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 some rule-of-thumb intuitions) to a-priori decide how well expression (1) will hold for a given system. For example, many key results in nonequilibrium stat mech 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 up-side 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..).