depends only on E and V , as Ω(E, V ).
We have to find the number of states within the interval
E to E + δE and in the volume V . Thus, Ω has to be
(32)
proportional to the total volume V and that of the energy
space. Since we assume that the particles are classical and
that the interactions are negligible, they can occupy the
same space. For the N particle system, we end up with
(33)
Ω(E, V ) ∝ V N χ(E),
(37)
E,V
where χ(E) does not depend on volume.
What we have obtained is the condition of equilibrium
Now, the calculation of the equation of state is straightin the case where all the state variables E, V and N are forward. We have
allowed to vary
S = kB ln Ω = kB N ln V + kB ln χ(E) + constant
T1 = T2 , p1 = p2 , µ1 = µ2 .
(34)
which gives the pressure
Also, we have obtained the first law of thermodynamics
N
for infinitesimal reversible processes
or pV = N kB T.
(38)
p = kB T
V
dE = dQ − dW = T dS − pdV + µdN.
(35) Chemists prefer the form
This is often called the fundamental thermodynamic relapV = nRT,
tion. It should be noted that the differential of the internal
energy consists terms AdB, of which one is always an ex- where n = N/NA is the number of moles of the gas particles
tensive variable and the other intensive. This observation and R = kB NA is the gas constant. Note that equation of
holds generally.
state (38) (nor R) does not depend on the kind of the
molecules constituting the gas.
Note also that for thermally isolated systems
dS =
In order to acquire complete knowledge on the
macrostate of the ideal gas, one should also calculate the
equation of state for the internal energy E from equation
dQ
=0
T
which is the definition of an adiabatic process.
4.2 Equations of State
Relations (20) and (23) are of particular interest
1
∂S ∂S
1
=
,
= X.
T
∂E ∂x
T
1
∂S
∂ ln χ(E)
=
= kB
.
T
∂E
∂E
This means that the mean energy E = E(T ) is function of
only temperature, not volume V . The explicit dependence
requires the knowledge on χ(E). We will return to its
(36) determination later.
If the gas is composed of m different species of molecules,
Namely, they constitute a way to find relations between
the equation of state is still (38) where
temperature, generalized forces (like p and µ) and exterm
m
nal parameters (like N and V ). The relations are called
X
X
N=
Ni and p =
pi .
equations of state and are important since they connect
i=1
i=1
quantities that are readily measurable. We will list here
some examples of well known equations of state.
Here
pi = Ni kB T /V
Ideal gas
is the partial pressure of the ith gas.
Let us consider a gas of N monatomic particles. We
assume that the gas in confined in a container of volume
Virial expansion of real gases
V , and that the interactions and quantum effects between
the particles are negligible. This is called classical ideal gas.
The interactions between the constituent molecules beIdeal gas is a good approximation for dilute gases that have come important in real gases. Nevertheless, the equation
12
Surface tension
of state of any gas can be written in a general series of the
particle density ρ = N/V as
Surface tension is a property of a surface of a liquid
h
i
that
allows it to resist external forces. It has temperature
p = kB T ρ + B2 (T )ρ2 + B3 (T )ρ3 + · · · ,
(39) dependence
!n
T
σ = σ0 1 −
where Bk is the kth virial coefficient.
TC
where the temperature is measured in ◦ C, TC and n are
dependent on the liquid, and σ0 is the tension at T = 0◦ C.
van der Waals equation
The interactions between the molecules of real gases are
Electric polarization
• repulsive at short distances (due to the Pauli principle); every particle needs at least the volume b which
implies V & N b.
When a piece of material is in an external electric field
E we define
D = 0 E + P,
• attractive at large distances due to the induced dipole
where
momenta. The pressure decreases when two particles
are separated by the attraction distance. The probability of this is ∝ ρ2 .
If the material is homogeneous and dielectric it has
!
b
E,
P= a+
T
repulsive
1.5
1.0
attractive
where a and b are almost constant and a, b ≥ 0.
0.5
1.0
1.5
2.0
2.5
- 0.5
r  rm
Curie’s law
When a piece of paramagnetic material is in magnetic
field H we write
-1.0
"
VLJ (r) = ε
r 12
m
r
P = electric polarization
= atomic total dipole moment density.
This can be modelled with the Lennard-Jones potential
V LJ
2.0
D = electric flux density
−2
r 6
m
r
B = µ0 (H + m),
#
,
where
B = magnetic flux density
where ε is the depth of the potential and rm is the distance
where the potential is at minimum.
m = magnetic polarization
We improve the ideal gas equation of state
= atomic total magnetic moment density.
p0 V 0 = N kB T
Polarization obeys roughly Curie’s law
so that
m=
V 0 = V − N b and p = p0 − aρ2 = true pressure.
ρC
H,
T
where C is a constant related to the individual atom.
Thus,
4.3 Thermodynamic Potentials
(p + aρ2 )(V − N b) = N kB T
Consider a conservative mechanical system, such as a
spring, where work can be stored in the form of potential
energy and subsequently retrieved. Under certain circumstances the same is true for thermodynamic systems. Energy can be stored in a thermodynamic system by doing
work on it through a reversible process. This work can later
be retrieved in the form of work. This kind of energy that
can be stored and retrieved in the form of work is called
free energy. We will show in the following that there are as
many free energies as there are combinations of constraints.
Stretched wire
Consider a wire at rest with length L0 . The stretching
of the wire to length L causes the tension
σ = E(T )(L − L0 )/L0 ,
where E(T ) is the temperature dependent constant of elasticity.
13
The free energies are analogous to the potential energy of
a spring and are, thus, called thermodynamic potentials.
This leads to a useful set of so called Maxwell’s relations
!
!
∂T
∂p
=−
∂V
∂S
S,N
V,N
!
!
∂T
∂µ
=
∂N
∂S
S,V
V,N
!
!
∂p
∂µ
=−
∂N
∂V
Internal energy
The internal energy of a thermodynamic system is determined as a sum of all forms of energy intrinsic to the
system. According to the first law (35),
dE = T dS − pdV + µdN,
S,V
which means that S, V and N are the natural variables of
the internal energy14
S,N
In an irreversible process,
T ∆S > ∆Q = ∆E + ∆W,
E = E(S, V, N ).
The first law implies the relations
!
!
∂E
∂E
=T ,
= −p ,
∂S
∂V
V,N
S,N
which means that
∆E < T ∆S − p ∆V + µ ∆N.
∂E
∂N
!
If S, V and N stay constant in the process then the internal
energy decreases. Thus we can deduce that
= µ.
S,V
In an equilibrium with given S, V and N the internal
energy is at the minimum.
Because E, S, V and N are extensive, we have that
We consider a reversible process in a thermally isolated
system
E(λS, λV, λN ) = λE(S, V, N ),
ΔL
where λ is an arbitrary constant characterizing the size of
the system.
Assume then that we make infinitesimal changes to the
internal energy, i.e. S → S + S, V → V + V and N →
N + N . Now
E(S + S, V + V, N + N ) = E(S, V, N )
!
!
!
∂E
∂E
∂E
+
S +
V +
∂S
∂V
∂N
V,N
S,N
E(S + S, V + V, N + N ) = E(S, V, N ) + E(S, V, N ).
We end up with the Euler equation for homogeneous functions
!
!
!
∂E
∂E
∂E
E=S
+V
+N
,
∂S
∂V
∂N
S,V
V2
F
We partition ∆W into the components
"
#
Z
work due to the
p dV =
change of the total
"volume
#
work done by the
∆Wfree =
.
gas against the
force F
Now
or,
E = T S − pV + µN
V1
equilibrium position
S,V
S,N
p2
N.
On the other hand, we have that
V,N
p1
(40)
∆Wfree
=
∆W1 + ∆W2 = p1 ∆V1 + p2 ∆V2
=
(p1 − p2 )∆V1 = (p1 − p2 )A ∆L
which is the fundamental equation.
= −F ∆L.
The combinations of parameters in the above equation
According to the first law we have
are such that they lead to an exact differential for E. Thus,
Z
the second derivatives of E have to be independent on the
∆E = ∆Q − ∆W = ∆Q − p dV − ∆Wfree
order of differentiation, e.g.
= ∆Q − ∆Wfree .
∂2E
∂2E
=
.
Because now ∆Q = 0, we have
∂V ∂S
∂S∂V
14 Often
∆E = −∆Wfree = F ∆L,
internal energy is also denoted with U .
14
i.e. when the variables S, V and N are kept constant the
change of the internal energy is completely exchangeable
with the work. ∆E is then called free energy and E thermodynamic potential.
so the natural variables of A are T , V and N . We immediately obtain the partial derivatives
∂A
S = −
∂T V,N
Note If there are irreversible processes in an isolated
∂A
system (V and N constants) then
p = −
∂V T,N
∆Wfree ≤ −∆E.
∂A
µ =
.
∂N T,V
Helmholtz free energy
Due to the fact that dA is an exact differential, we obtain
Internal energy is the appropriate thermodynamic po- the Maxwell relations
tential in processes where the S, V and N are kept con
stant. Additionally, the internal energy can be determined
∂S
∂p
=
as a function of these natural variables, and all thermody∂V T,N
∂T V,N
namic properties of the system can be calculated by taking
∂µ
∂S
partial derivatives of the internal energy with respect to its
= −
∂N T,V
∂T V,N
natural variables. Often, the experimental circumstances
are such that the macrostate of the studied system is more
∂p
∂µ
=
−
.
conveniently represented with different set of independent
∂N T,V
∂V T,N
state variables. The change in the variable set made with
the Legendre transformation.
Similar to the case with internal energy, we have for an
For example, let us look at the function f (x, y) of two irreversible change
variables. We denote
∆A < −S ∆T − p ∆V + µ ∆N,
∂f (x, y)
z = fy =
∂y
i.e. when the variables T , V and N are constant the system
drifts to the minimum of the free energy. Correspondingly
and define the function
∆Wfree ≤ −∆A,
g = f − yfy = f − yz.
when (T, V, N ) is constant.
Now,
dg
Helmholtz free energy is suitable for processes that are
done at constant volume and temperature. The constant
temperature can be achieved by embedding the studied
system into heat bath. Generally, a bath is an equilibrium
system, much larger than the system under consideration,
which can exchange given extensive property with the system15 .
= df − y dz − z dy = fx dx + fy dy − y dz − z dy
= fx dx − y dz.
Thus we can take x and z as independent variables of the
function g, i.e. g = g(x, z). Obviously
y=−
∂g(x, z)
.
∂z
Enthalpy
Corresponding to the Legendre transformation f → g,
there is the inverse transformation g → f
Using the Legendre transform
E →H =E−V
f = g − zgz = g + yz.
∂E ∂V
= E + pV
S,N
In a process where T , V and N are constant, it is worth- We move from the variables (S, V, N ) to the variables
(S, p, N ). The quantity
while to study the transformed potential
H = E + pV
∂E
E →A=E−S
∂S V,N
is called enthalpy.
or
Now,
A = E − TS
dH = T dS + V dp + µ dN.
which is the (Helmholtz) free energy.
15 In the case of heat bath, the exchanged property is entropy, resulting in the constant temperature of the system determined by the
bath.
Now
dA = −S dT − p dV + µ dN,
15
From this we can read the partial derivatives
∂H
T =
∂S p,N
∂H
V =
∂p S,N
∂H
µ =
.
∂N S,V
Corresponding Maxwell
∂T
∂p S,N
∂T
∂N S,p
∂V
∂N S,p
Gibbs’ function
The Legendre transformation
∂E
∂E
E →G=E−S
−V
∂S V,N
∂V S,N
defines the Gibbs function or the Gibbs free energy
G = E − T S + pV.
Its differential is
relations are
∂V
=
∂S p,N
∂µ
=
∂S p,N
∂µ
=
.
∂p S,N
dG = −S dT + V dp + µ dN,
so the natural variables are T , p and N . For the partial
derivatives we can read the expressions
∂G
S = −
∂T p,N
∂G
V =
∂p T,N
∂G
.
µ =
∂N T,p
In an irreversible process one has
∆Q = ∆E + ∆W − µ ∆N < T ∆S.
Now ∆E = ∆(H − pV ), so that
∆H < T ∆S + V ∆p + µ ∆N.
We see that
In a process where S, p and N are constant spontaneous
changes lead to the minimum of H, i.e. in an equilibrium
of a (S, p, N )-system the enthalpy is at the minimum.
From these we obtain the Maxwell relations
∂S
∂V
= −
∂p T,N
∂T p,N
∂µ
∂S
= −
∂N T,p
∂T p,N
∂V
∂µ
=
.
∂N T,p
∂p T,N
The enthalpy is a suitable potential for an isolated system in a pressure bath (p is constant).
In an irreversible process
Let us look at an isolated system in a pressure bath.
Now
dH = dE + d(pV )
∆G < −S ∆T + V ∆p + µ ∆N,
holds, i.e. when the variables T , p and N stay constant
the system drifts to the minimum of G.
and
dE = dQ − dW + µ dN.
Correspondingly
Again we partition the work into two components:
∆Wfree ≤ −∆G,
dW = p dV + dWfree .
when (T, p, N ) is constant.
The Gibbs function is suitable for systems which are allowed to exchange mechanical energy and heat.
Now
dH = dQ + V dp − dWfree + µ dN
and for a finite process
Z
Z
Z
∆H ≤ T dS + V dp − ∆Wfree + µ dN.
Grand potential
The Legendre transform
∂E
∂E
−N
E →Ω=E−S
∂S V,N
∂N S,V
When (S, p, N ) is constant one has
∆H ≤ −∆Wfree
defines the grand potential
Ω = E − T S − µN.
i.e. ∆Wfree is the minimum work required for the change
∆H.
Its differential is
Note Another name of enthalpy is heat function (in constant pressure).
dΩ = −S dT − p dV − N dµ,
16
so the natural variables are T , V and µ.
and
Cp = T
The partial derivatives are now
∂Ω
S = −
∂T p,µ
∂Ω
p = −
∂V T,µ
∂Ω
.
N = −
∂µ T,V
∂S
∂T
=
p,N
∂H
∂T
(41)
p,N
Compressibility and expansion
Typically, one can control T and p in the experiments
(instead of S and V ). If we consider the isobaric change
in volume as a function of temperature, we can define the
volume heat expansion coefficient
1 ∂V
αp =
.
V ∂T p,N
We get the Maxwell relations
∂S
∂p
=
∂V T,µ
∂T V,µ
∂N
∂S
=
∂µ T,V
∂T V,µ
∂p
∂N
=
.
∂µ T,V
∂V T,µ
Similarly, in isothermic volume change we define the compressibility
1 ∂V
κT = −
.
V
∂p T,N
In the case of adiabatic compression, we define the adiabatic compressibility
1 ∂V
.
κS = −
V
∂p S,N
In an irreversible process
∆Ω < −S ∆T − p ∆V − N ∆µ,
The sound propagation is related to adiabatic compression of the gas, rather than isothermic, resulting in the
relation
r
1
cS =
ρm κ S
between the velocity of sound and the adiabatic compressibility. In the above formula, we have denoted the mass
density of the gas with ρm . One can show (exercise) that
the two compressibilities are related by
holds, i.e. when the variables T , V and µ are kept constant
the system moves to the minimum of Ω.
Correspondingly
∆Wfree ≤ −∆Ω,
when (T, V, µ) is constant.
The grand potential is suitable for systems that are allowed to exchange heat and particles.
κT = κS + V T
αp2
.
Cp
4.4 Application of Thermodynamics
Free expansion
Specific heat
Consider thermally insulated rigid container with two
Consider an increase in temperature by ∆T due to ab- compartments. Assume that the volume V1 is filled with
sorption of heat ∆Q by the system. We define the specific gas at temperature T1 . Open the valve separating the comheat or heat capacity as
partments so that the gas is free to expand and to fill the
whole container with volume V2 .
∆Q
C=
.
∆T
If the increase in temperature is done without the change
in volume, the process is called isochoric. According to the
1st law, we have for infinitesimal reversible processes
V
1
D Q = D W = 0
dE = T dS = dQ
which results in specific heat at constant volume
∂S
∂E
CV = T
=
∂T V,N
∂T V,N
For (isobaric) processes in constant pressure, we have
We are interested in the temperature T2 after the equilibrium is reached. It is worthwhile to remember, that right
after the valve is opened, the system is in non-equilibrium
state and that the expansion of the gas is irreversible. Now,
the system is adiabatically insulated and thus
dH = T dS
∆Q = 0.
17
Also, the gas does not do work in the process, i.e.
We consider now a reversible isenthalpic (and dN = 0)
process initial → final. Here
∆W = 0.
dH = T dS + V dp = 0,
According to the first law,
so
∆E = 0.
dS = −
V
dp.
T
(∗)
This means that due to the conservation of energy we
Now T = T (S, p), so that
have that
E(T2 , V2 ) = E(T1 , V1 ).
∂T
∂T
dT =
dS +
dp.
Remember, that for ideal gas we had that E(T ) does not
∂S p
∂p S
depend on volume. Thus, the temperature does not change
in the free expansion of the ideal gas, i.e. T1 = T2 . One On the other hand
can show that for a real gas (like a van der Waals gas) with
T
∂T
=
,
interactions, the free expansion results in cooling, i.e.
∂S p
Cp
T2 < T1 .
where Cp is the isobaric heat capacity (see Eq. (41)).
Using the Maxwell relation
∂V
∂T
=
∂p S
∂S p
Joule-Thomson (throttling) process
Consider then a gas flow in a pipe with thermally insulated walls.
and the partial derivative relation
gas flow
p1
V1
p2
V2
∂V
∂S
∂T
∂S p
∂T
∂V p
=
p
we can write
dT =
porous plug
T
T
dS +
Cp
Cp
∂V
∂T
dp.
p
Introduction of a constriction to the flow, such as a
Substituting into this the differential dS in constant enporous plug, results in a constant pressure difference p1 >
thalpy (∗) we get so called Joule-Thomson coefficients
p2 across the constriction. In a steady state p1 and p2 are
#
"
temporal constants, and the process is again irreversible.
∂T
T
V
∂V
Given the temperature T1 before (on the left hand side of
=
−
.
∂p H
Cp
∂T p T
the plug), we are interested in the temperature T2 after
the plug. When an infinitesimal amount of matter passes
Defining the heat expansion coefficient αp so that
through the choke the work done by the system is
dW = p2 dV2 + p1 dV1 .
1 ∂V
αp =
,
V
∂T p
V1
V2
we can rewrite the Joule-Thomson coefficient as
Initial state Vinit
0
Final state
0
Vfinal
∂T
V
µJT =
=
(T αp − 1).
The work done by the system is
∂p H
Cp
Z
We see that when the pressure decreases the gas
∆W = dW = p2 Vfinal − p1 Vinit .
• cools down, if T αp > 1.
According to the first law we have
• warms up, if T αp < 1.
∆E = Efinal − Einit = ∆Q − ∆W = −∆W,
so that
For ideal gases ∂T
= 0 holds. For real gases ∂T
is
∂p
∂p
H
H
positive below the inversion temperature, so the gas cools
Thus in this process, the enthalpy H = E + pV is con- down. In the case of van der Waals gas, one can show that
stant, i.e. the process is isenthalpic,
i
1 h 2a
µ
=
−
b
,
JT
∆H = Hfinal − Hinit = 0.
Cp RT
Einit + p1 Vinit = Efinal + p2 Vfinal .
18
where one assumes that the gas has low density implying
RT v a and v b (v = V /n is the molar volume and
n = N/NA ).
done with a single additional variable called the degree of
reaction ξ so that
dNj = νj dξ.
Joule-Thomson effect is often used in the process of liquefying gases. In order to achieve cooling, one has to work
in such temperature and pressure range that µJT > 0.
At room temperature, all gases except hydrogen, helium
and neon cool upon expansion by the Joule-Thomson process. For example, for helium one has to pre-cool it first
(e.g. with liquid hydrogen) below 34 K. Also, the throttling process is applied in many thermal machines, such as
refrigerators, air conditioners, and heat pumps.
When ξ increments by one, one reaction of the reaction
formula from left to right takes place.
Convention: When ξ = 0 the reaction is as far left as it
can be. Then we always have
ξ ≥ 0.
The differential for the Gibbs potential can now be written as
X
X
dG =
µj dNj = dξ
νj µj .
Chemical reaction
j
j
Chemical reactions involve large numbers of molecules We define
breaking and reforming their chemical bonds. The chemiX
∂G
∆r G ≡ ∆r ≡
=
νj µj .
cal bonds characterize the manner by which the atoms are
∂ξ p,T
j
joined to form molecules16 . Examples include:
∆r is thus the change in the Gibbs function per one reac• Covalent bond, caused by the mutual sharing of election.
tron pairs, such as in H2 .
Since (p, T ) is constant G has a minimum at an equilib• Ionic bond, molecule held together by electrostatic at- rium. The equilibrium condition is thus
traction of two oppositely charged ions, e.g. NaCl.
X
∆r Geq =
νj µeq
j = 0.
• Polar bond, covalent bond but unequal sharing of elecj
trons which causes permanent dipole moment of the
molecule, e.g. H2 O.
In a non-equilibrium dG/dt < 0, so if ∆r > 0 we must have
dξ/dt < 0, i.e. the reaction proceeds to left and vice versa.
Regardless of the type of bonds, the macroscopic properWe assume that the components obey the ideal gas equaties of all chemical reactions can be described by thermotion of state. Then one can show17
dynamics. In the following, we will discuss only reactions
involving electrically neutral species of molecules.
µj = kB T [φj (T ) + ln p + ln xj ],
Chemical equilibrium is reached when the rate of proth
duction equals the rate of depletion for each species of wherePxj = Nj /N is the concentration of j component,
Nj ,
molecules involved. Reactions themselves do not stop, even N =
at equilibrium. In general, the chemical reaction formula
µ0j
5
is written as
− ηj − ln T,
φj (T ) =
X
k
T
2
B
0=
ν j Mj .
and µ0j and ηj are constants. So,
j
Here νj ∈ I are the stoichiometric coefficients and Mj
stand for the molecular species.
∆r G = k B T
(burning of hydrogen sulphide)
The equilibrium condition can now be written as
P
Y ν
xj j = p− j νj K(T ),
→
2 H2 S + 3 O2 ←2 H2 O + 2 SO2 .
A
B
H2 S O2
−2 −3
C
H2 O
2
P Y
ν
νj φj (T ) + kB T ln p νj
xj j .
j
Example Consider for example the chemical reaction
j
Mj
νj
X
j
D
SO2
2
where
K(T ) = e−
P
j
νj φj (T )
Typically, the chemical reactions take place in constant is the equilibrium constant of the reaction. The equilibrium
pressure and temperature. Thus, the relevant thermody- condition is called the law of mass action.
namic potential is the Gibbs’ potential. It turns out that
The reaction heat is the change of heat ∆r Q per one
the Gibbs’ free energy can be generalized to such situa- reaction to right. A reaction is
tions where the system is out of equilibrium. This can be
16 More
info in the courses Solid State Physics and Condensed Matter Physics.
19
17 For the rigorous derivation, one needs the concepts of quantum
statistics, which will be introduced later. For now the derivation of
this formula is omitted.
5. Ensemble Theory
• Endothermic, if ∆r Q > 0 i.e. the reaction takes heat.
The preceding chapter was devoted in studying the
macroscopic consequences of statistical physics in the thermodynamic limit. We argued that the whole of the classical thermodynamics can be derived from the knowledge of
the functional dependence of the number of microstates on
the macroscopic parameters, e.g. Ω = Ω(S, V, N ). In this
chapter, we will consider the way how the actual explicit
dependence can be resolved.
• Exothermic, if ∆r Q < 0 i.e. the reaction releases heat.
We write ∆r G as
∆r G = −kB T ln K(T ) + kB T
X
νj ln pxj .
j
Now
∆Q =
∆E + ∆W = ∆E + p ∆V = ∆(E + pV )
=
5.1 Phase Space of a Classical System
∆H,
The microstate of a classical system can be determined up to arbitrary accuracy, at least in principle.
This is achieved by measuring the instantaneous positions and momenta of all particles constituting the system.
This requires 3N position coordinates (q1 , q2 , . . . , q3N ) and
3N momentum coordinates (p1 , p2 , . . . , p3N ). The space
spanned by the position coordinates is often referred to as
the configuration space, and that spanned by the momentum coordinates as the momentum space. Together, the
6N coordinates form the so-called phase space.
since ∆p = 0.
