Exact power spectra of Brownian motion with solid friction

TL;DR

This paper derives exact, closed-form expressions for the power spectrum of Brownian motion with solid friction, revealing a crossover between stick and slip regimes influenced by external forces.

Contribution

It introduces an alternative, exact Laplace transform-based solution for the velocity autocorrelation and power spectrum of Brownian motion with Coulomb friction, expanding on previous spectral methods.

Findings

Closed-form power spectrum expressions derived

Identification of stick-slip crossover behavior

Enhanced understanding of friction effects on Brownian dynamics

Abstract

We study a Langevin equation describing the Brownian motion of an object subjected to a viscous drag, an external constant force, and a solid friction force of the Coulomb type. In a previous work [H. Touchette, E. Van der Straeten, W. Just, J. Phys. A: Math. Theor. 43, 445002, 2010], we have presented the exact solution of the velocity propagator of this equation based on a spectral decomposition of the corresponding Fokker-Planck equation. Here, we present an alternative, exact solution based on the Laplace transform of this equation, which has the advantage of being expressed in closed form. From this solution, we also obtain closed-form expressions for the Laplace transform of the velocity autocorrelation function and for the power spectrum, i.e., the Fourier transform of the autocorrelation function. The behavior of the power spectrum as a function of the dry friction force and external forcing shows a clear crossover between stick and slip regimes known to occur in the presence of solid friction.