Stochastic Brownian motors with small potential energy fluctuations
Abstract
We suggest a new approach to description of driven diffusive systems in which fluctuations of nano-particle potential energy, being a source of directed motion, are considered as small. The base of the approach is the method of the Green’s functions, which allow writing analytical expressions for the sought-for average nano-particle velocity. These expressions are essentially simplified under the assumption of high temperatures. Within this high-temperature approximation, we consider a system (Brownian motor) with small harmonic oscillations of the particle potential energy of a saw-tooth shape. The obtained representation for the average motor velocity is the product of two functions. One of them depends on the fluctuation frequency and the other includes geometrical parameters of the system (that is of the steady and perturbing potential profiles). It is shown that the frequency dependence of the average velocity is a nonmonotonic function with the maximum position determining by the inverse characteristic diffusion time. In its turn, the average motor velocity as a function of the asymmetry parameter of the nonfluctuating component of the potential profile and the phase shift of the fluctuating one is sign-changing. This result means that there exists a possibility to govern the direction of the motor motion by the phase shift of harmonic oscillations relative to a fixed saw-tooth potential component. The intervals of both the phase shift and the asymmetry parameter values, in which the particle can move to the right or to the left, are diagrammed. The regularities obtained are of great importance to choose the values of parameters which provide effective regimes of Brownian motors operation.References
1. Trakhtenberg L.I., Gerasimov G.N. Metal Containing Polymers: Cryochemical Synthesis, Structure and Physico-Chemical Properties. Metal/Polymer Nanocomposites. (New York: Wiley & Sons, 2005).
2. Suzdalev I.P. Nanotechnology: physicochemistry of nanoclusters, nanostructures and nanomaterials. (Moscow: KomKniga, 2006). [in Russian].
3. Physico-Chemical Phenomena in Thin Films and at Solid Surfaces. Ed. by L.I. Trakhtenberg, S.H. Lin, O.J. Ilegbusi. (Amsterdam: Elsevier, 2007).
4. Romanovsky Yu.L., Tikhonov A.N. Molecular energy transducers of the living cell. Proton ATP synthase: a rotating molecular motor. Phys. Usp. 2010. 53: 893. [in Russian]. https://doi.org/10.3367/UFNe.0180.201009b.0931
5. Pamme N. Continuous flow separations in microfluidic devices. Lab Chip. 2007. 7:1644. https://doi.org/10.1039/b712784g
6. Cha T.G., Pan J., Chen H., Salgado J., Li X., Mao C., Choi J.H. A synthetic DNA motor that transports nanoparticles along carbon nanotubes. Nature nanotechnology. 2013. 9(1):39. https://doi.org/10.1038/nnano.2013.257
7. Cheetham M.R., Bramble J.P., McMillan D.G.G., Bushby R.J., Olmsted P.D., Jeuken L.J.C., Evans S.D. Manipulation and sorting of membrane proteins using patterned diffusion-aided ratchets with AC fields in supported bilayers. Soft Matter. 2012. 8: 5459. https://doi.org/10.1039/c2sm25473e
8. Reimann P. Brownian Motors: Noisy Transport far from Equilibrium. Phys. Rep. 2002. 361: 57. https://doi.org/10.1016/S0370-1573(01)00081-3
9. Astumian R.D. Adiabatic Theory for Fluctuation-Induced Transport on a Periodic Potential. J. Phys. Chem. 1996. 100(49): 19075. https://doi.org/10.1021/jp961614m
10. Rozenbaum V.M. High-temperature brownian motors: Deterministic and stochastic fluctuations of a periodic potential. JETP Letters. 2008. 88(5): 342. https://doi.org/10.1134/S0021364008170128
11. Astumian R.D., Bier M. Fluctuation driven ratchets: molecular motors. Phys. Rev. Lett. 1994. 72(11): 1766. https://doi.org/10.1103/PhysRevLett.72.1766
12. Rozenbaum V.M. Mechanical motion in nonequilibrium nanosystems. Nanomaterials and Supramolecular Structures: Physics, Chemistry, and Applications. (London: Springer, 2009).
13. Rozenbaum V.M., Korochkova T.Ye., Chernova A.A., Dekhtyar M.L. Brownian motor with competing spatial and temporal asymmetry of potential energy. Phys. Rev. E. 2011. 83(5): 051120. https://doi.org/10.1103/PhysRevE.83.051120
14. Korochkova T.Ye., Shkoda N.G., Chernova A.A., Rozenbaum V.M. Exact analytical solutions in the theory of Brownian motors and pumps. Surface. 2012. 4(19): 19. [in Russian].
15. Korochkova T.Ye., Shapochkina I.V., Rozenbaum V.M. Influence of inertia to passive and active nanoparticles transport along the phase interface. Him. Fiz. Tehnol. Poverhni. 2013. 4(4): 427. [in Russian].
16. Tsomyk O.Ye., Korochkova T.Ye., Rozenbaum V.M. Molecular rotor as a high-temperature Brownian motor. Him. Fiz. Tehnol. Poverhni. 2016. 7(4): 444. https://doi.org/10.15407/hftp07.04.444
17. Rozenbaum V.M., Shapochkina I.V., Lin S.H., Trakhtenberg L.I. A theory of slightly fluctuating ratchets. JETP Lett. 2017. 105(8): 521. https://doi.org/10.1134/S0021364017080069
18. Ulyanov V.V. Integral methods in quantum mechanics. (Kharkov: Vyshcha shkola, 1982). [in Russian].
19. Gardiner C.R. Handbook of Stochastic Methods. (Springer, Berlin, 1985, 2nd ed).
20. Rosenbaum V.M. Brownian motion and surface diffusion. In: Physics and chemistry of the surface. Book 1. Physics of the surface. 2(Part VI, Chapter 23). (Kyiv: Chuiko Institute of Surface Chemistry of NAS of Ukraine, 2015). [in Russian].
21. Rozenbaum V.M., Shapochkina I.V., Sheu S.-Y., Yang D.-Y., Lin S.H. High-temperature ratchets with sawtooth potentials. Phys. Rev. E. 2016. 94(5): 052140. https://doi.org/10.1103/PhysRevE.94.052140
22. Rozenbaum V.M., Makhnovskii Y.A., Shapochkina I.V., Sheu S.-Y., Yang D.-Y., Lin S.H. Adiabatically Slow and Adiaba tically Fast Driven Ratchets. Phys. Rev. E. 2012. 85: 041116. https://doi.org/10.1103/PhysRevE.85.041116
23. Faucheux L.P., Bourdieu L.S., Kaplan P.D., Libchaber A. Optical thermal ratchet. Phys. Rev. Lett. 1995. 74: 1504. https://doi.org/10.1103/PhysRevLett.74.1504
24. Robilliard C., Lucas D., Grynberg G. Modelling a ratchet with cold atoms in an optical lattice. Appl. Phys. A. 2002. 75(2): 213. https://doi.org/10.1007/s003390201333
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.