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Generalized Langevin equation with multiplicative trichotomous noise; pp. 113–127

Full article in PDF format | doi: 10.3176/proc.2011.2.04

Erkki Soika, Romi Mankin, Jaanis Priimets


The influence of noise flatness and memory-time on the dynamics of a generalized Langevin system driven by an internal Mittag-Leffler noise and by a multiplicative trichotomous noise is studied. In the asymptotic limit at a short memory time the dynamics corresponds to a system with a pure power-law memory kernel for a viscoelastic type friction. However, at long and intermediate memory times the behaviour of the system has a qualitative difference. In particular, a critical memory time and a critical memory exponent have been found, which mark dynamical transitions in the resonant behaviour of the system. The obtained results show that the model considered is quite robust and may be of interest also in cell biology.


  1. Gammaitoni, L., Hänggi, P., Jung, P., and Marchesoni, F. Stochastic resonance. Rev. Mod. Phys., 1998, 70, 223–287.

  2. Mankin, R., Laas, K., Laas, T., and Reiter, E. Stochastic multiresonance and correlation-time-controlled stability for a harmonic oscillator with fluctuating frequency. Phys. Rev. E, 2008, 78, 031120(1)–(11).

  3. Mankin, R., Laas, T., Soika, E., and Ainsaar, A. Noise-controlled slow-fast oscillations in predator-prey models with the Beddington functional response. Eur. Phys. J. B, 2007, 59, 259–269.

  4. Sauga, A. and Mankin, R. Addendum to “Colored-noise-induced discontinuous transitions in symbiotic ecosystems”. Phys. Rev. E, 2005, 71, 062103(1)–(4).

  5. Mankin, R., Haljas, A., Tammelo, R., and Martila, D. Mechanism of hypersensitive transport in tilted sharp ratchets. Phys. Rev. E, 2003, 68, 011105(1)–(5).

  6. Ibrahim, R. A. Excitation-induced stability and phase transition: a review. J. Vib. Control, 2006, 12, 1093–1170.

  7. Mankin, R., Soika, E., Sauga, A., and Ainsaar, A. Thermally enhanced stability in fluctuating bistable potentials. Phys. Rev. E, 2008, 77, 051113(1)–(9).

  8. Magnasco, M. O. Forced thermal ratchets. Phys. Rev. Lett., 1993, 71, 1477–1481.

  9. Reimann, P. Brownian motors: noisy transport far from equilibrium. Phys. Rep., 2002, 361, 57–265.

10. Tammelo, R., Mankin, R., and Martila, D. Three and four current reversals versus temperature in correlation ratchets with a simple sawtooth potential. Phys. Rev. E, 2002, 66, 051101(1)–(5).

11. Götze, W. and Sjögren, L. Relaxation processes in supercooled liquids. Rep. Prog. Phys., 1992, 55, 241–376.

12. Carlsson, T., Sjögren, L., Mamontov, E., and Psiuk-Maksymowicz, K. Irreducible memory function and slow dynamics in disordered systems. Phys. Rev. E, 2007, 75, 031109(1)–(8).

13. Weber, S. C., Spakowitz, A. J., and Theriot, J. Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm. Phys. Rev. Lett., 2010, 104, 238102(1)–(4).

14. Gu, Q., Schiff, E. A., Grebner, S., Wang, F., and Schwarz, R. Non-Gaussian transport measurements and the Einstein relation in amorphous silicon. Phys. Rev. Lett., 1996, 76, 3196–3199.

15. Kou, S. C. and Xie, X. S. Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule. Phys. Rev. Lett., 2004, 93, 180603(1)–(4).

16. Min, W., Luo, G., Cherayil, B. J., Kou, S. C., and Xie, X. S. Observation of a power-law memory kernel for fluctuations within a single protein molecule. Phys. Rev. Lett., 2005, 94, 198302(1)–(4).

17. Porrà, J. M., Wang, K.-G., and Masoliver, J. Generalized Langevin equations: anomalous diffusion and probability distributions. Phys. Rev. E, 1996, 53, 5872–5881.

18. Lutz, E. Fractional Langevin equation. Phys. Rev. E, 2001, 64, 051106(1)–(4).

19. Burov, S. and Barkai, E. Fractional Langevin equation: overdamped, underdamped, and critical behaviors. Phys. Rev. E, 2008, 78, 031112(1)–(8).

