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Rocking Subdiffusive Ratchets: Origin, Optimization and Efficiency

Published online by Cambridge University Press:  24 April 2013

I. Goychuk  and
V. O. Kharchenko
I. Goychuk*
Affiliation:
Institute of Physics, University of Augsburg, Universitätstr. 1, D-86135 Augsburg, Germany & Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany
V. O. Kharchenko
Affiliation:
Institute of Applied Physics, NAS of Ukraine, 58 Petropavlovskaya str., 40030 Sumy, Ukraine
*
⋆ Corresponding author. E-mail: [email protected]
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Abstract

We study origin, parameter optimization, and thermodynamic efficiency of isothermal rocking ratchets based on fractional subdiffusion within a generalized non-Markovian Langevin equation approach. A corresponding multi-dimensional Markovian embedding dynamics is realized using a set of auxiliary Brownian particles elastically coupled to the central Brownian particle (see video on the journal web site). We show that anomalous subdiffusive transport emerges due to an interplay of nonlinear response and viscoelastic effects for fractional Brownian motion in periodic potentials with broken space-inversion symmetry and driven by a time-periodic field. The anomalous transport becomes optimal for a subthreshold driving when the driving period matches a characteristic time scale of interwell transitions. It can also be optimized by varying temperature, amplitude of periodic potential and driving strength. The useful work done against a load shows a parabolic dependence on the load strength. It grows sublinearly with time and the corresponding thermodynamic efficiency decays algebraically in time because the energy supplied by the driving field scales with time linearly. However, it compares well with the efficiency of normal diffusion rocking ratchets on an appreciably long time scale.

Keywords

anomalous Brownian motiongeneralized Langevin equationmemory effectsviscoelasticityratchet transportstochastic thermodynamics
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 2: Anomalous diffusion , 2013 , pp. 144 - 158
Copyright
© EDP Sciences, 2013

