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AbstractBy analytical methods we study the large time properties of the solution of a simple one-dimensional model of stochastic Stokes' drift. Semi-explicit formulae allow us to characterize the behaviour of the solutions and compute global quantities such as the asymptotic speed of the center of mass or the effective diffusion coefficient. Using an equivalent tilted ratchet model, we observe that the speed of the center of mass converges exponentially to its limiting value. A diffuse, oscillating front attached to the center of mass appears. The description of the front is given using an asymptotic expansion. The asymptotic solution attracts all solutions at an algebraic rate which is determined by the effective diffusion coefficient. The proof relies on an entropy estimate based on homogenized logarithmic Sobolev inequalities. In the travelling frame, the macroscopic profile obeys to an isotropic diffusion. Compared with the original diffusion, diffusion is enhanced or reduced, depending on the regime. At least in the limit cases, the rate of convergence to the effective profile is always decreased. All these considerations allow us to define a notion of efficiency for coherent transport, characterized by a dimensionless number, which is illustrated on two simple examples of travelling potentials with a sinusoidal shape in the first case, and a sawtooth shape in the second case. © 2008 Elsevier B.V. All rights reserved.
CitationBlanchet A, Dolbeault J, Kowalczyk M (2008) Travelling fronts in stochastic Stokes’ drifts. Physica A: Statistical Mechanics and its Applications 387: 5741–5751. Available: http://dx.doi.org/10.1016/j.physa.2008.06.011.
SponsorsA.B. and J.D. have been partially supported ECOS-CONICYT # C05E09. J.D. wishes to thank the members of the DIM for their hospitality. A.B. acknowledges the support of the KAUST investigator award. M.K. has been partially supported by: FONDECYT 1050311, Nucleo Milenio P04-069-F, FONDAP and ECOS-CONICYT # C05E05. All computations have been performed with Mathematica (TM).