Eur. Phys. J. E (2017) 40: 60
https://doi.org/10.1140/epje/i2017-11549-9

Regular Article

Single particle Brownian motion with solid friction

¹ School of Physical Sciences, Jawaharlal Nehru University, 110067, New Delhi, India
² Beverly and Raymond Sackler School of Physics and Astronomy, Tel Aviv University, 69934, Ramat Aviv, Israel
³ Faculty of Engineering, Holon Institute of Technology, Golomb 52, 5810201, Holon, Israel

^* e-mail: purijnu@gmail.com
^** e-mail: bricki@netvision.net.il

Received: 19 December 2016
Accepted: 19 May 2017
Published online: 8 June 2017

Abstract

We study the Brownian dynamics of a solid particle on a vibrating solid surface. Phenomenologically, the interaction between the two solid surfaces is modeled by solid friction, and the Gaussian white noise models the vibration of the solid surface. The solid friction force is proportional to the sign of relative velocity. We derive the Fokker-Planck (FP) equation for the time-dependent probability distribution to find the particle at a given location. We calculate analytically the steady state velocity distribution function, mean-square velocity and diffusion coefficient in d-dimensions. We present a generic method of calculating the autocorrelations in d-dimensions. This results in one dimension in an exact evaluation of the steady state velocity autocorrelation. In higher dimensions our exact general expression enables the analytic evaluation of the autocorrelation to any required approximation. We present approximate analytic expressions in two and three dimensions. Next, we numerically calculate the mean-square velocity and steady state velocity autocorrelation function up to d = 3 . Our numerical results are in good agreement with the analytically obtained results.