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I've never really understood how exactly Liouville's theorem about time-conservation of phase space volume can be reconciled with the second law of thermodynamics. Recently I came across this popular article,

http://www.necsi.edu/projects/baranger/cce.pdf

"

**Chaos, Complexity, and Entropy"**by Michel Baranger. In this paper, from page 13 on, he explains this 'paradox' with chaos: slightly different initial conditions in phase space 'fractalize' the volume, which remains constant due to Liouville. But by coarse graining we approach this fractalized volume by a solid volume ("bagging the mess"), and it is

*this*coarse-grained volume which is bigger than the fractalized volume. Hence the second law of thermodynamics.

Is anyone familiar with this view? At the end Baranger comments on the subjectivity of this coarse-grained volume and relates it to the laws of big numbers.

Any comments are appreciated :)

(edit: no idea why the text color has become blue, but never mind)