Statistical Mechanics – Mini Partition Function

The partition function is perhaps the single most important quantity in statistical mechanics. It is the means by which we translate microphysical knowledge of a thermodynamic system into macrophysical knowledge. The partition function is defined as the summation

\displaystyle Z=\sum_\alpha e^{-\beta E_\alpha }

where E_\alpha is the energy associated with the \alphath microstate, and the coefficient appearing in the exponent, thermodynamic \beta, is inversely proportional to the system’s temperature. If we know the microphysics of the system – that is, the set of energies \{E_\alpha\} available to it – we can evaluate the partition function, and in doing so make available to us further knowledge of the system, such as its mean energy

\displaystyle U=-\frac{\partial\ln Z}{\partial\beta},

its entropy

S=\beta U+\ln Z,

the probability that it occupies state \alpha

\displaystyle p_\alpha=\frac{1}{Z}e^{-\beta E_\alpha},

and any other thermodynamic variables that arise naturally out of the system under study.

Typically, we will be concerned with large populations of particles. You might expect that this makes quantising the energy levels \{E_\alpha\} difficult, and in general this is true; in principle, every one of the million million million million or so particles in any given blob of stuff can interact with each and every other one of its million million million million neighbours. We could be talking about 1047 different interactions!

However, it is sometimes reasonable to treat the particles in a system as non-interacting. In this model, the energy of one particle does not affect the energy of any of the other particles. Such a treatment radically simplifies the evaluation of the partition function. Here’s how:

The total energy E_\alpha of a system of particles is the sum of the energy of each particle. We imagine that the ith particle has access to energy states \{\epsilon_{n_i}\}. This means each microstate \alpha is defined by the set of numbers \{n_i\}, which gives a list of the states occupied by each particle:

\alpha\equiv\{n_1,n_2,...,n_N\}

E_\alpha=\epsilon_{n_1}+\epsilon_{n_2}+...+\epsilon_{n_N}

The partition function Z then becomes a sum over the set of all individual particle microstates:

\displaystyle Z=\sum_{n_1,n_2,...,n_N} e^{-\beta\left(\epsilon_{n_1}+\epsilon_{n_2}+...+\epsilon_{n_N}\right)}

The exponent can be split into a product:

\displaystyle Z=\sum_{n_1,n_2,...,n_N} e^{-\beta \epsilon_{n_1}} e^{-\beta \epsilon_{n_2}}...e^{-\beta \epsilon_{n_N}}

When carrying out this sum over all the indices \{n_i\}, we could choose a particular value of, say, n1, carry out the summation over all the other indices, then choose another value of n1, sum over all other indices again, and so on. When we are summing over all remaining indices, the value of e^{-\beta \epsilon_{n_1}} is constant. This allows us to express the sum as

\displaystyle Z=\sum_{n_1}e^{-\beta \epsilon_{n_1}}\sum_{n_2,...,n_N}e^{-\beta \epsilon_{n_2}}...e^{-\beta \epsilon_{n_N}}

Similar reasoning tells us we can do this for every index, such that the partition function becomes

\displaystyle Z=\sum_{n_1}e^{-\beta\epsilon_{n_1}}\sum_{n_2}e^{-\beta\epsilon_{n_2}}...\sum_{n_N}e^{-\beta\epsilon_{n_N}}

But if every particle is identical, in that it has access to the same set of energy levels \{\epsilon_n\}, each one of these sums is identical:

\displaystyle Z=\left(\sum_n e^{-\beta\epsilon_n}\right)^N

So the partition function describing the composite system is simply a mini partition function, corresponding to a system comprising a single particle, raised to the power of N, the total number of particles in the system. It is therefore useful to introduce the single-particle partition function z, defined by

\displaystyle z=\sum_n e^{-\beta\epsilon_n}

where \{\epsilon_n\} is the set of energies available to each individual particle. It is good to establish this relation now, so that it can be used in the future. In many circumstances it is a useful approximation, but there is actually something seriously wrong with its formulation, something which has nothing to do with non-interactivity. We have ignored the implications of the particle’s indistinguishability. By this, we do not mean that our modern-day experimental apparatus is incapable of telling, say, one electron from another, but that this is fundamentally impossible. Strictly speaking, this renders the derivation above invalid, since in saying that we could write the list

\alpha\equiv\{n_1,n_2,...,n_N\},

we assumed we could tell one particle from another. But that’s a discussion for another day! For now, the expression above will be adequate, and we’ll use it to study some simple systems.

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