Homework Help: Microcanonical vs. canonical ensembles

Pacopag · Apr 11, 2008

1. The problem statement, all variables and given/known data
Suppose we have a system of particles and we wish to calculate the thermodynamic variables (e.g. energy, entropy, pressure, etc.)
Will the result depend on whether we consider the microcanonical or the canonical ensemble?
I want to calculate the partition function to get the TD variables, but the way in which we need to calculate the partition function depends on the ensemble that we consider.

2. Relevant equations

3. The attempt at a solution

genneth · Apr 11, 2008

The result will be consistent.

Pacopag · Apr 11, 2008

Great! Thanks.

Pacopag · Apr 11, 2008

Does it matter which ensemble I use? micro/canoncal?

genneth · Apr 12, 2008

No -- use which ever one you are more comfortable with.

Pacopag · Apr 12, 2008

When I do the same problem with either ensemble, I get very different looking expressions for, say, the entropy. I guess what you're saying is that if I do the correct manipulations I should always be able to cast one expression in the form of the other?

genneth · Apr 12, 2008

Yep --- they are completely compatible. After all, they are both just special cases of the general maximum entropy principle.

Pacopag · Apr 12, 2008

Great. Thanks

Snugg1 · May 8, 2008

A further Question

Hi,

I agree with what you have said about the values being the same. I have calculated the mean square deviation of the energy in terms of Cv, using the Boltzmann Distribution, and I am trying to figure out what criterion is needed to to have the same energy values for both canonical & microcanonical ensembles. Any thoughts?

marcusl · May 8, 2008

Microcanonical is used when no energy or particles are is exchanged between subsystems or with the outside (no heat transfer through container walls, for instance). Canonical allows energy exchange but the number of particles must remain constant. Grand canonical also allows for change in particles (diffusion, etc.).

reilly · May 10, 2008

The two distribtions are substantially different. The microcanonical distribution deals with a single, fixed system energy, while the canonical distribution does not. However, a partition of the microcanonical system can yield canonical sub-systems -- what conditions would need to hold for this to be true?

Also, think about whether a canonical system can be composed of a set of microcanonical systems. Think about entropy in this context.
Regards,
Reilly Atkinson