Brownian motion with dry friction on a randomly vibrated plate: inhomogeneous friction thresholds, the static–dynamic threshold distinction, and quantitative theory–experiment reconciliation

A small object (solid or droplet) on a horizontally vibrated plate whose random acceleration drives it against dry friction performs a modified Brownian motion that de Gennes (2005) analyzed by scaling arguments, predicting an effective diffusion coefficient suppressed as $D\propto\Gamma^3/\Delta^4$ in the friction-dominated (partly stuck) regime. The source leaves three issues unresolved: the effect of spatially inhomogeneous friction thresholds (trapping islands), the case of distinct static and dynamic thresholds ($\Delta_s\neq\Delta_d$), and experimental verification of the scaling predictions. Later literature has resolved much of this: the exact white-noise theory of the homogeneous model exists (velocity propagator, correlation functions, power spectra, d-dimensional diffusion coefficient), colored-noise and non-Gaussian extensions have been studied, distinct static and dynamic thresholds have been given a rigorous mathematical framework (piecewise deterministic Markov processes) with phase-duration statistics, and experiments on vibrated solids, granular probes and hysteretic drops have been performed. What remains open: the effective diffusivity and trapping-time statistics for spatially varying thresholds, including the boundary between diffusive and trapped behavior; the diffusivity and scaling laws for $\Delta_s\neq\Delta_d$, for which a framework but no result exists; and the quantitative reconciliation of theory with the measured diffusivity scalings and small-velocity deviations.

Background

Classical Brownian motion is described by the Langevin equation, in which a randomly fluctuating force — representing thermal agitation or any chaotic environment — is balanced by viscous friction, a drag force proportional to velocity. De Gennes (2005) proposed a variant in which the dissipation is dry (Coulomb) friction, the threshold behavior familiar from solid–solid contact: an object on a surface does not move until the applied force exceeds a threshold, and once moving it is resisted by a force of roughly constant magnitude $\Delta$ opposing the velocity, written $-\Delta\,\sigma(v)$ with $\sigma(v)=\mathrm{sign}(v)$. For a liquid drop the analogous dissipation is contact angle hysteresis — the difference between the advancing and receding contact angles at the drop's contact line (the boundary where liquid, solid, and air meet) — which likewise acts as a threshold force; de Gennes explicitly included this 'dry friction for drops'. The physical picture is a small solid or droplet on a horizontal plate vibrating randomly, so that in the object's frame the driving is a random acceleration; idealized as Gaussian white noise of power $\Gamma$, the model equation is

$$m\dot v = -\Delta\,\sigma(v) + \xi(t),\qquad \langle\xi(t)\xi(0)\rangle = m^2\Gamma\,\delta(t).$$

Because dry friction fully arrests the object after a finite time rather than exponentially fast, the stochastic dynamics differs qualitatively from the Ornstein–Uhlenbeck process (the Gaussian velocity process produced by linear viscous friction): the stationary velocity distribution develops a cusp at $v=0$, and part of the probability sits in a 'stuck' state. De Gennes analyzed the motion 'at two levels' — scaling laws and a propagator calculation (the propagator is the transition probability density of the velocity process, obtainable from the associated Fokker–Planck equation, the partial differential equation governing the probability density) — and predicted two regimes: below a crossover threshold $\Delta^*$ the motion is essentially ordinary Langevin diffusion, while for $\Delta>\Delta^*$ (the partly stuck regime, where any viscous drag is negligible compared with the dry friction) the long-time mean-square displacement still grows linearly in time but with a strongly reduced effective diffusion coefficient $D\propto\Gamma^3/\Delta^4$. A similar scaling had been obtained by Caughey and Dienes (1961) for a first-order nonlinear system with white-noise input, and Hayakawa (2005) independently derived the stationary velocity distribution. De Gennes suggested three physical realizations: micron-size solid particles whose multi-asperity contacts retain a dry-friction threshold against thermal noise; macroscopic objects on randomly vibrated sheets; and drops with contact angle hysteresis on vibrated substrates.

De Gennes' analysis left three issues unresolved: (1) the threshold $\Delta$ was taken spatially uniform, whereas real plates have position-dependent frictional properties — an 'island' where the local threshold exceeds the available driving can trap the object, and the effect of such inhomogeneity on the effective diffusion coefficient and on the statistics of trapping times was not addressed; (2) the static threshold $\Delta_s$ (the force needed to start motion from rest) and the dynamic threshold $\Delta_d$ (the friction force during motion) were assumed equal, whereas real contacts usually have $\Delta_s>\Delta_d$, and the consequences for the diffusion laws were not explored; (3) the scaling predictions awaited experimental verification, which would require a theory beyond scaling that keeps the finite correlation time of the plate acceleration and its full probability distribution.

