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Biased Tracer Diffusion in Hard-Core Lattice Gases: Some Notes on the Validity of the Einstein Relation

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Instabilities and Nonequilibrium Structures IX

Part of the book series: Nonlinear Phenomena and Complex Systems ((NOPH,volume 9))

Abstract

In this presentation we overview some recent results on biased tracer diffusion in lattice gases. We consider both models in which the gas particles density is explicitly conserved and situations in which the lattice gas particles undergo continuous exchanges with a reservoir, which case is appropriate, e.g., to adsorbed monolayers in contact with the vapor phase. For all these models we determine, in some cases exactly and in other ones — using a certain decoupling approximation, the mean displacement of a tracer particle (TP) driven by a constant external force in a dynamical background formed by the lattice gas particles whose transition rates are symmetric. Evaluating the TP mean displacement explicitly we are able to define the TP mobility, which allows us to demonstrate that the Einstein relation between the TP mobility and the diffusivity generally holds, despite the fact that in some cases diffusion is anomalous. For models treated within the framework of the decoupling approximation, our analytical results are confirmed by Monte Carlo simulations. Perturbance of the lattice gas particles distribution due to the presence of a biased TP and the form of the particle density profiles are also discussed.

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References

  1. S. Havlin and D. Ben-Avraham, Advan. Phys. 36, 695 (1987)

    Article  ADS  Google Scholar 

  2. J. P. Bouchaud, A. Comtet, A. Georges and P. Le Doussal, Ann. Phys. (NY) 201, 285 (1990)

    Google Scholar 

  3. J. P. Bouchaud and A. Georges, Phys. Repts. 195, 129 (1991)

    MathSciNet  Google Scholar 

  4. J. Drake and J. Klafter, Dynamics in Confined Molecular Systems, Physics Today 43, 46 (1990)

    Article  ADS  Google Scholar 

  5. J. Drake, J. Klafter and P. Levitz, Chemical and Biological Microstructures as Probed by Dynamic Processes, Science 251, 1574 (1991); J. M. Drake, P. Levitz, J. Klafter, N. J. Turro, K.S. Nitsche and K.F. Cassidy, Phys. Rev. Letters 61, 865 (1988)

    Article  ADS  Google Scholar 

  6. J. M. Drake, P. Levitz and J. Klafter, Time Resolved Studies of Optical Probes in Confining Spaces, in: ”Dynamical Processes in Condensed Molecular Systems” Eds. J. Klafter, J. Jortner and A. Blumen (World Scientific, Singapore, 1989)

    Google Scholar 

  7. S.F. Buriatsky, G. Oshanin and A.I. Chernoutsan, in: Low Dimensional Conductors and Superconductors, Springer Research Reports in Physics, eds. A. Ovchinnikov and I. Ukrainsky (Springer, Berlin, 1991), p.121

    Google Scholar 

  8. G. Oshanin, M. Moreau and S.F. Burlatsky, Adv. Colloid and Interface Sci. 49, 1 (1994)

    Article  Google Scholar 

  9. G. Oshanin and S.F. Burlatsky, J. Stat. Phys. 65, 1109, 1095 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. A. Blumen, Nuovo Cimento B 63, 50 (1981)

    Article  ADS  Google Scholar 

  11. A. Blumen, J. Klafter and G. Zumofen, in: Optical Spectroscopy of Glasses, ed. I. Zschokke (Reidel, Dordrecht, 1986), p.199

    Chapter  Google Scholar 

  12. S.F. Buriatsky, G. Oshanin and A. Chernoutsan, Phys. Lett. A 149, 47 (1990)

    Article  ADS  Google Scholar 

  13. M.F. Shlesinger, H. Scher and J.G. Bendler, Phys. Today 1, 26 (1991)

    Google Scholar 

  14. H. Scher and E.W. Montroll, Phys. Rev. B 12, 12, 2455 (1975)

    Article  ADS  Google Scholar 

  15. G. Oshanin, S.F. Burlatsky, M. Moreau and B. Gaveau, Chem. Phys. 177, 803 (1993)

    Article  Google Scholar 

  16. Y. Pomeau, A. Pumir and W. Young, Phys. Fluids A 1, 462 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. T.E. Harris, J. Appl. Prob. 2, 323 (1965)

