Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Relaxation Processes in Many Particle Systems — Recurrence Relations Approach

Discontinuity, Nonlinearity, and Complexity 2(1) (2012) 43--56 | DOI:10.5890/DNC.2012.11.002

Anatolii V. Mokshin

Department of Computational Physics, Kazan Federal University, Kazan, 420008, Russia

Download Full Text PDF



The general scheme for the treatment of relaxation processes and temporal autocorrelations of dynamical variables for many particle systems is presented in framework of the recurrence relations approach. The time autocorrelation functions and/or their spectral characteristics, which are measurable experimentally (for example, due to spectroscopy techniques) and accessible from particle dynamics simulations, can be found by means of this approach, the main idea of which is the estimation of the so-called frequency parameters. Model cases with the exact and approximative solutions are given and discussed.


We would like to thank M. Howard Lee (University of Georgia, USA) for very useful discussions.


  1. [1]  Zwanzig, R. (2001), Nonequilibrium Statistical Mechanics, Oxford University Press, Oxfrod.
  2. [2]  Mokshin, A.V., Yulmetyev, R.M., and Hänggi, P. (2005), Simple measure of memory for dynamical processes described by a generalized langevin equation, Physical Review Letters, 95, 200601(1)-200601(4).
  3. [3]  Uchaikin, V.V. (2003), Self-similar anomalous diffusion and Levy-stable laws, Physics-Uspekhi, 46, 821-849.
  4. [4]  Risken, H. (1989), The Fokker-Planck Equation. Methods of Solution and Applications, 2nd ed., Springer-Verlag, Berlin.
  5. [5]  Gammaitoni, L., Hänggi, P., Jung, P., and Marchesoni, F. (1998), Stochastic resonance, Review of Modern Physics, 70, 223-287.
  6. [6]  Berthier, L., et al. (2011), Dynamical Heterogeneities in Glasses, Colloids, and Granular Media, Oxford University Press, Oxford.
  7. [7]  Berne, B.J. and Harp, G.D. (1970), On the Calculation of Time Correlation Functions, Advances in Chemical Physics XVII, 63, 63-227.
  8. [8]  Kubo, R. (1957), Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, Journal of the Physical Society of Japan, 12, 570-586.
  9. [9]  Zausch, J. et al. (2008), From equilibrium to steady state: The transient dynamics of colloidal liquids under shear, Journal of Physics: Condensed Matter, 20, 404210.
  10. [10]  Rahman, R. (1964), Correlations in the motion of atoms in liquid argon, Physical Review A, 136, 405-411.
  11. [11]  Lee, M.H. (2007), Why irreversibility is not a sufficient condition for ergodicity, Physical Review Letters, 98, 190601.
  12. [12]  Callen, H.B., Welton, T.R. (1951), Irreversibility and generalized noise, Physical Review, 83, 34-40.
  13. [13]  Hansen, J. P., McDonald, I. R. (2006), Theory of Simple Liquids, Academic Press, New York.
  14. [14]  Mokshin, A.V., Chvanova, A.V., and Khusnutdinoff, R.M. (2012), Mode-coupling approximation in a fractionalpower generalization: Particle dynamics in supercooled liquids and glasses, Theoretical and Mathematical Physics, 171, 541-552.
  15. [15]  Götze, W. (2009), Complex Dynamics of Glass-Forming Liquids - A Mode-Coupling Theory, Oxfrod University Press Inc., New York.
  16. [16]  Balucani, U., Lee, M.H., Tognetti, V. (2003), Dynamical correlations, Physics Report, 373, 409-492.
  17. [17]  Lee, M.H. (1983), Derivation of the generalized Langevin equation by a method of recurrence relations, Journal of Mathematical Physics, 24, 2512-2514.