Much of this program has since been carried out for the homogeneous model. The exact white-noise theory is known: Touchette, Van der Straeten and Just (2010) solved the time-dependent Fokker–Planck equation exactly for pure dry friction and in series form for dry plus viscous friction and a constant external force, obtaining the velocity propagator and correlation function (and correcting some of de Gennes' formulas); the exact power spectra (Touchette, Prellberg and Just 2012), first-passage properties (the statistics of the time for the velocity to first reach a prescribed level) and large-deviation properties (the exponential rarity of atypical fluctuations) (Chen and Just 2014), spatial displacement statistics (Menzel and Goldenfeld 2011), and the diffusion coefficient in $d$ dimensions (Das, Puri and Schwartz 2017) followed. Colored noise — random forcing with a finite correlation time $\tau$ — was treated approximately by Geffert and Just (2017) and computationally by Garnier and Mertz (2023), who showed that the white-noise approximation, while reliable for velocity statistics, can be badly wrong for the diffusivity unless $\tau$ is extremely small. For distinct static and dynamic thresholds, Garnier, Lu and Mertz (2023) formulated the problem as a piecewise deterministic Markov process (PDMP — a stochastic process alternating between deterministic motion phases and randomly triggered transitions; here motion restarts from the stuck state when the forcing exceeds $\Delta_s$), proved ergodicity, and characterized the distributions of the durations of stuck and moving phases. On the experimental side, the Chaudhury group measured drift and diffusion of small blocks and drops on white-noise-vibrated surfaces (2002–2011), Gnoli, Puglisi and Touchette (2013) realized the model with a rotator probe with bearing friction driven by a granular gas (a cloud of macroscopic grains fluidized by vibration, acting as a noise bath), and thermal-noise realizations at the micron and nanoscale have been probed (nanodroplet depinning; thermal-force contact spectroscopy). Notably, the measured diffusivity scaling with noise power (about $\Gamma^{1.61}$ for biased objects, versus $\approx\Gamma^{1.74}$ in single-threshold Langevin simulations) is not reproduced by the existing white-noise or colored-noise theory, and the granular experiments show deviations at small velocities where real bearings depart from ideal Coulomb friction.

Problem Statement

The problem is to complete the theory of de Gennes' dry-friction Brownian motion for realistic friction laws: characterize the effective long-time transport of an object of mass $m$ on a randomly vibrated plate, driven by a plate acceleration with noise power $\Gamma$ and correlation time $\tau$ and resisted by dry friction, in the three extensions that the original treatment and its exact successors left open.

(1) Spatially inhomogeneous threshold. Replace the constant $\Delta$ by a random threshold field $\Delta(x)$ with stated statistics (a periodic field enters only as a limiting consistency check). Determine how the statistics of $\Delta(x)$ modify the effective diffusion coefficient; establish when the long-time motion remains genuinely diffusive (mean-square displacement growing linearly in time) and when it becomes subdiffusive (sublinear growth) or the object becomes permanently trapped in regions (islands) where the local threshold exceeds the available driving — including how the answer depends on whether the driving is bounded or unbounded (under the unbounded Gaussian driving pinned above, sticking is transient and the object eventually escapes any island, so the trapping-time distribution and its tails are the central quantitative observable, while the bounded-driving case is to be addressed qualitatively); and derive the statistics of trapping (sticking) times.

(2) Distinct static and dynamic thresholds. For $\Delta_s\neq\Delta_d$ (typically $\Delta_s>\Delta_d$: motion starts only when the driving exceeds $\Delta_s$, while the friction during motion is $\Delta_d$), derive the effective diffusion coefficient and the modification of the scaling laws — for example, the replacement of the white-noise partly-stuck result $D\propto\Gamma^3/\Delta^4$ — as functions of $\Delta_s$, $\Delta_d$, $\Gamma$ and $\tau$. A mathematical framework and the stuck/moving phase-duration statistics already exist; the diffusivity itself and its scaling laws do not.

(3) Quantitative theory–experiment reconciliation. Identify the ingredients (finite correlation times, non-Gaussian acceleration distributions, departures of real friction laws from ideal Coulomb behavior at small velocities) needed for the theory to reproduce the measured diffusivities, their noise-power scalings, and the small-velocity velocity statistics in the existing experimental realizations — vibrated small objects, granular probes with bearing friction, and hysteretic drops.

These three parts serve one common objective: a complete, experimentally validated account of noise-driven diffusion with realistic dry friction. The quantities sought are the effective diffusion coefficient, the regime boundaries between diffusive, subdiffusive and trapped behavior, the sticking/trapping-time statistics, and the scaling exponents. For part (3) the target is to show that the identified ingredients reproduce the measured diffusivities, their noise-power scalings and the small-velocity statistics.

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