    Article  Google Scholar 

  18. D.G. Levitt, Phys. Rev. A 8, 3050 (1973)

    Article  ADS  Google Scholar 

  19. J.K. Percus, Phys. Rev. A 9, 557 (1974)

    Article  ADS  Google Scholar 

  20. S. Alexander and P. Pincus, Phys. Rev. B 18, 2011 (1978)

    Article  ADS  Google Scholar 

  21. T.M. Ligget, Interacting Particle Systems, (Springer, New York, 1985)

    Book  Google Scholar 

  22. M.J.A.M. Brummelhuis and H.J. Hilhorst, J. Stat. Phys. 53, 249 (1988); see also: Marco Brummelhuis, Application of Random Walk: Tracer Diffusion, Lattice Covering, and Damage Spreading, Ph. D. Thesis, Leiden University, The Netherlands, 1991

    Article  MathSciNet  ADS  Google Scholar 

  23. Ya. Sinai, Theory Probab. Appl. 27, 247 (1982)

    MathSciNet  Google Scholar 

  24. H. Kesten, Physica A 138, 299 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. S. Havlin, J. Stat. Phys. 58, 653 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. J. Klafter, G. Zumofen and A. Blumen, Chem. Phys. 177, 821 (1993); M. F. Shlesinger, J. Klafter and G. Zumofen, Above, Below and Beyond Brownian Motion, American J. of Phys. 67, 1253 (1999)

    Article  Google Scholar 

  27. S.D. Druger, A. Nitzan, and M.A. Ratner, J. Chem. Phys. 79, 3133 (1983); S.D. Druger, M.A. Ratner, and A. Nitzan, Phys. Rev. B 31, 3939 (1985)

    Article  ADS  Google Scholar 

  28. A.K. Harrison and R. Zwanzig, Phys. Rev. A 32, 1072 (1985)

    Article  ADS  Google Scholar 

  29. R. Granek and A. Nitzan, J. Chem. Phys. 90, 3784 (1989)

    Article  ADS  Google Scholar 

  30. O. Benichou, J. Klafter, M. Moreau and G. Oshanin, Phys. Rev. E 62, 3327 (2000)

    Article  ADS  Google Scholar 

  31. E. E. Polymeropoulos and J. Brickmann, Ann. Rev. Biophys, 14, 315 (1985)

    Article  Google Scholar 

  32. S. Chandra, Superionic Solids, Principles and Applications, (North-Holland, Amsterdam, 1981)

    Google Scholar 

  33. S. Bhattarcharya, J.P. Stokes, M.W. Kim, and J.S. Huang, Phys. Rev. Lett. 55, 1884 (1985)

    Article  ADS  Google Scholar 

  34. G.S. Grest, I. Webman, S.A. Safran, and A.L.R. Bug, Phys. Rev. A 33, 2842 (1986)

    Article  ADS  Google Scholar 

  35. A.L.R. Bug and Y. Gefen, Phys. Rev. A 35, 1301 (1987)

    Article  ADS  Google Scholar 

  36. C.M.C. Gambi, M.G. Giri, M. Carlà, D. Senatra, and A. Chittofrati, Phys. Rev. E 56, 4356 (1997)

    Article  ADS  Google Scholar 

  37. H. Cheradarne, J.L. Souquet and J.M. Latour, Mater. Res. Bull. 15, 1173 (1980)

    Article  Google Scholar 

  38. B.L. Papke, R. Dupon, M.A. Ratner and D.F. Shriver, Solid State Ion. 5, 685 (1981)

    Article  Google Scholar 

  39. C.A. Vincent, Prog. Solid State Chem. 17, 145 (1987)

    Article  Google Scholar 

  40. J.L. Lebowitz and J.K. Percus, Phys. Rev. 155, 122 (1967)

    Article  ADS  Google Scholar 

  41. P. Ferrari, S. Goldstein and J.L. Lebowitz, Diffusion, Mobility and the Einstein Relation, in: Statistical Physics and Dynamical Systems, eds.: J. Fritz, A. Jaffe and D. Szasz (Birkhäuser, Boston, 1985) p.405

