Excerpts of Some
Statistical Mechanics
Lectures Found on the Web
Statistical Mechanics
in the
Canonical Ensemble
Outline of the
Formalism
and
Some Examples
(similar to parts of
Chs. 6 & 7
in Reif’s book)
The Canonical Ensemble:
Outline of the General Formalism
1  UkT
pr  e
Z
r
Z  e

Ur
kT
r
• This is THE CANONICAL DISTRIBUTION & gives
“the probability that a system in contact with a heat
bath at temperature T should be in a particular state”.
• r labels the states of the system. At low temperatures, only
the lowest states have any chance of being occupied. As the
temperature is raised, higher lying states become more and
more likely to be occupied.
• In this case, in contact with the heat bath, all microstates are
therefore not equally likely to be populated.
Canonical Distribution
1
pr  e
Z

Ur
kT
Z  e

Ur
kT
r
• Often, there are huge numbers of microstates that can all
have the same energy. This is called DEGENERACY.
• In this case, the summations are over each individual
energy level rather than sum over each microstate.
1
p(U r )  g (U r )e
Z
U
 r
kT
Z   g (U r )e

Ur
kT
Ur
• The sum is now over each different energy Ur & g(Ur) is
the number of states possessing the energy Ur. The
probability is that of finding the system with energy Ur.
Entropy in the Canonical Ensemble
• The system of interest is in equilibrium with a heat bath for which
the energy of the system fluctuates & the probability of finding
any particular microstate is variable. How can the entropy be
calculated for such a system? If the entropy S can be
found, all thermodynamic variables can be calculated.
• The system is in a heat bath made up of (M-1) replica
subsystems to the one of interest.
• Each subsystem may be in one of
many microstates. The number of
subsystems in the ith microstate is ni.
• The number of ways of arranging n1
systems of microstate 1, n2 systems
of microstate 2, n3….is:
M!
W
 ni !
i
Entropy
S   k  pi ln pi
i
• This is the general definition of entropy &
holds even if the probabilities of each
individual microstate are different.
1
1
S   k  pi ln pi   k  ln  k ln W
W
i
i 1 W
W
Entropy in The Canonical Ensemble
S   k  pi ln pi
i
1
pi  e
Z

Ui
kT
Z  e

Ui
kT
i
• The general Definition of Entropy, in combination
with The Canonical Distribution allows the
calculation of the system thermodynamic properties:
Ui
ln pi  
 ln Z
kT
 Ui

S   k  pi ln pi  k  pi 
 ln Z 
i
i
 kT

1
U
S   piU i k ln Z  pi   k ln Z
T i
T
i
Helmholtz Free Energy
U
S   k ln Z
T
F  U  TS   kT ln Z
• Ū  Average value of the system
Internal Energy.
• (Ū - TS)  Average value of the
Helmholtz Free Energy, F.
• The Partition Function Z is much more than a
mere normalizing factor. Z: Acts as a “Bridge”
linking microscopic physics (quantum states) to
the energy & so to all macroscopic properties of
a system.
Helmholtz Free Energy
F  U  TS  kT ln Z
• F is a state function. Now, the calculation of some
thermodynamic properties of the system.
• Ignore the fact that Ū is an average & let U = Ū.
• Use the definitions for various thermodynamic variables.
For an infinitesimal, quasistatic reversible change:
F  U  TS
dF  dU  TdS  SdT
dF  dQR  PdV  TdS  SdT
dF  TdS  PdV  TdS  SdT   PdV  SdT
dF   PdV  SdT
F  U  TS   kT ln Z
• Using the properties of partial derivatives gives:
 F 

  P
 V T
 F 
 T ln Z 
P  
  k

 V T
 V T
F 
T ln Z 
 F 



   S S  
  k

 T V
 T V
 T V
• So, the energies of the microstates of the
system are linked to thermodynamic variables
such as pressure & entropy.
Various Measurable Parameters:
2nd Derivatives of the Helmholtz Free Energy F
• Elastic Moduli are the stress/strain or force/unit
area divided by the fractional deformation:
 F 
 P 
K  V 

