A path integral formalism for non-equilibrium Hamiltonian statistical systems

TL;DR

This paper introduces a path integral formalism for non-equilibrium Hamiltonian systems using a manifold of quasi-equilibrium densities, enabling practical computation and analysis of steady states.

Contribution

It develops a novel path integral approach based on a generalized Boltzmann principle for non-equilibrium systems, linking thermodynamics, quantum analogies, and ensuring unique steady-state solutions.

Findings

Path integral formalism for non-equilibrium systems proposed

Connections established with quantum theory and Wiener processes

Existence of unique steady-state distribution under certain conditions

Abstract

A path integral formalism for non-equilibrium systems is proposed based on a manifold of quasi-equilibrium densities. A generalized Boltzmann principle is used to weight manifold paths with the exponential of minus the information discrepancy of a particular manifold path with respect to full Liouvillean evolution. The likelihood of a manifold member at a particular time is termed a consistency distribution and is analogous to a quantum wavefunction. The Lagrangian here is of modified generalized Onsager-Machlup form. For large times and long slow timescales the thermodynamics is of Oettinger form. The proposed path integral has connections with those occuring in the quantum theory of a particle in an external electromagnetic field. It is however entirely of a Wiener form and so practical to compute. Finally it is shown that providing certain reasonable conditions are met then there exists a unique steady-state consistency distribution.