Tuesday, 1 November 2011

Everything I've ever learnt is broken..?

Hmmm....

So here's my gripe from earlier today. We all know that the canonical ensemble has energies assigned according to the Boltzmann distribution. We also know that this is derived by maximising the entropy subject to the constraint of constant energy... but isn't the canonical distribution the one which minimised the free energy?

The trick is this: to derive the Boltzmann distribution we consider a subsystem, treating the rest of the (much larger) system as a reservoir at fixed total energy, and maximise the total entropy. In this way we get the distribution which minimises the free energy of any given subsystem: voila! This also makes it obvious why the microcanonical and canonical ensembles must become equivalent as the size of the system becomes larger: neat!

Now the apparent contradiction is gone, and I understand the ensembles a little bit better: everything isn't broken.
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3 comments:

  1. Seth Olsen2 November 2011 at 11:59

    I'm not sure I understand the problem (I can't find the earlier gripe to which you refer).

    Is a reservoir needed? You can justify the canonical distribution subjectively as the least biased inference based on a well-defined average energy (see paper below). The Boltzmann distribution is just the simplest distribution which is completely specified by its average energy (once the sample space is known).

    Jaynes, E. (1957). Information Theory and Statistical Mechanics. Physical Review, 106(4), 620–630.

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  2. James3 November 2011 at 10:17

    My gripe is essentially the first paragraph.

    I suppose the problem could be rephrased like this: at what point do I go from treating a system (not in contact with a reservoir) as a microcanonical system to a canonical system? The probability distributions are different in the limit of very small systems, but the same for large systems.

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  3. James7 November 2011 at 09:04

    Here's a question: If I have an ensemble with fixed chemical potential, volume and energy (I believe this is called a 'grand microcanonical ensemble'), what is the equilibrium distribution?

    As far as I can see, the Lagrange multiplier arguments lead to the Boltzmann distribution again...

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