Effect of Coulombic friction on spatial displacement statistics

Abstract

The phenomenon of Coulombic friction enters the stochastic description of dry friction between two solids and the statistic characterization of vibrating granular media. Here we analyze the corresponding Fokker-Planck equation including both velocity and spatial components, exhibiting a formal connection to a quantum mechanical harmonic oscillator in the presence of a delta potential. Numerical solutions for the resulting spatial displacement statistics show a crossover from exponential to Gaussian displacement statistics. We identify a transient intermediate regime that exhibits multiscaling properties arising from the contribution of Coulombic friction. The possible role of these effects during observations in diffusion experiments is shortly discussed.

Rectification of asymmetric surface vibrations with dry friction: An exactly solvable model

“…In fact, vibrations with PSN statistics should in principle be easier to implement than Gaussian noise, since the fluctuations appear on a separate timescale. Dry friction has recently been shown to be relevant in particular to biological soft matter systems, which opens up the possibility of realizing the transport properties of our model in a nanoscale system [19,[30][31][32]. The thermal environment would provide an additional noise source, but overall the qualitative transport features should remain unchanged since thermal noise does not exert any net force.…”

“…(3) converges to Brownian motion with dry friction, which has recently been investigated both theoretically and experimentally [15][16][17][18][19][20][21][22][23]. No directed motion results in this case unless an additional bias, such as a constant force, acts on the object.…”

Asymptotic Derivation of Langevin-like Equation with Non-Gaussian Noise and Its Analytical Solution

“…Non-linear frictions are ubiquitous in nature [55][56][57] and are known to appear in systems such as granular [58][59][60], biological [61][62][63][64] and atomic-surface ones [65][66][67]. We note that non-linear frictions can be discontinuous functions with respect to velocity in general (e.g., Coulombic friction), and their singular effects on stochastic properties have been interesting topics [31][32][33][34]51,[68][69][70][71][72][73][74][75][76][77]. Indeed, as will be shown in the next section, the distribution function can be strongly singular around the peak.…”

Acoustic observations of near‐bed sediment concentration and flux statistics above migrating sand dunes

“…Their model involves a constant downstream‐directed mean force, a Coulomb‐type friction force, and a random (white noise) fluctuating force. Similar models have also been proposed and studied for randomly forced grains in containers, as a prototype for granular gases [ Kawarada and Hayakawa , ; Menzel and Goldenfeld , ]. The Fan et al [] model predicts the following equilibrium distribution for the horizontal momenta of an ensemble of bed load particles: where the subscript i denotes direction (streamwise or cross stream), F i is the mean force, Δ is the friction force, and 2 D i is the variance of the random fluctuating force.…”