  18. [18]  Feldman, Yu., Puzenko, A. and Ryabov, Ya. (2006), Dielectric relaxation phenomema in complex materials, In: Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, vol. 133, Part A, edited by Coffey,W.T. and Kalmykov, Yu.P.
  19. [19]  Lee, M.H. (2000), Generalized Langevin equation and recurrence relations, Physical Review E, 62, 1769-1772.
  20. [20]  Lee, M.H. (2000), Heisenberg, Langevin, and current equations via the recurrence relations approach, Physical Review E, 61, 3571-3578.
  21. [21]  Jr. Florencio, J. and Lee, M.H. (1985), Exact time evolution of a classical harmonic-oscillator chain, Physical Review A, 31, 3231-3236.
  22. [22]  Abramowitz, M. and Stegun, I.S. (1972), Handbook of Mathematical Functions, Dover, New York.
  23. [23]  Mokshin, A.V., Yulmetyev, R.M., Khusnutdinoff, R.M., and Hänggi P. (2006), Analysis of the dynamics of liquid aluminium: Recurrent relation approach, Journal of Physics Condensed Matter, 19(4), 046209(1)-046209(16).
  24. [24]  Jr. Florencio, J. and Lee, M.H. (1987), Relaxation functions, memory functions, and random forces in the onedimensional spin 1/2-XY and transverse Ising models, Physical Review B, 35, 1835-1840.
  25. [25]  Rubin, R.J. (1963), Momentum autocorrelation functions and energy transport in harmonic crystals containing isotopic defects, Physical Review, 131, 964-989.
  26. [26]  Lee, M.H., Hong, J. and Jr. Florencio, J. (1987), Method of recurrence relations and applications to many-Body systems, Physica Scripta, T19, 498-504.
  27. [27]  Yulmetyev, R.M., Mokshin, A.V., Hänggi, P., and Shurygin, V.Yu. (2002), Dynamic structure factor in liquid cesium on the basis of time-scale inveriance of relaxation processes, JETP Letters, 76, 147-150.
  28. [28]  Yulmetyev, R.M., Mokshin, A.V., Hänggi, P. and Shurygin, V.Yu. (2001), Time-scale invariance of relaxation processes of density fluctuation in slow neutron scattering in liquid cesium, Physical Review E, 64, 057101(1)- 057101(4).
  29. [29]  Bansal, R. and Pathak, K.N. (1974), Sum rules and atomic correlations in classical liquids, Physical Review A, 9, 2773-2782.
  30. [30]  Yulmetyev, R.M., Mokshin, A.V., Scopigno, T. and Hänggi, P. (2003), New evidence for the idea of time-scale invariance of relaxation processes in simple liquids: the case of molten sodium, Journal of Physics: Condensed Matter, 15, 2235-2257.
  31. [31]  Khusnutdinoff, R.M., Mokshin, A.V. and Yulmetyev, R.M. (2009), Molecular dynamics of liquid lead near its melting point, Journal of Experimental and Theoretical Physics, 108, 417-427.
  32. [32]  Mokshin, A.V., Yulmetyev, R.M. Khusnutdinoff, R.M. and P. Hänggi (2006), Collective dynamics in liquid aluminum near the melting temperature: Theory and computer simulation, Journal of Experimental and Theoretical Physics, 103, 841-849.
  33. [33]  Mountain, R.D. (1966), Spectral Distribution of Scattered Light in a Simple Fluid, Review of Modern Physics, 38, 205-214.
  34. [34]  Wierling, A. (2012), Dynamic structure factor of linear harmonic chain - a recurrence relation approach, The European Physical Journal B, 85, 1-9.
  35. [35]  Wierling, A., Sawada, I. (2010), Wave-number dependent current correlation for a harmonic oscillator, Physical Review E, 82, 051107(1)-051107(11).
  36. [36]  Wierling, A., Sawada, I. (2012), Dynamic Structure Factor for a Harmonic Oscillator and the Harmonic Oscillator Chain, Contributions to Plasma Physics, 52, 49-52.