Statistical Mechanics – How is the Partition Function Related to the Equipartition Theorem?

partition functionstatistical mechanicsthermodynamics

I see that the partition function is associated to the way particles are "partitioned" among energy levels. The equipartition theorem divides (partitions) average energy among degrees of freedom allowed to the particles.

Equating the average energy based on the partition function, given by
$$\langle E\rangle = -\frac{\partial \ln Z}{\partial \beta}\,,$$
with the average energy of the system given by the equipartition theorem, given by $\langle E\rangle = (f/2)kT$, where $f$ is the total number of degrees of freedom in the system, $\beta = 1/kT$, and $\langle E\rangle$ is the average energy of the system, we get
$$
-\frac{\partial \ln Z}{\partial \beta}
=
\frac{f}{2}kT\,.
$$
Simplifying, we get
$$
\ln Z = -\frac{f}{2}\ln(kT) + const.\,,
$$
or,
$$
Z = (kT)^{-f/2} \times const.
$$
So, is it correct to say that:

  1. the partition function $Z$ shows how the average energy is partitioned among the degrees of freedom at a particular temperature OR
  2. the "partitions" of energy associated with the partition function $Z$ are actually the degrees of freedom allowed to the particles at a particular temperature.

Best Answer

The partition function is related to the equipartition theorem (the theorem is derived from that function) but the word "partition" refers to a different thing in the two cases. Here is a general way to think of the partition function in any ensemble.

If we assign a "multiplicity" $A_i$ to microstate $i$, then the probability of microstate is $$ p_i = \frac{A_i}{\sum_i A_i} = \frac{A_i}{\text{partition function}} $$ The denominator is the partition function of the ensemble and the summation is over all microstates that are compatible with the macroscopic state, whether this is $E,V,N$ (microcanonical), $T$, $V$, $N$ (canonical) or other.

In the canonical ensemble we have $A_i = e^{-\beta E_i}$, where $E_i$ is the energy of the microstate. It is the special form of the energy and its dependence on the degrees of freedom that leads to equipartition of energy. More specifically, equipartition is achieved in the limit $1/\beta = k T \to \infty$

So, while the partition function and the equipartition theorem are related to each other, the partition function is a more general tool that describes the probability of microstate, while the equipartition theorem is a result that is based on the partition function in the limit of high temperature.

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