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References

Amblard, F., Maggs, A. C., Yurke, B., Pargellis, A. N., Leibler, S.. Subdiffusion and anomalous local viscoelasticity in actin networks. Phys. Rev. Lett. 77 (1996), 4470–4473. CrossRefGoogle ScholarPubMed
Barbi, F., Bologna, M., Grigolini, P.. Linear response to perturbation of non-exponential renewal processes. Phys. Rev. Lett., 95 (2005), 220601. CrossRefGoogle Scholar
Barkai, E.. Fractional Fokker-Planck equation, solution, and application. Phys. Rev. E, 63 (2001), 046118. CrossRefGoogle ScholarPubMed
Bartussek, R., Hänggi, P., Kissner, J. G.. Periodically rocked thermal ratchets. Europhys. Lett., 28 (1994), 459–464. CrossRefGoogle Scholar
N. N. Bogolyubov, Elementary example of establishing thermal equilibrium in a system coupled to thermostat. in On some statistical methods in mathematical physics. Acad. Sci. Ukrainian SSR, Kiev, 1945, pp. 115-137, in Russian.
Bouchaud, J.-P., Georges, A.. Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep., 195 (1990), 127–293. CrossRefGoogle Scholar
Burov, S. and Barkai, E., Critical exponent of the fractional Langevin equation. Phys. Rev. Lett., 100 (2008), 070601. CrossRefGoogle ScholarPubMed
Chen, Y. C., Lebowitz, J. L.. Quantum particle in a washboard potential. I. Linear mobility and the Einstein relation. Phys. Rev. B, 46 (1992), 10743–10750. CrossRefGoogle Scholar
Cole, K. S., Cole, R. H.. Dispersion and absorption in dielectrics I. Alternating current characteristics. J. Chem. Phys., 9 (1941), 341–352. CrossRefGoogle Scholar
Deng, W. H., Barkai, E., Ergodic properties of fractional Brownian-Langevin motion. Phys. Rev. E, 79 (2009), 011112. CrossRefGoogle ScholarPubMed
Ford, G. W., Kac, M., Mazur, P.. Statistical Mechanics of Assemblies of Coupled Oscillators. J. Math. Phys., 6 (1965), No. 4, 504-515. CrossRefGoogle Scholar
Gammaitoni, L., Hänggi, P., Jung, P., Marchesoni, F.. Stochastic Resonance. Rev. Mod. Phys., 70 (1998), 223-288. CrossRefGoogle Scholar
T. C. Gard. Introduction to Stochastic Differential Equations. Dekker, New York, 1988.
Gemant, A.. A method of analyzing experimental results obtained from elasto-viscous bodies., Physics, 7 (1936), 311–317. CrossRefGoogle Scholar
Goychuk, I., Hänggi, P., Non-Markovian stochastic resonance. Phys. Rev. Lett., 91 (2003), 070601. CrossRefGoogle ScholarPubMed
Goychuk, I., Hänggi, P., Theory of non-Markovian stochastic resonance. Phys. Rev. E, 69 (2004), 021104. CrossRefGoogle Scholar
Goychuk, I., Hänggi, P., Vega, J. L., Miret-Artes, S.. Non-Markovian stochastic resonance: three state model of ion channel gating. Phys. Rev. E, 71 (2005), 061906. CrossRefGoogle Scholar
Goychuk, I., Heinsalu, E., Patriarca, M., Schmid, G., Hänggi, P.. Current and universal scaling in anomalous transport. Phys. Rev. E, 73 (2006), 020101 (Rapid Communication). CrossRefGoogle Scholar
Goychuk, I. and Hänggi, P.. Anomalous escape governed by thermal 1/f noise., Phys. Rev. Lett., 99 (2007), 200601. CrossRefGoogle ScholarPubMed
Goychuk, I., Anomalous relaxation and dielectric response. Phys. Rev. E, 76 (2007), 040102(R). CrossRefGoogle ScholarPubMed
Goychuk, I.. Viscoelastic subdiffusion: from anomalous to normal. Phys. Rev. E, 80 (2009), 046125. CrossRefGoogle ScholarPubMed
Goychuk, I.. Subdiffusive Brownian ratchets rocked by a periodic force. Chem. Phys., 375 (2010), 450–457. CrossRefGoogle Scholar
I. Goychuk, P. Hänggi. Subdiffusive dynamics in washboard potentials: two different approaches and different universality classes. in J. Klafter, S. C. Lim, R. Metzler, editors. Fractional Dynamics, Recent Advances. World Scientific, Singapore, 2011, Ch. 13, pp. 307–329.
Goychuk, I.. Viscoelastic subdiffusion: generalized Langevin equation approach. Adv. Chem. Phys., 150 (2012), 187–253. Google Scholar
Goychuk, I.. Is subdiffusional transport slower than normal?. Fluct. Noise Lett., 11 (2012), 1240009. CrossRefGoogle Scholar
Goychuk, I., Kharchenko, V.. Fractional Brownian motors and stochastic resonance. Phys. Rev. E, 85 (2012), 051131. CrossRefGoogle ScholarPubMed
He, Y., Burov, S., Metzler, R., Barkai, E.. Random time-scale invariant diffusion and transport coefficients. Phys. Rev. Lett., 101 (2008), 058101. CrossRefGoogle ScholarPubMed
Heinsalu, E., Patriarca, M., Goychuk, I., Schmid, G., Hänggi, P.. Fractional Fokker-Planck dynamics: Numerical algorithm and simulations. Phys. Rev. E, 73 (2006), 046133. CrossRefGoogle ScholarPubMed
Heinsalu, E., Patriarca, M., Goychuk, I., Hänggi, P., Use and abuse of a fractional Fokker-Planck dynamics for time-dependent driving. Phys. Rev. Lett., 99 (2007), 120602. CrossRefGoogle ScholarPubMed
