Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

I am confused about phase-volume contraction in dissipative systems. Please help me catch the flaw in my understanding. From a macroscopic point of view I understand that a dynamic system tends to go to an equilibrium state or a limit cycle if its not chaotic. But now trying to understand it in phase space:

1) Consider a system in complete thermodynamic equilibrium. It is a non-dissipative system (it cannot dissipate anymore). Therefore it could be in any microstate allowed by constraints (e.g., conserved quantities) and by Liouville theorem the probability density does not change. Therefore isn't the phase space volume accessible to this system the whole phase space (allowed by constraints)?

2) Now consider a non-equilibrium system. Its location in the space has some peaked probability distribution based on the initial conditions, i.e., it occupies a small space volume, and when it approaches equilibrium the probability of finding it becomes uniform and spreads over the complete phase space. Doesn't that mean the phase volume has expanded?

share|improve this question
add comment

1 Answer

If the dissipative system has a thermodynamic equilibrium state, then in general, the set of microscopic initial conditions is larger than the set of microscopic states in the thermodynamic equilibrium state. Imagine a melting ice with a final state of water at 10°C. The initial state (some microscopic configuration corresponding to ice, or in general a set of microscopic configurations corresponding to ice) will be (almost) never visited anymore in the final equilibrium state. I say almost because mathematically it can happen, but the probability of that to happen due to statistical fluctuations is negligible in practice.

share|improve this answer
    
Do you mean 0 degree C? I mean are we talking about an isolated system of melting ice or one that gains energy to go to 10 C. Since my posing this question a few months ago I have realized that the term "dissipative" is sort of used in two ways: 1) The system loses energy... in that case I now understand that the phase space will shrink (lower energy fewer states) 2) Constant energy but equilibration with entropy generation. I was referring to this definition in my question. In such a case, e.g., ice melting to water at 0 C constant energy –  Sankaran Apr 9 '13 at 18:04
    
I still think that the phase volume must be increase. Of course I am going to assume ergodicity and coarse-graining to stay away from the irreversibility paradox –  Sankaran Apr 9 '13 at 18:07
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.