Thermodynamics of bits and batteries

It is not uncommon in nonequilibrium statistical mechanics to describe
a system by a Fokker-Planck equation obeying the
fluctuation-dissipation theorem. If one treats the theory of such
Fokker-Planck equations axiomatically, one finds that the finite
models of this theory are precisely the detailed-balanced, finite
Markov chains. In this setting, given a pair of systems each described
by a Markov chain, we present an axiom that describes which
detailed-balanced Markov chains on the product space are
physically-valid interactions. To every system, we associate a
real-valued function we call "internal action" on its probability
simplex. This function captures not only how much free energy a system
holds, but also how slowly this free energy relaxes to the equilibrium
value. For triples consisting of two systems and an interaction
between them, we conjecture an action processing inequality that
interactions reduce internal action. Together with the
Szilard-Landauer correspondence between energy and information, this
implies the impossibility of spontaneous copying of information from a
bit of low robustness to a bit of high robustness, i.e., a
thermodynamic no-cloning. We are able to prove this conjecture for the
simplest non-trivial case --- Markov chains with two states ---
through case enumeration and explicit calculation.