Heinsalu, E., Patriarca, M., Goychuk, I., Hänggi, P.. Fractional Fokker-Planck subdiffusion in alternating force fields. Phys. Rev. E, 79 (2009), 041137. CrossRefGoogle ScholarPubMed
B. D. Hughes. Random walks and random environments, Vols. 1,2. Clarendon Press, Oxford, 1995.
Jülicher, F., Ajdari, A., Prost, J.. Modeling molecular motors. Rev. Mod. Phys., 69 (1997), 1269–1282. CrossRefGoogle Scholar
Kharchenko, V., Goychuk, I.. Flashing subdiffusive ratchets in viscoelastic media. New J. Phys., 14 (2012), 043042. CrossRefGoogle Scholar
A. N. Kolmogorov. Dokl. Akad. Nauk SSSR, 26 (1940), 115–118 (in Russian), English transl. Wiener spirals and some other interesting curves in a Hilbert space, in V. M. Tikhomirov, editor. Selected Works of A. N. Kolmogorov, vol. I, Mechanics and Mathematics. Kluwer, Dordrecht, 1991, pp. 303-307.
Kubo, R.. The fluctuation-dissipation theorem. Rep. Prog. Phys., 29 (1966), 255–284. CrossRefGoogle Scholar
R. Kubo, M. Toda, and M. Hashitsume. Nonequilibrium Statistical Mechanics, 2nd ed. Springer, Berlin, 1991.
Kupferman, R.. Fractional kinetics in Kac-Zwanzig heat bath models. J. Stat. Phys., 114 (2004), 291–326. CrossRefGoogle Scholar
Lubelski, A., Sokolov, I. M., Klafter, J.. Nonergodicity mimics inhomogeneity in single particle tracking. Phys. Rev. Lett., 100 (2008), 250602. CrossRefGoogle ScholarPubMed
Lutz, E.. Fractional Langevin equation. Phys. Rev. E, 64 (2001), 051106. CrossRefGoogle ScholarPubMed
Magnasco, M. O.. Forced thermal ratchets. Phys. Rev. Lett., 71 (1993), 1477–1481. CrossRefGoogle Scholar
Mainardi, F., Pironi, P.. The fractional Langevin Equation: Brownian motion revisited. Extracta Mathematicae, 11 (1996), 140. Google Scholar
Makhnovskii, Yu. A., Rozenbaum, V. M., Yang, D.-Y., Lin, S. H., Tsong, T. Y.. Flashing ratchet model with high efficiency. Phys. Rev. E, 69 (2004), 021102. CrossRefGoogle ScholarPubMed
Mandelbrot, B. B., van Ness, J. W.. Fractional Brownian motions, fractional noises and applications. SIAM Review, 10 (1968), No. 4, 422-437. CrossRefGoogle Scholar
B. B. Mandelbrot. The Fractal Geometry of Nature. W.H. Freeman Company, New York, 1977.
Hänggi, P., Marchesoni, F.. Artificial Brownian motors: bontrolling transport on the nanoscale. Rev. Mod. Phys., 81 (2009), 387–442. CrossRefGoogle Scholar
Maxwell, J. C.. On the dynamical theory of gases. Phil. Trans. R. Soc. Lond., 157 (1867), 49–88. CrossRefGoogle Scholar
Metzler, R., Klafter, J.. The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep., 339 (2000) 1–77. CrossRefGoogle Scholar
Montroll, E. M., Weiss, G.H.. Random walks on lattices.2. J. Math. Phys., 6 (1965), No. 2, 167. CrossRefGoogle Scholar
Palmer, R. G., Stein, D. L., Abrahams, E., Anderson, P. W.. Models of hierarchically constrained dynamics for glassy relaxation. Phys. Rev. Lett., 53 (1984), 958–961. CrossRefGoogle Scholar
Reimann, P.. Brownian motors: noisy transport far from equilibrium. Phys. Rep., 361 (2002), 57–265. CrossRefGoogle Scholar
Scher, H., Montroll, E. M.. Anomalous transit time dispersion in amorphous solids. Phys. Rev. B, 12 (1975), 2455–2477. CrossRefGoogle Scholar
Seifert, U.. Efficiency of autonomous soft nanomachines at maximum power. Phys. Rev. Lett., 106 (2011), 020601. CrossRefGoogle ScholarPubMed
Sekimoto, K.. Kinetic characterization of heat bath and the energetics of thermal ratchet models. J. Phys. Soc. Jpn., 66 (1997) 1234–1237. CrossRefGoogle Scholar
Shlesinger, M. F.. Asymptotic solutions of continuous time random walks. J. Stat. Phys., 10 (1974), 421–434. CrossRefGoogle Scholar
Sokolov, I.M., Klafter, J.. From diffusion to anomalous diffusion: a century after Einstein’s Brownian motion. Chaos, 15 (2005), 026103. CrossRefGoogle ScholarPubMed
Sokolov, I. M., Klafter, J.. Field-induced dispersion in subdiffusion. Phys. Rev. Lett., 97 (2006), 140602. CrossRefGoogle ScholarPubMed
Sokolov, I. M., Heinsalu, E., Hänggi, P., Goychuk, I.. Universal fluctuations in subdiffusive transport. Europhys. Lett., 86 (2009), 30009. CrossRefGoogle Scholar
C. Van den Broeck, Kumar, N., Lindenberg, K.. Efficiency of isothermal molecular machines at maximum power. Phys. Rev. Lett., 108, (2012) 210602. Google Scholar
U. Weiss. Quantum Dissipative Systems, 2nd ed. World Scientific, Singapore, 1999.
Zwanzig, R.. Nonlinear generalized Langevin equations. J. Stat. Phys., 9 (1973), 215–220. CrossRefGoogle Scholar
R. Zwanzig. Nonequilibrium statistical mechanics. Oxford Univ. Press, Oxford, 2008.

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