# Is / (How is) the partition function related to the equipartition theorem?

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.

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 $$1/\beta = k T \to \infty$$
• @Walrus Technically you are correct but only in the limit $T\to\infty$. The whole point of the partition function is to express it as a function of the microscopic state: $Z = \frac{1}{h^{3N}N!}\int e^{-\beta E} d r_i^{3N} d p_i^{3N}$, with $E=\sum_i \frac{p_i^2}{2m} + V(r_1,\cdots r_N)$. This function conveys everything we can know about a system composed of $N$ interacting particles. Apr 6, 2023 at 19:20