 V
2
 V T
 V T
2
• Heat Capacity at constant volume:
 F 
 Q 
 S 
CV  
  T
  T  2 
 T V
 T V
 T V
2
Mean Internal Energy
• Ū  Thermal Average of the system Internal
Energy. The actual internal energy fluctuates
because the system is interacting with the heat bath.
• How large are the fluctuations? Are they important?
  (T ln Z ) 
U  F  TS   kT ln Z  kT 

 T
V
T ln Z 

2  ln Z
U   kT ln Z  kT  ln Z 
  kT
T V
T

Fluctuations in Internal Energy
• A measure of the departure from the mean is the
standard deviation, as it is in any statistical theory.


2
( U )  U  U  U  U
2


 U   
CV  
 
 T V  T


2
U e

Ui
kT
i
i
e
r

Ur
kT





V
2


2
( U )  U  U  U  U
2

 
CV  
 T


U e

Ui
kT
i
i
e
U
 r
kT
r
2


2
1
2
 
U U
2
kT


V

2
( U )  U  U  kT CV
2
2
2
2

The Variance
The relative fluctuation (U/ Ū)
gives the most useful information.
kT CV
U

U
U
2
• Ū & CV are extensive properties proportional to the size
of the system ~ N ( Number of particles in the system).
U
1

U
N
• For Typical Macroscopic Systems with ~1023
particles, the fluctuations (U/ Ū) ~ 10-11
U
1

U
N
• So, the fluctuations are tiny, which means that U & Ū
can be considered identical for all practical purposes.
• Based on this, it is clear that Macroscopic Systems
interacting with a heat bath effectively have their
energy determined by that interaction.
• Similar relationships can be found for other relative
fluctuations of properties of macroscopic systems.
Summary – Statistical Mechanics
• Microstate – The state of a system defined microscopically
– a complete description on the atomic scale.
• Macrostate – The state of a stystem of macroscopic size
specified by a few macroscopically observable quantities only.
• Statistical Weight (W or ) of a macrostate – is the
number of microstates compising the macrostate.
• Postulate of equal a priori probabilities – “for an
isolated system in a definite macrostate, the W microstates
comprising this macrostate occur with equal probability.
• Equilibrium Postulate – “For an isolated macroscopic
system, defined by U,V,N (which are fixed) and variable
parameters , equilibrium corresponds to those values of  for
which the statistical weight W(U,V,N, ) attains its maximum.”
Boltzmann Definition of Entropy
S (U ,V , N , )  k ln W (U ,V , N , )
Definition of Temperature
1  S (U ,V , N ) 


T 
U
V ,N
Definition of Pressure
 S (U ,V , N ) 
P  T

V

U , N
General Definition of Entropy
S   k  pi ln pi
i
Canonical Ensemble Distribution
• pi  Probability that the system at temperature T
is in the state i with energy Ui:
1 U
pi 
i
e kT
Z
• Partition Function  Sum over All Microstates:
Z  e
U i
kT
i
• Mean Energy:
 ln Z
U  kT
T
2
• Helmholtz Free Energy:
F  U  TS
Application
A Simple Model
of Paramagnetism
Section 6.3 (Spin = ½)
Section 7.8 (Spin = S)
Paramagnetic Materials
• This is a simple model to develop our
understanding of Statistical Mechanics but
it proves to be very significant.
• Paramagnets contain atoms which have
magnetic dipole moments (). These do not
interact with each other but can respond to an
applied external magnetic (B) field.
• The dipoles can be crudely thought of as
independent (atomic) bar magnets arranged
on a crystal lattice.
• The dipoles can be crudely thought of as
independent (atomic) bar magnets arranged on a
crystal lattice.
• Crude Picture: 
• In a B field each dipole can exist in one of two
states – aligned with the field (spin up) or anti
aligned (spin down).
• Spin up dipoles have an energy -B, spin down +B.
• We want to find out how the magnetization of the
material depends on the temperature and the
applied field.
• As all the dipoles are independent of each other
we really only need to look at the average
properties of one dipole. We can use all the other
dipoles as the heat bath – Canonical Ensemble.
• We have the two possible microstates and
energies already – get Partition Function Z1 for
our single dipole.
Z  e
U i
kT
e
 B
kT
e
 B
kT
i
x
 B 
 2 cosh
  2 cosh x
 kT 
B
kT
• We can now calculate the probability of spin up
versus spin down.
1 kTU
pi  e
Z
1
p 
e x
2 cosh x
1
p 
e x
2 cosh x
i
1 .0
0 .9
0 .8
Probability
0 .7
0 .6
Sp in alig ne d (U = - B)
Sp in an tia lign ed (U = + B)
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
0 .0
0 .5
1 .0
1 .5
X=B/kT
2 .0
2 .5
3 .0
• We can now calculate the mean magnetic
moment of our individual dipole.
1
p 
e x
cosh x
  p  p 

