There Are Times That Try Bunny's Brains

Loschmidt's paradox is that the laws of thermodynamics are time asymmetric because entropy always increases, but the underlying laws of physics are symmetric under time reversal. It should not therefore be possible to derive the second law of thermodynamics from first principles.

 This has lead to the suicide of as many physicists (Boltzmann and Ehrenfest among them) as Schroedinger's cat.  Recently friends Motl and the  Capitalist Imperialist Swine have wandered into the killing field.  The Pig plays the chalk, quoting Boltzmann

Of course, when the past is determined by the correct method – the method of retrodictions which is a form of Bayesian inference – we will find out that the lower-entropy states are exponentially favored. We won't be able to become certain about any property of the Universe in the past but some most universal facts such as the increasing entropy will of course follow from this Bayesian inference. In particular, the correctly "retrodicted past entropy" will more or less coincide with the "actual past" curve.

The secret sauce is the statement that in retrodictions

...the lower-entropy states are exponentially favored.

I'm putting that in my "remains to be demonstrated" file.

There are times in every blogger's life when a Bunny comes across something that is really really interesting, but does not have the angle or the understanding to really push to a conclusion.  "A proof of Clausius' theorem for time reversible deterministic microscopic dynamics", Journal of Chemical Physics, 134 (2011) 204113 by D.J. Evans, S.R. Williams and D. J. Searles is the ticket for Eli.  This trio has been thinking about how to establish thermodynamics on other than the normal ad hoc basis.  Yes bunnies, the laws of thermodynamics are observations, better put Ansatze, or as the Wikipedia puts it educated guesses that are verified later by its results.  The Clausius inequality, should the readers have forgotten is

for a cyclic process and is equivalent to the second law.

EW&S prove this is true only in the limit of infinite time, where the temperature is that of the "underlying equilibrium state" and only assuming T-mixing, ergodic consistency and the axiom of causality.  That means, of course, that there are quantum issues yet to be confronted, but certainly this appears valid for classical thermodynamics.

 In a later paper (JCP 137 (2012) 194109), the three authors conclude:

Our proof (of the zeroth law) is constructed using an array of previous results.  Out proof of the zeroth "law" is far more informative than the corresponding derivation using ergodic theory.  Combining the present proof with the observation that for an isolated mechanical system the energy is constant and our recent proof of the Clausius inequality, we see that all the so called laws of classical thermodynamics are mathematical results provable from the laws of mechanics supplemented by the axiom of causality and by the T mixing condition.  Theis might be regarded as changing the logical status of thermodynamics.

A second, less obvious result of our work is that for nonequilibrium systems entropy seems to play no role at all!  Its place is taken by dissipation.  The idea that one could use the Gibbs entropy in proving relaxation to equilibrium is obviously erroneous as discussed in a range of papers . . . . .

So why does the Gibbs approach to equilibrium work?

The present work points out, however, that entropy is not really necessary away from equilibrium.  It is only at, or very near to, equilibrium when dissipation is identically zero or so small that local thermodynamic equilibrium can be assumed, that entropy (and the Gibbs approach-ER) may be useful.  One may resort to other dynamical notions of entropy, cf. the Bolzmann entropy, or the one defined in the third paper of Ref. 21 to avoid these problems.

The infinite time is where the fun lies.  Remember Loschmidt?  In a purely mechanical universe, motion is time reversible, e.g. given a complete description of the initial condition one can postdict where all the particles were at earlier time and predict where they will be in the future, but classical thermodynamics claims this not to be the case, and statistical mechanics establishes that for large systems this will not be possible because of entropic driving forces.

About a decade ago Evans and collaborators showed experimentally, that for small systems, easily of the size of today's nanostructures, there can be momentary violations of the second law.  There would appear to be nothing really standing in the way of building an entangled system of such a size and that could really be the Mayan calendar for physics.