The next information may improve your “This Week’s Finds”:

The principle of minimum production of entropy was established by Ilya Prigogine. He did very clear that it is only valid in the linear regime of non-equilibrium thermodynamics.

As is well-known the principle does not apply far from equilibrium, where the equations of motion cannot be obtained from minimizing some given action.

Entropic extensions of mechanics have been known for about 30 years now, much before Oettinger work!

The GENERIC equation of motion is a special case of the

more general canonical theory equation first obtained by Keizer for random variable n

dn/dt = (dn/dt)_mech + (dn/dt)_diss + (dn/dt)_rand

The first term is the usual reversible term, the second term is a dissipative term obtained from entropy S

(dn/dt)_diss = Omega [ exp(-n^+ @S/@n) - exp(-n^- @S/@n)]

The third term, (dx/dt)_rand, accounts for fluctuations.

GENERIC ignores the fluctuation term (read page 27 of the book that you cited). And only covers dissipation in the Markovian approximation.

Their derivation of dS/dt >= 0 is another proof that this approach is approximate. In more general non-Markovian evolutions, entropy increases according to the second law S(t) >= S(0), but the increasing is not locally monotonic. I.e. the local law dS/dt >= 0 is not valid at *all* times.

It is only in the Markovian approximation in which dS/dt >= 0 is valid.

Oettinger applies this approximation for instance in his (6.72). Thus the resulting M matrix associated to his dissipative bracket [S,n] is only asymptotically valid in the ‘kinetic’ range t >> tau.

There is further limitations and oversimplifications of the GENERIC approach which I am not noticing here. Also their ‘understanding’ of irreversibility as CC is without mathematical basis. It is just another instance of what van Kampen named “mathematical funambulism”.

To learn more about Keizer canonical theory and modern generalizations read

http://www.canonicalscience.org/research/time.html

http://www.canonicalscience.org/research/canonical.html

http://www.canonicalscience.org/research/nanothermodynamics.html

NOTE: Using my correct Spanish name, your blog gives encoding error and stop from posting! I am forced to post with an incorrect name using American encoding!

## Re: This Week’s Finds in Mathematical Physics (Week 295)

You start to see why Raoul Bott’s training in electrical engineering helped his mathematics.

To see if entropy and energy are linked as you say, it might be worth exploring other varieties of entropy: Kolmogorov-Sinai entropy, topological entropy, volume entropy, etc. I wonder what, if anything, is the common essence.

I see that Roland Gunesch is now hosting the material that Chris Hillman had collected on entropy.