When the total amount matter is constant
dG = −S dT + V dp
holds in a reversible process and
G
G
1
G
S
V
=
dG − 2 dT = −
+
dT + dp
d
T
T
T
T2
T
T
H
V
= − 2 dT + dp,
T
T
The phase point (qi , pi ), where i = 1, 2, . . . , 3N is a representative point. Every representative point corresponds
to a possible state of the system. As it evolves in time,
a representative point carves a trajectory into the phase
space. The direction of the trajectory is determined by the
velocity vector v = (q̇i , ṗi ), whose components are solved
from Hamilton’s equations of motion
because G = H − T S. We see that
G
∂
.
H = −T 2
∂T T
p,N
Now
∂
∂T
∆r G
T
= −kB
∂H(qi , pi )
∂pi
∂H(qi , pi )
,
ṗi = −
∂qi
d
ln K(T ),
dT
q̇i =
so that
d
ln K(T ).
∆ r H = kB T
dT
This expression is known as the reaction heat.
2
(42)
(43)
where H(qi , pi ) ≡ H(q, p) is the Hamiltonian of the system.
Macroscopic parameters limit the accessible regions in
phase space. For example, the knowledge of the volume of
the system V immediately tells us which are the possible
values for the coordinates qi in the configuration space.
Moreover, if the system has the energy E, the trajectory
is restricted to the hypersurface
H(q, p) = E.
More realistically, the energy is determined in the interval
E to E + δE, and the trajectories are within the hypershell
defined by these limits.
Given the values of the relevant macroscopic state variables (such as E, V and N ), the system can be in any of
the possible microstates within the corresponding hypershell. We define the ensemble in phase space as a group
of mental copies of the given system, one for each possible
representative point.
At any given time t, the number of representative points
within the volume element d3N qd3N p around the point
(q, p) is given by
ρ(q, p; t)d3N qd3N p,
20
(44)
where ρ(q, p; t) is the density function. Accordingly, we
define the ensemble average of a given physical quantity
f (q, p) as
R
hf i =
f (q, p)ρ(q, p; t)d3N qd3N p
R
.
ρ(q, p; t)d3N qd3N p
We call the ensemble stationary if
5.2 Liouville’s Theorem and Microcanonical Ensemble
We consider the ”flow” of representative points in the
phase space. Let ω be an arbitrary volume in the phase
(45) space and σ the corresponding surface. The number of
points within the volume increases as
Z
∂
ρdω,
∂t ω
∂ρ
= 0.
∂t
where dω = d3N qd3N p. On the other hand, the flow out of
the volume can be written as
Z
Z
This implies time-independent averages hf i and, thus, the
ρv · n̂dσ =
∇ · (ρv)dω,
stationary ensemble describes a system in equilibrium.
σ
ω
where v is the velocity of the representative points, dσ is
the surface element, and n̂ the unit vector normal to the
element. We have also applied the divergence theorem,
where the divergence in the phase space is naturally given
by
3N X
∂
∂
(ρq̇i ) +
(ρṗi ) .
∇ · (ρv) =
∂qi
∂pi
i
Ergodicity
We can imagine two kinds of averages in the ensemble.
If we consider a single representative point, we can follow
it along its trajectory for a long time T and calculate the
time average
1
T →∞ T
Z
f = lim
T
f (q(t), p(t))dt.
The number of representative points is conserved, and
thus
Z
Z
On the other hand, we can look at the whole swarm of
∂
ρdω
=
−
∇ · (ρv)dω.
representative points and take the ensemble average (45)
∂t ω
ω
at any given time. If the ensemble is stationary, the hf i is
This has to hold for an arbitrary volume ω, so we end up
time-independent and
with the continuity equation of the ensemble
Z T
1
∂ρ
hf i = lim
hf idt.
+ ∇ · (ρv) = 0.
(46)
T →∞ T 0
∂t
0
By applying the definition of divergence, the continuity
The two averages in the previous equation are completely equation can be rewritten as
independent. Thus,
∂ρ
dρ
=
+ {ρ, H} = 0,
(47)
dt
∂t
hf i = hf i.
which is called Liouville’s theorem. In the above, we have
According to ergodic hypothesis 18 , the trajectory of a repre- denoted the classical Poisson bracket with
sentative point travels through every point in the ensemble
3N X
in a sufficiently long period of time. Thus, the time aver∂A ∂B
∂A ∂B
−
{A, B} =
ages of physical quantities have to be the same for every
∂qi ∂pi
∂pi ∂qi
i=1
member of the ensemble. Thus,
Liouville’s theorem means that the representative points in
hf i = f ,
the ensemble move in the phase space in the same manner
as an incompressible fluid in the physical space.
where the latter is calculated for a trajectory of any member of the ensemble.
Microcanonical ensemble
In the experiments, what we measure are long-time avLiouville’s theorem holds quite generally for classical syserages. In other word,
tems, whereas
∂ρ
The ensemble average of any physical quantity
=0
f is identical to the value one expects to ob∂t
tain by making an appropriate measurement on is merely a requirement for the equilibrium. Thus, in equithe given system.
librium we get
18 The proof of the ergodic hypothesis is not at all trivial and, thus,
omitted here. It was first proposed by L. Boltzmann (1871). Notice
also the similarity with the law of large numbers.
21
{ρ, H(q, p)} =
3N X
∂ρ
i=1
∂ρ
q̇i +
ṗi
∂qi
∂pi
= 0.
Consider then a system of N molecules confined to a
volume V with the total energy within the range E ≤
H(q, p) ≤ E + δE. One possible way to satisfy both of
the above conditions is to assume that the density is completely independent on the on the coordinates, i.e.
constant if E ≤ H(q, p) ≤ E + δE
ρ(q, p) =
0
otherwise
Let us try to justify this more rigorously in terms of the
phase space arguments presented above. The Hamiltonian
of the ideal gas is independent on the configuration space,
due to the lack of interactions, i.e.
This is the definition of microcanonical ensemble. Physically, this implies that the ensemble is uniformly distributed among all possible microstates. Thus, there is
equal a priori probability for a given member of the microcanonical ensemble of being in any of the accessible microstates. Microcanonical ensemble is, therefore, describing isolated systems in equilibrium. It is of great importance, since whenever we have two interacting systems A
and A0 the situation can reduced to the case of an isolated
system by studying the combined system A + A0 .
where m is the mass of the atom. The volume of the hypershell of the accessible states is given by
Z
Z
Z
3N
3N
N
ω= d q d p=V
d3N p.
3N
X
p2i
H(q, p) =
,
2m
i=1
P3N p2i
≤ E + δE defines a shell
The condition E ≤ i=1 2m
in the momentum
√ space, lying in between a sphere with
radius R(E) = 2mE and that of slightly larger radius
R(E + δE). In the limit δE E, the thickness of the shell
is approximately
It should be kept in mind that even though classical
r
physics allows in principle the determination of the coorm
.
R(E
+
δE)
−
R(E)
≈
δE
dinates qi and pi up to an arbitrary accuracy, in practice
2E
there always exists uncertainties ∆qi and ∆pi (e.g. due to
measurement precision). We denote the product of uncer- The volume integral in momentum space is then given by
the product of this thickness and the surface area of the
tainties of conjugated variables with
(momentum) hypersphere with radius R(E). In general,
h0 = ∆qi ∆pi .
the surface area Sn (R) of n-dimensional sphere with radius
R is given by
This means that the phase space is divided into volume
2π n/2 Rn−1
f
elements of size h0 (f = 3N is the number of degrees of
Sn (R) =
,
Γ(n/2)
freedom), each of which can be seen to contain one distinguishable microstate. Thus, the number of microstates where Γ is the gamma-function. Thus, we obtain
within the relevant hypershell of the phase space is, asympδE N (2πmE)3N/2
totically,
ω=
V
.
ω
E
[(3N/2) − 1]!
Ω= f,
h0
Now, the number of accessible microstates is
where
Z
Z
ω=
dω =
df qdf p
Ω(E, V, N ) =
and the integration is done over the hypershell. From the
knowledge of Ω, thermodynamics follows via entropy
S = kB ln Ω,
δE
ω
=
V N χ(E),
3N
h
Eh3N
where χ(E) = (2πmE)3N/2 /(E[(3N/2) − 1]!).
(48)
as in the previous chapters.
5.3 Canonical Ensemble
Often the system under study is not isolated, but interFundamentally, the simultaneous measurement of canon- acting with a much larger system (i.e. a bath or a reservoir)
ically conjugated variables is restricted by the Heisenberg fixing one or more of the systems macroscopic properties.
uncertainty principle
One example of such instances is when the system is immersed in a heat bath.
∆qi ∆pi ≥ ~.
(49)
We consider a small system A in thermal interaction with
This implies that the ultimate limit of accuracy is h0 → h. a heat bath A0 . In equilibrium, the two systems have the
same temperature T , but the energy E of the studied system is a variable. This kind of treatment is often beneficial,
Example: Ideal gas
since in experimental situations it is typically easier to conConsider the equilibrium of monatomic ideal gas consist- trol temperature, instead of energy.
ing of N particles and confined in volume V . Previously,
We assume that the bath contains an infinitely large
we argued in Eq. (37) that the number of accessible minumber of mental copies of the system A. These copies
crostates in ideal gas is
form the so-called canonical ensemble in which the macroΩ ∝ V N χ(E).
scopic state of the systems is determined by the parameters
22
T , V and N . The system A can in principle be any relatively small macroscopic system, or even a distinguishable
microscopic system.
Gibbs entropy
Then, we would like to know what is the probability
Pr that the system is in any one particular microstate r
of energy Er . In other words, we are not interested in
measuring any properties that depend upon the heat bath,
but want a statistical ensemble for the system that averages
over the relevant states of the bath.
The entropy of the whole system (system+bath) is again
given by Eq. (48). But, we are interested merely on the
properties of the small system A, and thus we have to generalize our definition of entropy to20
Z
d3N qd3N pρ(q, p) ln ρ(q, p)
X
= −kB
Pr ln Pr .
S = −kB
(53)
(54)
Consider that the common temperature of the system in
r
equilibrium with the heat bath is T . This determines the
total energy E (0) of the combined system A + A0 . Now,
This is the Gibbs entropy which is valid even for nonassuming that the system A is in a state with energy Er ,
equilibrium systems, as well as system that interact with
0
then the bath has such energy Er that
surroundings. The latter form contrasts the fact that there
is a limit in how small ”compartments” one can divide the
0
(0)
Er + Er = E = constant.
phase space (ultimately set by the Heisenberg uncertainty
principle), given by h3N . Thus,
The bath is much larger than the system, and thus
Er
=
E (0)
Pr = ρ(qr , pr )h3N .
Er0
1 − (0) 1.
E
(50)
We denote with Ω0 (Er0 ) the number of states of accessible
states of the bath with energy Er0 . Because the combined
system A + A0 is isolated and in equilibrium, all states
should be found with equal probabilities. Thus,
Pr ∝ Ω0 (Er0 ) = Ω0 (E (0) − Er ).
Because of Eq. (50), we have that19
We see immediately that in the case of an isolated system
in equilibrium, Pr = 1/Ω for all r and
S = kB ln Ω.
Now, we can show alternatively21 that the canonical distribution results from the maximization of the entropy with
the constraints (exercise)
X
hEi =
Er Pr = constant
r
0
ln Ω
(Er0 )
0
= ln Ω (E
(0)
)+
0
∂ ln Ω
∂E 0
X
(Er0
−E
(0)
) + ···
Pr = 1.
r
E 0 =E (0)
0
' constant − β Er .
Physical significance of canonical distribution
In equilibrium β 0 = β = 1/kB T , and we obtain
The internal energy for the system interacting with a
heat
bath is given by the expectation value
Pr ∝ exp(−βEr ).
P
∂
r Er exp(−βEr )
So, we see that the probability Pr is proportional to the
=−
ln Z(β).
hEi = P
∂β
r exp(−βEr )
Boltzmann constant. The probabilities should be normalized, so by P
summing over all possible states r of A we We recall from the chapter discussing the thermodynamic
should have r Pr = 1, and
potentials, that the relevant potential for processes in conexp(−βEr )
.
Pr = P
r exp(−βEr )
(51)
This is called the canonical distribution. The sum over
states normalizing the distribution,
Z(β) =
X
exp(−βEr ),
stant temperatures is the Helmholtz free energy A. Now,
∂(A/T )
hEi = A + T S =
,
∂(1/T ) N,V
which implies
(52)
A(T, V, N ) = −kB T ln Z(T, V ).
(55)
r
This is the most fundamental result of canonical ensemis often referred to as the partition function. Canonical ble theory, since now the all thermodynamic quantities are
ensemble consists of mental copies of A, all of which obey given in terms of the partition function.
the distribution (51).
20 Compare this with the proof of the H-theorem.
19 Again,
21 By using Lagrange’s multipliers, check e.g. the course of Analytical Mechanics.
we study the logarithm of Ω due to reasons of convergence.
23
and
Example: Classical ideal gas
gi exp(−βEi )
Pi = P
.
Again, we consider classical ideal gas as an example. We
i gi exp(−βEi )
recall that ρ(q, p) is the measure of the probability of finding a representative point in the vicinity of the phase space Often one can assume that the accessible energy levels are
very close to one another compared with the average energy
point (q, p). In canonical ensemble,
E. Accordingly, one can assume that the energy E is conρ(q, p) ∝ exp(−βH(q, p)),
tinuous within any reasonable interval (E, E + δE). Correspondingly, the probability that the given system from the
The number of distinct microstates in a volume element
canonical ensemble has the energy in this energy range is
dω was found to be22
dω
.
N !h3N
(56)
P (E) = P(E)dE ∝ exp(−βE)g(E)dE,
(58)
where P(E) is the probability density and g(E) is the density of states. We obtain that
Z ∞
Z(T, V ) =
exp(−βE)g(E)dE.
Thus, the partition function is
Z
1
Z(T, V ) =
exp(−βH(q, p))dω.
N !h3N
0
In the case of ideal gas,
Naturally, if δE is sufficiently small energy interval, we
have that the number of states within range (E, E + δE)
is
Ω ≈ g(E)δE.
N
X
p2i
H(q, p) =
2m
i=1
and the partition function can be written in the form
Since the Boltzmann factor is a monotonically decreas
N
1 V
1
ing function of E, and the density of states monotonically
3/2
N
Z(T, V ) =
(2πmkB T )
[Z1 (T, V )] .
=
increasing one, we have that
N ! h3
N!
(57)
∂ −βE
This means that because of the absence of the interparti=0
e
g(E) ∂E
cle interactions, the partition function can be written as a
E=E ∗
product of single molecule partition functions.
determines a maximum of P (E). This can rewritten as
Now, we can write
∂ ln g(E) 2
3/2
N
h
= β.
A(T, V, N ) = N kB T ln
−1 ,
∂E E=E ∗
V 2πmkB T
Now, we can write
3/2 N
h2
∂A
= kB T ln
µ=
∂N T,V
V 2πmkB T
∂S(E)
1
= .
S = kB ln g and
∂E
T
and
E=hEi
∂A
N kB T
P =−
=
.
Thus, we see that the most probable value is equal to the
∂V V,N
V
mean, i.e. E ∗ = hEi.
Also, we obtain the expression for the internal energy of
Consider then the mean
the classical ideal gas
R
Eg(E) exp(−βE)dE
∂
3
hEi = R
.
hEi = A + T S = −
ln Z
= N kB T.
g(E) exp(−βE)dE
∂β
2
Er
we are interested in the variance
Energy fluctuations in canonical ensemble
σ 2 = hE 2 i − hEi2 = −
∂hEi
= kB T 2 CV .
∂β
In many physical cases, the accessible energy levels can
be degenerate, meaning that there exists in general gi states We see that because CV and hEi are extensive variables,
belonging to the same energy value Ei . Thus,
one has
σ
1
X
∝√ .
Z(T, V ) =
gi exp(−βEi ),
hEi
N
i
22 The
additional factor N ! comes from the quantum mechanical
fact that the constituting particles in the system are indistinguishable.
We will return to this later when we talk about quantum statistics
more thoroughly.
24
Thus, in the limit N → ∞ we have that the energy of a system in the canonical ensemble is for all practical purposes
equal to the mean energy. This means that the situation is
more or less the same as in the microcanonical ensemble!
Moreover, if we make the Taylor expansion around E ∗ =
The derivation of the equipartition theorem embodies
hEi
also the so-called viral theorem by Clausius (1870). The
virial of the system
h
i S
1
−βE
2
X
ln e
g(E) ≈ − βhEi +
− 2 (E − hEi) + · · · .
V=
hqi ṗi i = −3N kB T.
(62)
kB
2σ
i
Thus, we see that
Immediately from Eq. (60), one obtains
P(E) ∝ e
−βE
g(E) ≈ e
−β(hEi−T S)
exp
(E − hEi)2
,
−
2σ 2
V = −2K,
meaning that the probability distribution is Gaussian
with
√
the mean hEi and the standard deviation σ = kB T 2 CV .
(63)
where K is the average kinetic energy of the system.
5.4 Applications of Canonical Ensemble
Equipartition theorem
A system of harmonic oscillators
Consider the expectation value
R ∂H −βH
xi ∂xj e
dω
∂H
R
i=
,
hxi
∂xj
e−βH dω
Harmonic oscillator is one of the most practical models in physics. We will now examine a system of N noninteracting oscillators, mainly to illustrate the canonical
ensemble formulation, but also for the later development
where xi and xj are any of the 6N generalized coordinates of the theory of black-body radiation and the theory of
(q, p), and H is the (classical) Hamiltonian of the system. lattice vibrations.
By using partial integration to the numerator, we obtain
First, consider the classical Hamiltonian of a single oshxi
cillator
∂H
i = δij kB T.
∂xj
1 2
1
mω 2 qi2 +
p .
2
2m i
The single oscillator partition function is then
Z Z
dqdp
kB T
Z1 (β) =
e−βH(q,p)
=
,
h
~ω
H(qi , pi ) =
We thus obtain
hpi
∂H
∂H
i = hpi q˙i i = kB T = −hqi ṗi i = hqi
i.
∂pi
∂qi
where ~ = h/2π. This is the classical counting of the
number of accessible microstates. Similar to the non3N
3N
interacting gas, the partition function of N oscillators is
X
X
hpi q̇i i = −
hqi ṗi i = 3N kB T.
(59) a product of N single oscillator partition functions24
i=1
i=1
N
kB T
N
Z(β) = [Z1 (β)] =
.
Hamiltonian of a system that is quadratic in its coordi~ω
nates can be written as
X
X
Again, from the knowledge of the partition function, the
H=
Aj Pj2 +
Bj Q2j ,
thermodynamics follows. For example,
j
j
~ω
,
A = N kB T ln
where Qj and Pj are canonically conjugate and Aj and Bj
kB T
constants of the problem. Clearly,
~ω
µ = kB T ln
,
X ∂H
∂H
kB T
Pj
+ Qj
= 2H.
(60)
∂Qj
∂Pj
p = 0,
j
kB T
This implies
S = N kB ln
+1 ,
~ω
1
hHi = f kB T,
(61)
2
hEi = N kB T.
where f is the number of non-zero coefficients Aj and Bj .
So, we see that each harmonic term in the Hamiltonian The internal energy agrees with the equipartition of energy,
makes a contribution of 12 kB T to the internal energy of because there are two independent quadratic terms for each
the system. This also results to a contribution of 12 kB to single-oscillator Hamiltonian.
the specific heat CV . This the theorem of equipartition of
It should be noted at this point that when we derived the
energy among the degrees of freedom of the system23 .
canonical distribution, we did not make assumptions on the
By summing over i, we obtain
23 There will corrections to this theorem in the quantum limit where
some of the degrees of freedom can be ”frozen” (cannot be excited)
due to the insufficient amount of available (thermal) energy.
25
24 Note, that the oscillators themselves are distinguishable, but the
excitations (i.e. photons/phonons) are distinguishable. We will return to this.
nature of the energy levels Er of the system. So, instead
of classical energies, we can calculate the thermodynamics
starting from the quantum mechanical eigenvalues for the
single oscillators, namely
1
En = n +
~ω.
2
Consider N such magnetic dipoles, each with magnetic
moment µ. We assume that the dipoles are not interacting
with one another, and that they are localized and classical in the sense that they are distinguishable and freely
orientable. The internal energy of the N dipole system is
given by its potential energy due to the external field H
E=
Accordingly, the single-oscillator partition function is
Z1 (β) =
N
X
Ei = −
i=1
∞
X
−1
~ω
.
e−β(n+1/2)~ω = 2 sinh
2kB T
n=0
N
X
µi · H = −
N
X
i=1
µH cos θi .
i=1
Again, due to the fact that the dipoles do not interact, we
can write
This results in the N -oscillator partition function
Z(β) = [Z1 (β)]N ,
X
Z1 (β) =
eβµH cos θ .
Z(β) = [Z1 (β)]N = e−(N/2)β~ω (1 − e−β~ω )−N .
θ
Again, we obtain
1
A = N ~ω + kB T ln(1 − e−~ω/kB T ) ,
2
The dipoles will obviously align themselves along the
field. The magnitude of the magnetization will be
∂A
N ∂
. (65)
ln Z1 (β) = −
Mz = N hµ cos θi =
β ∂H
∂H T
µ = A/N,
p = 0,
S = N kB
By denoting with dΩ = sin θdθdφ the elemental solid angle
representing the orientation of the dipole, we obtain
β~ω
−β~ω
−
ln(1
−
e
)
,
eβ~ω − 1
Z
1
~ω
hEi = N ~ω + β~ω
.
2
e
−1
(64)
It is interesting to notice, that the energy per oscillator
approaches the equipartition energy only when kB T ~ω.
In low temperatures, the internal energy is given by the
zero-point energy of the oscillators ~ω/2.
XE\
2π
π
Z
eβµH cos θ sin θdθdφ = 4π
Z1 (β) =
0
0
We have, thus, obtained the magnetization per dipole
µz =
Mz
= µL(βµH),
N
where
L(x) = coth x −
NÑΩ
3.0
1
x
is the Langevin function.
Classical
2.0
Μz  Μ
1.2
1.5
1.0
2.5
Schrödinger
Planck
1.0
sinh(βµH)
.
βµH
ΒΜH  3
0.8
0.5
0.6
0.5
1.0
1.5
2.0
2.5
3.0
k B T ÑΩ
0.4
Average energy (64) per oscillator, labelled as
Schrödinger oscillator. Note that if there were no zeropoint oscillations, the classical limit would be off by
− 12 N ~ω (Planck oscillator).
Statistics of paramagnetism
0.2
ΒΜH
0
2
4
6
8
10
12
14
We note that the parameter βµH gives the strength of
the field in terms of thermal energy. When µH kB T , we
have magnetic saturation
Paramagnetism is a form of magnetism caused by permaMz ' N µ.
nent dipole moments of the constituent atoms, generated
by the electron spin and angular momentum. Magnetiza- In the large temperature limit, µH k T and
B
tion of paramagnetic materials is created only in the presence of an external magnetic field H. The magnetization
N µ2
C
M
'
H ≡ H,
z
disappears when the field is removed.
3kB T
T
26
and the corresponding energies are ±ε = ±µB H. Thus,
which is the Curie law of paramagnetism and C is the
Curie constant.
Z(β) = [2 cosh(βε)]N ,
Quantum mechanics limits the number of possible orientations of the dipoles. Generally, the magnetic moment
and the total angular momentum l of the dipole are related
by
e
l,
µ= g
2m
where g(e/2mc) is the gyromagnetic ratio, g is Lande’s gfactor,
l2 = J(J + 1)~2 , J =
and
1 3 5
, , , . . . or 0, 1, 2 . . .
2 2 2
A = −N kB T ln[2 cosh(βε)],
ε
ε
ε
S = N kB ln 2 cosh
−
tanh
,
kB T
kB T
kB T
ε
E = −N ε tanh
,
kB T
ε
M = N µB tanh
,
kB T
2
∂E
∆
e∆/kB T (1 + e∆/kB T )−2 ,
C=
= N kB
∂T H
kB T
where ∆ = ε − (−ε) = 2µB H is the difference between
the two allowed dipole energies. Let us analyse these a
bit. First, we found the natural result E = −M H. In the
limit T → 0, S → 0 and E → −N ε. This is a result of
complete ordering of the dipoles along the field. At high
temperatures, the dipoles are randomly oriented and the
energy and magnetization vanish.
3 S(S + 1) − L(L + 1)
g= +
2
2J(J + 1)
where S and L are the spin and orbital quantum numbers
of the dipole. We obtain
µ2 = g 2 µ2B J(J + 1)
C Nk B
0.7
and
0.6
µz = gµB m , m = −J, −J + 1, . . . , J − 1, J
0.5
where µB = e~/2m is the Bohr magneton.
