Symmetry properties of brownian motors with fluctuating periodic potential energy

  • I. V. Shapochkina Belarusian State University
  • T. Ye. Korochkova Chuiko Institute of Surface Chemistry of National Academy of Sciences of Ukraine
  • V. M. Rozenbaum Chuiko Institute of Surface Chemistry of National Academy of Sciences of Ukraine


We consider the inertialess motion of a Brownian particle in a potential field described by an arbitrary periodic function of coordinate and time. The first terms of an expansion of the average particle velocity over the small parameter, which is the ratio of changes of the potential energy amplitude to the thermal energy, have been represented, that is, the expression for the high-temperature Brownian motor average velocity. The analysis of those expressions revealed the vector and shift symmetry as well as the hidden space-time symmetry of Cubero- Renzoni (D.Cubero, F.Renzoni). These symmetry types have been used to analyze space-time dependences of potential energy which prevent appearance of ratchet effect. We also study the additional types of symmetry, which are inherent in adiabatically slow and fast Brownian motors, as well as the conditions for the potential energy at which the average motor velocity becomes zero.


1. Reimann P. Brownian Motors: noisy transport far from equilibrium. Phys. Rep. 2002. 361(2–4): 57.

2. Hänggi P., Marchesoni F. Artificial Brownian motors: Controlling transport on the nanoscale. Rev. Mod. Phys. 2009. 81(1): 387.

3. Schadschneider A., Chowdhury D., Nishinari K. Stochastic Transport in Complex Systems: From Molecules to Vehicles. (Amsterdam: Elsevier, 2010).

4. Goychuk I. Molecular machines operating on the nanoscale: from classical to quantum. Beilstein. J. Nanotechnol. 2016. 7: 328.

5. Cubero D., Renzoni F. Brownian Ratchets: From Statistical Physics to Bio and Nano-motors. (Cambridge, UK: Cambridge University Press, 2016).

6. Rousselet J., Salome L., Ajdari A., Prost J. Directional motion of Brownian particles induced by a periodic asymmetric potential. Nature. 1994. 370: 446.

7. de Souza S.C.C., Van de Vondel J., Morelle M., Moshchalkov V.V. Controlled multiple reversals of a ratchet effect. Nature. 2006. 440: 651.

8. Gommers R., Bergamini S., Renzoni F. Dissipation-induced symmetry breaking in a driven optical lattice. Phys. Rev. Lett. 2005. 95: 0073003.

9. Kedem O., Lau B., Weiss E.A. How to drive a flashing electron ratchet to maximize current. Nano Lett. 2017. 17(9): 5848.

10. Dekhtyar M.L., Ishchenko A.A., Rozenbaum V.M. Photoinduced molecular transport in biological environments based on dipole moment fluctuations. J. Phys. Chem. B. 2006. 110(41): 20111.

11. Rozenbaum V.M., Chernova A.A. Near-surface Brownian motor with synchronously fluctuating symmetric potential and applied force. Surface Science. 2009. 603: 3297.

12. Rozenbaum V.M., Dekhtyar M.L., Lin S.H., Trakhtenberg L.I. Photoinduced diffusion molecular transport. J. Chem. Phys. 2016. 145: 064110.

13. Kanada R., Sasaki K. Thermal ratchets with symmetric potentials. J. Phys. Soc. Jpn. 1999. 68: 3759.

14. Reimann P. Supersymmetric ratchets. Phys. Rev. Lett. 2001. 86(22): 4992.

15. Denisov S., Flach S., Hänggi P. Tunable transport with broken spacetime symmetries. Phys. Rep. 2014. 538: 77.

16. Cubero D., Renzoni F. Hidden symmetries, instabilities, and current suppression in Brownian ratchets. Phys. Rev. Lett. 2016. 116: 010602.

17. Rozenbaum V.M. Brownian motors in the low-energy approximation: Classification and properties. J. Exp. Theor. Phys. 2010. 110(4): 653.

18. 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.

19. Rozenbaum V.M., Makhnovskii Y.A., Shapochkina I.V., Sheu S.Y., Yang D.Y., Lin S.H. Adiabatically slow and adiabatically fast driven ratchets. Phys. Rev. E. 2012. 85(4): 041116.

20. Riskin H. The Fokker-Plank Equation. Methods of Solution and Applications. (Berlin: Springer-Verlag, 1989).

21. Thomson W. The kinetic theory of the dissipation of energy. Proceedings of the Royal Society of Edinburgh. 1874. 8: 325.

22. Parrondo J.M.R. Reversible ratchets as Brownian particles in an adiabatically changing periodic potential. Phys. Rev. E. 1998. 57(6): 7297.

How to Cite
Shapochkina, I. V., Korochkova, T. Y., & Rozenbaum, V. M. (2017). Symmetry properties of brownian motors with fluctuating periodic potential energy. Surface, (9(24), 57-68.
Theory of surface chemical structure and reactivity.