Review of statistical mechanics

 

Ensembles

The central application of statistical mechanics rests on the Gibbs postulate:

The ensemble average of a property corresponds to the thermodynamic quantity.

For example, the average energy corresponds to the internal energy, the average pressure corresponds to the thermodynamic pressure, etc.

To accomplish the kind of averaging needed to calculate the pressure, for example, we must consider a large number of systems. The concept of a collection of systems, or an ensemble was first introduced by Gibbs. An ensemble consists of a very large number of systems, each constructed to be a replica on the macroscopic level. Corresponding to each ensemble there is a partition function that represents the average number of states accessible at a given temperature.

 

Symbol

Ensemble Name

Fixed Variables

W

Microcanonical

N, V, E

Q

Canonical

N, V, T

 

The magnitude of the partition function, W(E) for the microcanonical ensemble is the same as the degeneracy of the system at the given energy, E. The principle of equal a priori probabilities states that each and every quantum state of the system must be represented an equal number of times. Another way to put this is to state that in an isolated system (N, V, E, fixed) any one of the W possible quantum states is equally likely. The partition function W of the microcanonical ensemble is the same as the thermodynamic probability W introduced in the Boltzmann equation for the statistical entropy, S = k ln W. The same equation holds for the microcanonical ensemble, S = k ln W and thus represents a direct thermodynamic connection between the partition function and the entropy. The difficulty with application of this information is that it is very difficult to obtain a set of molecules at a constant energy. In spite of this experimental difficulty, the microcanonical ensemble is useful for illustrating the number of degenerate (equal energy) states in systems of interest. The conclusion from these investigations will be that the number of quantum states accessible to a system is vast and that provides the motivation for application of statistical techniques to the calculation of average quantities, fluctuations and transport properties.

The canonical ensemble is most used for the derivation of thermodynamic quantities. In this ensemble the number of molecules, temperature and volume are held constant. The partition function Q(N,V,T) is a number that corresponds to the number of thermally accessible levels at a given temperature. To obtain Q(N,V,T) we need a quantum mechanical model for the energy levels of atoms and molecules in the system. This is provided from the consideration of translation, rotation, and vibration that we have already reviewed.

 

Molecular motion: quantum mechanics provides the energy levels for statistical averaging

We consider the three types of molecular motion

  1. translation
  2. vibration
  3. rotation

 

From quantum mechanics we have

HY = EY

where H is the hamiltonian

 

 

where h is h/2p, that is, Planck's constant divided by 2p. The hamiltonian comprises both the kinetic and potential energy of the system. For the purposes of statistical mechanics it is the energy levels and their degeneracy that are of interest. The wavefunctions, Y do not appear in averages and are therefore not given below.

 

Translation

Translation is calculated using the particle-in-a-box treatment.

Setting the potential U(x, y, z) = 0, and for one dimension the hamiltonian is

The energy levels are

in one dimension. The extension to three dimensions is easily made.

 

Vibration

The classical Hooke's law potential is 1/2 kx2 and this is exactly what is used as the potential in the quantum mechanical hamiltonian. In one dimension this becomes

The energy levels are

where w = (k/m)1/2.

 

Rotation

The rigid rotor hamiltonian is

The energy levels are

 

I is the moment of inertia of the rotor.

The essence of statistical mechanics is to connect these quantum mechanical energy levels to the macroscopically measured thermodynamic energies, pressure, and entropy. There are two important aspects of these energy levels. First, there is a ladder of increasing energy states. Second, in some cases there is a degeneracy associated with the states. For the rigid rotor solutions the degeneracy is 2J + 1.