A piecewise deterministic Markov process approach modeling a dry friction problem with noise

TL;DR

This paper introduces a piecewise deterministic Markov process model for dry friction systems with noise, deriving equations and statistical tools to analyze static and dynamic phases, and establishing ergodicity and stationary measures.

Contribution

It presents a novel mathematical framework using PDMPs to model dry friction with noise, including derivation of Kolmogorov equations and stationary measures.

Findings

Derived Kolmogorov equations for the model

Proved ergodicity of the process

Provided formulas for stationary distributions and phase durations

Abstract

Understanding and predicting the dynamical properties of systems involving dry friction is a major concern in physics and engineering. It abounds in many mechanical processes, from the sound produced by a violin to the screeching of chalk on a blackboard to human infant crawling dynamics and friction-based locomotion of a multitude of living organisms (snakes, bacteria, scallops..) to the displacement of mechanical structures (building, bridges, nuclear plants, massive industrial infrastructures) under earthquakes and beyond. Surprisingly, even for low-dimensional systems, the modeling of dry friction in the presence of random forcing has not been elucidated. In this paper, we propose a piecewise deterministic Markov process approach modeling a system with dry friction including different coefficients for the static and dynamic forces. In this mathematical framework, we derive the corresponding Kolmogorov equations and related tools to compute statistical quantities of interest related to the distributions of the static (sticked) and dynamic phases. We show ergodicity and provide a representation formula of the stationary measure using independent identically distributed portions of the trajectory (excursions). We also obtain deterministic characterizations of the Laplace transforms of the probability density functions of the durations of the static and dynamic phases.