0.4
Now, the single-dipole partition function is
J
X
Z1 (β) =
0.3
0.2
eβgµB mH
0.1
m=−J
= sinh
1
1
βgµB mH .
1+
βgµB mH / sinh
2J
2J
0
1 ∂
Mz
=
ln Z1 (β) = (gµB J)BJ (βgµB mH),
N
β ∂H
BJ (x) =
1
1+
2J
3
4
5
6
k B T ¶
Interestingly,
where
2
The specific heat displays the Schottky anomaly, i.e. a
peak as function of temperature. Generally, the Schottky
anomaly occurs in systems with limited number of energy
levels.
The mean magnetic moment per dipole is
µz =
1
1
kB
=−
tanh−1
T
ε
1
1
1
coth 1 +
x −
coth
x
2J
2J
2J
E
,
Nε
which implies that T < 0 when E > 025 . Positive values of
energy are abnormal, since they correspond to a situation
is the Brillouin function of order J. Similar to classical
where the magnetization is in opposite direction to the apcase, we have saturation for strong fields. In the weak field
plied field. Yet, it has been realized in experiments with
limit, we also obtain the Curie law
nuclear spins in a crystal (with LiF-crystal by Purcell and
2 2
Pound in 1951). A sudden reversal of a strong magnetic
g µB J(J + 1)
CJ
µz '
H≡
H,
(66) field will result in negative magnetization and thus in neg3kB T
T
ative temperature, assumed that the relaxation time for
the mutual interaction among the spins is small compared
where CJ is the Curie constant.
with that for the spins and the lattice.
It should be noted that when J → ∞ the number of
We showed before that the heat will flow from the syspossible orientations of the dipoles become infinitely large
tem
with lower value of β to the one with a higher value.
and the problem reduces to its classical counterpart.
Interestingly, one can show that this holds also for negative
Peculiarities of spin systems
temperatures. In a sense then, systems at negative temConsider system with N non-interacting spin- 1 (J = 1 ) peratures are hotter than those at positive temperatures!
2
2
25 This
dipoles. Such a dipole has only two possible orientations,
27
should be compared with the exercise 3.1!
5.5 Grand Canonical Ensemble
so
In addition to energy, also the number N of particles in a
statistical system is hardly ever measured. Thus, the next
natural step is to regard both N and E as variables and
allow them to fluctuate.
S=
hEi
hN i
−µ
+ kB ln ZG .
T
T
Grand potential
Now, one can proceed in the same way as in the case of
In thermodynamics we defined the
canonical ensemble. One can either consider a small system
0
A immersed in the large reservoir A with which it can
Ω = E − T S − µN,
exchange energy and particles, or, maximize the entropy
so in the grand canonical ensemble the grand potential is
with the constraints
X
hEi = constant
Ω = −kB T ln ZG .
hN i = constant
With the help of this the grand canonical distribution can
be written as
Pr,N = eβ(Ω−Er +µN ) .
Pr = 1.
r
Notify The grand canonical state sum depends on the
variables T , V and µ, i.e.
In both cases, one ends up with the probability
exp(−β(Er − µN )
r,N exp(−β(Er − µN )
Pr,N = P
(67)
ZG = ZG (T, V, µ).
of observing the system with energy Er and N particles.
This is the grand canonical distribution. Similar to the
Density and energy fluctuations
canonical distribution, the studied system can be any relLet us evaluate the fluctuations in the number of partiatively small macroscopic system or a distinguishable micles around the mean
croscopic system. Moreover, the grand canonical ensemble
1 ∂hN i
consists of mental copies of system A obeying the grand
2
.
σN
= hN 2 i − hN i2 =
canonical distribution in the particle-energy bath.
β
∂µ T,V
We can write the grand canonical partition function.
X
ZG =
e−β(Er −µN ))
We obtain for the particle density n = hN i/V
2
σn2
σN
kB T ∂hN i
kB T ∂v
=
=
=
−
hni2
hN i2
hN i2
∂µ T,V
V
∂µ T
1
∝p
hN i
r,N
Number of particles and energy
where v = 1/n.
Now
P
r,N
hN i =
N e−β(Er −µN ))
ZG
1 ∂ ln ZG
β ∂µ
=
and
P
hEi =
=
r,N
It is straightforward to show that G = µN , implying
that the chemical potential is just another name for the
amount free Gibbs energy per particle. Thus, one obtains
(68) the Gibbs-Duhem equation
S
dµ = vdP −
dT.
(69)
hN i
Er e−β(Er −µN ))
By assuming that T is constant, we obtain
ZG
∂
ln
ZG
∂ ln ZG
kB T 2
+ kB T µ
.
∂T
∂µ
σn2
kB T
=
κT ,
2
hni
V
where κT = −(1/V )(∂V /∂P )T is the isothermal compressibility.
Entropy
Typically,
the relative fluctuations of the particle go as
p
1/ hN i. There are, however, exceptions. For example, in
phase transitions 26 the compressibility can become excessively large. Especially, at critical points27 the compressibility diverges leading to large fluctuations in the particle
According to the definition, the Gibbs entropy is
X
S = −kB
Pr,N ln Pr,N .
r,N
26 We will discuss more on phase transitions in the following section,
and also later in the course.
27 See definition later.
Now
ln Pr,N = −βEr + βµN − ln ZG ,
28
density. In this kind of situations the grand canonical en- GB (NB , P, T )
semble can lead to different result than the corresponding
∂GB
∂GA
canonical ensemble, and should be preferred due to the
dG =
dNA +
dNB = (µA −µB )dNA .
correct physical picture it gives.
∂NA T,P
∂NB T,P
We can repeat the calculation for the variance of the
energy. It gives
2
σE
2
= kB T C V +
∂hEi
∂N
2
Because Gibbs energy is minimized at the equilibrium, we
have that if µA > µB the number of molecules in phase B
will increase and that of A decrease. Similarly, if µA < µB
the NA will increase and NB decrease. When
2
σN
.
T,V
This means that the fluctuations in energy of the grand
canonical ensemble is equal to that of the canonical ensemble plus a contribution arising from the fluctuating N .
These fluctuations are typically negligible, but can also become large in phase transitions due to the second term in
the formula.
5.6 Phase Equilibrium and Phase Transitions
A big part of the success of statistical physics and thermodynamics has been gained in the study of phase transitions. Thermodynamic phases are such regions in the phase
diagram where the thermodynamic functions are analytic.
Phase transitions on the other hand arise at point, lines
or surfaces of the diagram where such properties are nonanalytic. The typical phases of matter are
µA (P.T ) = µB (P, T )
the Gibbs free energy is independent on NA and NB and
the phases are said to coexist. This equilibrium condition
draws a coexistence curve in (p, T )-plane. If three phases
are simultaneously in equilibrium, the is crossing of three
coexistence curves is called the triple point. The point
where the coexistence curve ends is called critical point.
c o e x is te n c e
c u rv e
T
1
2
p
Now, let us define pσ (T ) on the coexistence curve in
(p,
T )-plane, determining the phase boundary between the
• vapor phase is a low density gas obeying ideal-gas
phases
A and B. Accordingly,
equation of state, and possible corrections described
by the virial expansion (such as van der Waals equa∂µA
∂µB
dpσ
∂µB
dpσ
∂µA
tion),
+
+
=
.
∂T p
∂p T dT
∂T p
∂p T dT
• liquid phase is dense fluid with strong interactions beAccording to the Gibbs-Duhem equation (69), we have
tween the atoms,
• solid phase is static structure where the constituent
atoms are bound together either in regularly or irregularly.
S
∂µ
,
=−
N
∂T p
V
∂µ
=
.
N
∂p T
• plasma phase is similar to gas in which certain portion
of the atoms is ionized. No definite shape or volume,
We obtain the Clausius-Clapeyron equation
but unlike gas, can form structures in magnetic fields.
SB − SA
∆S
∆H
dpσ
=
=
=
,
(70)
In addition to these, there exists exotic phases in low
dT
VB − VA
∆V
T ∆V
temperatures that arise from the corrections in statistics
introduced by quantum mechanics. Such phases include where
the superfluid and supersolid phases in helium, and the
∆Q = T ∆S = ∆E + p∆V = ∆H
superconducting phase that is observed in numerous comis the phase transition heat or latent heat released in a
pounds.
process where the coexistence curve is crossed at constant
T and p.
Phase equilibrium
Consider two phases A and B contained in a cylinder
Phase transitions
at constant pressure P and temperature T . Assume that
Generally, when the free energy is an analytic function
the total number of particles is N = NA + NB . If the two
phases do not coexist at these P and T , the numbers NA only single phase exists. This is denoted in the phase diaand NB will change in order to approach equilibrium. This grams with the open spaces between the coexistence curves.
leads to the change in Gibbs energy G = GA (NA , P, T ) + When the parameters (in our case T or p) are changed so,
29
that the system reaches the coexistence curve, the free enAccording to the Clausius-Clapeyron equation
ergy becomes non-analytic and a phase transition occurs.
dp
∆Hls
In a phase transition the chemical potential
=
dT
T
∆Vls
∂G
µ=
we have
∂N p,T
is continuous. Instead
S=−
∂G
∂T
V =
∂G
∂p
> 0,
if ∆Vls > 0
1)
dp
dT
< 0,
if ∆Vls < 0
2)
.
p
d p
p
and
dp
dT
1 ) s o lid
d T
flu id
> 0
p
d p
d T
2 )
< 0
flu id
s o lid
T
are not necessarily continuous.
T
T
A transition is of first order, if the first order derivatives
(S, V ) of G are discontinuous and of second order, if the
We see that when the pressure is increased in constant
second order derivatives are discontinuous.
temperature the system
1) drifts ”deeper” into the solid phase,
Examples
2) can go from the solid phase to the liquid phase.
a) Vapor pressure curve
p
fu s io n c u rv e
c ritic a l
p o in t
C
s o lid
flu id
v a p o r
s u b lim a tio n
T
v a p o r p re ssu re
c u rv e
c u rv e
trip le p o in t
T
c) Sublimation curve
Now
dH = T dS + V dp = Cp dT + V (1 − T αp ) dp,
since S = S(p, T ) and using Maxwell relations and definitions of thermodynamic response functions
∂S
∂S
∂V
Cp
dS =
dp +
dT = −
dp +
dT.
∂p T
∂T p
∂T p
T
We consider the transition
The vapor pressure is small so dp ≈ 0, and
liquid → vapor.
Supposing that we are dealing with ideal gas we have
∆V = Vv =
Hs
= Hs0 +
Z
T
Cps dT
solid phase
Cpv dT
vapor (gas).
0
N kB T
,
p
Hv
= Hv0 +
Z
T
0
since
Let us suppose that the vapor satisfies the ideal gas state
equation. Then
Vl(iquid) Vv(apor) .
Because the vaporization heat (the phase transition heat)
∆Hlv is roughly constant on the vapor pressure curve we
have
dp
∆Hlv p
so
=
.
dT
N kB T 2
Integration gives us
∆Vvs =
N kB T
N kB T
− Vs ≈
,
p
p
dp
∆Hvs
p ∆Hvs
=
≈
,
dT
T ∆Vvs
N kB T 2
where ∆Hvs = Hs − Hv .
p = p0 e−∆Hlv /N kB T .
For a monatomic ideal gas Cp = 52 kB N , so that
0
∆Hvs
5
1
ln p = −
+ ln T −
N kB T 2
kB N
b) Fusion curve
Now
∆Vls = Vl(iquid) − Vs(olid)
Z RT
0
Cps dT 0
dT +constant.
T2
0
Here ∆Hvs
is the sublimation heat at 0 temperature and
pressure.
can be positive or negative (for example H2 O).
30
5.7 Gibbs Paradox and Correct Enumeration of Microstates
Up to this point, our discussions on enumeration of the
accessible microstates has been classical. According to
classical physics, each microstate corresponds to a point in
a phase space, spanned by the coordinates and momenta of
the constituent particles of the studied system. Now, classical physics consider elementary particles as distinguishable,
meaning that an pairwise exchange of phase-space coordinates of two particles leads to a new microstate. However,
this line of thinking results in overcounting of the accessible
states, which is traditionally demonstrated by the so-called
Gibbs paradox related to the entropy of mixing of two ideal
gases.
B
T
p
A
A
T
B
p
When the numbers of particles are large, we can use
Stirling’s formula and write
∆Smix = SA+B − SA − SB
= −kB (NA ln NA − NA ln N + NB ln NB − NB ln N )
≈ kB (ln N ! − ln NA ! − ln NB !).
Thus, we see that the problem would be solved by reducing
the entropies by a factor kB ln N(A,B) !.
This kind of a factor appears to the entropy when the
particles constituting the system are considered indistinguishable 28 . By correcting the counting of the states in
this way, one obtains results that agree with the reality, at
least at the classical limit. We will later show, independently, that this kind of reduction has to be made to the
number of accessible microstates in order to have classical
statistical physics as the true limit of quantum statistical
physics.
Entropy of mixing
A
The problem arises when the two gases are the same.
Equation for the entropy of mixing (71) still holds, implying that the entropy will increase after the opening of
the valve. However, since the two gases are the same, the
process should be reversible! This is the Gibbs paradox.
B
Consider two ideal gases (A and B) confined compartments separated by a valve in volume V . Suppose that
initially pA = pB = p and TA = TB = T . If we open
the valve, both gas expand adiabatically (∆Q = 0) to the
whole volume.
In a mixture of ideal gases every component satisfies the
state equation
pj V = Nj kB T.
The concentration of the component j is
Nj
pj
= ,
N
p
xj =
where the total pressure p is
X
p=
pj .
j
Each constituent gas expands in turn into the volume
V . Since pA = pB and TA = TB , we have Vj = V xj . The
change in the entropy is (exercise)
∆S =
X
Nj kB ln
j
V
Vj
or
∆Smix = −N kB
X
xj ln xj .
(71)
j
Now ∆Smix ≥ 0, since 0 ≤ xj ≤ 1, and the equality holds
only when either NA = 0 or NB = 0.
31
28 In
fact, this was done already in Eq. (56).
6. Quantum Statistics
where the matrix elements hm|ρ̂|ni ≡ ρmn (t) = am (t)a∗n (t).
One can show that all of the measurable information in
The ensemble theory developed in the preceding chapter
For
is extremely general. However when applied to quantum- the state vector is contained in the density operator.
29
example,
the
expectation
value
can
be
written
as
systems composed of indistinguishable entities, one has to
proceed with great care. One benefits in writing the ensemhÂi = Tr(Âρ̂)
(73)
ble theory in a manner that is more suitable to quantum
mechanical treatment, i.e. in the language of operators.
and the Schrödinger equation as
i
ρ̂˙ = − [Ĥ, ρ̂(t)].
~
6.1 Density Operator
Let us consider a quantum system characterized by the
Hamiltonian operator Ĥ. Recall from quantum mechanics,
that the (normalized) state vector |ψ; ti gives the maximal
description of the system. Quantum states that can be
represented in the above manner as vectors of a Hilbert
space, are called pure states. The time evolution of the
state vector is given by the Schrödinger equation
Ĥ|ψ; ti = i~
∂
|ψ; ti.
∂t
Then, let {|ni} be a complete set of orthonormal vectors,
i.e. the basis, in the Hilbert space under consideration.
Every vector can be written as a linear superposition of
the basis vectors, especially
X
|ψ; ti =
an (t)|ni,
n
where
an (t) = hn|ψ; ti .
Thus, the state of the system can be described equally well
with the coefficients an (t)
X
X
i~ȧn (t) =
am (t)hn|Ĥ|mi =
am (t)Hnm ,
m
(74)
The latter equation describing the time-evolution of the
density operator is called von Neumann equation. It should
be noted that von Neumann equation follows from the Liouville equation by the application of Dirac’s correspondence principle, i.e.
i
{H, ρ} → − [Ĥ, ρ̂].
~
The above discussion holds an implicit assumption that
the system is isolated, and that we have a complete knowledge of the preparation of the quantum state |ψ; ti. In
practice however, we have to settle for less. We can only
restrict the set of accessible states into a set, ensemble,
{|ψ k ; ti|k = 1, . . . , N }, whose members do not have to be
mutually orthogonal. Then, what we can say is that the
state k is prepared with probability P (k). Accordingly, we
generalize the definition of the density operator into
X
X
X
ρ̂(t) =
P (k)|ψ k ; tihψ k ; t| =
P (k)
ρkmn (t)|mihn|
k
≡
X
k
mn
ρmn (t)|mihn|.
mn
m
We see that the matrix elements of the density operator
where Hnm = hn|Ĥ|mi are the matrix elements of Ĥ. The are ensemble averages of the corresponding elements of the
coefficients an (t) have the standard interpretation as the states in the ensemble. Especially, the diagonal elements
X
X
probability amplitudes of the system to be in states |ni.
ρnn (t) =
P (k)ρknn (t) =
P (k)|an (t)|2
Since we assume that the state is normalized, we have that
k
k
X
2
|an (t)| = 1.
give the probabilities of finding a system in state n in the
n
ensemble. Notice the two type of averaging, one quantum
mechanical due to the probability amplitudes, and one staEach measurable quantity α of the system has a corretistical.
sponding Hermitian operator  in the Hilbert space H of
This describes a mixed state where we have an incomthe system. The possible outcomes of a measurement of
plete
knowledge on the state of the system. The density
the property α are the (real) eigenvalues of Â. The mean
operator
presented in this form contains both the statistical
outcome is given by the expectation value
and quantum mechanical information about the system. It
hÂi ≡ hψ; t|Â|ψ; ti.
should be noted that Eqs. (73) and (74) hold also for the
general definition.
A description of the system, equivalent to the one given by
the state vector, can be obtained by defining the so-called
Entanglement with the environment
density operator
X
One important aspect, that will be discussed more rigρ̂(t) = |ψ; tihψ; t| =
am (t)a∗n (t)|mihn|
orously later when we study non-equilibrium systems, is
m,n
X
29 It should be noted that trace operation is independent on the
=
ρmn (t)|mihn|,
(72) choice of the basis. Also, Trρ̂ = 1 which follows from the normalizamn
tion of the state vectors.
32
that the studied systems in general interact with their sur- will be studied later. In the case of an equilibrium system,
roundings, one example being a system immersed to a heat the density operator should be static. This implies that
bath. In such cases, the Hamiltonian of the whole system
∂ ρ̂
(system+environment+interactions) is generally of form30
i~
= [Ĥ, ρ̂] = 0.
∂t
ĤT = Ĥ + ĤB + Ĥint .
Thus, the density operator commutes with the Hamiltonian, which means that they can be diagonalized simultaOnly in the cases where Ĥint = 0, it is possible to write
neously. Let {|ni} be the eigenbasis of the Hamiltonian.
down the state vector of the bare system. In the presence
In this basis, the density operator is diagonal, i.e.
of interactions, the system and its environment become
X
entangled, meaning that the state vector cannot be written
ρ̂ =
ρn |nihn|.
as a tensor product of a system part and a bath part, i.e.
n
|Ψ; ti =
6 |ψ; ti ⊗ |ψB ; ti. Therefore, it is reasonable to speak
only about the state vector of the combined system31
Entropy
X
|Ψ; ti =
AnB (t)|ni|Bi.
Recall that the Gibbs entropy was defined as
nB
X
S = −kB
Pn ln Pn ,
The Schrödinger equation gives, in principle, the timen
evolution of the state vector of the combined system. But
even if could produce a solution32 , it would give an unnec- where Pr is the probability of finding a system in the state
essarily large amount of information about the bath. This r in the ensemble. Based on the above discussion, Pn = ρn
excess information is naturally interesting in its own right, and we can write
X
but at the moment we are only interested in the behaviour
S = −kB
ρn ln ρn
of the system. Therefore, it is practical and convenient
n
to ”trace out” the bath and hide its contributions into the
X
= −kB
hn|ρ̂ ln ρ̂|ni
probabilities P (k). This can be seen by considering the exn
pectation value of the operator  in systems Hilbert space
= −kB Tr ρ̂ ln ρ̂ .
hÂi = hΨ; t|Â|Ψ; ti
XX
The last equality is the general definition of entropy in
=
hΨ; t|nBi hnB|Â|n0 B 0 i hn0 B 0 |Ψ; ti
quantum mechanics. It should be noted that it is indenn0 BB 0
pendent on the choice of basis (since the trace is). Also,
≡ Trsys (Âρ̂sys ),
the definition bears great generality as it holds also in nonequilibrium systems.
where
XX
Microcanonical ensemble
ρ̂sys =
|n0 i hn0 B|Ψ; ti hΨ; t|nBi hn|.
B nn0
Recall, that in microcanonical ensemble the constituting
mental
copies of a system are characterized by fixed N in
is the reduced density operator of the system. Clearly,
a fixed volume V . The energy of the system is lying in
ρ̂sys = TrB (|Ψ; tihΨ; t|).
(75) the interval (E, E + δE), where δE E. The number
of accessible microstates is denoted with Ω(E, V, N ; δE),
We thus see that the density operator of the system is and each microstate occurs with equal probability (in acwritten as a partial trace of that of the combined system, cordance of the postulate of equal a priori probabilities).
Thus, the matrix elements of the density operator in the
taken over the environment variables only.
energy eigenbasis are of form
6.2 Equilibrium Density Operator
1/Ω for each accessible state
ρn =
0
for all other states
The density operator is an extremely powerful tool in the
study of both equilibrium and non-equilibrium quantum
We see immediately that Trρ̂ = 1, as it should for normalsystems, both closed and open. The non-equilibrium cases
ized states. We see that the entropy in the microcanonical
30 It should be noted that the environment can consists of a large
ensemble is given by the familiar formula
set of quantum systems, or even of a single quantum system.
31 Formally, we have to combine the Hilbert spaces H and H
A
B of
the interacting quantum systems with the tensor product HA ⊗ HB .
Accordingly in our case, Ĥ → Ĥ ⊗ IˆB , ĤB → Iˆ ⊗ ĤB , |ni|Bi ≡
|ni ⊗ |Bi, where |Bi are the basis vectors of HB and Iˆ (IˆB ) is the
identity operator of H (HB ).
32 This is a formidable task, since the bath contains in general a
macroscopic amount of degrees of freedom!
33
S = kB ln Ω.
We first emphasize that Ω is calculated quantum mechanically, so that the indistinguishability of the particles has
been taken into account right from the start. This means
that we should avoid paradoxes, such as the Gibbs’. Also,
if the number of quantum states within δE is Ω = 1 we 6.3 Systems of Indistinguishable Particles
obtain S = 0, i.e. the Nernst theorem (third law of therLet us then turn into quantum mechanical description
modynamics).
of a system of N identical particles. For simplicity, we will
Moreover, the case Ω = 1 corresponds to the pure state, deal first systems where the particles are non-interacting.
in which case the construction of the ensemble is quite The Hilbert space of the system is a tensor product of
superfluous because every system in the ensemble would single-particle Hilbert spaces H1
be in the same state.
H = H1 ⊗ H1 ⊗ · · · ⊗ H1 .
{z
}
|
N copies
Canonical ensemble
If |qi i are the position eigenstates in the single-particle
Hilbert spaces, one can write a general state of the N particle system as
Z
Z
|Ψi = · · · dq1 · · · dqN |q1 . . . qN iΨ(q1 , . . . , qN ),
Consider then the canonical ensemble where the
macrostate is characterized by constant parameters N , V
and T , and the energy E is a variable quantity. The density
matrix in energy basis is again diagonal, and
ρn =
e−βEn
, n = 0, 1, 2, . . .
Z
where
Z=
X
e−βEn
n
is the partition function. This means that the density operator can be written as
ρ̂ =
e−β Ĥ
,
−β
Ĥ
Tr e
where |q1 . . . qN i = |q1 i ⊗ · · · ⊗ |qN i, and Ψ(q1 , . . . , qN ) =
hq1 . . . qN |Ψi is the wave function. Now, because the particles do not interact, we can write the solution of the
time-independent Schrödinger equation (we denote q =
(q, . . . , qN ))
ĤΨE (q) = EΨE (q)
as
ΨE (q) = ψε1 (q1 ) ⊗ · · · ⊗ ψεN (qN )
{z
}
|
(76)
N times
with
E=
where
e−β Ĥ =
∞
X
j=0
εi .
i=1
j
(−1)j
N
X
(β Ĥ)
.
j!
The pairs (εi , ψi (i)) are the solutions of the single-particle
Schrödinger equation
The expectation values are thus of form
Tr Âe−β Ĥ
.
hÂi =
Tr e−β Ĥ
Ĥi ψi (i) = εi ψi (i),
(78)
where we have denoted ψi (i) = ψεi (qi ).
