Handle URI:
http://hdl.handle.net/10754/600077
Title:
Travelling fronts in stochastic Stokes’ drifts
Authors:
Blanchet, Adrien; Dolbeault, Jean; Kowalczyk, Michał
Abstract:
By 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.
Citation:
Blanchet 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.
Publisher:
Elsevier BV
Journal:
Physica A: Statistical Mechanics and its Applications
Issue Date:
Oct-2008
DOI:
10.1016/j.physa.2008.06.011
Type:
Article
ISSN:
0378-4371
Sponsors:
A.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).
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Full metadata record

DC FieldValue Language
dc.contributor.authorBlanchet, Adrienen
dc.contributor.authorDolbeault, Jeanen
dc.contributor.authorKowalczyk, Michałen
dc.date.accessioned2016-02-28T06:35:34Zen
dc.date.available2016-02-28T06:35:34Zen
dc.date.issued2008-10en
dc.identifier.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.en
dc.identifier.issn0378-4371en
dc.identifier.doi10.1016/j.physa.2008.06.011en
dc.identifier.urihttp://hdl.handle.net/10754/600077en
dc.description.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.en
dc.description.sponsorshipA.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).en
dc.publisherElsevier BVen
dc.subjectBrownian motionen
dc.subjectDrift velocityen
dc.subjectEffective diffusionen
dc.subjectMolecular motorsen
dc.subjectRatcheten
dc.subjectStochastic Stokes' driften
dc.subjectTransport coherenceen
dc.titleTravelling fronts in stochastic Stokes’ driftsen
dc.typeArticleen
dc.identifier.journalPhysica A: Statistical Mechanics and its Applicationsen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
dc.contributor.institutionCentre de Recherche en Mathematiques de la Decision, Paris, Franceen
dc.contributor.institutionUniversidad de Chile, Santiago, Chileen
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