2 cosh x
1
p 
e x
cosh x
(e  e ) 
x

x
2 cosh x
2 sinh x
   tanh x
• So the mean energy of an individual dipole is:
U   B tanh x
• We have the values necessary for the individual
dipole and because our dipoles do not interact all
other dipoles must behave similarly.
• A solid of N dipoles therefore has an energy:
U  N U   NB tanh x
• And a mean magnetic moment (or magnetization)
in the direction of the applied field of:
M  N   N tanh x
• Note that U = -MB
• The magnetization or the magnetic moment L
per unit volume:
M N tanh x
L
V

V
• In the limit of a tanh x  x weak field or high
temperature x<<1 and so:
L
Nx
V
N

B
VkT
2
1 .0
1 .0
0 .9
0 .8
Probability
0 .7
0 .6
0 .6
Spin Up
Spin Down
Magnetisation
0 .5
0 .4
0 .4
0 .3
0 .2
0 .2
0 .1
0 .0
0 .0
0 .0
0 .5
1 .0
1 .5
X=b/kT
2 .0
2 .5
3 .0
Magnetisation / N/V
0 .8
Reif, Figure 6-3-1
“Saturation
Magnetization”
“Curie’s Law” of Paramagnetism
M0  χH, χ  (N0μ2)/(kT)
χ  “Curie Susceptibility”
Curie’s “Law”
• The susceptibility  is the magnetization per
applied field intensity which for small
magnetizations is given by H = B/0.
L N  0
 
H
VkT
2
1

T
• This is Curie’s “Law”. It holds very well for
paramagnetic materials with weakly interacting dipoles.
• It works so well that it can be used for temperature
calibration. For example Cerium Magnesium Nitrate
obeys Curie’s “Law” to 0.01K!
 vs. T plot
•1/ = T/C gives a straight line of gradient C-1 and intercept zero
• T = C gives a straight line parallel to the x-axis at a constant value of
T, thus showing the temperature independence of the magnetic moment.
Curie “Law”
• Because single electrons are magnets, if you place them in a
magnetic field, they’ll align with the field. However the energy
difference between aligned with field and against field is
<< thermal energy at room temperature. So, there are random
orientations, with equal populations of alignment with/against field.
• As you lower T, energy difference becomes more important & the
population changes, with more aligned anti-parallel to the field.
• To explain this behavior, Curie invented a parameter – called the
“Magnetic Susceptibility”, χ, – which is a measure of how
attracted a sample is to a magnetic field. This is normally
measured as an apparent mass increase. As more electrons align
anti-parallel to the field at low temperature, χ increases. In fact,
χ is inversely proportional to the temperature. This is the
Curie “Law”: (1/χ) = CT
C = “The Curie Constant”
Magnetic Heat Capacity
U  N U   NB tanh x   NB tanh
B
kT
• The paramagnetic solid has an energy that depends on
temperature It therefore must have a magnetic heat capacity.
• Experimentally one measures the heat capacity at
constant magnetic field intensity H.