    Google Scholar 

  42. O. Bénichou, A.M. Cazabat, A. Lernarchand, M. Moreau and G. Oshanin, J. Stat. Phys. 97, 351 (1999)

    Article  ADS  MATH  Google Scholar 

  43. S.F. Burlatsky, G. Oshanin, M. Moreau and W.P. Reinhardt, Phys. Rev. E 54, 3165 (1996)

    Google Scholar 

  44. G. Oshanin, J. De Coninck, M. Moreau and S.F. Burlatsky, in: Instabilities and Non-Equilibrium Structures VII, ed. E Tirapegui, (Kluwer Academic Pub., Dordrecht), to be published; cond-mat/9910243

    Google Scholar 

  45. C. Landim, S. Olla and S.B. Volchan, Comrnun. Math. Phys. 192, 287 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. Q. Gu, E.A. Schiff, S. Grebner and R. Schwartz, Phys. Rev. Lett. 76, 3196 (1996)

    Article  ADS  Google Scholar 

  47. F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis and S. Leibler, Phys. Rev. Lett. 77, 4470 (1996)

    Google Scholar 

  48. E. Barkai and J. Klafter, Phys. Rev. Lett. 81, 1134 (1998)

    Article  ADS  Google Scholar 

  49. E. Barkai and V.N. Fleurov, Phys. Rev. E 58, 1296 (1998)

    Article  ADS  Google Scholar 

  50. J. Kurchan, J. Phys. A 31, 3719 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. L.F. Cugliandolo, D.S. Dean and J. Kurchan, Phys, Rev. Lett. 79, 2168 (1997)

    Article  ADS  Google Scholar 

  52. P.G. de Gennes, Recherche 7, 919 (1976)

    Google Scholar 

  53. Y. Gefen, S. Alexander and A. Aharony, Phys. Rev. Lett. 50, 77 (1983)

    Article  ADS  Google Scholar 

  54. H. Bottger and V.V. Bryskin, Phys. Status Solidi B 113, 9 (1982)

    Article  ADS  Google Scholar 

  55. M. Barma and D. Dhar, J. Phys. C 16, 1451 (1983)

    Article  ADS  Google Scholar 

  56. D. Dhar and D. Stauffer, Int. J. Mod. Phys. C 9, 349 (1998)

    Article  ADS  Google Scholar 

  57. S.R. White and M. Barma, J. Phys. A 17, 2995 (1984)

    Article  ADS  Google Scholar 

  58. Y. Gefen and I. Goldhirsch, J. Phys. A 18, L1037 (1985)

    Google Scholar 

  59. K. Nakazato and K. Kitahara, Prog. Theor. Phys. 64, 2261 (1980)

    Article  ADS  Google Scholar 

  60. K.W. Kehr and K. Binder, in: Application of the Monte Carlo Method in Statistical Physics, ed. K. Binder, (Springer-Verlag, Berlin, 1987) and references therein.

    Google Scholar 

  61. M. Brummelhuis and H. Hilhorst, Physica A 156, 575 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  62. M.-C. Desjonquéres and D. Spanjaard, Concepts in Surface Physics, (Springer Verlag, Berlin, 1996)

    Book  Google Scholar 

  63. R.A. Tahir-Kheli and R.J. Elliott, Phys. Rev. B 27, 844 (1983)

    Article  ADS  Google Scholar 

  64. H. van Beijeren and R. Kutner, Phys. Rev. Lett. 55, 238 (1985)

    Article  ADS  Google Scholar 

  65. S. Katz, J.L. Lebowitz and H. Spohn, Phys. Rev. B 28, 1655 (1983); J. Stat. Phys. 34, 497 (1984)

    Article  ADS  Google Scholar 

  66. S.N. Majumdar and M. Barma, Phys. Rev. B 44, 5306 (1991)

    Article  ADS  Google Scholar 

  67. S.N. Majumdar and M. Barma, Physica A 177, 366 (1991)

    Article  ADS  Google Scholar 

  68. Y. Levin, J.J. Arenzon and M. Sellitto, Europhys. Lett. 55, 767 (2001)

    Article  ADS  Google Scholar 

  69. O. Bénichou, A.M. Cazabat, J. De Coninck, M. Moreau and G. Oshanin, Phys. Rev. Lett. 84, 511 (2000); Phys. Rev. B 63, 235413 (2001)

    Article  ADS  Google Scholar 

  70. J. Lyklema, Solid-Liquid Interfaces, Fundamentals of Interface and Colloid Science, Vol. 2, (Academic-Harcourt-Brace, New York, 1995)

