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Journal of Statistical Physics

, Volume 158, Issue 5, pp 1100–1125 | Cite as

Brownian Motion of a Rayleigh Particle Confined in a Channel: A Generalized Langevin Equation Approach

  • Changho Kim
  • George Em KarniadakisEmail author
Article

Abstract

We study confined Brownian motion by investigating the memory function of a \(d\)-dimensional hypercube (\(d\ge 2\)), which is subject to a harmonic potential and suspended in an ideal gas confined by two parallel walls. For elastic walls and under the infinite-mass limit, we obtain analytic expressions for the force autocorrelation function and the memory function. The transverse-direction memory function possesses a nonnegative tail decaying like \(t^{-(d-1)}\), from which anomalous diffusion is expected for \(d=2\). For \(d=3\), the position-dependent friction coefficient becomes larger than the unconfined case and the increment is inversely proportional to the square of the distance from the wall. We also perform molecular dynamics simulations with thermal walls and/or a finite-mass hypercube. We observe faster decay due to the thermal wall (\(t^{-3}\) for \(d=2\) and \(t^{-5}\) for \(d=3\) under the fully thermalizing wall) and convergence behaviors of the finite-mass memory function, which are different from the unconfined case.

Keywords

Memory effects Long-time tail Finite-mass effects Anomalous diffusion Molecular dynamics 

Notes

Acknowledgments

This work was partially supported by the new DOE Center on Mathematics for Mesoscopic Modeling of Materials (CM4) and by the NSF (Grant DMS-1216437). Computations were performed at the IBM BG/Q with computer time provided by an INCITE grant.

