1. CHAPTER 9 Statistical Physics
9.0 Intro
9.1 Historical Overview
9.2 Maxwell Velocity Distribution
9.3 Equipartition Theorem
9.4 Maxwell Speed Distribution
9.5 Classical and Quantum Statistics
9.6 Fermi-Dirac Statistics
9.7 Bose-Einstein 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 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)

2. 9.0 : What are we dealing with?
Physics : object, state, equation of motion, …
~infinitely many objects in some cases (e.g., gas)
Practically it is neither possible to track the motions of the constituent particles nor interesting to know
Physical properties (states) defined in average sense. It is what we measure or see.
Need to introduce concepts of statistics and probability
Eventually, we want to learn quantum statistics (developed in 20C)

3. The fundamental assumption
Probability? Statistics?
Rolling n dices. p ~ (# of possible events)/6n
The fundamental assumption : each (accessible) ‘event’ has the same probability
Probability ⋉ # of events

4. Ideal gas system as an example - event
Consider a gas molecule. How is an ‘event’ defined for the molecule?
Event = microstate
Six parameters for a gas molecule — the position (x, y, z) and the velocity (vx, vy, vz)
These parameters define six-dimensional phase space
Small unit volume (ℏ/2𝑚) in the phase space defines an event. x v
microstate
(event)
∆𝑝∆𝑥=𝑚∆𝑣∆𝑥= ℏ 2
 Minimum volume ∆𝑣∆𝑥= ℏ 2𝑚
Do all the events have the same probability?  no!

5. Ideal gas system – isolated system
microstates for a system have the same probability if the system is isolated  no influence from outside
Isolated  energy is constant  states with the same energy are accessible
All the accessible states have the same probability  just count the # of states! x v
microstate (event)
accessible states
Does a molecule in a container has a fixed energy?  No
What went wrong?  molecule interacts with the container, more ‘accessible’ states  not an isolated system.
How are we going to deal with the situation?

6. Making an isolated system
(Container + molecule) is an isolated system  we can use the fundamental assumption for this system
However, we are only interested in the molecule, not the container. For example, what is the probability for the molecule to have (x,v)?
 count # of accessible states for the system for given (x,v) of the molecule
# of accessible states for the system = # of accessible states for the container
This is an isolated system and we know how to deal with it
Yet, this remains to be what we want to know 
How do we find # of states for the container?

7. Entropy & probability
Entropy S defined as 𝑑𝑆=𝛿 𝑄 𝑟𝑒𝑣 /𝑇 (in relation to the 2nd law of thermodynamics)
In statistical mechanics, 𝑆= 𝑘 𝐵 𝑙𝑛Ω or Ω= 𝑒 𝑆/ 𝑘 𝐵 where Ω is the number of microstates (for an isolated system)
From thermodynamics, 1 𝑇 = 𝜕𝑆 𝜕𝐸 or Ω~ 𝑒 𝐸/ 𝑘 𝐵 𝑇
Ω 𝑡𝑜𝑡 = Ω 𝑏𝑜𝑥 Ω 𝑝𝑡𝑙 = Ω 𝑏𝑜𝑥 ( the state for the molecule is fixed)
Combining above three gives 𝑝~ Ω 𝑏𝑜𝑥 ~ 𝑒 𝐸 𝑏𝑜𝑥 / 𝑘 𝐵 𝑇 ~ 𝑒 (𝐸 𝑡𝑜𝑡 −𝐸)/ 𝑘 𝐵 𝑇 ~ 𝑒 −𝐸/ 𝑘 𝐵 𝑇
Probability for a system (molecule) in thermal equilibrium with a reservoir (box) at a state is proportional to 𝑒 −𝐸/ 𝑘 𝐵 𝑇  Boltzmann factor
𝑝 𝑖 = 𝑒 − 𝐸 𝑖 / 𝑘 𝐵 𝑇 𝑗 𝑒 − 𝐸 𝑗 / 𝑘 𝐵 𝑇
Container has energy Etot-E
This has energy E

8. 9.1 : Historical Overview Benjamin Thompson (Count Rumford)
Put forward the idea of heat as merely the motion of individual particles in a substance
James Prescott Joule
Demonstrated the mechanical equivalent of heat
James Clark Maxwell
Brought the mathematical theories of probability and statistics to bear on the physical thermodynamics problems
Showed that distributions of an ideal gas can be used to derive the observed macroscopic phenomena
His electromagnetic theory succeeded to the statistical view of thermodynamics

9. Historical Overview Einstein
Published a theory of Brownian motion, a theory that supported the view that atoms are real
Bohr
Developed atomic and quantum theory