We denote with ni the number of particles in the state
with energy εi . Thus,
X
N=
ni
i
Grand canonical ensemble
and
Similarly, in the case of grand canonical ensemble, where
both E and particle number N are variables, we can write
ρ̂ =
e−β(Ĥ−µN̂ )
,
ZG
E=
X
ni εi .
i
(77)
where N̂ is the particle number operator and
X
ZG =
= e−β(Er −µN ) = Tr e−β(Ĥ−µN̂ )
r,N
is the grand partition function. Again, the ensemble average
Tr Âe−β(Ĥ−µN̂ )
hÂi =
.
ZG
Now it should be noted that, according to quantum mechanics, even if two particles are in different single-particle
states, an interchange between them should not lead into
a new microstate. This means that as long the numbers
ni remain the same, any permutation P among the N particles should result into same microstate. In this way, the
indistinguishability of the particles is guaranteed. We define the permutation P as the reordering of the coordinates
as
P Ψ(1, 2, . . . , N ) = Ψ(P 1, P 2, . . . , P N ).
For arbitrary P , we should have
|P Ψ|2 = |Ψ|2 .
34
(79)
The simplest way to form a wave function Ψ that obeys the have to be either symmetric or antisymmetric also in the
above condition is make a linear combination of all possible systems of interacting particles.
N ! permutations of the wave function ΨE .
There are two possibilities with Eq. (79) fulfilled:
Spin-Statistics Theorem
One can show by using relativistic quantum field theory, that there exists an intrinsic link between the spin
meaning that the wave function is symmetric, or
of the particle and the statistics it obeys. Namely, particles with integral spin must have symmetric wave func
+Ψ if P is an even permutation,
tions and are, therefore, bosons. Correspondingly, particles
PΨ =
−Ψ if P is an odd permutation
with half-odd integer spin have antisymmetric wave functions and are fermions. All particles in nature are either
which means that the wave function is antisymmetric. One
fermions or bosons. The fermions constitute the matter
can show that every permutation of coordinates can be dearound us: quarks (up, down, strange, charm, bottom,
composed into successive pairwise exchanges called transtop), leptons (electron, muon, tau, corresponding neutripositions. If the number of transpositions forming the
nos and antiparticles), and their compositions with odd
permutation is even (odd) the permutation is called even
number of constituents. Bosons are responsible for inter(odd). The sign of the permutation
actions: photons, gluons, W and Z bosons, Higgs boson;
but include also compositions of fermions with even num+1 if P is an even permutation,
σ(P ) =
ber of constituents.
−1 if P is an odd permutation
In fact, since the Hilbert spaces of quantum systems are
The symmetric wave functions are of form
in general complex vector spaces, one could in principle
have for every transposition PT
1 X
P ΨE (q).
(80)
ΨS (q) = √
N! P
PT Ψ = eiθ Ψ
P Ψ = Ψ for all P
The antisymmetric wave functions can be written as
1 X
ΨA (q) = √
σ(P )P ΨE (q)
N! P
ψ1 (1) ψ1 (2)
1 ψ2 (1) ψ2 (2)
=√ ..
..
N ! .
.
ψN (1) ψN (2)
···
···
···
···
ψN (N ) ψ1 (N )
ψ2 (N )
..
.
where θ is a real variable, which is connected to the spin s
of the particle by θ = 2πs. One immediately realizes that

for bosons
 +1
−1 for fermions
e2iπs = (−1)2s =
 2iπs
e
for anyons
(81)
Thus, anyons are such particles that can acquire any phase
when two of them are interchanged. For topological reasons, this can occur only in dimensions smaller than 3.
which is the so-called Slater determinant. One should no- When the spin of an anyon is a rational fraction, other than
tice that in the antisymmetric state, if we change two par- integer of half-integer, it is said to obey fractional statistics.
ticles in the same single-particle state, we should obtain Naturally, elementary particles cannot be anyons. Neverthe same state. This means that P ΨA = ΨA . But, since theless, the collective excitations, i.e. quasiparticles, of inexchange of two particles is an odd permutation, we should teracting elementary particles restricted to two dimensions
have P ΨA = −ΨA . Thus, it is impossible to have an an- can display fractional statistics, and are essential ingreditisymmetric wave function with two particles in the same ents e.g. in the theory of fractional Hall effect.
single-particle state. This is a result equivalent to Pauli’s
One interesting ”hot topic” at the moment is the search
exclusion principle for electrons.
of anyons obeying non-abelian statistics. This means that
The systems composed of particles obeying the exclusion instead of a phase factor, the many particle state is changed
principle are described by an antisymmetric wave function. by a matrix operation in a transposition, implying that
The statistics governing the particles is called Fermi-Dirac two successive transpositions do not in general commute.
statistics, and the constituent particles fermions. For sym- One candidate for this are the so-called Majorana fermions,
metric wave functions we have no such problems. The par- fermions that are their own anti-particles. Again, elementicle numbers ni are not restricted. The statistics gov- tary particles are not found to be Majorana fermions, but
erning such systems is Bose-Einstein statistics, and the they have been realized as (anyonic) quasiparticles in suconstituent particles are bosons. It should be noted that perconductors.
the Hilbert space of a many particle system not the whole
space spanned by the function ΨE but a subspace spanned
by either symmetric (bosons) or antisymmetric (fermions) 6.4 Non-Interacting Bosons and Fermions
state vectors.
We want to calculate the statistics for N non-interacting
Even though the presentation has been done for non- bosons or fermions, i.e. for quantum ideal gas. Recall, that
interacting particles, one can show that the wave functions the quantum state has ni particles in the single-particle
35
energy state with energy εi . The eigenstate of the Hamil- Also, the (occupation) number operator is the familiar
tonian of N non-interacting particles is then denoted with
n̂i = â†i âi
|n1 , n2 , . . . , ni , . . .i
(82)
and obeys the relation
resulting in
n̂i |n1 , n2 , . . . , ni , . . .i = ni |n1 , n2 , . . . , ni , . . .i,
X
Ĥ|n1 , n2 , . . . , ni , . . .i =
ni εi |n1 , n2 , . . . , ni , . . .i
with ni = 0, 1, . . .
i
Bose-Einstein distribution
with
N=
X
ni and E =
i
X
For bosons, all fillings nk of the eigenstate ψk are allowed. Each particle in the state contributes with energy
εk and chemical potential µ, so the single-particle grand
partition function is
ni εi .
i
It is straightforward to show that the basis
{|n1 , n2 , . . . , ni , . . .i} is complete
X
ˆ
|n1 , n2 , . . . , ni , . . .ihn1 , n2 , . . . , ni , . . . | = I,
k
ZG
= Tre−β(Ĥk −µn̂k ) =
∞
X
e−β(εk −µ)nk
nk =0
{ni }
=
where Iˆ is the identity operator, and orthonormal
1
1 − e−β(εk −µ)
,
where the trace is taken over the complete set of symmetric
many-body states. Thus, the grand partition function for
bosons is
Y
For such energy eigenstate, we can always define the cre1
boson
†
ZG
=
.
−β(εk −µ)
ation operator âi so that
1
−
e
k
hn01 , n02 , . . . |n1 , n2 , . . .i = δn1 n01 δn2 n02 . . .
â†i |n1 , n2 , . . . , ni , . . .i = C|n1 , n2 , . . . , ni + 1, . . .i,
The grand potential is
i.e. â†i creates one particle into the state i. Correspondingly, the annihilation operator obeys
â†i |n1 , n2 , . . . , ni , . . .i
ΩG = −kB T
boson
ln ZG
= kB T
X
−β(εk −µ)
ln 1 − e
.
k
= C 0 |n1 , n2 , . . . , ni − 1, . . .i
We recall that the average number of particles is given
by Eq. (68)
meaning that âi removes one particle from the state i.
∂ΩG
hN i = −
.
It is beneficial to use grand canonical ensemble, since we
∂µ
are dealing with both particle and energy exchange with This means that since the grand potential of the whole N
the bath. Again, because the particles do not interact we particle system is composed of a sum of grand potentials
can write the grand partition function ZG as a product of of the individual states, we can write the expected number
k
as
single-particle partition functions ZG
of particles in state ψk as
Y
k
∂Ωk
1
ZG =
ZG
.
hn̂k i = − G = β(ε −µ)
.
k
∂µ
e k
−1
We have thus seen that the number of particles in a singleparticle bosonic state obeys the Bose-Einstein distribution
Bosons
hn̂iBE =
Creation and annihilation operators for bosons
1
.
eβ(ε−µ) − 1
(83)
For bosons, the creation and annihilation operators obey
This describes the filling of single-particle eigenstates with
the commutation relations
non-interacting bosons.
X n\ BE
2.0
[âi , â†j ] = δij
[âi , âj ] =
[â†i , â†j ]
= 0.
T = 0.1 Μ
1.5
One can show that (this has been done for harmonic oscillator creation and annihilation operators in quantum mechanics courses)
√
âi |n1 , n2 , . . . , ni , . . .i = ni |n1 , n2 , . . . , ni − 1, . . .i
√
â†i |n1 , n2 , . . . , ni , . . .i = ni + 1|n1 , n2 , . . . , ni + 1, . . .i
X n\ MB
1.0
0.5
0.0
36
0.1
0.2
classical limit
0.3
0.4
0.5
H ¶ - ΜL  k B T
When the occupancy of a state is low, hni 1, we obtain
the Boltzmann distribution
where c and ai are complex coefficients and
0 1
0 −i
1
σ̂x =
, σ̂y =
, σ̂z =
1 0
i 0
0
hn̂iBE ≈ e−β(ε−µ) .
0
−1
The condition for this is ε − µ kB T 33 . As ε → µ we are the Pauli spin matrices.
obtain hn̂iBE → ∞. In systems of non-interacting bosons
The creation and annihilation operators are now
µ is always less than the lowest single-particle eigenenergy.
0 1
0 0
As a final remark, Eq. (64) corresponds to an average
σ̂+ =
, σ̂− =
0 0
1 0
excitation level of a quantum harmonic oscillator
1
hn̂iqho =
eβ~ω
−1
.
Fermi-Dirac distribution
Due to the antisymmetry of the wave function of
One can use this to show that the excitations inside a harmonic oscillator can be treated as particles obeying Bose- fermions, only nk = 0 and nk = 1 are allowed. This means
that the single-particle partition function is
Einstein statistics.
k
ZG
=
Fermions
1
X
e−β(εk −µ)nk = 1 + e−β(εk −µ) .
nk =0
Creation and annihilation operators for bosons
The grand partition function for the N fermion system is
For fermions, the creation and annihilation operators thus
Y
obey the anti-commutation relations
fermion
ZG
=
1 + e−β(εk −µ)
k
{âi , â†j } = âi â†j + â†j âi = δij
We obtain the average occupation in the state ψk to be
{âi , âj } = {↠, ↠} = 0.
i
j
P1
One can show that
âi |n1 , n2 , . . . , ni , . . .i
√
(−1)Si ni |n1 , n2 , . . . , ni − 1, . . .i
=
0
nk =0
hn̂k i = P
1
nk e−β(εk −µ)nk
nk =0
if ni = 1
otherwise
â†i |n1 , n2 , . . . , ni , . . .i
√
(−1)Si ni + 1|n1 , n2 , . . . , ni + 1, . . .i
=
0
e−β(εk −µ)nk
=
1
.
eβ(εk −µ) + 1
We have thus shown that the average occupation of a
single-particle state in a fermionic system obeys the FermiDirac distribution
if ni = 0
otherwise
f (ε) = hn̂iFD =
1
,
eβ(ε−µ) + 1
(84)
In the above, Si = n1 + n2 + · · · + ni−1 . The (occupation) where f (ε) is often called the Fermi function. Again, in
the limit εk − µ kB T , the occupation of the state ψk is
number operator
†
low and the distribution is given by the Boltzmann distrin̂i = âi âi
bution. In fermionic systems, the chemical potential can
and obeys again the relation
be greater or less than any eigenenergy; at low temperatures the chemical potential separates the filled states from
n̂i |n1 , n2 , . . . , ni , . . .i = ni |n1 , n2 , . . . , ni , . . .i,
the empty states, and is often called as the Fermi energy
εF = µ(T = 0).
with ni = 0, 1
X n\ FD
Example: Spin- 12 system
1.2
Spin- 21 particle is characterized by two quantum states,
spin-”up” and spin-”down”, denoted with
1
0
| ↑i =
, | ↓i =
.
0
1
1.0
X n \ MB
T=0
0.8
0.6
D¶~k B T
One can show that every Hermitian operator Ô in the twodimensional Hilbert space spanned by {| ↓i, | ↑i} can written as
Ô = cIˆ + a1 σ̂x + a2 σ̂y + a3 σ̂z ,
33 Rather interestingly, this is fulfilled usually at high temperatures,
because then there are more available states for occupation, leading
to large and negative µ.
37
0.4
0.2
0.0
classical limit
small T
0.5
1.0
1.5
2.0
¶ Μ
Note that only states within |ε − µ| . kB T are partially
filled.
7. Ideal Bose Gas
Maxwell-Boltzmann gas
We calculate the trace in the grand partition function
over the whole Hilbert space, i.e. over states ΨE . For each
value of N , we eliminate the overcounting by dividing with
N!
N
X 1 X
ZG =
e−βεk
eN βµ
(85)
N!
N
k
Y −β(εk −µ)
=
ee
.
We see that the distributions for both bosons and
fermions imply that in certain parameter range the statistics deviate from the classical Maxwell-Boltzmann statistics. For bosons, the standard textbook examples include
Bose-Einstein condensation, black body radiation and vibrations of the crystal lattice. The last example is vital
for understanding, e.g., the specific heats of metals at high
temperatures. Nevertheless, we will omit the discussion
here34 .
k
Degeneracy discriminant
The grand potential is
ΩG = −kB T
X
Before we continue the exploration of the resulting phenomena, we would like to have some kind of criterion to
separate between the classical and quantum realms.
e−β(εk −µ) .
k
Thus, we end up with the average number of particles in a
single-particle eigenstate with energy ε
hn̂iMB = e−β(ε−µ) ,
(86)
We have seen that the Maxwell-Boltzmann distribution
is valid when
hn̂k i 1
for all single-particle states k. This can be rewritten as
which is called the Maxwell-Boltzmann distribution. One
sees that this indeed is the limiting case of both the BoseEinstein and Fermi-Dirac distributions in the limit ε−µ kB T . It is worthwhile to notice, that the correct result was
obtained by tracing over the whole Hilbert space, and by
requiring the indistinguishability of the particles with the
factor 1/N !.
Interestingly, the non-interacting quantum particles fill
the single-particle states according to the same law
hn̂i =
1
eβ(ε−µ) + c
eβµ eβεk ≤ 1,
since min εk = 0. In the above, eβµ is often called the
fugacity, which describes the ease of adding particles to
the system. Now, recall that average value of particles can
be written as
∂ΩG
N = hN̂ i = −
∂µ
where the grand potential ΩG for the Maxwell-Boltzmann
gas is
X
ΩG = −kB T eβµ
e−βεk = −kB T eβµ Z1
,
where c = −1 for bosons, c = 1 for fermions and c = 0 for
Maxwell-Boltzmann statistics.
k
where Z1 (β) is the single-particle canonical partition function. In the classical limit, we obtained the result (57)
X
Z1 (β) =
e−βεk
k
V
≈ 3 (2πmkB T )3/2
h
≡ V λ−3
T ,
where λT =
length.
p
h2 /(2πmkB T ) is the thermal de Broglie wave
Thus, we can write the condition for classical statistics
as
ρλ3T 1,
(87)
where ρ = N/V is the density of the gas.
One gets an intuitive picture of the above condition by
considering the volume per particle as a sphere of radius r
1
4π 3
=
r .
ρ
3
34 Interested reader can refer to the material on the course of condensed matter physics.
38
In addition, it is beneficial to consider the particles composNow, the energy of a plane wave is given by the dispering the gas as quantum mechanical wave packets 35 . Next, sion relation 38
we make a rough estimate of the size of the wave packet.
~2 k 2
ε
=
.
k
We assume that the average energy of a particle is given
2m
by the thermal energy (equipartition theorem):
Thus, the number of plane waves in a small range of energy
h
p2
h
dε is
=
≈ kB T → p ≈ p
.
2
2m
λ
V
dk
h /(2πmkB T )
T
D(ε)dε = (4πk 2 )
dε
,
8π 3
dε
If we assume that the quantum mechanical uncertainty in
the momentum ∆p is less than the momentum p itself, we where the first term is the surface area of a sphere in kobtain according to Heisenberg’s uncertainty principle
space, the second term is the density of plane waves per
unit volume in k-space, and the third is the thickness of
∆x ≥ λT .
the spherical shell. The above equation defines the energy
Now, the thermal wave length λ can be considered as the density of states
T
minimum size of the wave packet.
Thus, we see that the classical limit occurs when the
wave packets do not overlap. In other words, this means
that the quantum correlations between the particles are
suppressed when the particles are small compared to the
volume they are occupying.
D(ε) =
2πV
(2m)3/2 ε1/2 .
h3
7.1 Bose-Einstein Condensation
One should note that the quantum effects reveal themWe saw in the previous section that the number of parselves when the density of the particles is high and/or the ticles in a single particle state of a many-boson system can
temperature of the system is low, together with small par- take any non-negative integer value.
ticle mass36 .
X n\
BE
2.0
Density of free particles in a box
1.5
Before we start the actual demonstration of the power
of quantum statistics, we note that one often has to study
statistical quantities of type
X
Fε
1.0
T decreases
0.5
ε
H ¶ - ΜL  k T
B
where Fε is a function defined by the single-particle energy
0.0
0.1
0.2
0.3
0.4
0.5
eigenstates. If the studied volume is large, the eigenstates
Especially interesting is the case in extremely low temlie close to one another and Fε can be taken as continuous functions of energy. Thus, the summations are easily peratures where all of the particles tend to go to the lowest
accessible quantum state. This results into a new form of
evaluated as integrals.
matter, Bose-Einstein condensate. Let us try to estimate
In the case of free particles, we assume that they are the temperature region where this macroscopic quantum
confined to a cube with V = L3 and subjected to peri- phenomenon occurs.
odic boundary conditions 37 , resulting in a solution of the
Now, the number of particles in the ground state (ε = 0)
Schrödinger equation in the form of plane waves
is
1
1
z
(88)
ψk = √ eik·r ,
n0 = hn̂0 i = −1
=
→ ∞,
V
z −1
1−z
with the wave vector k = (2π/L)(n1 , n2 , n3 ) where ni are
integers. Each state occupies the volume (2π/L)3 in the
k-space, so the density of plane waves in the k-space is
V
.
D(k) =
8π 3
because z → 1, when T → 0 (exercise). Thus, we obtain
for the total number of particles
N=
X
ε
1
z −1 eβε
−1
=
X
z
1
+
.
1−z
z −1 eβε − 1
ε6=0
35 See
Quantum mechanics.
is vital to acknowledge that the temperature and density
should be compared with each other. For example, in white dwarfs
one has a degenerate quantum gas even in temperatures of the order
107 K.
37 Check the course of Solid state physics or Condensed matter
physics for more details on this calculation.
36 It
39
Now, when the volume V is large, the spectrum of single
particle energy states is almost continuous. Thus, one can
38 Dispersion relation simply means the dependence of the energy
on the wave vector.
approximate the summation with an integral39
Z ∞
X
1
D(ε)
N=
=
n
+
dε
0
−1
βε
−1
z e −1
z eβε − 1
0
ε
= n0 +
We thus see, that
ρλ3T = g3/2 (z),
n0 3
ρλ3T =
λ + g3/2 (1),
V T
In other words, when
V
g3/2 (z).
λ3T
1
Γ(ν)
∞
Z
0
Typically, the density of the gas is kept constant and the
temperature is lowered. The temperature below which the
chemical potential µ is zero, i.e. z = 1, is
2/3
ρ
2π~2
.
Tc =
mkB
2.612
V kB T
= kB T ln(1 − z) −
g5/2 (z)
λ3T
which implies
p=−
∂ΩBE
∂V
=
1
ρ
there will be a macroscopic occupation in the ground state,
forming a Bose-Einstein condensate. Again, the condensate starts to arise when the thermal wave packets describing the bosons start to overlap.
∞
X
xν−1 dx
zn
.
=
z −1 ex − 1 n=1 nν
Similarly, one can show that the grand potential is
ΩBE
when ρλ3T ≥ 2.612.
λ3T ≥ 2.612
We have also identified the Bose-Einstein functions
gν (z) =
when ρλ3T < 2.612
kB T
g5/2 (z).
λ3T
(89) This is called the critical temperature, below which the condensate starts to form. Below the Tc , the relative fraction
Similarly, we can calculate the internal energy of the sys- of particles in the condensate is
3/2
tem
n0
2.612
T
=1− 3 =1−
.
(90)
Z ∞
N
λT ρ
Tc
εD(ε)
3
V
E=
dε = kB T 3 g5/2 (z)
z −1 eβε − 1
2
λT
0
T,N
n N
Thus, we obtain for ideal Bose gas
p=
1.4
2E
3V
n0  N
1.2
1.0
In the classical limit we have nl 1 for all states l.
Thus, n0 is negligible compared with V g3/2 (z)/λ3T and
ρλ3T = g3/2 (z).
0.8
0.4
Since µ < 0, we have that z = 0 when T → ∞, which is in
accordance with our condition (87) for classical statistics.
ne  N
0.6
coexistence
region
0.2
0.0
0.0
0.5
1.0
1.5
2.0
T T c
Also, when T → 0 the fugacity z → 1. So, the fugacity
We see that at Tc the gas goes through a phase transiobtains values 0 < z < 1, and within this interval the Bosetion,
and starts to coexist in two phases. The particles in
Einstein function is monotonically increasing function with
the
ground
state form the condensed phase, characterised
limits
by the order parameter n0 /N 40 . The particles in the excited states form a normal phase with relative fraction of
g3/2 (0) = 0
particles given by
g3/2 (1) = 2.612.
ne
n0
=1−
.
N
N
When the fugacity z = 1 (i.e. µ(T, N ) = 0, the density ρ
One can also imagine that the density of the gas is inand temperature T obey
creased at constant temperature, resulting in the critical
density
ρλ3T = g3/2 (1) = 2.612,
ρc = 2.612 × λ−3
T .
This implies that n0 = 0. If we still increase the density, Now, the critical temperature leads together with Eq. (89)
or decrease the temperature, the increase in the quantity in critical pressure
ρλ3T must originate from the particles occupying the ground
kB Tc
state.
pc = 3 g5/2 (1).
λT
39 Notice, that we removed the grand potential of the ground state
since it would be taken into account incorrectly by the density of
states, giving a zero weight.
40 We will return to the concept of order parameter when we study
more about phase transitions.
40
C
other hand, the particles occupying the excited states are
identified as the normal component. This can be denoted
as
V
ρ = ρs + ρn
j = js + jn
..
.
T
T
C
It should, nevertheless, be stressed that the quantitative
The phase transition is of first order, meaning that the agreement with the experiments requires the inclusion of
interactions to the model.
first derivative of the Gibbs’ energy is discontinuous.
BEC in ultra-cold atomic gases
Experimental realisation
Helium-4
Helium is a peculiar element, since it resists to go into
solid phase at standard pressure, even at the lowest temperatures. This is because of the small mass of the helium atom that causes so large zero-point motion that it
is enough to prevail over the weak inter-atomic attractive
(van der Waals) forces. Consequently, the boiling point of
Helium is extremely low, 4.2 kelvins. Helium has two stable isotopes 3 He and 4 He. 3 He is a fermion and, thus, we
concentrate here on the 4 He-atoms, which are bosons.
Helium-4 goes through a phase transition at 2.19 K. This
is visible as the discontinuity in the specific heat CV .
C
The first demonstration of gaseous Bose-Einstein condensation was made in dilute gases of alkali atoms in
199541 . In these experiments, the gases were trapped and
cooled with magneto-optical traps, which effectively confine the gas in a three dimensional harmonic potential.
The experiment is performed many times with different
final temperatures, and once the temperature is reached
the trapping magnetic field is turned off and the subsequent ballistic motion of the atoms is observed. Atoms in
a high temperature gas will obey thermal distribution and
expand in thermal manner. The atoms below the condensation temperature are in the ground state of the harmonic
trap, and will hardly move after the field is turned off.
V
H e II
H e I
The onset of Bose-Einstein condensation in rubidium,
taken from the experiments on rubidium atoms at Wiemann and Cornell’s group at JILA-NIST.
