Stochastic Stokes’ drift

Classical Stokes’ drift is the small time-averaged drift velocity of suspended nondiffusing particles in a fluid due to the presence of a wave. We consider the effect of adding diffusion to the motion of the particles, and show in particular that a nonzero time-averaged drift velocity exists in general even when the classical Stokes’ drift is zero. Our results are obtained from a general procedure for calculating ensembleaveraged Lagrangian mean velocities for motion that is close to Brownian, and are verified by numerical simulations in the case of sinusoidal forcing. PACS numbers: 02.50-r, 05.40+j, 05.60+w A travelling wave in a fluid gives suspended particles a small drift velocity known as Stokes’ drift [1, 2, 3]. When there is more than one wave, the drift velocity is calculated by summing the contributions from each wave [4, 5]. In this letter we consider the influence of diffusion on the magnitude and direction of the drift velocity. As in the classical (diffusionless) case, the amplitude of the travelling wave is assumed small compared to its wavelength; 1 a non-zero drift velocity appears at second order in the amplitude. In the presence of more than one wave, the classical Stokes’ drift can sum to zero. Diffusion then produces a nonzero drift velocity whose magnitude and direction depends on the diffusivity of the suspended particles. Several mechanisms for the directed motion of small particles without a net macroscopic force have been proposed in the last 10 years [6, 7, 8, 9, 10]. Interest in such ‘ratchet’ effects has been motivated by the search for the mechanisms of biological motors, such as the conversion of chemical energy into directed motion by protein molecules, and by possible applications, such as the separation of particles in solution based on their diffusion coefficients. In both these cases small particles are believed to follow dynamics that are overdamped (first derivative in time) and noise-dominated. A drift velocity dependent on the size of suspended particles in solution has been produced experimentally using an asymmetric periodic potential turned on and off periodically [11]. Published theoretical models [12, 13] combine a periodic asymmetric potential in one dimension with non-white fluctuations. In this letter we consider motion in arbitrary dimensions that is diffusion-dominated. There is also a small deterministic forcing whose amplitude will be used as an expansion parameter; a drift velocity appears at second order and depends on the diffusivity. Thus diffusion due to microscopic motions, for example diffusion of particles in solution, can be exploited using a carefully-chosen combination of forcings to produce a net motion that depends on the diffusivity. We illustrate the effect with sinusoidal forcing and compare our calculations with numerical results in one and two space dimensions. It is possible to arrange the wave motions so that particles of different diffusivities have a time-averaged drift velocity in different directions, resulting in what we call ‘fan-out’. This may have applications for sorting particles according to their diffusivities. We show numerically that the fan-out can have an angular range of more than 180 degrees. We first develop an expansion scheme for motion that is overdamped and diffusiondominated. Consider a stochastic process X ≡ (Xt)t≥0 taking values in IR m and satisfying