The Prandtl-Tomlinson model of friction with stochastic driving

Abstract

We consider the classical Prandtl-Tomlinson model of a particle moving on a corrugated potential, pulled by a spring. In the usual situation in which pulling acts at constant velocity $\dot{\gamma }$, the model displays an average friction force $\sigma$ that relates to $\dot{\gamma }$ (for small $\dot{\gamma })$ as $\dot{\gamma }\sim (\sigma -{\sigma }_c{)}^{\beta }$, where ${\sigma }_c$ is a critical friction force. The possible values of $\beta$ are well known in terms of the analytical properties of the corrugated potential. We study here the situation in which the pulling has, in addition to the constant velocity term, a stochastic term of mechanical origin (i.e, the total driving is a function of $\dot{\gamma }t$). We analytically show how this term modifies the force-velocity dependence close to the critical force, and give the value of $\beta$ in terms of the analytical properties of the corrugation potential and the scaling properties of the stochastic driving, encoded in the value of its Hurst exponent.

8 pages, 8 figures

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