Nonequilibrium Boltzmann distribution

Just as the equilibrium probability distribution can be expressed in terms of local energy $U(x)$ via the Boltzmann law:

So Rattling tell us that we can express non-equilibrium steady-states in terms of local Rattling value $\mathcal{R}(x)$ as:

This is a very powerful claim that has been sought by the nonequilibrium stat mech research community for decades, and so it comes at a price: the “$\approx$” in this expression is a poorly-controlled approximation. It generally works for “typical” systems, but that notion so far lacks a formal definition. Some of the core questions in our research are thus understanding the assumptions and the practical generality of this approximation. #formal #general

The down-side of this is that currently, we don’t have a clear formal method (beyond heuristics) to a priori decide how well expression (1) will hold for a given system. For example, many key results in nonequilibrium statistical mechanics are derived from perturbative expansion for systems close to equilibrium, which is not the case here, and we have not identified any a priori small parameter that yields (1) in the limit. The upside is that this allows us to circumvent a core no-go theorem that held back progress in this area. In practice, we show that this expression holds for many systems arbitrarily far from equilibrium—regime where many other methods fail—thus allowing us to theoretically make predictions and engineer behaviors in these (see Examples).