CHAPTER 9
Statistical Physics

underpins thermodynamics, ideal gas (a classical physics
model), ensembles of molecules, solids, liquids … the
universe


9.1 Justification for its need !
9.2 Classical distribution functions as examples of
distributions of velocity and velocity2 in ideal gas
9.3 Equipartition Theorem
9.4 Maxwell Speed Distribution


9.5 Classical and Quantum Statistics
9.6 Black body radiation, Liquid Helium, Bose-Einstein
condensates, Bose-Einstein statistics,
9.7 Fermi-Dirac Statistics …

Ludwig Boltzmann, who spent much of his life studying statistical
mechanics, died in 1906 by his own hand. Paul Ehrenfest, carrying on his
work, died similarly in 1933. Now it is our turn to study statistical
mechanics. Perhaps it will be wise to approach the subject cautiously.
- David L. Goldstein (States of Matter, Mineola, New York: Dover, 1985)
1
First there was classical physics with a cause
(or causes)
Newton’s three force laws, first unification in physics
Lagrange around 1790 and Hamilton around 1840 added significantly to the
computational power of Newtonian mechanics.
Pierre-Simon de Laplace (1749-1827)
Made major contributions to the theory of probability and well known clockwork
universe statement:
It should be possible in principle to have perfect knowledge of the universe. Such
knowledge would come from measuring at one time the position and velocities of
every particle of matter and then applying Newton’s law. As they are cause and
effect relations that work forwards and backwards in time, perfect knowledge can
be extended all the way back to the beginning of the universe and all the way
forward to its end.
So no uncertainty principle allowed …
2
then there was the realization that one does not
always need to know the cause (causes), can do
statistical analyses instead
Typical problem, flipping of 100 coins,
One can try to identify all physical condition before the toss, model the
toss itself, and then predict how the coin will fall down
if all done correctly, one will be able to make a prediction on how many
heads or tails one will obtain in a series of experiments
Statistics and probabilities would just predict 50 % heads 50% tails by
ignoring all of that physics,
The more experimental trials, 100,000 coin tosses, the better this
prediction will be borne out
3
Speed distribution of
particles in an ideal
gas in equilibrium,
instead of analyzing
what each individual
particle is going to
do, one derives a
distribution function,
determines the
density of states,
and then calculates
the physical
properties of the
system (always by
the same
procedures)
<KE>= <p2>/2m
There is one characteristic kinetic energy (or speed) distribution for each value of T, so
we would like to have a function that gives these distribution for all temperatures !!! 4
Path to statistical physics from classical to quantum for
bosons and fermions
Benjamin Thompson (Count Rumford) 1753 – 1814

Put forward the idea of heat as merely the kinetic energy of individual particles in
an ideal gas, speculation for other substances.
James Prescott Joule 1818 – 1889

Demonstrated the mechanical equivalent of heat, so central concept of
thermodynamics becomes internal energy of systems (many many particles at
once)
5
Beyond first or second year college physics
James Clark Maxwell 1831 – 1879, Josiah Willard Gibbs 1839 – 1903, Ludwig
Boltzmann 1844 – 1906 (all believing in reality of atoms, tiny minority at the time)
Brought the mathematical theories of probability and statistics to bear on the
physical thermodynamics problems of their time.
Showed that statistical distributions of physical
properties of an ideal gas (in equilibrium – a
stationary state) can be used to explain the observed
classical macroscopic phenomena (i.e. gas laws)
Gibbs invents notation for vector calculus, the form in which we use Maxwell’s
equations today
Maxwell’s electromagnetic theory succeeded his work on statistical foundation of
thermodynamics – so he was a genius twice over.
6
and then there came modern physics …
Einstein 1905
 PhD thesis, the correct theory of Brownian motion, a theory that
required atoms to be real, because there are measurable
consequences to their motion, (also start of quantitative
nanoscience as size of a common sugar molecule (1 nm) was
determined correctly)
Bose (with Einstein’s generalization) 1924
 Statistics of indistinguishable particles that are bosons (photons:
Bose, all other bosons: Einstein’s generalization)
Fermi and Dirac independently 1926
 Statistics of indistinguishable particles that are fermions
7
9.2: Maxwell Velocity and Velocity2 Distribution

internal energy in an ideal gas depends only on the movements
of the entities that make up that gas.

