Phase-space geometry of the generalized Langevin equation

TL;DR

This paper explores the geometric structure of the generalized Langevin equation's phase space, revealing how system preparation influences phase space complexity and structure, with implications for modeling heat bath interactions.

Contribution

It introduces a geometric perspective on the generalized Langevin equation, showing how phase space structure depends on system preparation and persists over time.

Findings

Phase space structure varies with system preparation.

Systems assumed to have existed forever have simpler phase space.

Differences in phase space persist even at long times.

Abstract

The generalized Langevin equation is widely used to model the influence of a heat bath upon a reactive system. This equation will here be studied from a geometric point of view. A dynamical phase space that represents all possible states of the system will be constructed, the generalized Langevin equation will be formally rewritten as a pair of coupled ordinary differential equations, and the fundamental geometric structures in phase space will be described. It will be shown that the phase space itself and its geometric structure depend critically on the preparation of the system: A system that is assumed to have been in existence for ever has a larger phase space with a simpler structure than a system that is prepared at a finite time. These differences persist even in the long-time limit, where one might expect the details of preparation to become irrelevant.