Canonical Ensemble

Question

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?

Answer 1

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.

Answer 2

1. What you do for larger systems is enlarge the reservoir until it dominates. The expression one obtains is then valid in this limit. The very definition of the word "temperature" involves this idealization.

2. The reservoir is a fiction introduced for the purpose of calculation. One could indeed use a physically large reservoir, such as the Pacific Ocean or something like that (with devices to make sure any heat exchanged is also rapidly conducted throughout the reservoir), but one could equally well use a small entity equipped with a thermostat. The thermostat has the job of making the small entity stay at the same temperature, no matter how much heat it exchanges with your chosen system. The claim is that the system will not care about exactly what is going on inside the reservoir, so the calculation based on a physically large reservoir also applies to this case.