Explaining order in non-equilibrium steady states

Statistical mechanics explains that systems in thermal equilibrium spend a greater fraction of their time in states with apparent order because these states have lower energy. This explanation is remarkable, and powerful, because energy is a "local" property of states. While non-equilibrium steady states can similarly exhibit order, there can be no local property analogous to energy that explains why, as Landauer argued 50 years ago. However, recent experiments suggest that a broad class of non-equilibrium steady states satisfy an approximate analogue of the Boltzmann distribution, with tantalizing possibilities for basic and applied science.

 

I will explain how this analogue can be viewed as one of several approximations of Markov chain stationary distributions that arise throughout network science, random matrix theory, and physics. In brief, this approximation "works" when the correlation between a Markov chain's effective potential and the logarithm of its exit rates is high. It is therefore important to estimate this correlation for different classes of Markov chains. I will discuss recent results on the correlation exhibited by reaction kinetics on networks and dynamics of the Sherrington–Kirkpatrick spin glass, as well as highly non-reversible Markov chains with i.i.d. random transition rates. (Featuring joint work with Dana Randall and Frank den Hollander.)