Partition Function in Statistical Mechanics

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Partition Function

Copyright © 2011 by Robert Finkel

Much of the routine calculating in statistical mechanics is done by standard manipulations of the partition function Z.

This page gives prescriptions for finding average energy and standard deviations using the partition function.

Some Notation

Introduce a customary notation for factor by representing it with the Greek letter beta,

Also note that some systems attain the same energy level in multiple ways. These are said to be degenerate levels.

For example, a photon may have the same energy for two different polarizations. Let level be achievable in ways.

We can then group of these degenerate levels together so that the canonical distribution now becomes,

This is the more general form of the canonical distribution for discrete energy levels. The version that applies to continuous

energy distributions is called the Maxwell distribution..

· A solution of isomers A and B with corresponding energies .is in thermal equilibrium, .

Level B has two forms with the same energy (said to be doubly degenerate). Write an expression for the ratio of the

A population to the B population.

Partition Functions

Again, the partition function for the canonical distribution is ,.

Most statistical information can be derived from various manipulations of the partition function. For example,

two very useful relations are:

and

where s is the standard deviation of energy.

· A collection of N electrons with magnetic moments µ is placed in a weak magnetic field B. An approximate

partition function for this system is

Z = exp(Nm²B²b²/2)

Find the average energy for this system. (Knowledge of magnetism not needed.)
[ans -Nm²B²/kT ]

Independent Systems and Dimensions

When two independent systems have entropies and, the combination of these systems has a total entropy S

given by. Entropy is said to be “additive” or “extensive.” The partition function for the composite is known

to be the product of the independent Z’s. The rule can be extended to any number of independent systems.

The composite Z for K independent systems is

The same product rule for Z applies when you consider independent motions or independent dimensions.

For example, a molecule that has independent modes of rotation, vibration and translation may be expressed in obvious notation as

Similarly, if the dynamics can be easily separated into independent motions along axes x, y, and z, we can write

We are often concerned with independent particles. If Z₁ is the partition function for a single distinguishable particle,

then the partition function for N such particles is simply

· Given that the partition function for an ideal gas of N classical particles moving in one dimension (x-direction) in

a rectangular box of sides L_x, L_y, and L_z is

Find the partition function for the gas in three dimensions and for N particles. Use your result to calculate the average energy.