How the arrow of time emerges from incomplete knowledge: a path-integral approach

TL;DR

This paper presents a path-integral approach to derive irreversible behavior and the arrow of time from time-reversible Hamiltonian mechanics by considering incomplete knowledge and phase space exploration.

Contribution

It introduces a path-integral formulation of lack-of-fit reduction in non-equilibrium thermodynamics, deriving the GENERIC framework from Hamiltonian mechanics without fitting parameters.

Findings

Dissipation emerges from ignoring degrees of freedom.

Derived a formula for diffusion constant in an ideal gas.

Generalized information measures to non-Boltzmann entropies.

Abstract

How does the arrow of time (dissipative, irreversible behavior) emerge from time-reversible Hamiltonian mechanics? Two ingredients are needed: the underlying system must be ergodic or phase-mixing, and our knowledge of the system must be incomplete. When the detailed dynamics explores its phase space and stays close to a submanifold parametrized by a reduced set of state variables, the lack-of-fit reduction method reveals that the effective equations for those reduced variables are necessarily irreversible. To make this precise, we present a path-integral formulation of the lack-of-fit reduction in non-equilibrium thermodynamics, which shows how the GENERIC framework (reversible Hamiltonian part plus irreversible gradient flow) emerges from purely Hamiltonian mechanics without any fitting parameters. The formulation is based on the Onsager-Machlup variational principle, and it yields reduced dynamical equations by minimizing the information discrepancy between the detailed and reduced evolutions. Subsequently, the reduction method is illustrated on the Kac--Zwanzig model, confirming that dissipation emerges solely from ignoring degrees of freedom, and on diffusion, where a formula for the diffusion constant in an almost ideal gas is derived. We also show how to generalize the Fisher information matrix and Kullback--Leibler divergence to arbitrary concave entropies via the principle of maximum entropy, including non-Boltzmann-Gibbs cases such as the Tsallis--Havrda--Charvat entropy.