10. 9.2: Maxwell Distribution of Molecular Speed
Ideal gas in a box.
What is the expected # of particles with the speed between 𝑣 and 𝑣+𝑑𝑣?
Molecules do not interact with each other. We can focus on one particular molecule
Six parameters (position (x, y, z) and velocity (vx, vy, vz)) for the molecule  a state is a small box in the six-dimensional phase space
Position can be neglected because the energy depends only on the velocities (remember 𝑒 −𝐸/ 𝑘 𝐵 𝑇 ?)
We can focus only on this
and ignore the rest

11. Velocity Distribution
Define the velocity distribution function 𝑓( 𝑣 ) ( because it is the velocity that defines a state)
𝑓( 𝑣 ) 𝑑 3 𝑣 = probability of finding the particle with the velocity within 𝑑 3 𝑣 at 𝑣 . ( 𝑑 3 𝑣 =𝑑 𝑣 𝑥 𝑑 𝑣 𝑦 𝑑 𝑣 𝑧 )
𝑓( 𝑣 ) 𝑑 3 𝑣 ∝ 𝑒 −𝑚 𝑣 2 /2 𝑘 𝐵 𝑇  𝑓 𝑣 𝑑 3 𝑣 =𝐶 𝑒 −𝛽𝑚 𝑣 2 /2 where 𝛽= 𝑘 𝐵 𝑇
All we need to is to find C from 𝑓 𝑣 𝑑 3 𝑣 =1
Because v2 = vx2 + vy2 + vz2,
Rewrite this as the product of three factors.
 𝐶= 𝐶′ 3

12. 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, it equals to 1
then &
The velocity distribution function 𝑓 𝑣 is

13. 9.3: Maxwell Speed Distribution
It is useful to turn the velocity distribution into a speed distribution as we often do not care about the direction.
F(v) dv = the probability of finding a particle with speed between v and v + dv.
The volume of the spherical shell is 4πv2 dr. 
Maxwell speed distribution:
It is only valid in the classical limit.

14. Maxwell Speed Distribution
The most probable speed v*, the mean speed , and the root-mean-square speed vrms are all different.

15. Maxwell Speed Distribution
Most probable speed (at the peak of the speed distribution):
Mean speed (average of all speeds):
Root-mean-square speed (associated with the mean kinetic energy):
Standard deviation of the molecular speeds:
σv in proportion to
Kinetic energy :

16. 9.4: Equi-partition Theorem
‘In equilibrium, a mean energy of ½ kT per molecule is associated with each independent quadratic term in the molecule’s energy.’
Examples:
U = ½ kx2 + p2/2m  kT
U = (px2 + py2 + pz2)/2m  3kT/2

17. Equipartition Theorem – Ideal gas
In a monatomic ideal gas, each molecule has
There are three degrees of freedom.
Mean kinetic energy is
In a gas of N helium molecules, the total internal energy is
The heat capacity at constant volume is
For the heat capacity for 1 mole,
The ideal gas constant R = 8.31 J/K.

18. Equipartition Theorem - Rigid Rotator
For diatomic gases, consider the rigid rotator model.
The molecule rotates about either the x or y axis.
The corresponding rotational energies are
There are five degrees of freedom (three translational and two rotational).

19. Molar Heat Capacity Why does it turn on?  two more from here
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
Why does it turn on?

20. Boltzmann factor summary
Reservoir and system A
are in thermal contact.
Isolated system
(Heat) Reservoir
at T
The system that
we are interested in A
Reservoir is so large compared to system A
that its temperature does not change.
The system A can be at states with different energies
The probability for the system at a state with energy EA is proportional to exp(-bEA)

21. 9.5: Classical vs. Quantum Statistics
So far, we have classical distribution (statistics), which is valid when T is high and density is low (QM effect is weak)
When T is low (kBT < DE), the effect of the discrete energy levels shows up (remember the UV side of the Blackbody radiation?). For example, it is the reason why heat capacity CV  0 when T  0. (rotation and vibration contributions in the previous page)
There is more! When density is very high or T is low, there is a good chance that more than 1 particle occupy the same quantum (micro) state.
The distribution is strongly dependent on the the character of the particles (Fermion or Boson). It drastically changes the number of states and , as a result, the classical theory has to be modified.

