Skip to main content
SpringerLink
Log in

Diffusion in a Medium with Nonlinear Friction

  • Published:
International Journal of Thermophysics Aims and scope Submit manuscript

Abstract

The Langevin equation describing the Brownian motion of particles is studied in the case when the friction depends nonlinearly on the particle velocity. The intensity \(D(\upsilon )\) of the thermal random force driving the particles then, in equilibrium systems, also depends on the velocity, and the equation of motion should contain an additional ‘spurious’ force proportional to the derivative of \(D(\upsilon )\). This force is chosen in the kinetic representation, for which the stationary probability density for velocities is the Maxwell distribution and simultaneously the generalized Einstein relation is obeyed. A general formula for the diffusion coefficient of the particle is obtained and then specified for various models of dissipative forces studied in the literature, in particular, for the power-law friction. In this case the scaling of the diffusion coefficient with the model parameters is obtained also by a simple dimensional analysis, for systems both at equilibrium and far from it, when \(D\) is usually assumed to be independent of \(\upsilon \).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. P. Langevin, C. R. Acad. Sci. (Paris) 146, 530 (1908)

    MATH  Google Scholar 

  2. W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering (World Scientific, Singapore, 2004)

  3. A. Einstein, Ann. Phys. (Leipz.) 17, 549 (1905)

    Article  ADS  MATH  Google Scholar 

  4. P. Reimann, Phys. Rep. 361, 57 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. M. Schienbein, H. Gruler, Bull. Math. Biol. 55, 585 (1993)

    Article  MATH  Google Scholar 

  6. F. Schweitzer, W. Ebeling, B. Tilch, Phys. Rev. Lett. 80, 5044 (1998)

    Article  ADS  Google Scholar 

  7. W. Ebeling, F. Schweitzer, B. Tilch, BioSystems 429, 17 (1999)

    Article  Google Scholar 

  8. U. Erdmann, W. Ebeling, L. Schimansky-Geier, F. Schweitzer, Eur. Phys. J. B 15, 105 (2000)

    Article  ADS  Google Scholar 

  9. F. Schweitzer, W. Ebeling, B. Tilch, Phys. Rev. E 64, 021110 (2001)

    Article  ADS  Google Scholar 

  10. U. Erdmann, W. Ebeling, V.S. Anishchenko, Phys. Rev. E 65, 061106 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  11. U. Erdmann, W. Ebeling, L. Schimansky-Geier, A. Ordemann, F. Moss, arXiv:q-bio.PE/0404018 (2004)

  12. W. Ebeling, L. Schimansky-Geier, Eur. Phys. J. Spec. Top. 157, 17 (2008)

    Article  Google Scholar 

  13. B. Lindner, E.M. Nicola, Eur. Phys. J. Spec. Top. 157, 43 (2008)

    Article  Google Scholar 

  14. D.F. Tavares, E.D. Leonel, R.N. Costa Filho, Physica A 391, 5366 (2012)

    Article  ADS  Google Scholar 

  15. B. Lindner, J. Stat. Phys. 130, 523 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. H. Risken, The Fokker–Planck Equation, 2nd edn. (Springer, Berlin, 1984)

    MATH  Google Scholar 

  17. B. Lindner, New J. Phys. 9, 136 (2007)

    Article  ADS  Google Scholar 

  18. Yu.L. Klimontovich, Phys. Usp. 37, 737 (1994)

    Google Scholar 

  19. I. Kaplansky, An Introduction to Differential Algebra (Hermann, Paris, 1957)

    MATH  Google Scholar 

  20. R. Kubo, Rep. Prog. Phys. 29, 255 (1966)

    Article  ADS  Google Scholar 

  21. K. Itô, Proc. Imp. Acad. (Tokyo) 20, 519 (1944)

    Article  MATH  MathSciNet  Google Scholar 

  22. K. Itô, Mem. Am. Math. Soc. 4, 51 (1951)

    Google Scholar 

  23. R.L. Stratonovich, Vestnik Moskov. Univ., Ser. 1: Mat. Meh. 1, 3 (1964) [in Russain]

  24. R.L. Stratonovich, SIAM J. Control 4, 362 (1966)

    Article  MathSciNet  Google Scholar 

  25. P. Hänggi, Helv. Phys. Acta 51, 183 (1978)

    MathSciNet  Google Scholar 

  26. P. Hänggi, Helv. Phys. Acta 53, 491 (1980)

    MathSciNet  Google Scholar 

  27. Yu.L. Klimontovich, Statistical Theory of Open Systems (Kluwer, Dordrecht, 1995)

  28. B. Lindner, New J. Phys. 12, 063026 (2010)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

This work was supported by the Agency for the Structural Funds of the EU within the Projects NFP 26220120021, 26220120033, 26110230061, and by the Grant VEGA 1/0370/12.

Author information

Authors and Affiliations

  1. Department of Physics, Technical University of Košice, Park Komenského 2, 04200 , Kosice, Slovakia

    V. Lisý & J. Tóthová

  2. Department of Mathematics and Physics, University of Security Management, Kukučínova 17, 04001 , Kosice, Slovakia

    L. Glod

Authors
  1. V. Lisý
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Tóthová
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Glod
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Lisý.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lisý, V., Tóthová, J. & Glod, L. Diffusion in a Medium with Nonlinear Friction. Int J Thermophys 35, 2001–2010 (2014). https://doi.org/10.1007/s10765-013-1501-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10765-013-1501-4

Keywords

Search

Navigation