Bridging statistical mechanics and thermodynamics away from equilibrium: a data-driven approach for learning internal variables and their dynamics

TL;DR

This paper introduces a machine learning framework that links microscopic statistical mechanics to macroscopic thermodynamics through internal variables, ensuring physical consistency and predictive capability in non-equilibrium systems.

Contribution

It develops IB-VONNs, a novel data-driven method combining information bottleneck, normalizing flows, and neural networks to learn thermodynamically consistent internal variables and their dynamics.

Findings

Successfully applied to colloidal systems governed by Langevin dynamics.

Verified framework with an analytically solvable colloidal particle in an optical trap.

Extended the approach to a phase-transforming system lacking a statistical mechanics foundation.

Abstract

Thermodynamics with internal variables is a common approach in continuum mechanics to model inelastic (i.e., non-equilibrium) material behavior. While this approach is computationally and theoretically attractive, it currently lacks a well-established statistical mechanics foundation. As a result, internal variables are typically chosen phenomenologically and lack a direct link to the underlying physics which hinders the predictability of the theory. To address these challenges, we propose a machine learning approach that is consistent with the principles of statistical mechanics and thermodynamics. The proposed approach leverages the following techniques (i) the information bottleneck (IB) method to ensure that the learned internal variables are functions of the microstates and are capable of capturing the salient feature of the microscopic distribution; (ii) conditional normalizing flows to represent arbitrary probability distributions of the microscopic states as functions of the state variables; and (iii) Variational Onsager Neural Networks (VONNs) to guarantee thermodynamic consistency and Markovianity of the learned evolution equations. The resulting framework, called IB-VONNs, is tested on two problems of colloidal systems, governed at the microscale by overdamped Langevin dynamics. The first one is a prototypical model for a colloidal particle in an optical trap, which can be solved analytically, and thus ideal to verify the framework. The second problem is a one-dimensional phase-transforming system, whose macroscopic description still lacks a statistical mechanics foundation under general conditions. The results in both cases indicate that the proposed machine learning strategy can indeed bridge statistical mechanics and thermodynamics with internal variables away from equilibrium.