Canonical Ensemble Formulation In Statistical Mechanics

The purpose of statistical mechanics is to determine the macroscopic properties of a substance by looking at it from a molecular level, or more precisely, its quantum microstates. Macroscopic properties include pressure, temperature, internal energy, and entropy. The substance used could be a pure substance or a mixture of two or more pure substances.

Statistical mechanics employs the ensemble method. An ensemble is a very large collection of systems that all have certain fixed macroscopic properties. When we say that the collection is very large, if you were to actually physically produce each system, it couldn’t be done because you wouldn’t have enough materials to do so. Some text books even talk of having infinity many systems. The ensemble also behaves according to two postulates.

First Postulate: The instantaneous average of a property of the ensemble is the same as the time average of any system of that same property.

Second Postulate: The systems of the ensemble are distributed with equal probability over all possible microstates.

The second postulate then implies that the macrostate with the largest number of microstates is the most probable to occur.

Before going into more depth about statistical mechanics in regards to thermodynamic properties, let’s look at an example of the ensemble formulation with something easily familiar to most people, the rolling of two dice. When two dice are rolled, there are thirty-six different possible outcomes that can occur: each of the six numbers of one die can be paired with each of the six numbers from the other die. The sum of the two dice can range from two to twelve. Some of these values are more likely to occur. To roll a twelve, there is only one possible combination, two sixes, but to roll a seven, there are six possible combinations. The diagram below shows all of the possible combinations.

Suppose we have an ensemble where each system is a room where someone continually rolls two dice. Each system is numbered from 1 to n_E, the number of systems. Suppose the macroscopic property of the ensemble is the average value of sum of the two dice in each system. The microstate would then be the list of each sum 8, 3, 9, 7, 4, 3, 10, 7, 5, 11 7, 6, 4, 9, 2, 7, 8,…, where the order in the list corresponds to the ordering of the systems. So that means if two lists have the same number of each value (2 to 12), but the ordering is different, then those two lists are in different microstates.

From the first postulate, the average of the sum of the dice of all of the systems is the same of the time average over a considerable amount of time of the sum of the dice of any single system. From the second postulate, each possible combination of dice is equally probable, so ideally, out of many rolls, one thirty-sixth of them should belong to each combination of dice. Since that should be the case, based on all thirty-six combinations, the average of them is seven. So then the instantaneous average of the sum of dice at any time should be seven.

In statistical mechanics, there are three main ensemble formulations. They are listed with their respective constraints.

Microcanonical Ensemble: The volume, internal energy, and number of particles of each species are held fixed.

Canonical Ensemble: The volume, temperature, and the number of particles of each species are held fixed.

Grand Canonical Ensemble: The volume, temperature, and chemical potential of each species are held fixed.

Here we will look at the Canonical Ensemble Formulation. The diagram below shows what the ensemble might look like. A very large control mass is divided up into many systems which are numbered from 1 to n_E, the number of systems. Each system has the same volume, V, same temperature, T, and same number of particles of each species, N_a, N_b, etc.

The energy level of each system is denoted by E_j. The energy levels are based on quantum states which are discrete so then the energy levels of the systems are discrete. So then if we could list all of the energy levels and put them in order, say in the diagram below, then E_j is the jth energy level.

j	E_j, kJ/kg
0	0.000
1	0.226
2	0.314
3	0.547
4	0.711
5	0.920
6	1.029
etc.	etc.

So throughout the rest of this article, j will be the index for the energy levels of a system. From there, it is easy to note that g_j is the degeneracies associated with E_j and n_j is the number of systems with E_j. The collection of all the n_j's, denoted as {n_j}, is the set of all occupation numbers or more simply, the occupation set. U_E will be the total internal energy of the ensemble. With the information now given, we can state the constraints on the ensemble.

ENSEMBLE CONSTRAINTS

Obviously, adding the number of systems at each energy level has to add up to the total number of systems and adding the energy from each system has to add up to the total internal energy in the ensemble.

To use the Second Postulate, we need to know the total number of microstates, denoted as W, for any given {n_j}. This is given as:

Also from the Second Postulate, the macrostate having the largest number of microstates is the most probable. So we will want to maximize W for {n_j}. However, it will be easier to maximize ln(W) instead of W itself.

Since n_j and n_E are large, we can use Stirling's Approximation

for ln(n_j!) and ln(n_E!)

To maximize ln(W), take its differential and set it equal to zero. Remember that n_E and g_j are constant.

Maximize ln(W) subject to the ensemble constraints using Lagrange Multipliers. First take the differential of each constraint.

Next, multiply each of the resulting equations by the Lagrange Multipliers, alpha and beta, respectively.

Add Eqn. 2 and Eqn. 3 and subtract Eqn. 1

The maximum of ln(W) must hold for all possible values of dn_j, so then

Solve for n_j.

n_j^* corresponds to the n_j's that maximizes ln(W).

Again, from the first constraint:

Take the ratio

Q is the Canonical Partition Function. The next steps are first to determine beta and second to relate Q to other thermodynamic properties.

Since U_E is fixed, the canonical ensemble can be thought of as a system in the microcanonical ensemble. Using the resulta from the microcanonical ensemble formulation we have

Where S_E is the total entropy of the canonical ensemble and k_B is Boltzmann's constant. Substitute the expression for n_j using Eqn. 4 and omitting the asterisk. All of the details are listed below.

The ensemble average of any property is the sum of the value of that certain property for each system, which is also at a particular energy level, multiplied by the probability that system is at that energy level. The probability that a system is at a particular energy level is given by Eqn. 4.