Why is phase space volume conserved? Why couldn't you build a freezer that cuts the velocity of each particle in half? f(x)=1/2 x is a one-to-one function. No information is lost.
It's just something we've discovered. The laws of physics as we understand them can be expressed in the language of Hamiltonian mechanics and any Hamiltonian system must obey Liouville's theorem.
Physics is reversible, which means one can define a notion of "volume" which is conserved under the actual evolution of the system (this is much closer to a tautology than you would guess from the post).
In order for physics to be reversible, implementing your f would require making some other change, elsewhere in the system (or else some larger deviation from physics as we know it). Say, you would have to increment a counter somewhere marking "Number of times velocity has been halved: X"
If this were how physics worked, our notion of "volume" would just be changed (it would get multiplied by 2^X for each fixed X, and then added up across different values of X). And then notions of "entropy" and "temperature" would be changed as well, and order would be restored.
There are some conditions under which "temperature" and "entropy" correspond to other physical quantities, defined in terms of kinetic energy and so forth. This is not physical law, and this would just cease to be such a case.
This topic is nicely explained by Stephen Hawking too, in "a brief history of time", in which he proofs how that implies that even if we have CPT symmetry (and even if we had just T symmetry), the thermodynamical arrow of time and the cognitive arrow of time (subjective time) need to have the same direction, always.
Today's post, The Second Law of Thermodynamics, and Engines of Cognition was originally published on 27 February 2008. A summary (taken from the LW wiki):
To form accurate beliefs about something, you really do have to observe it. It's a very physical, very real process: any rational mind does "work" in the thermodynamic sense, not just the sense of mental effort. Engines of cognition are not so different from heat engines, though they manipulate entropy in a more subtle form than burning gasoline. So unless you can tell me which specific step in your argument violates the laws of physics by giving you true knowledge of the unseen, don't expect me to believe that a big, elaborate clever argument can do it either.
Discuss the post here (rather than in the comments to the original post).
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