Path integral approach to random motion with nonlinear friction

TL;DR

This paper applies a path integral method to analytically solve a nonlinear Langevin equation modeling stick-slip motion, revealing two fundamental motion types that explain complex frictional behaviors.

Contribution

It introduces an analytical solution to a nonlinear Langevin equation using path integrals, clarifying the physical nature of sliding and stick-slip motions.

Findings

Identifies two classes of optimal paths: sliding and stick-slip.

Shows the model explains complex solid/solid friction behaviors.

Provides a framework for analyzing nonlinear frictional systems.

Abstract

Using a path integral approach, we derive an analytical solution of a nonlinear and singular Langevin equation, which has been introduced previously by P.-G. de Gennes as a simple phenomenological model for the stick-slip motion of a solid object on a vibrating horizontal surface. We show that the optimal (or most probable) paths of this model can be divided into two classes of paths, which correspond physically to a sliding or slip motion, where the object moves with a non-zero velocity over the underlying surface, and a stick-slip motion, where the object is stuck to the surface for a finite time. These two kinds of basic motions underlie the behavior of many more complicated systems with solid/solid friction and appear naturally in de Gennes' model in the path integral framework.