American Institute of Mathematical Sciences

Advanced
October  2011, 16(3): 963-971. doi: 10.3934/dcdsb.2011.16.963

The flashing ratchet and unidirectional transport of matter

Dmitry Vorotnikov 1,

1. 

CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal

Received  July 2010 Revised  March 2011 Published  June 2011

We study the flashing ratchet model of a Brownian motor, which consists in cyclical switching between the Fokker-Planck equation with an asymmetric ratchet-like potential and the pure diffusion equation. We show that the motor indeed performs unidirectional transport of mass, for proper parameters of the model, by analyzing the attractor of the problem and the stationary vector of a related Markov chain.
Keywords: Markov chain, Flashing ratchet, Brownian motor, Fokker-Planck equation, periodic solution, transport..
Mathematics Subject Classification: 35B10, 35Q84, 60J10, 60J70, 82C7.
Citation: Dmitry Vorotnikov. The flashing ratchet and unidirectional transport of matter. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 963-971. doi: 10.3934/dcdsb.2011.16.963
References:
[1]

P. Amengual, A. Allison, R. Toral and D. Abbott, Discrete-time ratchets, the Fokker-Planck equation and Parrondo's paradox,, Proc. Royal Society London A, 460 (2004), 2269.  doi: 10.1098/rspa.2004.1283.  Google Scholar

[2]

R. D. Astumian, Thermodynamics and kinetics of a Brownian motor,, Science, 276 (1997), 917.  doi: 10.1126/science.276.5314.917.  Google Scholar

[3]

D. Astumian and P. Hänggi, Brownian motors,, Phys. Today, 55 (2002), 33.  doi: 10.1063/1.1535005.  Google Scholar

[4]

J. Dolbeault, D. Kinderlehrer and M. Kowalczyk, Remarks about the flashing rachet,, in:, 362 (2004), 167.   Google Scholar

[5]

D. Heath, D. Kinderlehrer and M. Kowalczyk, Discrete and continuous ratchets: from coin toss to molecular motor,, Discr. Cont. Dyn. Sys. Ser. B, 2 (2002), 1.   Google Scholar

[6]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1.  doi: 10.1137/S0036141096303359.  Google Scholar

[7]

D. Kinderlehrer and M. Kowalczyk, Diffusion-mediated transport and the flashing ratchet,, Arch. Rat. Mech. Anal., 161 (2002), 149.  doi: 10.1007/s002050100173.  Google Scholar

[8]

P. Palffy-Muhoray, T. Kosa and W. E, Brownian ratchets and the photoalignment of liquid crystals,, Braz. J. Phys., 32 (2002), 552.  doi: 10.1590/S0103-97332002000300016.  Google Scholar

[9]

B. Perthame and P. E. Souganidis, Asymmetric potentials and motor effect: a large deviation approach,, Arch. Rat. Mech. Anal., 193 (2009), 153.  doi: 10.1007/s00205-008-0198-1.  Google Scholar

[10]

A. D. Polyanin and A. V. Manzhirov, "Handbook of Mathematics for Engineers and Scientists,", Chapman & Hall/CRC, (2007).   Google Scholar

[11]

P. Reimann and P. Hänggi, Introduction to the physics of Brownian motors,, Appl. Phys. A, 75 (2002), 169.  doi: 10.1007/s003390201331.  Google Scholar

show all references

References:
[1]

P. Amengual, A. Allison, R. Toral and D. Abbott, Discrete-time ratchets, the Fokker-Planck equation and Parrondo's paradox,, Proc. Royal Society London A, 460 (2004), 2269.  doi: 10.1098/rspa.2004.1283.  Google Scholar

[2]

R. D. Astumian, Thermodynamics and kinetics of a Brownian motor,, Science, 276 (1997), 917.  doi: 10.1126/science.276.5314.917.  Google Scholar

[3]

D. Astumian and P. Hänggi, Brownian motors,, Phys. Today, 55 (2002), 33.  doi: 10.1063/1.1535005.  Google Scholar

[4]

J. Dolbeault, D. Kinderlehrer and M. Kowalczyk, Remarks about the flashing rachet,, in:, 362 (2004), 167.   Google Scholar

[5]

D. Heath, D. Kinderlehrer and M. Kowalczyk, Discrete and continuous ratchets: from coin toss to molecular motor,, Discr. Cont. Dyn. Sys. Ser. B, 2 (2002), 1.   Google Scholar

[6]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1.  doi: 10.1137/S0036141096303359.  Google Scholar

[7]

D. Kinderlehrer and M. Kowalczyk, Diffusion-mediated transport and the flashing ratchet,, Arch. Rat. Mech. Anal., 161 (2002), 149.  doi: 10.1007/s002050100173.  Google Scholar

[8]

P. Palffy-Muhoray, T. Kosa and W. E, Brownian ratchets and the photoalignment of liquid crystals,, Braz. J. Phys., 32 (2002), 552.  doi: 10.1590/S0103-97332002000300016.  Google Scholar

[9]

B. Perthame and P. E. Souganidis, Asymmetric potentials and motor effect: a large deviation approach,, Arch. Rat. Mech. Anal., 193 (2009), 153.  doi: 10.1007/s00205-008-0198-1.  Google Scholar

[10]

A. D. Polyanin and A. V. Manzhirov, "Handbook of Mathematics for Engineers and Scientists,", Chapman & Hall/CRC, (2007).   Google Scholar

[11]

P. Reimann and P. Hänggi, Introduction to the physics of Brownian motors,, Appl. Phys. A, 75 (2002), 169.  doi: 10.1007/s003390201331.  Google Scholar

[1]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[2]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485
[3]

Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016
[4]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[5]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056
[6]

Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008
[7]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028
[8]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028
[9]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250
[10]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215
[11]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165
[12]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic & Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009
[13]

Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65
[14]

Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079
[15]

Simon Plazotta. A BDF2-approach for the non-linear Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2893-2913. doi: 10.3934/dcds.2019120
[16]

Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028
[17]

Florian Schneider, Andreas Roth, Jochen Kall. First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic & Related Models, 2017, 10 (4) : 1127-1161. doi: 10.3934/krm.2017044
[18]

Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845
[19]

Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic & Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011
[20]

Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences