1. Classical Statistical Mechanics in the Canonical Ensemble: Application to the Classical
Ideal Gas

2. Canonical Ensemble in Classical Statistical Mechanics
As we’ve seen, classical phase space for a system with f degrees of freedom is f generalized coordinates & f generalized momenta (qi,pi).
The classical mechanics problem is done in the Hamiltonian formulation with a Hamiltonian energy function H(q,p).
There may also be a few constants of motion (conserved quantities):
energy, particle number, volume, ... 2

3. The Canonical Distribution in Classical Statistical Mechanics
The Partition Function has the form:
Z ≡ ∫∫∫d3r1d3r2…d3rN d3p1d3p2…d3pN e(-E/kT)
A 6N Dimensional Integral!
This assumes that we’ve already solved the classical mechanics problem for each particle in the system so that we know the total energy E for the N particles as a function of all positions ri & momenta pi.
E  E(r1,r2,r3,…rN,p1,p2,p3,…pN) 3

4. P(E) ≡ e[- E/(kBT)]/Z CLASSICAL Statistical Mechanics:
Let A ≡ any measurable, macroscopic quantity. The thermodynamic average of
A ≡ <A>. This is what is measured. Use probability theory to calculate <A> :
P(E) ≡ e[- E/(kBT)]/Z
<A>≡ ∫∫∫(A)d3r1d3r2…d3rN d3p1d3p2…d3pNP(E)
Another 6N Dimensional Integral!

5. Relationship of Z to Macroscopic Parameters
Summary of the Canonical Ensemble
(Derivations are in the book! Results are general & apply whether it’s a classical or a quantum system!)
Mean Energy: Ē  E = -∂(lnZ)/∂β
<(ΔE)2> = [∂2(lnZ)/∂β2]
β = 1/(kBT), kB = Boltzmann’s constant.
Entropy: S = kBβĒ + kBlnZ
An important, frequently used result! 5

6. Summary of the Canonical Ensemble: Helmholtz Free Energy:
F = E – TS = – (kBT)lnZ
Note that this gives: Z = exp[-F/(kBT)]
dF = S dT – PdV, so
S = – (∂F/∂T)V, P = – (∂F/∂V)T
Gibbs Free Energy:
G = F + PV = PV – kBT lnZ.
Enthalpy:
H = E + PV = PV – ∂(lnZ)/∂β 6

7. Summary of the Canonical Ensemble:
Mean Energy:
Ē = – ∂(lnZ)/∂ = - (1/Z)(∂Z/∂)
Mean Squared Energy:
<E2> = (rprEr2)/(rpr) = (1/Z)(∂2Z/∂2)
nth Moment:
<En> = (rprErn)/(rpr) = (-1)n(1/Z)(∂nZ/∂n)
Mean Square Deviation:
<(ΔE)2> = <E2> - (Ē)2 = ∂2lnZ/∂2 = -∂Ē/∂.
Constant Volume Heat Capacity
CV = (∂Ē/∂T)V = (∂Ē/∂)(d/dT) = - kBT2∂Ē/ ∂ 7

8. The Classical Ideal Gas Z ≡ ∫∫∫d3r1d3r2…d3rN d3p1d3p2…d3pN e(-E/kT)
So, in Classical Statistical Mechanics, the
Canonical Probability Distribution is:
P(E) = [e-E/(kT)]/Z
Z ≡ ∫∫∫d3r1d3r2…d3rN d3p1d3p2…d3pN e(-E/kT)
This is the tool we will use in what follows.
As we.ve seen, from the partition function
Z all thermodynamic properties can be
calculated: pressure, energy, entropy…. 8

9. Canonical Ensemble in Classical Statistical Mechanics.
Consider an Ideal Gas from the point of
view of microscopic physics. It is the
simplest macroscopic system.
Therefore, its useful use it to introduce the
use of the
Canonical Ensemble in Classical
Statistical Mechanics.
The ideal gas Equation of State is
PV= nRT
n is the number of moles of gas. 9

10. We’ll do Classical Statistical Mechanics,
but very briefly, lets consider the simple
Quantum Mechanics of an ideal gas & the
take the classical limit.
From the microscopic perspective, an ideal
gas is a system of N non interacting
Particles of mass m confined in a volume
V = abc. (a, b, c are the box’s sides)
Since there is no interaction, each molecule
can be considered a “Particle in a Box” as
in elementary quantum mechanics. 10

11. where nx, ny, nx = integers
Since there is no interaction, each
molecule can be considered a “Particle
in a Box” as in elementary quantum
mechanics.
The energy levels for such a system
have the form:
where nx, ny, nx = integers 11

12. Z = (q)N (2) (1) The energy levels for each molecule in the
Ideal Gas have the form: l
(nx, ny, nx = integers)
(1)
The Ideal Gas molecules are non
interacting, so the gas Partition Function
has the simple form:
Z = (q)N (2)
where q  One Particle Partition Function 12

13. Ideal Gas Partition Function: Z = (q)N (2)
q  One Particle Partition Function
Using (2) in the
Canonical Ensemble
formalism gives the
expressions on the
right for: mean energy
E, equation of state P
& entropy S: 13

14. The Partition Function for the 1- dimensional
particle in a “box” under the assumption that the
energy levels are so closely spaced that the sum
becomes an integral over phase space can be written:
(3)
For the 3 – dimensional particle in a “box”, th
3 dimensions are independent so that the
Partition Function can be written as the product
of 3 terms like equation (3). That is:
(4) 14

15. Now, using the Canonical Ensemble expressions from before:
Mean Energy & the
Equation of State
can be obtained (per
mole):
To obtain, for one mole of gas:

16.  “Gibbs’ Paradox” The Entropy can also be obtained:
(5)
As first discussed by Gibbs, Entropy in
Eq. (5) is NOT CORRECT!
Specifically, its dependence on particle
number N is wrong!
 “Gibbs’ Paradox”
in the first part of Ch. 7!