Define a velocity distribution function
.
= the probability of finding a particle with velocity
between
.
where
is similar to the product of a wavefunction with its complex conjugate (in
3D), from it we can calculate expectation values (what is measured on
average) by the same integration procedure as in previous chapters !!
8
Maxwell Velocity Distribution

Maxwell proved that the velocity probability distribution function is
proportional to exp(−½ mv2 / kT), special form of exp(-E/kT) – the
Maxwell-Boltzmann statistics distribution function.
Therefore
where C is a
proportionality factor and β ≡ (kT)−1. k: Boltzmann constant,
which we find everywhere in this field

Because v2 = vx2 + vy2 + vz2 then

Rewrite this as the product
of three factors.
Is the product of the three functions gx, gy gz which are just for one
variable (1D) each
9
Maxwell Velocity Distribution

g(vx) dvx is the probability that the x component of a gas
molecule’s velocity lies between vx and vx + dvx.
if we integrate g(vx) dvx over all of vx and set it equal to 1,
we get the normalization factor

The mean value (expectation value) of vx
Full Widths at
Half Maximum
e-0.5 = 0.607 g(0)
That is similar to the expectation value of momentum in the square wells
10
Maxwell Velocity2 Distribution

The mean value of vx2, also an expectation value that is a simple
function of x
This is not zero because it
is related to kinetic energy,
remember the expectation
value of p2 was also not
zero
1.3806488(13)×10−23
J K−1
8.6173324(78)×10−5
eV K−1
It relates the human invented
energy scale (at the individual
particle level) to the absolute
temperature scale (a physical thing)
gas constant R divided by Avogadro’s number NA
11
Maxwell Velocity2 Distribution

The results for the x, y, and z velocity2 components are identical.

The mean translational kinetic energy of a molecule:

Equipartion of the kinetic energy in each of 3 dimension a particle
may travel, in each degree of freedom of its linear movement

this result can be generalized to the equipartition
theorem
12
9.3: Equipartition Theorem
Equipartition Theorem:

For a system of particles (e.g. atoms or molecules) in equilibrium
a mean energy of ½ kT per system member is associated with
each independent quadratic term in the energy of the system
member.

That can be movement in a direction, rotation about an axis,
vibration about an equilibrium position, …, 3D vibrations in a
harmonic oscillator

Each independent phase space coordinate:
degree of freedom
13
Equipartition Theorem

In a monatomic ideal gas, each molecule has

There are three degrees of freedom.
Mean kinetic energy is 3(1/2 kT) = 3/2 kT
In a gas of N helium atoms, the total internal energy is



CV = 3/2 N k
For the heat capacity for 1 mole

The ideal gas constant R = 8.31 J/K

14
As predicted, only 3
translational
degrees of freedom
2 more (rotational)
degrees of freedom
discrepancies due to quantized vibrations, not
due to high particle density
2 more (vibrational)
degrees of freedom
plus vibration, which
also adds two times
1/2 kBT
We get excellent agreement for the noble gasses, they are just single
particles and well isolated from other particles
15
Molar Heat Capacity

The heat capacities of diatomic gases are temperature dependent,
indicating that the different degrees of freedom are “turned on” at
different temperatures.
Example of H2
16
The Rigid Rotator Model

For diatomic gases, consider the rigid rotator model.

The molecule rotates about either the x or y axis.
The corresponding rotational energies are ½ Ixωx2 and ½ Iyωy2.
There are five degrees of freedom, three translational and two
rotational. (I is rotational moment of inertia)


17
Two more degrees of freedom, ½ Ixωx2
and ½ Ixωx2
Two more degrees of
freedom, because
there are kinetic and
potential energy, both
are “quadratic” (both
have variables that
appear squared in a
formula of energy is a
degree of freedom, ½
m (dr/dt)2 and ½ κ r2
18
Using the Equipartition Theorem

In the quantum theory of the rigid rotator the allowed energy
levels are

From previous chapters, the mass of an atom is largely confined
to its nucleus
Iz is much smaller than Ix and Iy. Only rotations about x and y are
allowed at reasonable temperatures.




Model of diatomic molecule, two atoms connected to each other
by a massless spring.
The vibrational kinetic energy is ½ m(dr/dt)2, there is kinetic and
potential energy ½ κ r2 in a harmonic vibration, so two extra
degrees of freedom
There are seven degrees of freedom (three translational, two
rotational, and two vibrational for a two-atom molecule in a gas).
19
not that simple
six degrees of
freedom
according to classical physics, Cv
should be 3 R = 6/2 kBT NA for solids
and independent of the temperature
We will revisit this problem when we have learned of quantum distributions, concept of
phonons, which are quasi-particle that are not restricted by the Pauli exclusion principle
20
Maxwell’s speed (v) distribution
Slits have small
widths, size of it
defines dv (a small
speed segment of
the speed
distribution)
21
9.4: Maxwell Speed Distribution ΙvΙ

Maxwell velocity distribution:
Where


let’s turn this into a speed distribution.
F(v) dv = the probability of finding a particle with speed
between v and v + dv.
One cannot derive F(v) dv (i.e. a distribution of a scalar entity)
simply from f(v) d3v (the velocity distribution function, i.e. a
distribution of vectors and their components), we need idea of
phase space for this derivation
22
Maxwell Speed Distribution

Idea of phase space, to count how many states there are
Suppose some distribution of particles f(x, y, z) exists in normal
three-dimensional (x, y, z) space.