B
 dQ 
 U 
CH  
tanh
 
   NB
T
kT
 T  H  T  H
• The paramagnetic compound is weakly magnetic, so
B = 0 H
so B is also constant.
Magnetic Heat Capacity

B
 B 
2  B 
C H   NB
tanh
 Nk 
 sech 

T
kT
 kT 
 kT 
2
CH / Nk
0.4
Magnetic Heat Capacity
0.2
0.0
0
1
2
3
1/X = kT/ B
4
5
This is also called The Schottky Heat Capacity
 B 
2  B 
C H  Nk 
 sech 

 kT 
 kT 
2
• In fact, this is a general result for the heat
capacity in any two level system.
CH / Nk
0.4
Magnetic Heat Capacity
0.2
0.0
0
1
2
3
1/X = kT/ B
4
5
Isolated Paramagnetic Solid
• Now consider a very similar problem to the one just
discussed. Now, constrain (fix) the total energy U of the
isolated system.
• N total dipoles, n spin-up aligned with the applied B
field: 
• U is obviously a function of n.
U ( n)  ( N  n) B  nB  B( N  2n)
• A given energy U(n) corresponds to a given number of
n spin up atoms with statistical weight:-
N!
W ( n) 
( N  n)! n!
• So, the Entropy is given by:
N!


S ( n)  k ln W ( n)  k ln

 ( N  n)! n! 
• For large N (~1023) Stirling’s Approximation
can be used:
S ( n)  k N ln N  n ln n  ( N  n) ln( N  n)
• The Temperature of the spin system is given by:
1 S S ( n) S ( n) dn 1 S ( n)




T U U ( n)
n dU  n
• Further manipulation gives:
1 S S ( n) S ( n) dn



T U U ( n)
n dU
• Also, we know that:
U ( n)  B( N  2n)
1
 1 S ( n)  k  N  n 
• So that:


ln

T 2 B n
2 B  n 
• Now, solve for n, to find the density of spin up atoms:
n
1 x
 e
N Z1
Negative Temperatures?
• From the previous derivation,
1
 1 S ( n)  k  N  n 


ln 

T 2 B n
2 B  n 
• Manipulate this to obtain:
1  k  N  n
k
 n 

ln
ln


T 2 B  n  2 B  N  n 
• Note that, if n < N/2 then more than half the
dipoles are anti-parallel to the field & we also
get the surprising result that
T becomes negative!
What is a Negative Temperature?
• First note that, as the temperature T , the
populations of spin-up & spin-down particles become equal!
• We just saw that if n < N/2 (more
spins down than up),
1 .0
T becomes negative!
0 .9
0 .8
This is called a population inversion.
• For a negative temperature the
entropy & statistical weight must
be decreasing functions of E.
Probability
• A negative temperature state
must therefore be “hotter”
than T , because it is a
higher energy state of the
system than T  !
0 .7
0 .6
Sp in alig ne d (U = - B)
Sp in an tia lign ed (U = + B)
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
0 .0
0 .5
1 .0
1 .5
X=B/kT
2 .0
2 .5
3 .0
What is Negative Temperature?
• For a negative temperature the entropy and statistical weight
must be decreasing functions of E.
• This can happen if the system possess a state of finite
maximum energy – such as our paramagnet with U=NB.
• No systems exist where this happens for all particular
aspects (I.e. vibrational energies, electronic energies and
magnetic energies).
• However, if one such aspect or subsystem is effectively
decoupled from the others, so they do not interact, that
subsystem may be considered to reach internal equilibrium
without being in equilibrium with the others.
• This is the case for magnetic systems where the relaxation
times between atomic spins is much quicker than the
relaxation between spins and the vibrational modes of the
lattice.
Negative Temperature
• In the paramagnet, the lowest possible energy is
U = -NB and the highest U = +NB. These are both
unique microstates so S = 0.
• In between we can only reach states with positive
energy with a negative temperature!!!
1.0
1.0
0.5
Energy / NB
Energy / NB
System Energy
System Energy
0.5
0.0
-0.5
-1.0
0.0
-0.5
-1.0
-5
-4
-3
-2
-1
0
1
Temperature
2
3
4
5
-20
-10
0
1 / Temperature
10
20