On 1/9/2014 2:26 PM, Jesse Mazer wrote:
Liouville's theorem is derived in deterministic classical mechanics. If you take a volume of phase space, each point in that volume is a specific microstate, and if you evolve each microstate forward for some time T using the deterministic equations of physics, you get a later set of microstates which occupy their own volume in phase space. Liouville's theorem just says the two volumes must be equal. It only becomes statistical if you interpret the original set of microstates as representing your own uncertainty--if you just know the original macrostate, you may choose to consider the statistical ensemble of microstates compatible with that macrostate, then they will give the "volume of phase space" that you start with. But that's just an extra layer of interpretation, Liouville's theorem itself is not really statistical.

Right. And entropy, the log of the number of possible states, only increases when "possible states" is defined by some macroscopic constraint. There is always a finer microstate definition of possible states such that the number and the entropy don't increase. Entropy is a consequence of coarse-graining.


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