The distance of the particles at the point (x, y, z) to the origin is


the probability of finding a particle between
.
23
Maxwell Speed Distribution

Radial distribution function F(r), of finding a particle between r and r
+ dr {sure not equal to f(x,y,z) as we want to go from coordinates to length of
the vector, a scalar}

The volume of any spherical shell is 4πr2 dr.
now replace the 3D space coordinates x, y, and z
with the velocity space coordinates vx, vy, and vz

Maxwell speed distribution:
It is only going to be valid in the classical limit, as a few particles would have speeds in
excess of the speed of light.
note speed distribution function is different to velocity distribution function, but
both have the same Maxwell-Boltzmann statistical factor
24
Maxwell Speed Distribution

The most probable speed v*, the mean speed
mean-square speed vrms are all different.
, and the root-
25
Maxwell Speed Distribution

Most probable speed (at the peak of the speed distribution), simply plot
the function, take the derivative and set it zero, derive the
consequences:

Average (mean) speed, will be an expectation value that we calculate
from on an integral on the basis of the speed distribution function
26
average (mean) of the square of the speed, will be an expectation value that we
calculate from another integral on the basis of the speed distribution function
We define root mean square speed on its basis
which is of course associated with the
mean kinetic energy
We can also calculate the spread (standard deviation) of the speed
distribution function in analogy to quantum mechanical spreads
Note that σv in proportional to
So now we understand the whole function, can make calculations for all T
27
Straightforward: turn speed distribution into kinetic energy (internal energy of ideal
gas) distribution
28
Number
of
particles
with
energy in
interval E
and E +
dE
So we recover the equipartition
theorem for a mono-atomic gas
29
9.5: Needs for Quantum Statistics
If molecules, atoms, or subatomic particles are fermions, i.e. most of
matter, in the liquid or solid state, the Pauli exclusion principle
prevents two particles with identical wave functions from sharing
the same space. The spatial part of the wavefunction can be
identical for two particles in the same state, but the spin part f the
wavefunction has to be different to fulfill the Pauli exclusion
principle.
 If the particles under consideration are indistinguishable and
Bosons, they are not subject to the Pauli exclusion principle, i.e.
behave differently

There are only certain energy values allowed for bound systems
in quantum mechanics.

There is no restriction on particle energies in classical physics.
30
Classical physics Distributions

Boltzmann showed that the statistical factor exp(−βE) is a characteristic
of any classical system in equilibrium (in agreement with Maxwell’s
speed distribution)
{quantities other than molecular speeds may affect the energy of a given state (as we
have already seen for rotations, vibrations)}

Maxwell-Boltzmann statistics for classical system:

The energy distribution for classical system:
β ≡ (kBT)−1
A is a normalization factor, problem specific

n(E) dE = the number of particles with energies between E and E + dE.

g(E) = the density of states, is the number of states available per
unit energy range.
FMB gives the relative probability that an energy state is occupied at
31
a given temperature.

Classical / quantum distributions

Characteristic of indistinguishability that makes quantum
statistics different from classical statistics.

The possible configurations for distinguishable particles in either
of two (energy or anything else) states:
State 1
State 2
AB
A
B
B
A
AB

There are four possible states the system can be in.
32
Quantum Distributions

If the two particles are indistinguishable:
State 1 State 2
XX
X
X
XX

There are only three possible states of the system.

Because there are two types of quantum mechanical particles, two
kinds of quantum distributions are needed.
Fermions:



Particles with half-integer spins, obey the Pauli principle.
Bosons:

Particles with zero or integer spins, do not obey the Pauli principle.
33
Multiply each state with its number of
microstates for distinguishable particles –
sum it all up and you get the distribution
of classical physics particles
Ignore all microstates for
indistinguishable particles – sum it all up,
that would be the distribution for bosons
Ignore all microstates and states that
have more than one particle at the same
energy level, - sum it all up, that would be
the distribution of fermions
Serway et al, chapter 10 for details
Realize, there must be three different
distribution functions !!
34
Quantum Distributions

