The flashing ratchet and unidirectional transport of matter

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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

Abstract
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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 and 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.
[2]	R. D. Astumian, Thermodynamics and kinetics of a Brownian motor, Science, 276, (1997), 917-922. doi: 10.1126/science.276.5314.917.
[3]	D. Astumian and P. Hänggi, Brownian motors, Phys. Today, 55 (2002), 33-39. doi: 10.1063/1.1535005.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	A. D. Polyanin and A. V. Manzhirov, "Handbook of Mathematics for Engineers and Scientists," Chapman & Hall/CRC, Boca Raton, FL, 2007.
[11]	P. Reimann and P. Hänggi, Introduction to the physics of Brownian motors, Appl. Phys. A, 75 (2002), 169-178. doi: 10.1007/s003390201331.

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.
[2]	R. D. Astumian, Thermodynamics and kinetics of a Brownian motor, Science, 276, (1997), 917-922. doi: 10.1126/science.276.5314.917.
[3]	D. Astumian and P. Hänggi, Brownian motors, Phys. Today, 55 (2002), 33-39. doi: 10.1063/1.1535005.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	A. D. Polyanin and A. V. Manzhirov, "Handbook of Mathematics for Engineers and Scientists," Chapman & Hall/CRC, Boca Raton, FL, 2007.
[11]	P. Reimann and P. Hänggi, Introduction to the physics of Brownian motors, Appl. Phys. A, 75 (2002), 169-178. doi: 10.1007/s003390201331.

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