References

  1. 1.
    Alder, B.J., Wainwright, T.E.: Studies in molecular dynamics. I. General method. J. Chem. Phys. 31, 459 (1959)CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids, Reprint Edn. Oxford University Press, Oxford (1989)Google Scholar
  3. 3.
    Berkowitz, M., Morgan, J.D., McCammon, J.A.: Generalized Langevin dynamics simulations with arbitrary time-dependent memory kernels. J. Chem. Phys. 78, 3256–3261 (1983)CrossRefADSGoogle Scholar
  4. 4.
    Calderoni, P., Dürr, D., Kusuoka, S.: A mechanical model of Brownian motion in half-space. J. Stat. Phys. 55, 649–693 (1989)CrossRefADSzbMATHGoogle Scholar
  5. 5.
    Carbajal-Tinoco, M.D., Lopez-Fernandez, R., Arauz-Lara, J.L.: Asymmetry in colloidal diffusion near a rigid wall. Phys. Rev. Lett. 99, 138–303 (2007)CrossRefGoogle Scholar
  6. 6.
    Carof, A., Vuilleumier, R., Rotenberg, B.: Two algorithms to compute projected correlation functions in molecular dynamics simulations. J. Chem. Phys. 140, 124–103 (2014)CrossRefGoogle Scholar
  7. 7.
    Dekker, H.: Long-time tail in velocity correlations in a one-dimensional Rayleigh gas. Phys. Lett. 88A, 21–25 (1982)CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Despósito, M.A., Viñales, A.D.: Subdiffusive behavior in a trapping potential: mean square displacement and velocity autocorrelation function. Phys. Rev. E 80, 021–111 (2009)CrossRefGoogle Scholar
  9. 9.
    Dürr, D., Goldstein, S., Lebowitz, J.L.: A mechanical model of Brownian motion. Commun. Math. Phys. 78, 507–530 (1981)CrossRefADSzbMATHGoogle Scholar
  10. 10.
    Dürr, D., Goldstein, S., Lebowitz, J.L.: A mechanical model for Brownian motion of a convex body. Z. Wahrscheinlichkeitstheorie verw. Gebiete 62, 427–448 (1983)CrossRefzbMATHGoogle Scholar
  11. 11.
    Epstein, P.S.: On the resistance experienced by spheres in their motion through gases. Phys. Rev. 23, 710–733 (1924)CrossRefADSGoogle Scholar
  12. 12.
    Español, P., Zúñiga, I.: Force autocorrelation function in Brownian motion theory. J. Chem. Phys. 98, 574–580 (1993)CrossRefADSGoogle Scholar
  13. 13.
    Grebenkov, D.S., Vahabi, M., Bertseva, E., Forró, L., Jeney, S.: Hydrodynamic and subdiffusive motion of tracers in a viscoelastic medium. Phys. Rev. E 88, 040–701(R) (2013)Google Scholar
  14. 14.
    Green, M.S.: Brownian motion in a gas of noninteracting molecules. J. Chem. Phys. 19, 1036–1046 (1951)CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Hauge, E.H., Martin-Löf, A.: Fluctuating hydrodynamics and Brownian motion. J. Stat. Phys. 7, 259–281 (1973)CrossRefADSzbMATHGoogle Scholar
  16. 16.
    Houndonougbo, Y.A., Laird, B.B., Leimkuhler, B.J.: A molecular dynamics algorithm for mixed hard-core/continuous potentials. Mol. Phys. 98, 309–316 (2000)CrossRefADSGoogle Scholar
  17. 17.
    Hynes, J.T.: Nonlinear fluctuations in master equation systems. I. Velocity correlation function for the Rayleigh model. J. Chem. Phys. 62, 2972–2981 (1975)CrossRefADSGoogle Scholar
  18. 18.
    Hynes, J.T., Kapral, R., Weinberg, M.: Microscopic theory of Brownian motion: Mori friction kernel and Langevin-equation derivation. Physica 80A, 105–127 (1975)CrossRefADSGoogle Scholar
  19. 19.
    Jeney, S., Lukić, B., Kraus, J.A., Franosch, T., Forró, L.: Anisotropic memory effects in confined colloidal diffusion. Phys. Rev. Lett. 100, 240–604 (2008)CrossRefGoogle Scholar
  20. 20.
    Kawai, S., Komatsuzaki, T.: Derivation of the generalized Langevin equation in nonstationary environments. J. Chem. Phys. 134, 114–523 (2011)Google Scholar
  21. 21.
    Kim, C., Karniadakis, G.E.: Microscopic theory of Brownian motion revisited: the Rayleigh model. Phys. Rev. E 87, 032–129 (2013)Google Scholar
  22. 22.
    Kim, C., Karniadakis, G.E.: Time correlation functions of Brownian motion and evaluation of friction coefficient in the near-Brownian-limit regime. Multiscale Model. Simul. 12, 225–248 (2014)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Kneller, G.R.: Generalized Kubo relations and conditions for anomalous diffusion: physical insights from a mathematical theorem. J. Chem. Phys. 134, 224106 (2011)CrossRefADSGoogle Scholar
  24. 24.