T
T
C
Due to the shape of the CV -curve, the transition
was named λ-transition and the corresponding Tc the λtemperature. Interestingly, the shape of the CV curve resembles that of a Bose gas. Moreover, the critical temperature for Bose-Einstein condensation, calculated with the
4
He parameters, gives Tc = 3.13 K, which is not that far
from the observed transition temperature. This lead people
to believe that the transition in 4 He was a manifestation
of Bose-Einstein condensation.
7.2 Blackbody Radiation (Photon Gas)
Matter emits and absorbs radiation. In equilibrium, the
rates of these processes are equal, which is called the detailed balance. Now, consider a perfect absorber, a blackbody, that can absorb all radiation at all frequencies. What
will its emission spectrum look like? First, recall that the
electromagnetic radiation consists of plane waves, similar
to Eq (88). Since photons are massless, the dispersion relation is ωk = ck, where c is the speed of light. There are
two modes for each wave vector k, one for each transverse
polarization. Experimentally, it is impossible to have a perfect blackbody. Closed cavity with a small opening, and
in a constant temperature produces nevertheless a good
approximation.
However, the 4 He is not an ideal liquid. There exists
strong repulsive interactions between 4 He-atoms at short
distances, and attractive (although weak) ones at longer
distances. Nevertheless, the theoretical basis of the description of the fluid below the critical temperature is based
in the so-called two-fluid model. Below the critical temperature, the particles occupying the ground state are thought
The radiation spectrum of a blackbody was one of
to form a superfluid, a state of matter which behaves like
the key ignitions for the theory of quantum mechanics.
fluid with zero viscosity and zero entropy. The phase
41 Wiemann and Cornell with rubidium atoms, and Ketterle with
with a superfluid component is commonly called heliumII, whereas the completely normal 4 He is helium-I. On the sodium atoms. They shared the physics Nobel prize for this in 2001.
41
Namely, the classical physics predicts the so-called ultra- Thus, the average total energy in the field is43
violet catastrophe, radiation at infinite powers. This reZ ∞
~ω
sults from the fact that each mode of the radiation field
dωD(ω) β~ω
hEi =
.
e
−1
0
can have the equipartition energy kB T . Moreover, in a 3dimensional cavity, the number of modes is proportional to We thus see that the energy density can be written as
the square of the frequency. So, the radiated power should
Z ∞
ω3
~
E
follow the Rayleigh-Jeans law, and be proportional to the
dω β~ω
= 2 3
.
squared frequency, leading to unlimited radiated power at
V
π c 0
e
−1
high frequencies. This is clearly in contradiction with the
This implies the energy density per unit frequency
real world42 , except for the limit of low frequencies.
• an assembly of quantum harmonic oscillators with energies εs = ~ωs (ns + 21 ), where ns = 0, 1, 2, . . . and ωs
is the angular frequency of the oscillator.
• a gas of identical and indistinguishable quanta, photons, with energies ~ωs . Each photon is a boson, and
each mode can support any number ns = 0, 1, 2, . . . of
photons.
equipart
ition
~ω 3
The correct description to this problem was given by
,
(91)
u(ω) = 2 3 β~ω
th
π c (e
− 1)
Max Planck in the beginning of the 20 century, when he
proposed that the emitted energies are not distributed conwhich is Planck’s law of radiation. The spectrum of a pertinuously, but come in lumps proportional to the frequency,
fect absorber is determined solely by its temperature, not
i.e.
by its shape, material nor structure. At the room temεk = ~ωk .
perature, most of the energy radiated by the blackbody
Let us apply Planck’s ideas and consider a cavity of vol- is infra-red, not perceived by the human eye. Only after
ume V at temperature T . There exists two conceptually about 800 K, the black body starts to radiate visible light.
different, but practically identical viewpoints, as the field
uH ΩL
can be seen as
Ω max H T L
T3
T2 <
In both cases, one the same expectation value
T1 <
1
.
hns i = β~ωs
e
−1
Ω
Ω
Notice that this is the Bose-Einstein distribution with
µ = 0. Since we are dealing with an equilibrium gas, the
We see that the maximum of the energy density follows
free energy should be minimized in order to find the equi- the Wien displacement law
librium number of photons. There are no conservation laws
ωmax = constant × T
for photons, like those of mass, charge, lepton number, angular momentum and such. Thus, there is no constraints
imposed in the minimization, giving just (∂A/∂N )T,V = 0.
In the limit of small frequencies, one obtains
This implies the disappearance of the chemical potential.
kB T
D(ω)
In other words, the total number of particles in the present
u(ω) ≈ 2 3 ω 2 = kB T
,
π
c
V
case is indefinite.
We see that there exists a complete correspondence be- which is the Rayleigh-Jeans law, equivalent to the equipartween ”an oscillator in eigenstate ns with eigenenergy tition energy density: kB T per classical oscillator.
~ωs (ns + 21 )” and ”the occupation of the energy level ~ωs
In general, the total energy density obeys the Stefanby ns photons”. From this point on, regardless of the used Boltzmann law
viewpoint, the treatments are the same.
4
E
= σT 4 ,
The number of single-particle plane waves in a small
V
c
range dω is given by
43 Notice that here we neglect the (infinite) zero-point energy. This
is the so-called renormalization of the energy, justified by stating that
2V
dk
D(ω)dω = (4πk 2 )
dω
,
only the differences in the energy can be measured. Those readers
3
8π
dω
that have been observant, noticed that in the ”photon picture” there
where the factor two is due to the number of modes per k. seems to be no such term as the zero-point energy. Nevertheless, this
is just a sloppy notation, since according to quantum field theory, the
We thus obtain the density of states per unit frequency
zero-point fluctuations in the oscillator model are present as vacuum
fluctuations in the photon model. The vacuum fluctuations arise
as constant creation of photon–anti-photon pairs, which can interact
before annihilation, giving rise to the vacuum energy equivalent to
the zero-point energy of the oscillators.
V ω2
D(ω) = 2 3 .
π c
42 You
do not get a sun-tan by opening the door of a hot oven!
42
where
8. Ideal Fermi Gas
4
π 2 kB
W
σ=
= 5.67 · 10−8 2 4
3
2
60~ c
m K
In this chapter, we will study the gas of indistinguishable, non-interacting fermions. According to the previous
We can calculate also other properties of the photon gas. chapters, we have obtained the grand potential
X
The free energy is given by the grand potential (because
fermion
Ω
=
−k
T
ln(1 + ze−βε )
B
G
µ=0
ε
Z
∞
D(ω) ln(1 − e−β~ω) .
and the average number of particles
X
X
1
.
N
=
hn̂
i
=
ε
−1
βε
We obtain by the change of variable β~ω = x, and by
z e +1
ε
ε
integration by parts
Unlike bosons, the fugacity of fermion gas is unrestricted,
4σ
1
4
i.e. 0 ≤ z < ∞. Because of the Pauli principle no phenomA = − V T = − E.
3c
3
ena, like the Bose-Einstein condensation, where a macroscopic amount of particles go into the same single-particle
The entropy of the gas is given by
state can occur. Nevertheless, the fermion gas has its own
peculiarities that arise from the characteristic quantum na∂A
16σ
S=−
=
V T 3,
ture of its constituents, and those will be studied here.
∂T
3c
Similar to boson gases, one can change the summations
and the pressure
of statistical variables into integrations for large enough
volumes V . This leads to energy density of states
4σ 4
∂A
=
T .
p=−
∂V
3c
2πV
D(ε) = gs 3 (2m)3/2 ε1/2 ,
h
We thus see that the pressure of the photon gas is one third
of its energy density, i.e.
where gs is the number of different spin states (for spin- 1
A = ΩG + µN = ΩG = kB T
0
2
particle gs = 2). Accordingly, we obtain
1E
p=
.
3V
p
gs
= 3 f5/2 (z)
kB T
λT
This is a well known result of the radiation theory.
One of the most important examples of the blackbody and
gs
ρ = 3 f3/2 (z).
radiation is the 2.7 K cosmic microwave background, which
λT
is thought to be the remnant from the Big Bang.
In the above, the Fermi-Dirac functions are defined as
Z ∞
∞
X
zn
1
xν−1 dx
=
(−1)(n−1) ν .
fν (z) =
−1
βε
Γ(ν) 0 z e + 1 n=1
n
The internal energy is given by
Z ∞
εD(ε)
3
gs V
E=
dε = kB T 3 f5/2 (z),
−1
βε
z e +1
2
λT
0
giving also for the ideal Fermi gas the relation
p=
2E
.
3V
We see that in the classical limit
ρλ3T
= f3/2 (z) 1.
gs
This means that fν (z) ≈ z. Thus, we obtain the familiar
classical results
pV = N kB T , E =
3
N kB T
2
and so on. For the values ρλ3T /gs ∼ 1, one has to rely on
numerical calculations in expanding the series in the FermiDirac functions. In the quantum limit ρλ3T /gs → ∞, one
43
can write the thermodynamics in a closed form, and the On immediately sees, that the leading terms are the idengas is completely degenerate meaning that
tical to the ground-state results. From the average energy,
we can calculate the specific heat
1 for ε < µ0
nFD =
CV
π 2 kB T
0 for ε > µ0
=
.
N kB
2 εF
Here µ0 = εF is the chemical potential at T = 0, often
This implies that for T TF the specific heat is much less
referred to as the Fermi energy. All single particle states
than the classical value 23 N kB .
up to Fermi energy are completely filled (according to the
This last result is a consequence of the fundamental
Pauli principle), and those above the Fermi energy are comproperty of the Fermi gas. Due to the Pauli principle,
pletely empty.
the single-particle states far below the Fermi energy are
The value of the Fermi energy can be calculated from
completely filled. It means that the particles occupying
the density, because
such states cannot be (easily) excited because there are no
Z εF
empty states nearby, and therefore do not contribute to
3/2
gs (2εF m)
V
N=
D(ε)dε =
the specific heat, or for example to electric conductivity.
2
3
6π ~
0
For metals, the Fermi temperature is typically high (of the
order of 104 K, meaning that the low temperature approxi 2 2/3 2
6π ρ
~
mation is justified at the room temperature. Thus, the low
⇒ εF =
.
gs
2m
temperature ideal Fermi gas serves as an excellent starting
44
Similarly, one can calculate the average energy at T = 0, point of the study of metallic properties .
or the so-called ground-state energy, as
8.1 Magnetism in an Ideal Fermi Gas
√
Z εF
2gs V m3/2 5/2
εF ,
E0 =
εD(ε)dε =
Let us study the behaviour of the ideal Fermi gas under
5π 2 ~3
0
the influence of external magnetic field B. Previously on
page 26, we showed that if orientations the spins of the
which implies the average energy per particle
fermions are distributed according to the classical BoltzE0
3
mann distribution, the net magnetic moment Mz becomes
= εF .
saturated
at low temperatures (compared with the magniN
5
tude of the field). This implies that the magnetic suscepOften needed quantities are Fermi temperature, momentum tibility χ(T ) at constant temperature becomes zero
and wave vector, which are defined respectively as
∂Mz
= 0.
χ(T ) = µ0 lim
εF
B→∞
∂B T
TF =
kB
√
pF = 2mεF
If one takes into account the Fermi statistics, one obtains
pF
a very different behaviour, especially at low temperatures.
.
kF =
~
If we want to study properties like specific heat and entropy, we have to have a finite temperature. If we still
can assume that the temperature is low, meaning that
T µ/kB , we expect that the deviations from the ground
state are not that large and that we need only the first
terms from the so-called asymptotic Sommerfeld expansion
π2 1
(βµ)ν
fν =
1 + ν(ν − 1)
(92)
Γ(ν + 1)
6 (βµ)2
7π 4 1
+ ν(ν − 1)(ν − 2)(ν − 3)
+
·
·
·
.
360 (βµ)4
The Sommerfeld expansion gives us the low temperature
corrections to the thermodynamic quantities, such as
2
E
3
5π 2 kB T
= εF 1 +
+ ···
N
5
12
εF
Pauli paramagnetism
For simplicity, we assume that we are dealing with spin- 21
particles at low temperatures (T TF ). This kind of situation occurs, for example, with the electron gas in metals.
The energy of such a particle in external field is
( 2
p
− µB B when µ is parallel to B
2m
ε=
p2
when µ is antiparallel to B,
2m + µB B
where µ is the intrinsic magnetic moment of the particle
and µB = e~/2m is the Bohr magneton. Thus, we see that
the introduction of the magnetic field divides the spins into
two populations,
Now, since the studied Fermi gas is strongly degenerated
(T TF ), the occupations in the two different populations
are given by the Fermi distributions
np± =
2
2E
2
5π 2 kB T
p=
= ρεF 1 +
+ ··· .
3V
5
12
εF
1
eβ(εp ±µB B−εF )+1
,
44 This fact is exhaustively discussed in the condensed matter
physics course, and will not be studied further here.
44
where εp = p2 /2m and εF = µ(T = 0).
At finite temperatures, one should apply the Sommerfeld expansions to resolve the susceptibility. We omit the
All the states up to the Fermi energy εF are filled, which
implies that the wave numbers kF± for the two populations calculation here, and just state that the lowest order corrections to the low and high temperature susceptibilities
are different and can be determined from the condition
are
~2 kF+2

2 + µB B = εF
2


2m
 χ0 1 − π12 kεBFT
when T is low
~2 kF−2
χ=
3
− µB B = εF .


T
 χ∞ 1 − 2ρλ
2m
when T is high,
5/2
¶
respectively. The high temperature correction is proportional to (TF /T )3/2 and tends to zero as T → ∞.
Landau diamagnetism
In addition to the (para)magnetism associated to the
orientations of the spins, there exists additional form of
¶F
magnetism which arises from the quantization of the kinetic energy of charged particles associated with their motion perpendicular to the direction of the field. This is
k
k Fk F+
diamagnetic in character, meaning that it creates a field
opposite to the external field B, implying negative suscepClearly, the number of states in the population with the tibility χ(T ). According to the Bohr-van Leeuwen theorem,
parallel alignment is larger. In terms of the Fermi wave the phenomenon of diamagnetism cannot be explained in
terms of classical physics (exercise).
number, we obtain the population occupations
Z
+
kF
N+ =
0
Z
−
kF
N− =
Consider an ideal Fermi gas formed by particles with
charge q. In the presence of an external magnetic field, the
single-charge Hamiltonian is written as
2
1
− i~∇ − qA ,
Ĥ =
2m
k +3
4πk D(k)dk = V F 2
6π
2
4πk 2 D(k)dk = V
0
kF−3
.
6π 2
Thus, we obtain a net magnetic moment
where
M = µB (N− − N+ )
µB V −3
=
(k − kF+3 )
6π 2 F
3/2
µB V 2m
=
(εF + µB B)3/2 − (εF − µB B)3/2 .
2
2
6π
~
B = ∇ × A.
We assume that the magnetic field points in z-direction, i.e.
B = Bz, which allows us to choose the Landau gauge 45 for
the vector potential
A = (0, B x̂, 0).
Realistic values for fields produced in laboratories are of
the order B . 10 T, which corresponds to µB B/kB . 1
K. This is much smaller than the Fermi temperatures in
metals, and thus we can assume that µB B εF . This way
we have
3/2
V
2m
1/2
µ2B BεF .
M≈
2π 2 ~2
Thus, we obtain the Hamiltonian
2
p̂2x
1
p̂2
Ĥ =
+
p̂y − qB x̂ + z .
2m 2m
2m
As a result, a charged particle has a helical trajectory
whose axis is parallel to the external field, and whose proWe end up with the weak field Pauli susceptibility (per jection to the (x, y)-plane is a circle.
unit volume
We see that the Hamiltonian does not depend on the
µ0 ∂M
3
µ2
= µ0 ρ B ,
B→0 V ∂B
2
εF
operator ŷ, meaning that Ĥ and p̂y can be diagonalized
in the same basis. This means that in the y-direction the
Hamiltonian describes a free particle and altogether in the
which holds for T TF and µB B εF . Thus, we see
(x, y)-plane we have
that instead of being zero, the susceptibility is a constant
2
determined by the density. This should be compared to
p̂x
1
~ky
2
+ mωc x̂ −
,
Ĥxy =
the classical (high-temperature) Curie law, Eq. (66) (with
2m 2
mωc
g = 2 and J = 12 ), which implies
χ0 = lim
χ∞
45 Recall that a given field can be created with different choices for
potentials. The behaviour of a particle in the field is independent on
the choice for the potential. This is called gauge invariance.
µ0 µ2B
=ρ
.
kB T
45
i.e. a harmonic oscillator Hamiltonian with a displacement
p
in the x-coordinate by X = ky l02 where l0 = ~/qB is
the magnetic length. We have also defined the cyclotron
frequency ωc = qB/m. The eigenvalues of the harmonic
oscillator Hamiltonian are not dependent on the translations in the coordinates. Thus, the eigenenergies are of
form
1
~ωc , n = 0, 1, 2, . . .
En = n +
2
These are the so-called Landau levels of a charged particle
in a magnetic field.
In the classical limit, we have that f1/2 (z) ≈ z ≈ ρλ3T .
This implies
ρµ0 µ2eff
χ∞ = −
.
3kB T
We thus see that at high temperatures the diamagnetism
obeys the Curie law, but with the opposite sign. In the
quantum limit (T TF but still µeff B krmB T ), we
obtain
ρµ0 µ2eff
.
χ0 (T ) ≈ −
2εF
We have applied the first term of the asymptotic SommerBy using the periodic boundary conditions, we obtain in feld expansion (92). Note that the asymptotic results χ0
and χ∞ are precisely one third in magnitude of the paramthe (free) y-direction
agnetic results, given that the particles are electrons. This
2πny
gives us the intuition that the diamagnetic contribution
, ny = 0, ±1, ±2, . . .
ky =
Ly
to the materials response to a magnetic field is negligible
compared with the paramagnetic contribution.
and in the harmonic coordinate x
8.2 Relativistic Electron Gas in White
Dwarfs
0 ≤ X < Lx .
These restrict the possible values for ny :
The first application of Fermi statistics was related to
the
equilibrium studies of white dwarf stars. White dwarfs
⇒ Ns ≡ nmax
y
have burned most of their hydrogen in the nuclear fusion
(2H → He). When this energy release becomes absent, the
where q = Ze, Φ0 = h/e is the flux quantum, and Φ is the gravitational force is unbalanced and the star collapses into
magnetic flux through the system. We thus see that each a white dwarf.
Landau level is degenerate, with exactly one state for each
Consider then a typical idealization of white dwarf, conflux quantum and for each elementary charge.
sisting of a ball of helium with a mass M ≈ 1030 kg and
e
Φ
Lx Ly
= Z Lx Ly B = Z ,
=
2
2πl0
h
Φ0
mass density ρm ≈ 1010 kgm−3 . The temperature within
the white dwarf is of the order T = 107 K. This means that
the thermal energy per helium atom is
The total energy of the ideal charged Fermi gas is
εn =
p2z
+ En ,
2m
3
kB T ≈ 1 keV Eion ≈ 10 eV,
where the kinetic energy associated with the linear motion
2
along the field is assumed to be a continuous variable. Now,
where Eion is the ionization energy of a helium atom. Thus,
we can calculate the grand partition function
the helium atoms in white dwarfs are completely ionized.
X
ln ZG =
ln(1 + ze−βε )
Now, the mass of the helium consists mostly of that
ε
of the nucleus, and can be approximated as (4 He is more
Z ∞
Lz
abundant)
=
dpz
h
−∞
X
∞ M ≈ N (me + 2mp ) ≈ 2N mp ,
2
qB ×
Lx Ly
ln 1 + ze−β(~ωc (n+1/2)+pz /2m)
.
h
where N is the number of electrons, me is the mass of the
n=0
electron, and mp = 1.673 · 10−27 kg is the mass of the
Similar to Eq. (65), we can calculate the magnetization as proton. This gives us the approximate electron density
1 ∂
N
ρm
3
M=
ln ZG
.
ρ=
=
≈ 1036 electrons/m .
β ∂B
V
2m
z,V,T
p
After somewhat lengthy calculation (which we skip here), This gives us the Fermi energy of the electron gas
we obtain the low-field (µeff B kB T ) susceptibility
εF ≈ 106 eV ≈ mc2 = 0.512 · 106 eV.
3/2 2
(2πm) µeff
χ(T ) = −µ0
f1/2 (z),
Thus, the kinetic energies of the electrons are of the order of
3h3 β 1/2
their relativistic rest energy, meaning that in white dwarfs
where µeff = q~/2m (this is the Bohr magneton if q = we are dealing with relativistic Fermi gas. Additionally,
e). Notice that the susceptibility is diamagnetic in nature, the Fermi energy gives TF ≈ 1010 K, which means that the
irrespective of the sign of the charge of the particle.
electrons are (almost) completely degenerate.
46
In the equilibrium of the white dwarf, the outward force where
due to the pressure p of the relativistic degenerate electron
3/2
1/2 9π
~c
gas balances the gravitational pull
≈ 0.52 · 1057 mp
Mc = m p
512
Gm2p
dp
GM (r)
1/3 1/3
,
=−
~ 9π
Mc
dr
r2
Rc =
≈ 4700 km.
mc 8
mp
where G = 6.673 · 10−11 Nm2 /kg2 is the gravitational constant, and the mass at distance r from the star center is
For the radius of the star we obtain
4
3
1/3 "
1/3 #
M (r) = ρm r ,
M
M
3
1−
.
R = Rc
Mc
Mc
where we assume that the density is constant within the
star. By simple integration, we obtain the pressure at the
We see that the white dwarf has the maximum mass M =
center of the star
Mc , which it cannot exceed without collapsing to a neutron
8π
star or a black hole. This is called the Chandrasekhar limit,
2 2 2
p=
Gmp ρ R ,
3
and even more detailed calculations give Mc ≈ 1.44M ,
where we have assumed that the pressure vanishes at the where M is the mass of the Sun.
surface.
On the other hand, the pressure can be calculated from
the ground state properties of the relativistic electron gas.
First of all, recall that the relativistic energy of the electron
can be written as
p
εp = (mc2 )2 + (cp)2 .
This can be rewritten in terms of the wave vector as
p
εk = c~ k 2 + kc2 ,
where p = ~k and kc = mc/~ is the Compton wave vector.
Now, in the relativistic limit kF kc and one can write
the grand potential as
X
ΩG = −kB T
ln(1 + ze−βε )
ε
Z
∞
D(ε) ln(1 + ze−βε )dε
≈ −kB T
Z0 ∞
= −kB T
4πD(k) ln(1 + ze−βε(k) )dk
Z 0∞
V
1
dε
k 3 dk
2
−1
βε(k)
3π 0 z e
+ 1 dk
V ~c 4
≈−
k − kc2 kF2 + · · · ,
12π 2 F
=−
where we have applied partial integration. The grand potential gives the pressure
∂ΩG
~c 4
p=−
=
k − kc2 kF2 + · · · .
∂V T,N
12π 2 F
This should obey the previous balance condition, resulting in
8π
~c 4
Gm2p ρ2 R2 =
kF − kc2 kF2 + · · · .
2
3
12π
By substituting kF3 = 3π 2 ρ, we obtain
M
Mc
2/3
=1−
R
Rc
2 Mc
M
2/3
,
47
9. Entropy, Information and Arrow
of Time
as the direction of time in which the entropy increases.
Loschmidt paradox
Before the discussion on the main topic of this chapter, a
word of caution is in order. The discussion in this chapter
Let us dwell on this last statement. Consider a film first
will be subjective and, at some places, somewhat specula- played forwards and then backwards. Normally, we can
tive. Therefore, the contents should be read with an open separate between the directions. A classic example is the
mind and with a scientific doubt.
falling of a drinking class to the floor and it’s shattering
Entropy is perhaps the most influential, but at the to small fragments. The reverse (the re-organization of the
same time controversial, concept that arises from statis- fragments) seems irrational to us. Our minds are formed
tical physics. So far, we have encountered three different so that we can separate between the directions of time in
ways to determine the entropy. It is hard to tell exactly irreversible processes. But, the laws of nature are timewhat entropy is, but we should nevertheless try to develop reversible. The motions of atoms look reasonable whether
an intuition for it46 . In this chapter, we will review the the time is running backwards or forwards. It is then
definitions, and also look closer at some of the myriad of quite a mystery how the irreversibility emerges from the
reversible behaviour of things. Johann Loschmidt, a conimplications that result.
temporary of Boltzmann, was the first to see this contrabetween the second law and the reversible laws of
9.1 Thermodynamic Entropy: Irre- diction
nature, hence it was named the Loschmidt paradox.
versibility and Arrow of Time
One really cannot overvalue the second law of thermodynamics. Heat cannot spontaneously flow from cold regions
to hot regions. It is impossible to build a perpetual motion
machine. After P. W. Bridgman, there have been nearly
as many formulations of the second law as there have been
discussions of it. But, possibly the deepest philosophical
message is two-fold. The second law introduces the concept of entropy, that gives us the way to characterize all
processes in nature; In an isolated system, only such processes are allowed that increase the entropy47 . As such,
this is among the most fundamental statements we have
about nature. But, it tells us more. It points us the arrow
of time, separates the future from the past.
