Unifying Theories for Nonequilibrium Statistical Mechanics

Abstract

The question of deriving general force/flux relationships that apply out of the linear response regime is a central topic of theories for nonequilibrium statistical mechanics. This work applies an information theory perspective to compute approximate force/flux relations and compares the result with traditional alternatives. If it can be said that there is a consensus on the form of response theories in driven, nonequilibrium transient dynamics, then that consensus is consistent with maximizing the entropy of a distribution over transition space. This agreement requires the problem of force/flux relationships to be described entirely in terms of such transition distributions, rather than steady-state properties (such as near-equilibrium works) or distributions over trajectory space (such as maximum caliber). Within the transition space paradigm, it is actually simpler to work in the fully nonlinear regime without relying on any assumptions about the steady-state or long-time properties. Our results are compared to extensive numerical simulations of two very different systems. The first is a the periodic Lorentz gas under constant external force, extended with angular velocity and physically realistic inelastic scattering. The second is an $\alpha$-Fermi-Pasta-Ulam chain, extended with a Langevin thermostat that couples only to individual harmonic modes. Although we simulate both starting from transient initial conditions, the maximum entropy structure of the transition distribution is clearly evident on both atomistic and intermediate size scales. The result encourages further development of empirical laws for nonequilibrium statistical mechanics by employing analogies with standard maximum entropy techniques -- even in cases where large deviation principles cannot be rigorously proven.