    Google Scholar 

  71. S.F. Burlatsky, G. Oshanin, M. Mogutov and M. Moreau, Phys. Lett. A 166, 230 (1992)

    Article  ADS  Google Scholar 

  72. J. Crank, Free and Moving Boundary Problems, (Oxford University Press, Oxford, 1987)

    MATH  Google Scholar 

  73. The authors wish to thank Professor J.L. Lebowitz for posing us this question and encouraging our analysis.

    Google Scholar 

  74. D.B. Abraham, P. Collet, J. De Coninck and F. Dunlop, Phys. Rev. Lett. 65, 195 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  75. [76] D.B. Abraham, P. Collet, J. De Coninck and F. Dunlop, J. Stat. Phys. 61, 509 (1990)

    Article  ADS  Google Scholar 

  76. J. De Coninck and F. Dunlop, J. Stat. Phys. 47, 827 (1987)

    Article  ADS  Google Scholar 

  77. G. Oshanin, A.M. Cazabat, J. De Coninck and M. Moreau, Phys. Rev. E 58, R20 (1998)

    Article  ADS  Google Scholar 

  78. G. Oshanin, A.M. Cazabat, J. De Coninck and M. Moreau, J. of Mol. Liquids 76, 195 (1998)

    Article  Google Scholar 

  79. R. Arratia, Z. Ann. Probab. 11, 362 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  80. R.F. Steiner, J. Chem. Phys 22, 1458 (1954)

    Article  ADS  Google Scholar 

  81. T.L. Hill, J. Chem. Phys 20, 1259 (1952)

    Article  ADS  Google Scholar 

  82. B.D. Hughes, Random Walks and Random Environments, (Oxford Science Publ., Oxford, 1995)

    Google Scholar 

  83. P. Flajolet and A.M. Odlyzko, SIAM Journal on Discrete Math. 3, 216 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  84. Ajay and R.G. Palmer, J. Phys. A 23, 2139 (1990)

    Article  ADS  Google Scholar 

  85. T.J. Newman, Phys. Rev. B 59, 13754 (1999)

    Article  ADS  Google Scholar 

  86. Z. Toroczkai, Int. J. Mod. Phys. B11, 3343 (1997)

    Google Scholar 

  87. R.K.P. Zia and Z. Toroczkai, J. Phys. A 31, 9667 (1998)

    Article  ADS  MATH  Google Scholar 

  88. M.J.A.M. Brummelhuis and H.J. Hilhorst, Physica A 156, 575 (1989)

    Article  MathSciNet  ADS  Google Scholar 

  89. O. Benichou and G. Oshanin, Phys. Rev. E 66, 031101 (2002)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Physique Théorique des Liquides, Université Paris 6, 4 Place Jussieu, 75252, Paris, France

    G. Oshanin & M. Moreau

  2. Laboratoire de Physique de la Matière Condensée, Collège de France, 11 Place M.Berthelot, 75252, Paris Cedex 05, France

    O. Bénichou

  3. United Technologies Research Center, United Technologies Corporation, 411 Silver Lane, 129-21, East Hartford, CT, 06108, USA

    S. F. Burlatsky

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  1. G. Oshanin
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  2. O. Bénichou
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  3. S. F. Burlatsky
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  4. M. Moreau
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Editor information

Editors and Affiliations

  1. Facultad de Ingeniería, Universidad de los Andes, Santiago, Chile

    Orazio Descalzi

  2. Instituto de Física, Universidad Católica de Valparaíso, Valparaíso, Chile

    Javier Martínez

  3. Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile

    Sergio Rica

  4. Laboratoire de Physique Statistique, Ecole Normale Supérieure, Paris, France

    Sergio Rica

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Oshanin, G., Bénichou, O., Burlatsky, S.F., Moreau, M. (2004). Biased Tracer Diffusion in Hard-Core Lattice Gases: Some Notes on the Validity of the Einstein Relation. In: Descalzi, O., Martínez, J., Rica, S. (eds) Instabilities and Nonequilibrium Structures IX. Nonlinear Phenomena and Complex Systems, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0991-1_2

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  • DOI: https://doi.org/10.1007/978-94-007-0991-1_2

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