Fermi-Dirac distribution:
where

Bose-Einstein distribution:
Where


In each case Bi (i = 1 or 2) is a normalized factor which depends on the
problem.
Both distributions reduce to the classical Maxwell-Boltzmann
distribution when Bi exp(βE) is much greater than 1, this happens at low
densities (i.e. in a dilute gas at moderately high temperatures, i.e. room
temperature
35
Classical and Quantum Distributions
For
photons
in cavity,
Planck, A
= 1, α = 0
E is quantized in units of h
if part of a bound system
36
Quantum Distributions
has to do with
specific
normalization
factor



If all three
normalization
factors = 1,
just for
comparison
The exact forms of normalization factors for the distributions depend on the
physical problem being considered.
Because bosons do not obey the Pauli exclusion principle, more bosons can
fill lower energy states (are actually attracted to do so)
All three graphs coincide at high energies – the classical limit.
Maxwell-Boltzmann statistics may be used in the classical limit when
particles are so far apart that they are distinguishable, can be tracked by
37
their paths
When there are so many
states that there is a very
low probability of occupation
also if the particles are
heavy (macroscopic),
i.e. a bunch of classical
physics particle, Bohr’s
correspondence
38
principle again
Anything to do with solids, when high probability of occupancy of energy
states, e.g. electrons in a metal, which are fermions
39
anything to do with
liquids, when high
probability of
occupancy of energy
states
Bose-Einstein
4
condensate for 2
He
at 2.17 K superfluidity
(explained later on)
https://www.youtube.com/watch?v=2Z6UJbwxBZI
40
degeneracy of
the first exited
state in H atom
n
l
ml
ms up
ms down
2
0
0
+1/2
-1/2
2
1
1
+1/2
-1/2
2
1
0
+1/2
-1/2
2
1
-1
+1/2
-1/2
g functions, density of states, how many states there are per unit energy
value, in other words: the degeneracy if we talk about a hydrogen atom
g functions are problem specific !!
41
revisited
Einstein’s assumptions in 1907, atoms vibrate independently
of each other
(starting from zero point energy, due
to uncertainty principle)
he used Maxwell-Boltzmann statistics because there are so
many possibly vibration states that only a few of the
available states will be occupied, (and the other distribution
functions were not known at the time)
A. Einstein, "Die Plancksche Theorie der Strahlung und die
Theorie der spezifischen Wärme", Annalen der Physik 22
(1907) 180–190
i.e. at high temperatures is approaches the classical value of 2 degrees of freedom
42
with ½ kT each times 3 vibration direction (Bohr’s correspondence principle once more)
To account for different bond strength, different spring constants
hetero-polar bond in diamond much stronger than metallic bond in lead and
aluminum, so much larger Einstein Temperature for diamond (1,320 K) >> 50100 K for typical metals
43
Peter Debye lifted the
assumption that atoms
vibrate independably,
similar statistics, Debye
temperature TD
even better modeling with
phonons, which are
pseudo-particle of the
boson type
44
Blackbody Radiation
Blackbody Radiation
 Intensity of the emitted radiation is
Use Bose-Einstein distribution because photons are bosons with
spin 1 (they have two polarization states)
 For a free particle in terms of momentum in a 3D infinitely deep
well:
now our particles are measles

E = pc = hf so we need the equivalent of this formulae in terms of
momentum (KE = p2 / 2m)
45
Phase space again
Density of states in cavity, we can
assume the cavity is a sphere, we
could alternatively assume it is any
kind of shape that can be filled with
cubes …
46
Bose-Einstein Statistics

The number of allowed energy states within “radius” r of a sphere is

Where 1/8 comes from the restriction to positive values of ni and 2 comes
from the fact that there are two possible photon polarizations.
Resolve Energy equation for r, and substitute into the above equation for Nr

Then differentiate to get the density of states g(E) is

Multiply the Bose-Einstein factor in:
For photons, the normalization factor is 1, they are created and destroyed as needed
47
Bose-Einstein Statistics

Convert from a number distribution to an energy density
distribution u(E).

