On a structure preserving closure of Langevin dynamics

TL;DR

This paper extends the STIV framework to construct thermodynamically consistent macroscopic models from Langevin dynamics, ensuring gradient flow structure and second law compliance for arbitrary probability densities.

Contribution

It generalizes the STIV method to arbitrary densities, guarantees thermodynamic consistency, and demonstrates convergence to true densities in complex systems.

Findings

Numerical convergence to true probability densities in multiple systems

Extension of STIV to non-Gaussian densities while maintaining gradient flow

Reformulation of STIV in terms of observable averages obeying gradient flow

Abstract

Given a particle system obeying overdamped Langevin dynamics, we demonstrate that it is always possible to construct a thermodynamically consistent macroscopic model which obeys a gradient flow with respect to its non-equilibrium free energy. To do so, we significantly extend the recent Stochastic Thermodynamics with Internal Variables (STIV) framework, a method for producing macroscopic thermodynamic models far-from-equilibrium from the underlying mesoscopic dynamics and an approximate probability density of states parameterized with so-called internal variables. Though originally explored for Gaussian probability distributions, we here allow for an arbitrary choice of the approximate probability density while retaining a gradient flow dynamics. This greatly extends its range of applicability and automatically ensures consistency with the second law of thermodynamics, without the need for secondary verification. We demonstrate numerical convergence, in the limit of increasing internal variables, to the true probability density of states for both a multi-modal relaxation problem, a protein diffusing on a strand of DNA, and for an externally driven particle in a periodic landscape. Finally, we provide a reformulation of STIV with the quasi-equilibrium approximations in terms of the averages of observables of the mesostate, and show that these, too, obey a gradient flow.