Experimental Investigation of the Drift and Diffusion of Small Objects on a Surface Subjected to a Bias and an External White Noise: Roles of Coulombic Friction and Hysteresis

Abstract

We study the stochastic motion of a small solid block or a small water drop on a flat solid support in the presence of an external noise and a bias. The bias is caused either by inclining the plane of the support, as is the case with the solid block, or by creating a gradient of wettability, as is the case with a water drop. Both the solid block and the water drop exhibit drifted Brownian-like motion. There are, however, differences between the motion described here and that of a classical drifted Brownian motion, in that the Coulombic friction (for solid on solid) or wetting hysteresis (for water drops on a solid) accounts for a significant resistance to motion in addition to the kinematic friction. Although the displacement distribution here is non-Gaussian, the variance of the distribution increases with time, indicating that the overall motion follows simple diffusion. The diffusivity and the mobility of the solid object are considerably lower than the values expected when the diffusion is governed by only kinematic friction. The experimental diffusivity increases with the power of the noise with an exponent of 1.61, which is close to that (1.74) of an analysis based on the Langevin equation when the Coulombic friction is taken into account in addition to the kinematic friction. The ratio of diffusivity and mobility increases slightly sublinearly with the power of the noise with an exponent of about 0.8. The experimentally observed relaxation time of the process is, however, considerably smaller than the Langevin relaxation time. When the experimental ratio of diffusivity and mobility is taken into account in the distribution function of the displacement, the later quantity becomes amenable to an analysis that is similar to the conventional fluctuation relations.

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

“…(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.…”

Dynamical phase transition in drifted Brownian motion

“…Here, µ is the drift, W t is the standard Brownian motion (BM) on R with W 0 = 0, acting as a noise, and σ > 0 is the noise amplitude. This model represents in the simplest case a particle moving at constant velocity µ, perturbed by a Gaussian white noise originating from thermal noise or background vibrations [29]. The variable X t can also be interpreted as the log-return of a stock price with mean µ and volatility σ [30], or as the random charge dissipated in a resistor when applying a linearly-increasing voltage in time, in which case W t is a Nyquist noise [31].…”

Focusing by blocking: Repeatedly generating central density peaks in self-propelled particle systems by exploiting diffusive processes

“…On the one hand, this can for example be the crawling or swimming machinery of motile biological microorganisms such as bacteria and algal cells [12][13][14][15][16][17][18]. On the other hand, self-propelled particles were artificially realized for instance in the form of granular hoppers [19][20][21][22] or self-propelled droplets [23][24][25]. A particularly interesting example are Janus particles that allow localized heating by light illumination of only one half of their body [26][27][28][29][30]; or they catalyze chemical reactions on only part of their surface [31][32][33].…”