For all photons in the range E to E + dE

Using E = hf and |dE| = (hc/λ2) dλ

In the SI system, multiplying by c/4 is required.
and world wide fame for Satyendra Nath Bose 1894 – 1974 !
48
Liquid Helium


Has the lowest boiling point of any element (4.2 K at 1 atmosphere
pressure) and has no solid phase at normal pressure.
The density of liquid helium as a function of temperature.
49
Liquid Helium

The specific heat of liquid helium as a function of temperature
Thermal conductivity goes to infinity
at lambda point, so no hot bubbles
can form while the liquid is boiling,



The temperature at about 2.17 K is referred to as the critical
temperature (Tc), transition temperature, or lambda point.
As the temperature is reduced from 4.2 K toward the lambda point,
the liquid boils vigorously. At 2.17 K the boiling suddenly stops.
What happens at 2.17 K is a transition from the normal phase to
the superfluid phase.
50
Liquid Helium


The rate of flow increases dramatically as the temperature is
reduced because the superfluid has an extremely low viscosity.
Creeping film – formed when the viscosity is very low and some
helium condenses from the gas phase to the glass of some beaker.
51
Liquid Helium


Liquid helium below the lambda point is part superfluid and part
normal.
As the temperature approaches absolute zero, the superfluid
approaches 100% superfluid.

The fraction of helium atoms in the superfluid state:

Superfluid liquid helium is referred to as a Bose-Einstein
condensation.
not subject to the Pauli exclusion principle because (the
most common helium atoms are bosons
all particles are in the same quantum state
52
https://www.youtube.com/watch?v=2Z6UJbwxBZI
53
9.7: Fermi-Dirac Statistics





EF is called the Fermi energy.
When E = EF, the exponential term is 1.
FFD = ½
In the limit as T → 0,
At T = 0, fermions occupy the lowest energy levels.
Near T = 0, there is no chance that thermal agitation will kick a
fermion to an energy greater than EF.
54
Fermi-Dirac Statistics
T=0
T>0

As the temperature increases from T = 0, the Fermi-Dirac factor “smears out”.

Fermi temperature, defined as TF ≡ EF / k.
T = TF

.
T >> TF
When T >> TF, FFD approaches a decaying exponential of the Maxwell Boltzmann
statistics.
At room temperature, only tiny
amount of fermions are in the
region around EF,i.e. can
contribute to elecric current, …
55
Classical Theory of Electrical Conduction

Paul Drude (1900) showed on the basis of the idea of a free
electron gas inside a metal that the current in a conductor should
be linearly proportional to the applied electric field, that would be
consistent with Ohm’s law.
His prediction for electrical conductivity:

Mean free path is

Drude electrical conductivity:

.
56
Classical Theory of Electrical Conduction
From Maxwell’s speed distribution

According to the Drude model, the conductivity should be
proportional to T−1/2.

But for most metals is very nearly proportional to T−1 !!

This is not consistent with experimental results.

l and τ make only sense for a realistic microscopic model, so
whole approach abandoned, but free electron gas idea kept, just a
different kind of gas
57
58
59
60
61
All condensed matter (liquids and solids) problems are statistical quantum
mechanics problems !!
Quantum condensed matter physics problems are typically low temperature
problems
Ideal gasses can be modeled classically, because they have very low matter
densities
62
63
Quantum Theory of Electrical Conduction

The allowed energies for electrons are

The parameter r is the “radius” of
a sphere in phase space.
The volume is (4/3)πr 3.
The exact number of states up
to radius r is
.


64
Quantum Theory of Electrical Conduction

Rewrite as a function of E:


At T = 0, the Fermi energy is the energy of the highest occupied
level.
Total of electrons

Solve for EF:

The density of states with respect to energy in terms of EF:
65
Quantum Theory of Electrical Conduction

At T = 0,

The mean electronic energy:

Internal energy of the system:

Only those electrons within about kT of EF will be able to absorb thermal
energy and jump to a higher state. Therefore the fraction of electrons
capable of participating in this thermal process is on the order of kT / EF.
66
Quantum Theory of Electrical Conduction

In general,
Where α is a constant > 1.

The exact number of electrons depends on temperature.

Heat capacity is

Molar heat capacity is
67
Quantum Theory of Electrical Conduction





Arnold Sommerfeld used correct distribution n(E) at room
temperature and found a value for α of π2 / 4.
With the value TF = 80,000 K for copper, we obtain cV ≈ 0.02R,
which is consistent with the experimental value! Quantum theory
has proved to be a success.
Replace mean speed in Eq (9,37) by Fermi speed uF defined
from EF = ½ uF2.
Conducting electrons are loosely bound to their atoms.
these electrons must be at the high energy level.
at room temperature the highest energy level is close to the
Fermi energy.
We should use
68
Quantum Theory of Electrical Conduction


Drude thought that the mean free path could be no more than
several tenths of a nanometer, but it was longer than his
estimation.
Einstein calculated the value of ℓ to be on the order of 40 nm in
copper at room temperature.

The conductivity is

Sequence of proportions.
69
70

Rewrite Maxwell speed distribution in terms of energy.

For a monatomic gas the energy is all translational kinetic
energy.
where
71