    Kneller, G.R., Hinsen, K.: Computing memory functions from molecular dynamics simulations. J. Chem. Phys. 115, 11097–11105 (2001)CrossRefADSGoogle Scholar
  25. 25.
    Kneller, G.R., Hinsen, K., Sutmann, G.: Mass and size effects on the memory function of tracer particles. J. Chem. Phys. 118, 5283–5286 (2003)CrossRefADSGoogle Scholar
  26. 26.
    Kou, S.C.: Stochastic modeling in nanoscale biophysics: subdiffusion within proteins. Ann. Appl. Stat. 2, 501–535 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kou, S.C., Xie, X.S.: Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule. Phys. Rev. Lett. 93, 180–603 (2004)CrossRefGoogle Scholar
  28. 28.
    Kubo, R.: The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255–284 (1966)CrossRefADSGoogle Scholar
  29. 29.
    Kusuoka, S., Liang, S.: A classical mechanical model of Brownian motion with plural particles. Rev. Math. Phys. 22, 733–838 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Li, T., Raizen, M.G.: Brownian motion at short time scales. Ann. Phys. (Berlin) 525, 281–295 (2013)CrossRefADSGoogle Scholar
  31. 31.
    Linz, P.: Analytical and numerical methods for Volterra equations. Studies in applied and numerical mathematics. Society for Industrial and Applied Mathematics. http://dx.doi.org/10.1137/1.9781611970852 (1985)
  32. 32.
    Mazo, R.M.: Momentum-correlation function in a Rayleigh gas. J. Chem. Phys. 35, 831–835 (1961)CrossRefADSMathSciNetGoogle Scholar
  33. 33.
    Min, W., Luo, G., Cherayil, B.J., Kou, S.C., Xie, X.S.: Observation of a power-law memory kernel for fluctuations within a single protein molecule. Phys. Rev. Lett. 94, 198–302 (2005)CrossRefGoogle Scholar
  34. 34.
    Montgomery, D.: Brownian motion from Boltzmann’s equation. Phys. Fluids 14, 2088–2090 (1971)CrossRefADSzbMATHGoogle Scholar
  35. 35.
    Morgado, R., Oliveira, F.A.: Relation between anomalous and normal diffusion in systems with memory. Phys. Rev. Lett. 89, 100–601 (2002)CrossRefGoogle Scholar
  36. 36.
    Mori, H.: Transport, collective motion, and Brownian motion. Progr. Theoret. Phys. 33, 423–455 (1965)CrossRefADSzbMATHGoogle Scholar
  37. 37.
    Pechukas, P.: Generalized Langevin equation of Mori and Kubo. Phys. Rev. 164, 174–175 (1967)CrossRefADSGoogle Scholar
  38. 38.
    Porrà, J.M., Wang, K.G., Masoliver, J.: Generalized Langevin equations: anomalous diffusion and probability distributions. Phys. Rev. E 53, 5872–5881 (1996)CrossRefADSGoogle Scholar
  39. 39.
    Shin, H.K., Kim, C., Talkner, P., Lee, E.K.: Brownian motion from molecular dynamics. Chem. Phys. 375, 316–326 (2010)CrossRefADSGoogle Scholar
  40. 40.
    Snook, I.: The Langevin and Generalised Langevin Approach to the Dynamics of Atomic, Polymeric and Colloidal Systems. Elsevier Science, Amsterdam (2007)Google Scholar
  41. 41.
    Spohn, H.: Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 53, 569–615 (1980)CrossRefADSMathSciNetGoogle Scholar
  42. 42.
    Szász, D., Tóth, B.: A dynamical theory of Brownian motion for the Rayleigh gas. J. Stat. Phys. 47, 681–693 (1987)CrossRefADSzbMATHGoogle Scholar
  43. 43.
    Taniguchi, S., Iwasaki, A., Sugiyama, M.: Relationship between Maxwell boundary condition and two kinds of stochastic thermal wall. J. Phys. Soc. Jpn. 77, 124–004 (2008)Google Scholar
  44. 44.
    Tehver, R., Toigo, F., Koplik, J., Banavar, J.R.: Thermal walls in computer simulations. Phys. Rev. E 57, R17 (1998)CrossRefADSGoogle Scholar
  45. 45.
    Viñales, A.D., Despósito, M.A.: Anomalous diffusion: exact solution of the generalized Langevin equation for harmonically bounded particle. Phys. Rev. E 73, 016–111 (2006)CrossRefGoogle Scholar
  46. 46.
    Wang, G.M., Prabhakar, R., Sevick, E.M.: Hydrodynamic mobility of an optically trapped colloidal particle near fluid-fluid interfaces. Phys. Rev. Lett. 103, 248–303 (2009)Google Scholar
  47. 47.
    Wang, K.G., Tokuyama, M.: Nonequilibrium statistical description of anomalous diffusion. Phys. A 265, 341–351 (1999)CrossRefGoogle Scholar
  48. 48.
    Yamgaguchi, T., Kimura, Y., Hirota, N.: Molecular dynamics simulation of solute diffusion in Lennard–Jonnes fluids. Mol. Phys. 94, 527–537 (1998)CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

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