Consider the previous example on the mixing of two ideal
gases (see p. 31). As we open the valve, the constraints for
the two gases are changed. In the statistical physics language we have learned to use, the number of accessible
microstates becomes increased and the systems are not in
equilibrium. As the new equilibrium is achieved, one notices that the thing that characterizes this process is the
second law, and the quantity that changes is the entropy48 .
One attempt to resolve this issue is to state that the
whole Universe was in a low entropy state right after the
Big Bang. This way the familiar definitions of the past and
the future come about in terms of the increasing entropy in
the Universe. As a result, we would be witnessing a process
where the Universe is moving towards equilibrium as time
increases. In this equilibrium, in the words of Helmholtz
(1854), all energy will become heat, all temperatures will
become equal, and everything will be condemned to a state
of eternal rest. This ultimate fate of the Universe has been
named as the heat death of the Universe.
9.2 Disorder
Other way to interpret the entropy is as a measure of
the disorder in the system. Consider again the entropy
of mixing. We obtained (see Eq. (71) and the following
discussion) that for the mixture of two gases of indistinguishable particles, the entropy of mixing can be written
as (we assume that in the beginning VA = VB = V /2 and
NA = NB = N/2)
∆Smix = Smixed − Sunmixed
The increase in entropy calls for irreversibility. For ex
N/2 (V /2)N/2
V
ample, it is impossible to restore the initial equilibrium of
− 2kB ln
= 2kB ln
(N/2)!
(N/2)!
the two gases, both in separate compartments, simply by
closing the valve. In general, the ever-growing entropy tells
= N kB ln 2.
us that one direction of time is preferred over the other. It
defines, at least in the thermodynamical sense, the future We thus see that entropy increases by kB ln 2 every time we
insert a particle into either of the two compartments, pro46 The same statement holds also for energy and time. Both of them
are hard to define exactly, but we have learned to be fluent in using vided that we do not look which compartment we choose.
them to describe phenomena around us. This chapter is intended to This is a simplest possible instance of the counting entropy
be give guidelines to do the same in the case of entropy.
47 Strictly, processes that do not decrease the entropy.
48 As a side note, in this example it is important to notice the
inadequacy of the thermodynamical definition of entropy
Scounting = kB ln(# of configurations).
In the case of mixing entropy, each particle has two possible compartments, so N particles can be arranged into
2N different configurations. Generally, if there are m ≥ 2
where Q is the heat flow into the system. The circumstances we set
possibilities for a particle, the added entropy is accordingly
up for the system were such that heat is not exchanged between the
gas mixture and the surroundings, thus the change in entropy cannot kB ln m. This idea will be exploited in the last section of
result from such a process.
this chapter, when we discuss about information entropy.
∆S ≥
Q
,
T
48
The above discussion holds in cases where there exists
discrete amount of choices for the new particle. Discrete
choice arises naturally in quantum statistics where the
states are quantized. Generally, as we have already discussed, the equilibrium entropy measures the logarithm
of the number of different states the system can be in.
For example, the previous equilibrium statistical mechanics (Boltzmann) entropy
(Taken from K. Maruyama et al., Rev. Mod. Phys 81, 1
(2009).)
The process goes as follows. (a) we insert a partition that
divides the volume in two equal compartments. (b) the
demon measures the position of the molecule, and records
the measurement (whether the molecule is on the right or
on the left of the partition). (c) the demon connects a
load to the partition, working now as a piston, on the side
where she has measured the molecule. (d) by keeping the
S = kB ln Ω(E)
(93)
system at constant temperature, the demon can let the
measures the number of microstates with macroscopic en- gas expand isothermally and quasistatically (no change in
ergy E, with phase-space volume h3N allocated to each internal energy), resulting in work
Z V
Z V
state.
V −1 dV = kB T ln 2.
pdV = kB T
W =Q=
V /2
Maxwell’s Demon
Is it possible to extract work from the mixing entropy?
Consider a mixture of two ideal gases, A and B, and a
clever creature, the Maxwell’s demon, that works at the
valve between two compartments of the container. Suppose that the demon is able to control the valve in such
a way that it lets the A-molecules to pass through the
valve to the right side of the valve, and the B-molecules
to the left. Additionally, she would prevent the opposite
processes. As a result, each transfer of a molecule through
the valve decreases the entropy by kB ln 2, apparently violating the second law. The thing that is overlooked is that
the demon itself consumes entropy, which then balances
the entropy decrease as a whole. We will return to this in
the following section.
V /2
Thus, we see that the Szilard’s demonic engine extracts
heat Q from the surroundings and converts it completely
into mechanical work, apparently violating the second law.
The explanation of this requires further insights on our
concept of entropy.
9.3 Information Entropy
The Szilard’s realisation of Maxwell’s demon leads us to
our last and the most profound interpretation of the entropy, as a measure of our ignorance about a system. As
we have seen, in the entropy of mixing we add kB ln 2 to our
ignorance of the system every time we add a particle into
the container without looking where it went. Similarly, a
cycle in the Szilard’s engine transfers an amount kB T ln 2
of heat from surroundings into work, provided that one
knows on which side of the partition the molecule is. GenSzilard’s engine
erally, this implies that the entropy is not a property of
But first, let us return to the initial goal of the demon, the system, but is related our knowledge about the systhe extraction of work from entropy. Consider a simplified tem. Additionally, the second law then tells us that in the
version of the Maxwell’s demon, first presented in 1929 by equilibrium, our knowledge becomes minimal, reflected in
the Hungarian L. Szilard. This was the first time that the the equal probabilities of finding the system in one of its
significance of information in physics49 was pointed out.
allowed states.
The Szilard’s idea was to study one-molecule ideal gas in
volume V . Note, that we can consider the single molecule
Shannon entropy
as an ideal gas by forming an ensemble of one-molecule
In 1948, C. Shannon laid the foundation to information
gases and calculating the averages of various quantities as
theory by studying the information content of random variif it was an ideal gas with a large number of molecules.
ables. Interestingly, information and physics seem to have
deep connections. Consider a random variable X with the
possible outcomes {x1 , x2 , . . . xn }50 . Intuitively, we assume
that the information content I(X = xi ) related to an outcome xi depends only on the probability of the event. By
definition, we require that I is positive and additive. Also,
if we assume that we have another random (independent
on X) variable Y , the information content of measuring xi
and yj simultaneously should be the sum of the individual
information contents, i.e. I(X ∩ Y ) = I(X) + I(Y ). By
taking these into account, one unavoidably ends up with
the information content function
I(xi ) = − log P (xi ),
49 We
50 We assume here that X is a discrete random variable. Generalization to continuous variables is straightforward.
will go deeper into this connection in the following section.
49
where P (xi ) is the probability of observing the outcome xi .
The base of the log determines the units of information. In
the base of 2, the units are bits, and in the base of e the
units are nats.
The average information content
S = hI(xi )i = −
X
P (xi ) ln P (xi )
i
is the cornerstone of the theory of information. Interestingly, the average information content is exactly of the form
of Gibbs entropy
S = −kB hln pi i = −kB
X
pi ln pi ,
(94)
i
(Taken from S. Toyabe et al., Nat. Phys. 6, 988 (2010).)
which was our generalization of the Boltzmann entropy!
Thus, it was named as the Shannon entropy, according to
its founder. Note, that in information theory the unit of
entropy does not have the historical burden of having the
units J/K.
In 1951, L. Brillouin became the first to treat the entropy
and information on the same footing. The Gibbs entropy
can be seen as the maximum amount of information one
can obtain from the system by making a measurement. To
clarify this, consider an isolated system initially with entropy Si . Assume then that we acquire information about
the state of the system by making a measurement. The
gain in information is linked to the change in the entropy
by
I meas = Si − Sf > 0,
where Sf is the physical entropy after the measurement.
This means that the gain in information decreases physical entropy. In the Szilard’s engine, this occurs when we go
from (a) to (b), and we see that the entropy in the system
has turned into information inside the demon’s head. This
implies the duality of entropy, i.e. its having both thermodynamic and information theoretic aspects. If further
measurements are not made, we see that the second law
implies unavoidable decrease in the information about the
system. Thus indeed, entropy can be seen as the lack of
knowledge about the system.
In their experiment, Toyabe et al. studied system equivalent to particle stochastically jumping in a spiral-staircase
due to the thermal fluctuations51 . In order to prevent
downward jumps, the demon places a block behind the
particle whenever she observes an upward jump. Thus,
the particle climbs up the stairs without direct energy injection.
Naturally, the energy for useful work is extracted from
the heat bath setting up the temperature in the Szilard’s
engine. As we have already discussed, this apparently violates the second law because it does a perfect conversion of
heat Q into work W . Szilard’s original idea was that this
energy is returned to the heat bath by the measurement
device. However, Bennett showed in 1982 that the measurement can be done reversibly, i.e. without any change
in entropy, at least in principle.
But, there is still one thing that we have overlooked. At
the stage where the work is done, the gas expands isothermally and the demon looses her information about the location of the molecule. This is a logically irreversible process.
In general, logical reversibility means injective, i.e. one-toone, correspondence between logical and physical states.
When we process information the reversibility guarantees
that no information is lost. But, erasure of information
is irreversible! The mapping is not one-to-one, instead it
maps all physical states into the same logical state, which
we refer to as the standard state. It corresponds to a blank
sheet of paper. Landauer argued in his erasure principle
that the logical irreversibility must involve dissipation. The
minimum amount of dissipation equals the energy
kB T log 2
Exorcized demon
per bit. This being true, the demon’s efforts to violate
What about our demon then?
Well, as we have the second law have come to an end. In fact, just recently
seen, the decrease in the entropy of Szilard’s en- A. Bérut et al. (2012) have demonstrated experimentally
gine leads to an increase demon’s knowledge about that the Landauer limit of the information erasure can be
the location of the molecule.
Further on, this in- reached but not exceeded.
51 We will discuss more about fluctuations later when we consider
formation can be then converted into useful work,
as shown recently by S. Toyabe et al.
(2010). non-equilibrium statistics.
50
10. Interacting Systems
Quantum entropy
What about the Universe then? Is it erasing information
at the Landauer rate per bit towards the unavoidable heat
death, or is it computing reversibly its following step? It
is important to realise that non-zero entropies of isolated
systems are related to our ignorance about the state of the
system. The von Neumann definition,
The discussion so far has been about systems consisting of non-interacting entities. For a real connection between the theory and experiments, one has to include the
interparticle interactions operating in the system. The interactions are the reason why the equations in many-body
physics become intractable. One possibility to include the
interactions is in terms of the so-called cluster expansion.
S = −kB Tr(ρ̂ ln ρ̂),
(95)
In low-density gases, the non-interacting system can serve
is the fundamental version of the entropy, since it is quan- as an approximation. The physical quantities can then
tum mechanical, as is the nature. One should notice, how- be written in terms of series expansions in terms of twoever, that the von Neumann entropy is zero for pure states particle potentials, which provide the corrections to the
and non-zero for mixed states, but in both cases indepen- main terms arising from the non-interacting solution.
dent in time for isolated systems (exercise). This means
Another approach we are going to discuss is that of secthat if we can follow the microscopic evolution of a closed ond quantization.
system, our ignorance of the state does not change. The
ignorance arises only from our lack of knowledge on how
10.1 Cluster Expansions
the system was build.
Mayer et al. (1937 onwards) developed a systematic
Consider then the whole Universe. By definition, it is
isolated, and regardless of whether the Universe is in a method to carry out a cluster expansion in real gases obeypure or in a mixed state, its entropy has to be a constant! ing classical statistics. The generalization to interacting
This makes the heat death of the Universe seem consider- quantum gases was initiated by Kahn and Uhlenbeck, and
ably exaggerated. However, it is impossible to determine completed by Lee and Yang. The interplay between quanthe state of the whole Universe at microscopic level, let tum statistical effects and the effects due to interactions
alone calculate its future evolution from the Schrödinger leads to involved mathematics and will be omitted here.
equation. Therefore, the constancy of its entropy should
Consider a gas obeying the Hamiltonian
be out of reach of observations.
N
X
X
p2i
Consider then us making observations, which can only
+
uij ,
H=
2m i<j
cover a tiny fraction of the whole. Moreover, since the small
i=1
region of which we have knowledge, or want to study, is
unavoidably interacting with a part that is unknown to us, where uij = u(rij ) is the interaction potential between parwe cannot accurately call it isolated. Thus, if we measure ticles i and j, and rij = |ri −rj | is the distance between the
an increase in the thermodynamical entropy (in accordance particles. One example of such interaction is the Lennardwith the second law) it unavoidably has to be consequence Jones potential on p. 13.
of a decrease in entropy of the environment. This is the
In the classical limit, the canonical partition function is
only way to keep the total entropy of the Universe constant.
Z
1
dω −βH
Finally, when we derived the H-theorem in the exercises
Z(T, V, N ) =
e
3N
N
!
h
we introduced the transition rates between different states.
Z
Z
P
P
p2
These would not be present in an isolated system, and
1
i
−β i<j uij
3N
−β N
3N
i=1 2m
=
d
pe
d
re
can be seen as resulting from the (weak) coupling to the
N !h3N
environment. In addition, the equations are coarse-grained,
1
=
QN (T, V ),
i.e. we have wiped out the quantum coherence between the
N !λ3N
T
states. As a result, we see that the apparent irreversibility
that we observe in nature arises from our lack of exact where we have carried out the momentum integration and
knowledge on the coupling between the system and the defined the configuration integral
rest of the Universe, and on the coherences between the
Z
P
quantum states, thus resolving Loschmidt’s paradox.
QN (T, V ) = Q(T, V, N ) = d3N re−β i<j uij
Z
Y
References
= d3N r
e−βuij ,
i<j
This chapter is partly based on the following articles:
where the product is taken over all pairs of particles (ij)
(there are N (N − 1)/2 such pairs). For ideal gas, uij = 0
and
VN
Z(T, V, N ) =
N !λ3N
T
• K. Maruyama et al., Rev. Mod. Phys 81, 1 (2009).
• S. Toyabe et al., Nat. Phys. 6, 988 (2010).
• A. Bérut et al., Nature 483, 187 (2012).
• S. Popescu et al., Nat. Phys. 2, 754 (2006).
which is the same as Eq. (57) obtained previously.
51
Cluster expansion
The grand potential allows us to calculate thermodynamic quantities, such as the pressure
We would like to calculate the configuration integral QN .
∞
For this, we expand it in terms of Mayer functions
X
∂ΩG
bl (T )eβlµ
p
=
−
=
k
T
B
−βuij
∂V T,µ
λ3l
− 1.
fij = f (rij ) = e
T
l=1
and the average density of particles
∞
X
1 ∂ΩG
lbl (T )eβlµ
ρ=−
.
=
V
∂µ T,V
λ3l
T
u ij
2
u ij
l=1
1
By recalling the virial expansion (39)
f ij
pV = kB T [ρ + B2 (T )ρ2 + B3 (T )ρ3 + · · · ],
r ij
(96)
we can show that the virial coefficients Bk are now functions of the cluster integrals, e.g. (exercise)
-1
Z
We
1
3
−βu(r)
d
B
(T
)
=
−b
(T
)
=
r
1
−
e
2
2
see that the Mayer functions fij are bounded everywhere
2
and have the same range as the potential uij (we have
2
B3 (T ) = 4b2 − 2b3
used the Lennard-Jones potential). For the evaluation, we
B4 (T ) = −20b32 + 18b2 b3 − 3b4
make the expansion
Z
..
Y
.
QN = d3 r1 · · · d3 rN
(1 + fij ).
i<j
Ursell showed that the grand partition function can be
written in terms of cumulant expansion as
van der Waals equation
For very dilute gases, two-body clusters give the dominant contribution for the interactions. The interactions
between atoms in real gases can be well approximated by
the Lennard-Jones potential, as we have discussed. Accordingly (see p. 13), we assume that
∞
X
1
eβN µ QN
3N
N
!λ
T
N =0
X
Z
∞
1 βlµ
3
3
= exp
e
d r1 · · · d rl Ul (r1 , . . . , rl ) ,
l!λ3l
T
l=1
ZG =
• the interaction is hard core, i.e. strongly repulsive
when the distance between atoms r . rm .
where Ul (r1 , . . . , rl ) are called cluster functions or Ursell
functions. The cluster functions can be obtained by expanding the cumulant expansion in powers of λ−3
T exp(βµ),
leading into
• on the average, the interaction is attractive when
r & rm , but weak compared with the temperature,
i.e. βu(r) 1.
U1 (r1 ) = 1
Thus,
U2 (r1 , r2 ) = f12
U3 (r1 , r2 , r3 ) = f12 f13 + f12 f23 + f13 f23 + f12 f13 f23
..
.
e−βu(r) =
0
1 − βu(r)
when r . rm
when r & rm
The second virial coefficient becomes
Z ∞
B2 = 2π
drr2 1 − e−βu(r)
We thus see that Ul depends on all connected clusters of l
particles.
0
Virial expansion
≈b−
We define the cluster integral as
Z
1
bl =
d3 r1 · · · d3 rl Ul (r1 , . . . , rl ).
l!V
where
b=
Thus, the grand potential becomes explicitly proportional
to V :
ΩG (T, V, µ) = −kB T ln ZG (T, V, µ) = −V kB T
λ3l
T
2π 3
r
3 mZ
∞
a = −2π
drr2 u(r) > 0.
rm
∞
X
bl (T )eβlµ
l=1
a
,
T
. In the above approximation, we have ended up with the
van der Waals equation.
52
10.2 Second Quantization
where Ĥ1 is the non-interacting single-particle Hamiltonian, and Û is the potential operator for the two-particle
Another way to introduce the interactions to the manybody systems is by a formalism known as the second quan- interactions. By applying54the second quantization, we can
tization. In short, the distinction between the first and write the Hamiltonian as
X
1X
the second quantization is equivalent to that between the
Ĥ =
hi|Ĥ0 |jiâ†i âj +
hij|Û |kliâ†i â†j âl âk .
Hermite polynomials, and the creation and annihilation
2
ij
ijkl
operators together with number (Fock) states in the single
harmonic oscillator problem. The first quantization can be In the above, we have defined the matrix elements
seen as the quantization of the motion of the elementary
Z
particles. The second quantization quantizes also the elec- hi|Ĥ |ji = d3 rψ ∗ (r)Ĥ (r)ψ (r)
0
0
j
i
tromagnetic field, and even the particles. It is the starting
Z Z
Z
point of the microscopic (BCS-)theory of superconductivhij|Û |kli =
d3 r d3 r0 ψi∗ (r)ψj∗ (r0 )Û (r0 − r)ψl (r0 )ψk (r),
ity52 , the quantum field theories and the quantum electrodynamics, the theory of interaction between matter and
where ψi are the single-particle wave-functions defined in
radiation.
Eq. (78).
We have already introduced most of the essential conFor convenience, one often defines the so-called field opcepts.
In a many-body system, the wave functions
erators
are (anti-)symmetric superpositions of products of singleX
particle wave functions (see Eqs. (80-81)). The first step
ψ̂(r) =
ψi (r)âi ,
in second quantization is to change from the wave function
i
X
representation into the abstract number basis with the defψ̂ † (r) =
ψi∗ (r)â†i .
inition (82). This makes the calculations more convenient
i
since the number basis does not carry along the unphysical
information on the particle locations, which are not sensi- This way the Hamiltonian operator in the second quantible quantities since the particles are indistinguishable.
zation formalism can be written as
Z
Next, the vacuum state is defined as
Ĥ = d3 rψ̂ † (r)Ĥ0 (r)ψ̂(r)
|0, 0, . . .i ≡ |0i.
Z Z
1
This is a unique state that contains no particles. With
d3 rd3 r0 ψ̂ † (r)ψ̂ † (r0 )Û (r0 − r)ψ̂(r0 )ψ̂(r).
+
2
the application of the creation operators, one can write an
arbitrary number state in terms of the vacuum state as
We notice that this resembles an expectation value of the
Hamiltonian operator, but instead of wave functions we
1
† n1
† n2
â
â2
· · · |0i.
|n1 , n2 , . . .i = √
take calculate it with respect to the field operators. Hence
n1 !n2 ! · · · 1
the name second quantization.
We have also defined before the number operator of the ith
The field operators obey the (anti-)commutation relasingle-particle state as
tions
for all r and r0
n̂i = â†i âi .
[ψ̂(r), ψ̂(r0 )]± = [ψ̂ † (r), ψ̂ † (r0 )]± = 0
The number states are eigenstates of the number operators
[ψ̂(r), ψ̂ † (r0 )]± = δ(r − r0 ),
n̂i |n1 , . . . , ni , . . .i = ni |n1 , . . . , ni , . . .i,
where the eigenvalue ni is the number of particles in the where minus (plus) sign stands for (anti-)commutator for
corresponding single-particle state. Recall, that for bosons bosons (fermions). A straightforward calculation gives the
ni = 0, 1, 2, . . ., and for the fermions ni = 0, 1. The opera- number operator in terms of the field operators as
Z
tor for the total number of particles in the system is given
by
N̂
=
d3 rψ̂ † (r)ψ̂(r).
X
N̂ =
n̂i .
i
One can also show that
The second quantization equals to writing every operator
in terms of the creation and annihilation operators53 . Especially, we assume that the interacting many-body Hamiltonian operator is
Ĥ =
N
X
i=1
Ĥ0 (ri ) +
X
Û (ri − rj )
[Ĥ, N̂ ] = 0.
In general, an arbitrary many-body operator
 =
N
X
Â(ri ),
i=1
i<j
52 Further
54 We omit the lengthy derivation here. An interested reader can
find it from Fetter and Walecka: Quantum Theory of Many-Particle
Systems.
information, see Superconductivity course.
that creation and annihilation operators obey the (anti)commutation relations for bosons (fermions).
53 Recall,
53
can be written in second quantization as
X
 =
hi|Â|jiâ†i âj
is the momentum transfer during the collision.
In the case of half-integer spin particles, one has to label
the annihilation and creation operators also with the zcomponent of the spin. Finally, the Hamiltonian can be
written in the second quantization as
ij
Z
=
d3 rψ̂ † (r)Â(r)ψ̂(r).
Ĥ =
X
εk â†kσ âkσ +
kσ
Plane wave expansion
1 XXXX
U (k2 − k4 ) (97)
2V
k1 σ k2 λ k3
k4
δk1 +k2 ,k3 +k4 â†k1 σ â†k2 λ âk4 λ âk3 σ .
×
If we assume that there is no external potential, the nonTo simplify the notations, we have assumed that the pointeracting part of the Hamiltonian is of the form
tential does not change the spins of the particles. In such
~2 2
cases, one should include also the conservation of spin into
Ĥ0 = −
∇ .
2m
the calculation of the matrix elements of the interaction.
In the case of weak interactions, a good choice for the
single-particle states are plane waves
1
ψk (r) = √ eik·r .
V
The Fourier transform of the potential defines the socalled scattering amplitude 55
This gives the kinetic energy matrix elements (now i → k0
and so on)
Z
0
1
~2 k 2 ik·r
~2 0 2
hk |∇ |ki =
d3 re−ik ·r
e
= εk δkk0 ,
−
2m
V
2m
a(k) =
2 2
a λT and ρa3 1.
In this limit, the particles are slow, k ≈ 0 and the scattering
length is
Z
mU0
where U0 = d3 rU (r).
a=
4π~2
Correspondingly, the matrix elements of the potential
energy operator are
Z Z
1
hk1 k2 |Û |k3 k4 i = 2
d3 rd3 r0 e−i(k1 −k3 )·r
V
For bosons, we can omit the spin-summations in Eq. (97)
and we obtain in the low-temperature limit (recall k =
k2 − k4 = −(k3 − k1 ) = 0)
X
2πa~2 X † †
Ĥ =
εk â†k âk +
ak ak ak ak
mV
k
k
X † †
† †
+
ak ak0 ak ak0 + ak0 ak ak0 ak .
i(k4 −k2 )·r0
× U (r − r)e
Z
1
d3 re−ik·r U (r)
=
V
1
≡ U (k),
V
where
Z
U (k) =
m
U (k),
4π~2
which defines the length scale of the scattering process.
