Kramers equation and supersymmetry

TL;DR

This paper explores the supersymmetry in Hamilton's equations with noise and friction, revealing topological properties and providing new tools for analyzing reaction paths in complex systems.

Contribution

It introduces a supersymmetry framework applicable to time-dependent systems and develops practical methods for studying phase space currents and reaction paths.

Findings

Supersymmetry applies to both time-independent and time-dependent systems.

A 'reduced current' is defined to analyze phase space probability currents.

New strategies for computing reaction paths in multi-scale systems are proposed.

Abstract

Hamilton's equations with noise and friction possess a hidden supersymmetry, valid for time-independent as well as periodically time-dependent systems. It is used to derive topological properties of critical points and periodic trajectories in an elementary way. From a more practical point of view, the formalism provides new tools to study the reaction paths in systems with separated time scales. A 'reduced current' which contains the relevant part of the phase space probability current is introduced, together with strategies for its computation.