# Canonical Ensemble

I have a few doubts about the canonical ensemble. Let's consider a system A in contact with a heat reservoir.

1) To derive the probability of the system being in a state $E_r$, we consider the probability of the reservoir being in the state $E_t$ - $E_r$ where $E_t$ is the total energy of the system.Then we try to find that $E_r$ where the number of states of the reservoir is maximum.While this is fine for small systems, what do you do for larger systems?(You have to also start considering the number of states of the system at some point,right?)

2)The reservoir has a very large heat capacity and a very large number of degrees of freedom. Does this always mean it is physically larger than the system?

## 1 Answer

First, small or large has to be determined from a relative point of view. As long as the system much smaller than the reservoir, then the variation works fine and it is ok to ignore the higher order terms, giving us the standard $e^{-\beta E}$ factor.

Second, strictly speaking the number of degrees of freedom should not be directly related to the size of the system. However, in order for the reservoir to have sufficient contact with the system to act as the heat bath, then the size of the reservoir probably shouldn't be too small from a practical point of view.