Now, we are interested in the low-temperature behaviour,
meaning that we assume
k
where εk = ~2m
. Recall, that we have applied the periodic boundary conditions in the normalization of the plane
wave, which implies that the wave vectors k are quantized,
and thus we can apply the Kronecker delta function.
0
Low-temperature behaviour of interacting Bosegas
k6=k0
d3 rU (r)e−ik·r
By employing the properties of the annihilation and creation operators, one can write the Hamiltonian into form
(exercise)
X X
2πa~2
2N̂ 2 − N̂ −
n̂2k .
Ĥ =
εk n̂k +
mV
is the Fourier transformation whose inverse is defined as
1 X
U (r) =
U (k)eik·r .
V
k
k
k
Note, that the matrix elements give the probability amplitudes for the collision between single particles in momenThe energy eigenvalues are thus of form
tum states k3 and k4 . In the collision, total momentum
X
X has to be conserved, i.e. the momentum in the scattered
2πa~2
2
E{n
}
'
n
ε
+
2N
−
n2k ,
k
k k
states has to be equal to that of the colliding states
mV
k
k1 + k2 = k3 + k4 .
k
where the dependence of the eigenvalue on the distribution
between different wave vectors has been explicitly implicated.
This has been applied in the last equality. In the above,
k = k2 − k4 = −(k1 − k3 )
55 Compare
54
with Quantum Mechanics II.
For the ground state, the distribution set becomes
N for k = 0
nk =
0 for k 6= 0.
This means that the interactions lead to non-zero chemical
potential giving a correction to the fugacity
zc ' 1 + 4g3/2 (1)
This gives the ground state energy
E0 =
2πa~2 N 2
,
mV
pressure
p0 = −
∂E0
∂V
=
N
2πa~2 ρ2
,
m
and chemical potential
∂E0
4πa~2 ρ
µ0 =
=
.
∂N V
m
We see that the ground state properties of the interacting
Bose-gas at low temperatures are given by the scattering
length and density.
Generally, the thermodynamics can be calculated from
the partition function
X
Z(T, V, N ) =
exp(−βE{nk })
{nk }
=
X
exp − β
{nk }
X
k
n2 2πa~2 N 2
2 − 02
εk nk +
mV
N
.
The lowest order corrections to the ground state results,
that arise from the low but finite temperatures, can be
obtained by recalling the Eq. (90) for the ideal gas:
g3/2 (1) 1/3
λ3
n0
= 1 − 3c where λc =
N
λT
ρ
=1−
v
λ3T
1
vc =
.
where v =
vc
ρ
g3/2 (1)
By using this as an estimate for the relative population in
the ground state, one obtains the Helmholtz free energy
A(T, V, N ) = −kB T ln Z
= Aig (T, V, N ) +
2πa~2 N
m
1
2
v
+
− 2 ,
v vc
vc
where Aig is the free energy of the ideal Bose gas. The
thermodynamics is obtained from the free energy
∂(A/N )
2πa~2 1
1
∂A
=−
= pid +
+ 2
p=−
∂V T,N
∂v
m
v2
vc
T
∂A
4πa~2 1
1
µ=
= µig +
+
.
∂N T,V
m
v vc
These should be compared with the ground state results
when vc → ∞.
The condensation starts when v = vc and λT = λc ,
leading to
4πa~2
g3/2 (1)2
mλ6c
a
µc = 4g3/2 (1)kB Tc .
λc
pc = pig +
55
a
.
λc
11. Phase Transitions
The phenomena to which statistical physics has been
applied can be divide roughly to two categories. So far,
we have studied the first category, in which the constituting systems can be considered non-interacting. It includes specific heats of gases and solids, chemical reactions,
Bose-Einstein condensation, blackbody radiation, paramagnetism, elementary electron theory of metals, to name a
few. The most significant feature of this category is that
the thermodynamic functions of the systems involved are
smooth and continuous56 .
The second category can be characterised with discontinuous or singular thermodynamic functions, corresponding to phase transitions. It includes the condensation of
gases, the melting of solids, coexistence of phases, the behaviour of mixtures and solutions, ferromagnetism, antiferromagnetism, the order-disorder transitions, superfluid
transition from He-I to He-II, the normal to superconducting transition, and so on. These phenomena are characterized by the fact that the interactions between the constituents play an important role. The co-operation leads
to macroscopic significance at a particular critical temperature Tc . We try here to capture the aspects of these cooperative processes in terms of simplified models.
The regions with zero slope correspond to coexistence
of two or more phases with different densities, implying the onset of phase transition. The positive slopes
in the (p, v)-isotherms are non-physical, and result always from the approximations made in the derivation
of the equation of state58 . Such regions should be corrected with the Maxwell construction of equal areas.
• The exact partition function with finite N corrects the
singularities resulting from such corrections made for
the approximate isotherms at the limit N → ∞.
11.2 Condensation of a van der Waals Gas
Let us start with the simplest model for an interacting
gas, the van der Waals gas, obeying the equation of state
p=
a
RT
− ,
v − b v2
where v is the molar volume of the gas, and thus a and b
pertain to one mole of the gas. At the critical temperature,
both (∂p/∂v)T and (∂ 2 p/∂v 2 )T vanish. This leads to the
coordinates of the critical point
pc =
8a
a
, vc = 3b , Tc =
,
2
27b
27bR
11.1 General Remarks on Condensation
implying
RTc
8
We study a system with N particles, and assume that
K=
= .
pc vc
3
the potential can be written as a sum of two-particle terms

This allows us to write the equation of state in a universal
u(r) = ∞
for r ≤ σ

form by using reduced variables
∗
0 > u(r) > −ε for σ < r < r

∗
u(r) = 0
for r ≥ r .
p
v
T
pr =
, vr =
, Tr =
.
pc
vc
Tc
Thus, each particle can be seen as a hard sphere with the
diameter σ, surrounded with attractive potential of range
This implies that the reduced equation of state becomes
r∗ and depth −ε. Note, that the Lennard-Jones potential
meets these requirements up to a very good approximation.
3
(3vr − 1) = 8Tr .
p
+
r
We expect that the exact interparticle potentials lead to
vr2
phenomena that are, at least qualitatively, similar to those
obtained from our simplified form.
p
There are some generally accepted properties that the
exact partition function Z(T, V, N ) has to display the following:
critical point
• In the thermodynamic limit, f (v, T ) = N −1 ln Z =
−A/N kB T tends to depend only on v = V /N and T .
Thus, it is an intensive variable, and the thermodynamic pressure is then given by
∂f
∂A
= kB T
,
p(v, T ) = −
∂V T,N
∂v T
turning out to be a strictly non-negative quantity.
56 With
pc
T=Tc
T<Tc
p(T)
coexistence
curve
vl vc
• Function f is concave, implying57
∂p
≤ 0.
∂v T
57 This
T>Tc
vg
Below the critical temperature, the van der Waals
isotherms display regions with positive slope, i.e.
58 This and the following sentence will be clarified in the next section where we study these properties in terms of van der Waals gas.
a sole exception of BEC.
is relevant for the mechanical stability.
56
v
(∂p/∂v)T > 0. As stated, these regions should be corrected due to their non-physical nature. We replace them
with horisontal lines. In the phase diagram (((T, p)-plane),
these indicate the locations of phase transitions. The replacements are made in terms of Maxwell’s construction
of equal areas. The requirement for equal areas follows
simply from the requirements dp = 0 and dT = 0 for the
corrected isotherm, that imply the change in the (molar)
Gibbs energy
dg = −sdT + vdp = 0.
Where the actual value of constant is unimportant, and can
be used as the zero of energy. Interactions with Jij > 0
favour the parallel alignment, and can display ferromagnetism. If Jij < 0, the situation is reversed and we see the
possibility of antiferromagnetism.
The most important interactions are assumed to occur
between the nearest neighbours. As a result, the Hamiltonian of the N spin system can be written as
X
X
H=−
Jij si · sj − µB ·
si
i
hiji
The same result should be obtained by integrating the theoretical isotherm, i.e. the quantity dg = vdp, from the state where hiji indicates that the summation goes over all nearvl to state vg .
est neighbour pairs. Also, we have denoted the external
The locus of points vl (T ) and vg (T ) is called the coexis- magnetic field with B. In the following, we simplify this
tence curve. Within the region enclosed by this curve the even further by assuming that the strength of the interaction does not vary between the lattice sites, i.e. Jij = J.
gaseous and liquid phases mutually coexist.
Also, we will restrict the spin quantum number to the case
The critical behaviour of the van der Waals system difsi = 12 , and take into account only its z-component. This
fers in several respects from that of real systems. Neverallows us to treat the system classically, since all variables
theless, it is still used as the first approximation against
in the Hamiltonian commute. We end up with the Ising
which the more sophisticated theories are compared.
model
X
X
H = −J
σi σj − µB
σi ,
(98)
11.3 Ising Model of Phase Transitions
hiji
i
Next, we present a model that provides a unified theo- where σi = +1 for ”up” spin and −1 for ”down” spin.
retical basis for understanding (anti-)ferromagnetism, gasThe Ising model can be solved exactly in one and two
liquid and liquid-solid transitions, order-disorder transi- dimensions. Systems analogous to Ising model are, e.g.
tions in alloys, and so on. We will discuss the model in
terms of ferromagnetism, but the formulation can be gen• binary mixture composed of two species of atoms, A
eralized to the languages of the other physical phenomena.
and B, where each lattice point is occupied either type
of atom.
We assume that the atoms are bound to lattice sites,
implying that the spin degrees of freedom are independent
• lattice gas, where each lattice point is either occupied
on other degrees of freedom
by an atom or empty.
H ≈ Hspin + Hother .
Mean field approach to the Ising model
This implies that we can factorize the partition function
Before we solve the Ising model exactly, let us introduce an approximative way to handle the pairwise interactions. We make the mean field approximation, which
Because of the exchange interaction 59 , there is an energy
approximates the interaction energy between a lattice site
difference between the state of parallel spins and the one
and its nearest neighbours by assuming that the neighbours
of antiparallel spins, given by
all have spin hσi, the average spin per site. This leads to
the mean field Hamiltonian
ε↑↑ − ε↑↓ = −2Jij .
Z = Zspin Zother .
Because
H=−
1
(si + sj )2 − s2i − s2j
2
1
= S(S + 1) − s(s + 1)
2
1
if S = 1
4
=
− 34 if S = 0
si · sj =
N
N
N
X
X
1X
νJhσiσi − µB
σi = −
E(J, B)σi ,
2 i=1
i=1
i=1
where E(J, B) = 12 νJhσi + µB, and ν is the number of
nearest neighbours. Also, the prefactor 21 guarantees that
every pair is counted only once. The expectation value hσi
has to be determined in a self-consistent manner.
The partition function of a single spin is
X
Z=
eβEσi = 2 cosh(βE).
we can write the interaction energy as
εij = const. − 2Jij (si · sj ).
σi =±1
59 The exchange interaction arises as manifestation of the interplay
between the Coulomb repulsion and the Pauli principle between two
electrons. Interested reader can refer to the course of Condensed
matter physics.
57
Thus, the probability that the site i has the spin σi is
P (σi ) =
eβEσi
.
2 cosh(βE)
Again, the average magnetization of the lattice is
Exact calculation of one-dimensional Ising model
In practice, the only exactly solvable models are the one
and two dimensional Ising models. The one-dimensional
Ising model was first solved by Ising himself (1925). We
assume periodic boundaries of the spin chain, i.e.
M = N µhσi,
where
P
σi eβEσi
hσi = Pσi =±1 βEσi = tanh(βE).
σi =±1 e
σN +1 = σ1 .
This eliminates the undesired effects arising from the termination of the chain. Nevertheless, it does not affect the
thermodynamics of the (infinitely long) chain. The Hamiltonian of the system can be written as
If B = 0 we obtain the magnetization
1
hσi = tanh β νJhσi = tanh
2
νJhσi
.
2kB T
N
N
X
X
We see that hσi is the order parameter for the spin lattice.
H = −J
σi σi+1 − µB
σi
When temperature is high, the magnetization is zero since
i=1
i=1
the spins are randomly oriented. At lower temperatures,
the spins align spontaeously resulting in non-zero magnetization. We can determine the critical temperature Tc below where σ = ±1. Now, the partition function can be written
i
which the lattice becomes more and more ordered. This is as
calculated graphically by plotting hσi and tanh(αhσi) into
XX X
PN
PN
the same graph as a function of hσi. Here α = νJ/2kB T .
Z=
···
eβJ i=1 σi σi+1 +βµB i=1 σi
σ1
Α>1
=
1
XX
σ1
Α<1
- Σ0
Σ0
σ2
σN
···
σ2
N
XY
1
eβJσi σi+1 + 2 βµB(σi +σi+1 )
σN i=1
We define the 2 × 2 transition matrix
X Σ\
0
1
0
Tσσ0 = eβJσσ + 2 βµB(σ+σ ) ,
where σ, σ 0 = ±1. The partition function is then
XX X
Z=
···
Tσ1 σ2 Tσ2 σ3 · · · TσN σ1
-1
σ1
σ2
σN
X
The solutions are found at the intersections of the curves.
=
T N σ1 σ1 = Tr T N .
We find that when α < 1, there is only one intersection at
σ1
hσi = 0. Similary when α > 1 there exists three intersections at hσi = 0 and ±σ0 . One can show that the free The transition matrix
energy per site is
β(J+µB)
e
e−βJ
T =
e−βJ
eβ(J−µB)
G(J, 0)
−kB T ln 2
if
hσi = 0
=
1
−kB T ln[2 cosh( 2 βνJσ0 )] if hσi = ±σ0
N
which is clearly symmetric. Its eigenvalues are given by
q
Thus, hσi = ±σ0 are the possible equilibrium states since
2
βJ
−4βJ
λ± = e
cosh(βµB) ± sinh (βµB) + e
.
they minimize the free energy. There is the phase transition
(critical point) at α = 1, implying the critical temperature
Tc =
The trace is independent on the basis, and thus
νJ
.
2kB
N
Z = λN
+ + λ− .
We have thus shown that in the mean field approximaAlso,
tion, there is a phase transition at finite temperature for
N d-dimensional lattice. In the following, we will show that
λ−
.
ln Z = N ln λ+ + ln 1 +
the mean field theory leads to incorrect result in the case
λ+
of one-dimensional lattice. In general, one can show that
for d ≤ 3, the mean field calculation gives too high esti- In the thermodynamic limit (N → ∞),
mates for the critical temperature. For dimensions d ≥ 4,
N
the estimates are good but have only mathematical, rather
λ−
→ 0.
than physical, interest.
λ+
58
Thus,
Also, the specific heat diverges (logarithmically) at the critical point T = Tc , and the phase transition is continuous.
The two-dimensional Ising model is perhaps the simplest
exactly solvable model that displays a phase transition.
ln Z ≈ N ln λ+ ,
meaning that the larger eigenvalue determines the thermodynamic properties.
The free energy per spin is given by
11.4 Critical Exponents
G
kB T
=−
ln Z
N
N
q
= −J − kB T ln cosh(βµB) + sinh2 (βµB) + e−4βJ .
One way to classify phase transitions is the quantity m
called the order parameter. As in the case of magnetization, the order parameter usually is assigned to to the
expectation value of some observable of the system (such
as the average spin in the case of magnetization). AnaloAgain, thermodynamic quantities can be calculated from gously to µB/k T in the magnetization case, we identify
B
the free energy. In particular, the average spin per particle h as the ordering field coupled to the observable.60 In the
is
low-field limit (h → 0), m tends to limiting value m0 which
is 0 when T ≥ Tc and 6= 0 for T < Tc . We say that the
∂G/N
sinh(βµB)
σ ≡ hσi = −
=q
.
phase transition is continuous, if the order parameter ap∂µB T
sinh2 (βµB) + e−4βJ
proaches the zero value continuously. On the other hand,
if the change at the critical temperature is finite, the phase
61
We saw in the mean field approximation that the expec- transition is discontinuous. We consider here continuous
tation value σ behaves as an order parameter: σ = 0 corre- phase transitions.
sponds to completely stochastically oriented spins, whereas
It is typical for a continuous (second order) phase transi|σ| = 1 corresponds to the case of parallely aligned spins. tion that the system goes from high temperature phase to
As previously, the magnetization is
a low temperature phase with lower symmetry. This is referred to as the spontaneous symmetry breaking. Examples
M = N µσ.
include:
Clearly, when B → 0 also M → 0 for finite temperatures
T . This rules out the possibility of spontaneous break in
the symmetry between the opposite spins. Thus, we cannot
observe spontaneous magnetization, and hence the phase
transition, contrary to the mean field approach!
• Structural formation of crystal lattice.
• Ferromagnetic phase transition.
• Superconducting and superfluid transitions.
From the low-field susceptibility
N µ2 2J/kB T
∂M
=
χ0 (T ) = lim
e
,
B→0
∂B T
kB T
• Symmetry breaking of the electroweak interaction.
we see immediately that a ferromagnetic (J > 0) system
magnetizes strongly at low (but finite) temperatures. Nevertheless, when the external field is removed, the system
returns to the disordered state with σ = 0. Thus, the onedimensional Ising chain is a paramagnetic system without
any phase transitions. However, it does not obey the Curie
law (M 6∝ 1/T ) so it is not a Curie paramagnet.
The low-field susceptibility χ0 diverges when T → 0.
This means that there exists a phase transition at absolute
zero, corresponding to complete alignment of the spins.
The order parameter, various response functions (such
as the susceptibility) and correlation functions display nonanalytic behaviour, such as singularities, near the critical
point. The singular parts are with great accuracy proportional to some powers of the thermodynamic quantities h
and T − Tc . The critical exponents are defined as follows
• The behaviour of the order parameter is determined
by the exponent β
m(T ) ∼
0
(Tc − T )β
when T > Tc
when T < Tc .
Two-dimensional Ising model
• The divergence of the low-field susceptibility when
The two-dimensional Ising model can also be solved exT → Tc is given by the exponents γ (above) and γ 0
actly by generalizing the transition matrix method (On60 For a gas-liquid transition, one can choose the density difference
sager, 1944). The mathematics involved in the calculation are rather challenging, unfortunately. Therefore, we (ρl − ρc ) as the order parameter and the pressure difference (p − pc )
only give the results here. It turns out that in the two- as61the ordering field.
Typically, the order parameter is related to thermodynamic obdimensional case there is a phase transition at temperature servables that are given in terms of first derivatives of the free enTc =
ergy. In our previous terminology then, the continuous transitions
are second order transitions and discontinuous transitions first order
transitions.
2J
√ ≈ 2.269 J.
ln(1 + 2)
59
(below)
χ0 (T ) ∼
∼
∂m
∂h
reflected as disappearance of the odd powers of m0 . The
coefficients may be written as
X
X
X
φ0 (t, 0) =
qk tk , r(t) =
rk tk , s(t) =
sk tk , . . .
T,h→0
− Tc )−γ
−γ 0
(T
(Tc − T )
k≥0
when T > Tc
when T < Tc
k≥0
k≥0
The equilibrium occurs when the free energy is minimized.
The thermodynamic stability requires s(t) > 0.
Within experimental accuracy we have γ = γ .
We expect that the equilibrium is found at m0 = 0 when
• The dependence of the order parameter on the field is T > Tc , and at m0 6= 0 when T < Tc . This is obtained
given by the exponent δ
by choosing r(t) > 0 and r(t) < 0, respectively. Thus,
r(0) = 0. The lowest order approximation gives
m|T =Tc ∼ h1/δ when T = Tc , h → 0.
0
φ0 (t, m0 ) ≈ φ0 (t, 0) + r1 tm20 + s0 m40 + · · · .
• The singular part of the specific heat is determined by
Since we are considering phenomena near the critical point,
the exponents α and α0
we include only the terms written explicitly above.
2 ∂ G
Φ0
Ch (T ) ∼ −T
∂T 2 h
(T − Tc )−α when T > Tc
0
∼
(Tc − T )−α when T < Tc
In practice, we have α = α0 .
t > 0
One can show that the critical exponents for van der
Waals gas (exercise) and for the Ising model in the mean
field approximation are
t = 0 t < 0
-m s
β=
ms
m0
1
, γ = γ 0 = 1, δ = 3, α = α0 = 0.
2
We see that for t > 0 we have one minimum at m0 = 0.
In fact, all mean field theories give this set of exponents
For
t < 0, we obtain two minima whose locations are given
despite the possible structural differences. This suggests
by
that they belong to the same universality class.
p
∂φ0
= 0 ⇒ ms = ± r1 |t|/2s0 .
∂m0
11.5 Landau’s Theory of Phase Transitions
Region between these values corresponds to unstable
states. It should be corrected with a method similar to
In a continuous phase transition, the slope of the free en- Maxwell’s equal areas. Based on the above, the order paergy changes continuously and a symmetry is always bro- rameter depends on the temperature as
ken. As we have discussed, this can be seen as an appearm0 ∝ (Tc − T )1/2 ,
ance of an order parameter in the phase with lower symmetry. All such transitions can be described in terms of mean
field theory presented first by Landau (1937). We have al- implying that the critical exponent β = 1/2 for the mean
ready noticed that the mean field theory does not describe field theories.
all features of continuous phase transitions correctly, but
We can thus write near the critical point
nevertheless gives a good starting point for understanding
(
such transitions.
φ0 (t, 0)
for T > Tc
φ0 (t) ≈
r12
2
φ0 (t, 0) − 4T 2 s0 (Tc − T ) for T < Tc
Landau argued that the critical behaviour of a given sysc
tem can be determined by expanding its free energy in
powers of the zero-field order parameter m0 = m(h = 0) This implies that the critical exponents α = α0 = 0. Also,
as
it implies a jump r12 /2s0 Tc in the specific heat at the critical
point.
φ0 (t, m0 ) = φ0 (t, 0) + r(t)m20 + s(t)m40 + · · ·
T − Tc
Effects of the field and universality
where t =
, |t| 1.
Tc
When the field is non-zero, one benefits of making LegThe free energy should have the same symmetries as the endre transformation
Hamiltonian. Generally, the Hamiltonian remains unchanged (at zero field limit) when m0 → −m0 , which is
φh (t, m) = −hm + φh (t, 0) + r1 tm20 + s0 m40 + · · · .
60
With a straightforward calculation, one can show that the
critical exponents γ = γ 0 = 1. Quick inspection reveals
that the introduction of the field breaks the symmetry in
the system when T > Tc . This means that there is no more
continuous phase transition.
Thus, we have shown that the critical exponents of the
Landau theory, build only upon the analyticity of the free
energy and the symmetries of the Hamiltonian, result in exactly same critical exponents as the previous calculations
made with mean field theory! This means that while the
free energy itself depends on the structure of the system,
the critical exponents do not. This universality of the critical exponents suggests that we are dealing with a class of
systems that display qualitatively same critical behaviour,
despite the structural differences.
Breakdown of Landau theory
The mean field (Landau) theory neglects the fluctuations of the order parameter in large length scales. However, when one approaches the critical point these fluctuations diverge. One often refers them as critical fluctuations. When the fluctuations become larger than the mean
value for the order parameter, the mean field theory breaks
down. In other words, a mean field theory cannot be valid
arbitrarily close to the critical point.
To see this, consider correlations between particles in
the high temperature phase of the Ising model. They can
be characterized with the spin-spin correlation function
g(i, j) = hσi σj i − hσi ihσj i,
(99)
defined for the pair of spins at sites i and j. When i = j,
one obtains the fluctuations (standard deviation) of the
spin σ at site i. When the separation between i and j
increases, one expects that the corresponding spins get uncorrelated and hσi σj i → hσi ihσj i leading to g(i, j) → 0.
2
where σM
= hM 2 i − hM i2 . In the mean field theory, the
susceptibility diverges at the critical temperature. Implying the breakdown of the theory. In some systems, however,
the breakdown occurs only for very small |t|. Examples of
such systems include
• Superfluids: The normal-superfluid transition in He4
is a continuous phase transition which involves broken
gauge symmetry. Order parameter is the macroscopic
wave function Ψ of the condensed phase, and the free
energy is given as
φ(T, P ) = φ0 (T, P ) + r(T, P )|Ψ|2 + s(T, P )|Ψ|4 + · · · ,
where r(T, P ) = r0 (T, P )(T − Tc ) and r0 and s are
slowly varying functions of T and P .
