Canonical Ensemble Microstates

[jeffbarrington]

Hi, I've been looking at working with the canonical ensemble and getting the probabilities of a system being at a certain energy. For reference, I am following something of the form given under 'Canonical Ensemble' in this article: https://en.wikiversity.org/wiki/Statistical_mechanics_and_thermodynamics#Canonical_Ensemble

Now, I am not so sure about why we can say P_r = CΩ(E-E_r). This seems to imply that the small system lumped onto the side of the heat bath can only have one microstate associated with its small energy. This would be fine by me if it was so incredibly small that the energies are quantised quantum mechanically (in which case one energy would probably correspond to one microstate, but even then not always), but of course this theory came about before quantum mechanics got a hold and whoever thought it up was thinking of particles flying around in a tiny box or something like that, surely. In that case, could you not just reverse all the velocities of all of the particles and call that another microstate with the same energy? I'm just having trouble understanding quite how small this little system is (i.e. does it have like one or two particles in it or is it that there are a lot of particles but far fewer than the number in the reservoir?)

Sorry if this all sounds a bit confused.

Thanks in advance.

 

[gtoguy97]

No problem, this is a good question and something that has been debated for a long time! In the canonical ensemble, the small system lumped onto the side of the heat bath usually consists of many particles, but still far fewer than the number in the reservoir. This is because the system is assumed to be in equilibrium with the reservoir, meaning the fluctuations in energy are small.The idea is that, since fluctuations in energy are small, the microstates of the small system can be approximated by a single energy level. This means that the probability of the system being at a certain energy is determined by the number of microstates associated with that energy (Ω(E)) and a normalization constant (C). So, Pr = CΩ(E-Er), where Er is the average energy of the system.This is an approximation, and it does not always hold true. In some cases, such as when the system is small enough to be quantum mechanical, the probabilities of the system being at a certain energy will be determined by more than just a single energy level.