# Is it possible to derive the partition function from action?

According to thermal physics, any system, after an infinite amount of time, will obey the Boltzmann distribution. However as each particle follows a trajectory that minimises the action surely it should be possible to derive the Boltzmann distribution/partition function by minimising the actions

So my question is, is this possible and if so I would be interested to see how its done.

## 1 Answer

I assume you are referring to classical systems.

First of all, the assumption that any system will thermalize is not correct. In fact, it's quite easy to write down the action of a system that will never reach thermal equilibrium. A prominent example is the case of integrable systems, which will evolve by maintaining the initial value of their associated first integrals.

In equilibrium statistical mechanics, one usually starts by postulating that a certain system satisfies the so-called ergodic hypothesis. In simple terms, that is telling you that, under time evolution, your system will uniformly span its phase space. Therefore, you can replace a time average with a much simpler average over the phase space, which does not require you to solve the equations of motion (which would be typically impossible for a system with a large number of degrees of freedom).

So, in this sense, you start precisely by postulating that the information that is contained in the action, i.e. the information about the precise dynamics of a given system, is irrelevant when it comes to statistical averages. What might be highly nontrivial, is to show that a given dynamical system is actually ergodic. And of course, to prove that hypothesis, you do need the action of the system. For an example of dynamical systems whose ergodicity has been rigorously established, you can look at dynamical billiards.