• Superconductors: The condensed phase is a macroscopically occupied quantum state Ψ, serving as the
order parameter. In the normal-superconductor transition the phase of the order parameter changes. The
free energy is given by
Z
ψ(T ) = d3 r ψn (T ) + r(T )|Ψ(r)|2 + s(T )|Ψ((r))|4
1
|i~∇r Ψ(r)|2 .
+
2m
The Ginzburg-Landau theory of superconductivity is
obtained by extremizing the free energy with respect
to variations is Ψ∗ .
11.6 Scaling, Universality and Renormalization Group Approach
Even though the neglect of fluctuations in the mean field
theory leads into quantitatively incorrect results, it neverBy appyling the partition function (98), we obtain
theless produces the correct result that every physical sysX
tem belongs in a specific universality class, based on its
∂
ln Z = βµh
σi i = βhM i,
critical behaviour. In spite of the prediction, the mean
∂B
i
field theory does not provide any reason for universality.
P
where µ i σi is the net magnetization of the system. Sim- For that, we need the scaling theory.
ilarly,
∂2
Widom scaling
ln Z = β 2 (hM 2 i − hM i2 ).
∂B 2
We have described the singular behaviour in the vicinThus, we obtain for the susceptibility
ity of the critical point in terms of the critical exponents.
XX
∂hM i
The free energy can be written in terms of the regular and
χ=
= β(hM 2 i − hM i2 ) = βµ2
g(i, j).
∂B
singular terms as
i
j
This is an important result! It relates the susceptibility
φ(t, h) = φr (t, h) + φs (t, h),
with the fluctuations in the order parameter (magnetization). Also, it tells that the the fluctuations are intrin- implying that the critical exponents must come from the
sically related to the correlations between the individual singular part φs .
spins.
The idea behind the scaling hypothesis is that when
The mean field theory is assumed to be valid if it obeys the system approaches the critical point, the correlation
the so-called Ginzburg criterion
length 62 ξ becomes infinitely large resulting in diminished
2
σM
1 ⇒ kB T χ hM i2 ,
hM i2
62 Correlation length is the measure of the range of the correlations
in the system.
61
sensitivity to length transformations (changes of scale). In
other words, we say that there is no natural measure of
lenght in the system, except the atomic length. The system looks similar no matter what scale is used, and it is
called self-similar. This suggests that the singular part of
the free energy is a generalized homogeneous function, i.e.
it scales like
φs (λp t, λq h) = λφs (t, h),
when λ > 0 and p and q are independent on the system.
Direct calculations give the critical exponents in terms of
p and q (exercise):
1−q
p
q
δ=
1−q
2q − 1
0
γ=γ =
p
1
α = α0 = 2 − .
p
β=
These imply the scaling laws
α + 2β + γ = 2
β(δ − 1) = γ.
We thus see that only two of the critical exponents are
independent.
Kadanoff scaling and renormalization group
Unlike the Widom scaling hypothesis, the method presented by Kadanoff and developed by Wilson is based on
the microscopic properties of matter. We will present the
only the outlines of method:
• Combine the original microscopic state variables
blockwise into block variables.
• Determine the effective interactions between the
blocks. This is coarse graining of the system called
the block transform.
• The block transforms form a semigroup, so called
renormalization group. One can perform transforms
sequentially.
• Because in a system at the critical point has no natural length scale due to diverging correlation length,
the transformed systems look like copies of each other.
Thus, the critical point corresponds to a fixed point of
the transformations.
In general, the renormalization is a collection of techniques to remove the infinities arising from the theory. It
was first developed to deal with the infinite terms in the
perturbative expansions of the quantum electrodynamics.
Such infinities typically arise because when making the theory we have chosen to describe phenomena occuring at a
chosen scale of length, ignoring the degrees of freedom at
62
shorter scales. This kind of theories can be described in
terms of renormalization group, and are called effective
field theories. For example, practically all of the condensed
matter physics consists of effective field theories. Also, the
general relativity can be seen as an effective field theory of
(yet unknown) complete theory of quantum gravity.
12. Non-Equilibrium Statistics
This course has so far dealt with only statistical averages of physical quantities. We have shown that up to
very high accuracy these averages correspond to the relevant measurements made on systems in equilibrium. Nevertheless, there do exists deviations from the average values
called fluctuations. These fluctuations are generally small
in magnitude, but are important for several reasons:
• In the onset of the high degree spatial correlations in
the vicinity of the critical point.
• In understanding how a non-equilibrium state approaches equilibrium, and how the equilibrium persists.
interchangeably in the following). Since the studied particle is large compared to the vast number of smaller ones
constituting the fluid, its movement is very slow, and F (t)
is very rapidly fluctuating function in particle’s time scale.
Thus, we are not interested in the explicit time evolution
of the bath degrees of freedom. What we assume instead
is that in the above equation of motion those are averaged (traced out) quantities. As a consequence, the force
F (t) can be treated as a random function of time t. Additionally, we assume that in equilibrium (v = 0) there is
no preferred direction in space, and the ensemble average
vanishes, hF i = 0.
It is beneficial to write the random force as
F = hF iξ + ξ,
• In relating the dissipative properties of a system where ξ denote the rapidly fluctuating part around the
with the microscopic properties of the equilibrium ensemble average65 hF iξ . Naturally, hξi = 0. In the limit
(fluctuation-dissipation theorem).
of small hvi, one can expand hF i as
hF iξ = −mγhviξ .
In this final chapter, we wish to gain understanding on
the fluctuations in thermal equilibrium, the approach of the
equilibrium, and dissipation. It turns out that all these are We will show later that the coefficient γ is determined by
the properties of the function F (t) itself. For the moment,
intimately connected.
it will be taken as a phenomenological constant of friction.
One sees that the effect of the slowly varying component
12.1 Brownian Motion
hF iξ of the fluctuating force is causing relaxation of a parBrownian motion of a relatively massive particle im- ticle with an initial velocity v(0) towards the equilibrium
mersed in fluid will provide us the paradigm for describ- v = 0.
ing the equilibrium and non-equilibrium processes. Even
standard deviation Due to the rapidly fluctuating force,
in thermal equilibrium, the Brownian particle exhibits the velocity of the particle also fluctuates in time. Simirandom-like motion. This randomness is a consequence larly to the force, the velocity can be represented in terms
of the unavoidable interactions between the particle and rapid, zero-mean fluctuations v 0 around the ensemble avits surroundings.
erage velocity hvi as
ξ
The situation can be modelled with the Hamiltonian63
v = hviξ + v 0 .
H = Hsys + Hbath + Hint ,
The variations in hviξ are much slower than those in v.
where Hsys , Hbath and Hint are the Hamiltonians for the Altogether, the equation of motion is written as
system (the particle), its surroundings, and the interacdv
tion between them. We will consider, for simplicity, a one
= F(t) − mγv + ξ(t),
m
dt
dimensional particle, and we focus our attention on the location x of the particle. Hamilton’s equations of motion where we have assumed that γv = γhviξ 66 . The above
give
equation of motion is called the Langevin equation. The
Langevin
equation is a stochastic differential equation,
dx
=v
meaning that it contains a random term. The random sodt
lutions of such equations are generally called as stochastic
dv
dHsys
dHint
m
=−
−
≡ F(t) + F (t).
variables.
dt
dx
dx
The Langevin equation can be shown to hold also for
We have assumed above that the system and the bath inthe quantum mechanical operator x̂, in addition to the
teract (linearly) via the variable x. We denote with F
classical expectation value x = hx̂i discussed above. For
the force exerted on the massive particle by some exterfurther details on quantum Langevin equation, refer e.g.
nal potential, and with F the force that describes the inCardiner and Zoller, Quantum Noise.
teractions with the many other degrees of freedom in the
65 In the present case, the non-equilibrium ensemble is formed as
surrounding bath64 (we will use the words bath and fluid
63 Note
that this Hamiltonian describes a very general open system,
i.e. system interacting with a much larger bath.
64 The term noise is often used to describe the fluctuating force
F , since it describes the typically undesired deviations from the behaviour of an isolated system.
63
mental copies of the Brownian particle with different realisations of
the random variable ξ. We will continue of using the notation h·i
for the equilibrium (microcanonical, canonical, grand canonical) averages.
66 Fluctuations in v are slower than those of the force F that produces them. This is because of the large mass of the particle.
The informal derivation of the Langevin equation presented above shows explicitly how irreversibility arises in
Hamiltonian (quantum) mechanics. Consider the combined system of the particle and the bath (fluid). This
constitutes an isolated system, whose governing equations
are reversible and the energy should be conserved. Now,
the Langevin equation covers only the degrees of freedom of
the particle, neglecting the time evolution of the bath. The
effect of the bath on the particle is described with frictional
force −mγv, leading to dissipation of energy from the particle into the bath, and of the random force ξ(t) that causes
fluctuations in the particle degrees of freedom even at the
equilibrium.
Equilibrium fluctuations
We assume for now that the external potential is zero,
and that the particle is in thermal equilibrium with the
bath. Thus, we the Langevin equation (multiplied with x)
gives
d
dẋ
=m
(xẋ) − ẋ2 = −mγxẋ + xξ(t).
mx
dt
dt
Next, we take an ensemble average of both sides, and employ the equipartition theorem to 12 mhẋ2 i = 12 kB T . The
resulting differential equation is easily solvable, leading into
hxẋiξ = Ce−γt +
kB T
,
mγ
which is referred to as the Einstein relation.
We stress that the variance hx2 iξ is caused by the coupling to the environment. In the absence of the coupling,
the particle located initially at x0 stays there forever. However, the constant bombardment of the fluid molecules
cause random ”kicks” to the particle, thus causing the diffusive behaviour.
Correlation functions
Instead of assuming the thermal equilibrium, we can calculate directly the mean square velocity of the Brownian
particle (assume again that F = 0). First of all, by integrating the Langevin equation we obtain
Z
1 t −γ(t−t0 ) 0 0
e
ξ(t )dt
(101)
v(t) = v(0)e−γt +
m 0
It should be noted that the solution of the above equation
holds for a single realisation of ξ. Because ξ(t) is a random
variable, also v(t) and x(t) are random variables whose
properties are determined by ξ(t). This leads into
Z
2 −2γt t γt0
2
2
−2γt
hv (t)iξ =v (0)e
+ e
e hξ(t0 )iξ dt0
m
0
Z Z
1 −2γt t t 0 00 γ(t0 +t00 )
dt dt e
hξ(t0 )ξ(t00 )iξ .
+ 2e
m
0
0
The second term in the above equation is zero because
hξ(t)iξ = 0 for all t.
where C is a constant of integration. We thus see that
The function hξ(t0 )ξ(t00 )iξ appearing in the above equa−1
γ is a characteristic time constant of the system. If we tion is of greatest importance to us. It can be seen as the
assume that initially at t = 0 each particle in the ensemble temporal equivalent to the spin-spin correlation function
is at x(0) ≡ x0 = 0, we obtain C = −kB T /mγ. Thus,
defined in Eq. (99). It is a measure of the statistical correlation between the value of ξ at time t0 and its value at time
kB T
−γt
hxẋiξ =
(1 − e ),
t00 . We will refer to it as the (time) autocorrelation function
mγ
of the variable ξ. Generally, we will denote the temporal
and by integrating once more we have obtained the equi- correlation function of the variables A and B with
librium fluctuations (variance, x0 = 0)
CAB (t, t0 ) = hA(t)B(t0 )iξ .
2kB T
h(x(t) − x0 )2 iξ = hx2 iξ =
t − γ −1 (1 − e−γt ) . (100)
In order to proceed with the calculation, one needs thus
mγ
information over the time correlations of ξ. Typically in the
case of Brownian particle, the fluctuating force is assumed
When t γ −1 , we have
to be independent on its previous values67 . In addition, one
kB T 2
2
often assumes that ξ(t) is a stationary process, meaning
hx iξ ≈
t .
m
that its mean and variance does not change in shifts in
This means that at short time intervals the particle behaves time and space. Mathematically this is called Gaussian
as a free particle with the constant thermal velocity v = white noise process68 and expressed as delta-correlation
p
kB T /m. In the opposite limit, t γ −1 , we have
Cξ (t, t0 ) = gδ(t − t0 ),
2kB T
where δ(t) is Dirac’s delta function and g is a constant
hx2 iξ ≈
t.
mγ
given by the equipartition theorem.
Here one should recall the result from the very beginning
of the course, which stated that the variance of a random
walk grows as the number of steps N . Thus, in the long
time limit the particle executes a continuous random walk
and diffuses to the fluid with the diffusion constant
D=
kB T
,
mγ
64
This assumption results in
hv 2 (t)iξ = v 2 (0)e−2γt +
g
1 − e−2γt .
2m2 γ
67 This assumption of short memory is referred as the Markovian
approximation.
68 This terminology will become clear later when we discuss about
spectral density
When t → ∞ we have that hv 2 (t)i → g/2m2 γ. In this
limit, no matter what the initial velocity v(0) is, we expect
that the particle is in equilibrium with the bath. Thus, the
equipartition theorem gives
g = 2mγkB T.
We obtain for delta-correlated noise
kB T
1 − e−2γt .
hv 2 (t)iξ = v 2 (0)e−2γt +
m
By integrating Eq. (101), we obtain
Z t
1
0
v(0)
x(t) = x(0)+
1−e−γ(t−t ) )ξ(t0 )dt0 .
1−e−γt +
γ
mγ 0
Similar to v, we obtain (we set the origin at x(0) = 0)
!2
v(0)
hx2 (t)iξ =
1 − e−γt
γ
Z
2kB T t
ds 1 − e−γ(t−s) ) 1 − e−γ(t−s) )
+
mγ 0
!2
v(0)
=
1 − e−γt
γ
2
1
2kB T
t − (1 − e−γt ) +
(1 − e−2γt )
+
mγ
γ
2γ
!2
v(0)
=
1 − e−γt
γ
+
where W (x, x0 )δt is the probability that within a small time
interval δt the particle makes a transition from the displacement x into x0 . Note, that here we also make a Markovian approximation by assuming that the rate W (x, x0 )
does not depend on the history of the particle.
In the case of Brownian motion, we assume that only
those transitions are possible that are between nearby
states. This means that W (x, x0 ) is sharply peaked at
x0 = x, and rapidly goes to zero away from x. By writing
the master equation in terms of η = x − x0 , and making
a second order Taylor expansion around η = 0, one ends
with (exercise)
1 ∂2
∂
∂f
µ(x)f (x, t) +
D(x)f (x, t) ,
=−
2
∂t
∂x
2 ∂x
which is called the Fokker-Planck equation. In the above,
the first and second moments of the transition rate
W (x, x0 ) are given by
Z ∞
hδxi
= hvi
µ(x) =
ηW (x, x − η)dη =
δt
−∞
Z ∞
hδx2 i
D(x) =
η 2 W (x, x − η)dη =
.
δt
∞
In the case of Brownian particle, we obtained
µ(x) = hvi = 0
kB T
D(x) = 2
≡ 2D.
mγ
kB T
2kB T
t−
(1 − e−γt )(3 − e−γt )
2
mγ
mγ
Thus, the Fokker-Planck equation reduces in
When the initial velocity has the thermal equilibrium value
v 2 (0) = kB T /m, we recover the result (100).
∂2f
∂f
= D 2,
∂t
∂x
which is the one-dimensional diffusion equation. Its solution is a Gaussian
Fokker-Planck equation
Instead of solving the average behaviour of ensemble of
(x−x0 )2
1
Brownian particles, a more generalized way is to find the
f (x, t) = √
e− 4Dt ,
4πDt
rate at which the distribution of Brownian particles approaches the thermal equilibrium. The equation which
with the variance σ 2 = 2Dt. Again, the Gaussian solution
governs the decay of the non-equilibrium distribution is
suggest the interpretation of the diffusion process in terms
called the master equation. Here, we present a simplified
of continuous random walk.
version of it, named the Fokker-Planck equation.
Consider the displacement x(t) of a Brownian particle69 .
12.2 Fluctuation-Dissipation Theorem
Let f (x, t)dx be the probability that an arbitrary particle
In order to complete the discussion, we would still like
from the ensemble is found with the displacement between
a method how to relate the constant of friction γ with the
x and x + dx at any given time t. Naturally,
Z ∞
fluctuations. This is done with the fluctuation-dissipation
theorem, which basically tells that the decay of the equif (x, t)dx = 1.
−∞
librium fluctuations obey the same law than that for a
non-equilibrium state70 . We would like study how a therThe time-evolution of a probability density, such as
modynamic system responses when it is (weakly) forced
f (x, t), is typically governed by the master equation
out of equilibrium. Somewhat surprisingly, the calculation
Z ∞
∂f
turns out to be the simplest using the quantum mechanical
0
0
0
0
=
f (x , t)W (x , x) − f (x, t)W (x, x ) dx ,
density operator approach.
∂t
−∞
69 The following discussion can be done also for any other dynamical
random variable, such as v.
70 This way of stating the fluctuation-dissipation theorem is the
Onsager regression hypothesis.
65
Let us consider the changes to an observable  made where
Z ∞
by a weak external field F (t). The nature of the field is
1
arbitrary, it can be stochastic or deterministic. We assume
χ00AB (ω) =
dth[Â(t), B̂(0)]ieq eiωt ,
2~
−∞
that the field is coupled to the system via the observable
B̂. The Hamiltonian for the system is
and χ0AB (ω) is given by the Kramers-Krönig relation (defined later).
Ĥ = Ĥ − F (t)B̂,
0
where Ĥ0 is the equilibrium state Hamiltonian. Correspondingly, the density operator of the equilibrium state is
given by
exp(−β Ĥ0 )
ρ̂eq =
.
Tr exp(−β Ĥ0 )
Also, the equilibrium expectation value of an arbitrary
(hermitian) operator  is given by
On the other hand, the power spectrum related to temporal correlations between the observables  and B̂ is given
by Fourier transform of the correlation function (this is a
generalization of the Wiener-Khintchine theorem presented
below)
Z ∞
dthδ Â(t)δ B̂(0)ieq eiωt ,
SAB (ω) =
−∞
where the fluctuations are defined as δ Â(t) = Â(t) − hÂieq .
By employing the result (exercise)
Z ∞
The application of the small time-dependent field in- Z ∞
iωt
−β~ω
dth
B̂(0)
Â(t)i
e
dthÂ(t)B̂(0)ieq eiωt ,
=
e
eq
duces a small change to the density operator
hÂieq = Tr(Âρ̂eq ).
−∞
ρ̂(t) = ρ̂eq + δ ρ̂(t).
−∞
we obtain
The time-evolution is given by the von Neumann equation,
which can be approximated as
∂ ρ̂
i
i
= − [Ĥ, ρ̂(t)] ≈ −
[Ĥ0 , δ ρ̂(t)] − F (t)[B̂, ρ̂eq ] .
∂t
~
~
χ00AB (ω) =
1
1 − e−β~ω SAB (ω).
2~
This is the celebrated fluctuation-dissipation theorem that
relates the dissipation due to the perturbative field with the
spontaneous thermal fluctuations that occur in the equilibrium.
We see that in this approximation, the response is linear
with respect to the field. In the interaction picture71 one
can write the solution as
Z
0
0
i t
dt0 F (t0 )e−iĤ0 (t−t )/~ [B̂, ρ̂eq ]eiĤ0 (t−t )/~ .
δ ρ̂(t) =
~ −∞
The classical limit of the fluctuation-dissipation theorem
is given when ~ω kB T :
χ00AB (ω) =
The change in the expectation value (the response) of the
observable is then
hδ Â(t)i = hÂi − hÂieq = Tr Âδ ρ̂(t) .
ω
SAB (ω).
2kB T
Linear response theory
Generally, the linear response theory studies how the
One can show (exercise) that the response can be written system behaves when it is displaced from the equilibrium.
as the convolution
It assumes that the force F (t) driving the system out of
Z
equilibrium is weak, leading to
i t
hδ Â(t)i =
dt0 F (t0 )h[Â(t), B̂(t0 )]ieq ,
Z t
~ −∞
hδ x̂(t)i ≡ hx̂(t) − hx̂ieq i =
dt0 χ(t − t0 )F (t0 )
iĤ0 t/~
−iĤ0 t/~
−∞
Âe
. Thus, we can identify the
where Â(t) = e
Z ∞
linear response function (see the end of this chapter for
=
dτ χ(τ )F (t − τ ),
more information on linear response theory)
0
i
h[Â(τ ), B̂(0)]ieq ,
~
where hx̂ieq is the equilibrium value of the observable x̂,
and χ is the response function that characterises the variable’s response to the force field. The observable x̂ can
where we have employed the knowledge that the equilibhere be taken as the position of the Brownian particle, but
rium correlation depends only on the time interval τ .
can be in general any relevant observable that is randomWe see that the Fourier transform of the response be- ized by the force. The integral can be extended to −∞
comes
by assuming causality, i.e. χ(t) = 0 for t < 0. Thus, we
Z ∞
see that the response of x is a convolution of the response
hδ Â(ω)i =
dteiωt hδ Â(t)i = χ0AB (ω)+iχ00AB (ω) F (ω), function and the force. By taking a Fourier transform, we
−∞
obtain the linear response in frequency space
χAB (τ ) =
71 Unitary transformation ρ̂ = Û † ρ̂Û into the interaction picture
I
is obtained with Û = exp(−iĤ0 t/~).
hδ x̂(ω)i = χ(ω)F (ω).
66
The Fourier transform of the response function is often
referred to as the susceptibility. Note, that the response
function is real since the response δ x̂ is observable quantity.
Nevertheless, the susceptibility is complex
χ(ω) = χ0 (ω) + iχ00 (ω),
This is the Wiener-Khintchine theorem, that relates the
power spectral decomposition of the fluctuations in an observable with the Fourier transform of the autocorrelation
function of the fluctuations. In the above, we have assumed
that the state is stationary, meaning that
Cδx (t + τ, t) = Cδx (τ, 0) ≡ Cδx (τ ).
where
0
Z
∞
χ(t) cos(ωt)
χ (ω) =
−∞
∞
χ00 (ω) =
Z
χ(t) sin(ωt).
−∞
The real part of the susceptibility is sometimes (especially
when one studies linear response of electric circuits) called
the reactive response, and the imaginary part the dissipative response. Especially, in the case of Brownian particle
subjected to white noise, one can show that the susceptibility is
−1
χ(ω) = m(γ − iω)
,
Conclusions
This concludes the course of Statistical Physics. The
first part of the course lays the microscopic foundation to
the classical theory of thermodynamics. The quantum mechanical indistinguishability of identical particles were discussed in the beginning of the latter half of the course. New
phenomena that result from the quantum nature of particles, such as Bose-Einstein condensation and relativistic
One can show (omitted here) that the causality implies electron gas, were introduced. The inclusion of unavoidthat the reactive part can be written in terms of the dissi- able interactions between particles was studied in terms
pative response, or vice versa, as
of cluster expansions and second quantization. Finally, an
introduction to the more elaborate theories of phase transiZ ∞
ω0
2
0
tions and non-equilibrium statistics was given. Throughout
χ00 (ω 0 ) 02
dω
,
χ0 (ω) =
π 0
ω − ω2
the course, the aim has been in learning the fundamental
Z
2ω ∞ 0 0
ω0
statistical concepts that are needed in the more specialised
00
0
χ (ω) = −
dω ,
χ (ω ) 02
π 0
ω − ω2
advanced courses, such as Superconductivity, Condensed
Matter Physics, Quantum Optics in Electric Circuits, and
which are the Kramers-Krönig relations.
so on.
Power spectral density
Consider fluctuation δ x̂(t) = x̂(t) − hx̂i of an observable
x̂. It can be split into spectral components by taking the
(truncated) Fourier transform
Z
T /2
δ x̂(ω) =
dtδ x̂(t)eiωt .
−T /2
This describes the fraction of the fluctuation δ x̂(t) that
is oscillating with the frequency ω. Typically, the power
of the oscillations is related to the square of the variable,
which persuades us to define the equilibrium power spectral
density as
1
h|δ x̂(ω)|2 i.
S(ω) = lim
T →∞ T
Now, we can write72
Z
Z
0
1 T /2 T /2
dtdt0 eiω(t −t) hδ x̂(t0 )δ x̂(t)i
T →∞ T −T /2 −T /2
Z T
T − |τ | iωτ
= lim
dτ
e hδ x̂(τ )δ x̂(0)i
T →∞ −T
T
Z ∞
=
dτ eiωτ Cδx (τ ).
S(ω) = lim
−∞
72 Recall how the change of variables is made with multidimensional integrals!
67