20. Golding, J. and Cox, E. C. Physical nature of bacterial cytoplasm. Phys. Rev. Lett., 2006, 96, 098102(1)–(4).

21. Tolić-Nørrelykke, I. M., Munteanu, E.-L., Thon, G., Oddershede, L., and Berg-Sørensen, K. Anomalous diffusion in living yeast cells. Phys. Rev. Lett., 2004, 93, 078102(1)–(4).

22. Granek, R. and Klafter, J. Fractons in proteins: can they lead to anomalously decaying time autocorrelations. Phys. Rev. Lett., 2005, 95, 098106(1)–(4).

23. Metzler, R. and Klafter, J. The random walks’s quide to anomalous diffusion: a fractional dynamics approach. Phys. Rep., 2000, 339, 1–77.

24. Jeon, J.-H., Tejedor, V., Burov, S., Barkai, E., Selhuber-Unkel, C., Berg-Sørensen, K., Oddershede, L., and Metzler, R. In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett., 2011, 106, 048103(1)–(4).

25. Viñales, A. D. and Despósito, M. A. Anomalous diffusion induced by a Mittag-Leffler correlated noise. Phys. Rev. E, 2007, 75, 042102(1)–(4).

26. Soika, E., Mankin, R., and Ainsaar, A. Resonant behavior of a fractional oscillator with fluctuating frequency. Phys. Rev. E, 2010, 81, 011141(1)–(11).

27. Mankin, R. and Rekker, A. Memory-enhanced energetic stability for a fractional oscillator with fluctuating frequency. Phys. Rev. E, 2010, 81, 041122(1)–(10).

28. Astumian, R. D. and Bier, M. Mechanochemical coupling of the motion of molecular motors to ATP hydrolysis. Biophys. J., 1996, 70, 637–653.

29. Soika, E., Mankin, R., and Priimets, J. Response of a generalized Langevin system to a multiplicative trichotomous noise. In Recent Advances in Fluid Mechanics, Heat and Mass Transfer and Biology (Zemliak, A. and Mastorakis, N., eds). WSEAS, Puerto Morelos, Mexico, 2011, 87–93.

30. Kubo, R. The fluctuation-dissipation theorem. Rep. Prog. Phys., 1966, 29, 255–284.

31. Risken, H. The Fokker-Planck Equation. Springer–Verlag, Berlin, 1989.

32. Podlubny, I. Fractional Differential Equations. Academic Press, New York, 1999.

33. Viñales, A. D., Wang, K. G., and Despósito, M. A. Anomalous diffusion of a harmonic oscillator driven by a Mittag-Leffler noise. Phys. Rev. E, 2009, 80, 011101(1)–(6).

34. Mankin, R., Ainsaar, A., and Reiter, E. Trichotomous noise-induced transitions. Phys. Rev. E, 1999, 60, 1374–1380.

35. Mankin, R., Tammelo, R., and Martila, D. Correlation ratchets: four current reversals and disjunct “windows”. Phys. Rev. E, 2001, 64, 051114(1)–(4).

36. Doering, C. R., Horsthemke, W., and Riordan, J. Nonequilibrium fluctuation-induced transport. Phys. Rev. Lett., 1994, 72, 2984–2987.

37. Ming, Y., Li, C., Da-Jin, W., and Xiang-Lian, L. Phenomenon of repeated current reversals in Brownian ratchet. Chinese Phys. Lett., 2002, 19, 1759–1762.

38. Li, J., Luczka, J., and Hänggi, P. Transport of particles for a spatially periodic stochastic systems with correlated noises. Phys. Rev. E, 2001, 64, 011113(1)–(10).

39. Schliwa, M. and Woehlke, G. Molecular motors. Nature, 2003, 442, 759–765.

40. Shapiro, V. E. and Loginov, V. M. “Formulae of differentiation” and their use for solving stochastic equations. Physica A, 1978, 91, 563–574.

41. Kempfle, S., Schafer, I., and Beyer, H. Fractional calculus via functional calculus: theory and applications. Nonlinear Dyn., 2002, 29, 99–127.

42. Burov, S., Jeon, J.-H., Metzler, R., and Barkai, E. Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking. Phys. Chem. Chem. Phys., 2011, 13, 1800–1812.

43. Mankin, R., Laas, K., and Sauga, A. Generalized Langevin equation with multiplicative noise: temporal behavior of the autocorrelation functions. Phys. Rev. E, 2011, 83, 061131(1)–(10).

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