22. Classical Distributions
Boltzmann showed that the statistical factor exp(−βE) is a characteristic of any classical system.
 this is regardless of how quantities other than molecular speeds may affect the energy of a given state
(Maxwell)-Boltzmann factor for classical system:
The energy distribution for classical system:
n(E) dE = the number of particles with energies between E + dE
g(E) = the density of states, is the number of states available per unit energy range
FMB tells the relative probability that an energy state is occupied at a given temperature

23. Counting : distinguishable & indistinguishable
Indistinguishability of identical particles changes the way we count the number of states.
It is what makes quantum statistics different from classical statistics (of course, when it matters).
Example : The possible configurations for distinguishable and indistinguishable particles in either of two energy states:
The probability of each is one-fourth (0.25) or one-third (0.33)
State 1
State 2 AB A B
State 1
State 2 XX X

24. Fermion vs Boson ‘an event’ refers to a state for the whole system
That is not all! Fermions are subject to the Pauli exclusion principle and two Fermions cannot occupy the same state.
Bosons Fermions
3 events are possible for Bosons but only 1 event is possible for Fermions. Each event has equal probability (fundamental assumption)
Fermions:
Particles with half-spins. Two or more Fermions are not allowed to occupy the same state (Pauli exclusion principle).
Bosons:
Particles with zero or integer spins that do not obey the Pauli principle. Multiple particles are allowed to occupied the same state
State 1
State 2 XX X
State 1
State 2 X
‘state’ is for a single
particle
‘an event’ refers to a state for the whole system

25. Quantum Distributions
Consider a system with two identical Bosons with 3 possible states
event1
event2
event3
event4
event5
event6 2e e
What is the expected
# of particles n(E)
at each state?
Etotal e 2e 2e 3e 4e P
Ce0
Ce-be
Ce-2be
Ce-2be
Ce-3be
Ce-4be
Bose-Einstein distribution:
where (BBE normalization constant)
For Fermions, Fermi-Dirac distribution :
where
Both distributions reduce to the classical Maxwell-Boltzmann distribution when exp(βE) is much greater than 1.
Density of state or DOS

26. Quantum Distributions
The normalization constants for the distributions depend on the physical system being considered.
Because Bosons do not obey the Pauli exclusion principle, more bosons can fill lower energy states.
Three graphs coincide at high energies – the classical limit.
 Maxwell-Boltzmann statistics may be used in the classical limit.

27. 9.6: Fermi-Dirac Statistics
Electrons determine the properties of solids
Electrons are Fermions, and their density is very high.
 Fermi-Dirac distribution is extremely important.
EF is called the Fermi energy.
When E = EF, FFD = ½.
In the limit as T → 0,
At T = 0, fermions occupy the lowest energy levels available to them.
Near T = 0, there is little chance that thermal agitation will kick a fermion to an energy greater than EF.

28. Fermi-Dirac Statistics
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 simple decaying exponential or Maxwell-Boltzmann.

29. Failure of Classical Theory for Electrons
With the assumption that electrons are like ideal gas in solid, Paul Drude (1900) could explain various physical properties solids (such as electrical conductivity).
Failure of the Drude picture :
Predicted
measured
Heat capacity
(electron)
3R/2
~0.02R at RT
Heat conductivity
~T-1/2
~T-1
What is the problem?

30. Electron in solid as free particle in a box
We would like to calculate the density of states
The allowed energies for electrons are (see example 6.10)
Rewrite this as E = r2E1
The parameter r is the “radius” of a sphere in phase space.
The “volume” is πr 3.
The exact number of states up to radius r is
spin

31. Density of states for free electron in 3D
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:

32. 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.

33. Electron distribution in metal
In general,
where α is a constant > 1
The exact number of electrons depends on temperature.
Heat capacity is
Molar heat capacity is
kBTF = EF
TF = 80,000 K for Cu
uF = 1.6x106 m/s

34. 9.7: Bose-Einstein Statistics for blackbody
Not as important as Fermi-Dirac
Blackbody Radiation can be derived using BE statistics.
We consider radiation in metallic box (as we considered earlier in blackbody radiation)
Then, we calculate the density of states as we did in FD distribution
The density of states g(E) is
The Bose-Einstein factor:
Use the Bose-Einstein distribution because photons are bosons with spin 1.

35. Bose-Einstein Statistics for blackbody
Convert from a number distribution to an energy density distribution u(E).
Multiply by a factor E/L3
For all photons in the range E to E + dE
Using E = hc/λ and |dE| = (hc/λ2) dλ
In the SI system, multiplying by c/4 is required.

36. 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:

37. Superfluid transition
The specific heat of liquid helium as a function of temperature:
The temperature at about 2.17 K is referred to as the critical temperature (Tc), transition temperature, or lambda point.
What happens at 2.17 K is a transition from the normal phase to the superfluid phase.

38. Superfluid & transition
Superfluid liquid helium is referred to as a Bose-Einstein condensation.
not subject to the Pauli exclusion principle
all particles are in the same quantum state
By using BE statistics with BBE = 1, we get
Rearrange this,
The result is T ≥ 3.06 K.
The value 3.06 K is an estimate of Tc.

39. Bose-Einstein Condensation in Gases
Coulomb interaction among gas particles makes it difficult to obtain high densities for Bose-Einstein condensation.
In 1995, laser cooling and magnetic trap were used to cool gas of 87Rb atoms to about 20 nK. What remained was an extremely cold, dense cloud with Tc = 170 nK.