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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), 2269-2284. doi: 10.1098/rspa.2004.1283.  Google Scholar

[2]

R. D. Astumian, Thermodynamics and kinetics of a Brownian motor, Science, 276, (1997), 917-922. doi: 10.1126/science.276.5314.917.  Google Scholar

[3]

D. Astumian and P. Hänggi, Brownian motors, Phys. Today, 55 (2002), 33-39. doi: 10.1063/1.1535005.  Google Scholar

[4]

J. Dolbeault, D. Kinderlehrer and M. Kowalczyk, Remarks about the flashing rachet, in: "Partial Differential Equations and Inverse Problems," (2004), 167-175, Contemporary Mathematics, 362, American Mathematical Society, Providence, RI.  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), 1-15.  Google Scholar

[6]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. 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), 149-179. 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), 552-563. 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), 153-169. 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, Boca Raton, FL, 2007.  Google Scholar

[11]

P. Reimann and P. Hänggi, Introduction to the physics of Brownian motors, Appl. Phys. A, 75 (2002), 169-178. 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), 2269-2284. doi: 10.1098/rspa.2004.1283.  Google Scholar

[2]

R. D. Astumian, Thermodynamics and kinetics of a Brownian motor, Science, 276, (1997), 917-922. doi: 10.1126/science.276.5314.917.  Google Scholar

[3]

D. Astumian and P. Hänggi, Brownian motors, Phys. Today, 55 (2002), 33-39. doi: 10.1063/1.1535005.  Google Scholar

[4]

J. Dolbeault, D. Kinderlehrer and M. Kowalczyk, Remarks about the flashing rachet, in: "Partial Differential Equations and Inverse Problems," (2004), 167-175, Contemporary Mathematics, 362, American Mathematical Society, Providence, RI.  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), 1-15.  Google Scholar

[6]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. 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), 149-179. 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), 552-563. 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), 153-169. 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, Boca Raton, FL, 2007.  Google Scholar

[11]

P. Reimann and P. Hänggi, Introduction to the physics of Brownian motors, Appl. Phys. A, 75 (2002), 169-178. doi: 10.1007/s003390201331.  Google Scholar

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