Probability currents as principal characteristics in the statistical mechanics of non-equilibrium steady states

Journal of Statistical Mechanics: Theory and Experiment

Volumn , Issue 7, 2007, Pages

  Zia, R K P ^a   Schmittmann, B ^a  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

Author keywords

Driven diffusive systems (theory); Stationary states (theory); Stochastic processes (theory)

Indexed keywords

EID: 34547691286     PISSN: 17425468     EISSN: 17425468     Source Type: Journal    
DOI: 10.1088/1742-5468/2007/07/P07012     Document Type: Article

References (87)

1. Schmittmann B and Zia R K P 1995 Statistical Mechanics of Driven Diffusive Systems (Phase Transitions and Critical Phenomena vol 17) ed C Domb and J L Lebowitz (London: Academic)
2. Mukamel D 2000 Phase Transitions in Non-Equilibrium Systems (Soft and Fragile Matter: Nonequilibrium Dynamics, Metastability and Flow) ed M E Cates and M R Evans (Bristol: Institute of Physics Publishing)
3. Zia R K P and Schmittmann B 2006 J. Phys. A: Math. Gen. 39 L407
4. Kolmogorov A N 1936 Math. Ann. 112 155
5. Kelly F P 1979 Reversibility and Stochastic Networks (New York: Wiley)
6. Blythe R A 2001 PhD Thesis University of Edinburgh http://www.ph.ed.ac. uk/cmatter/links/rab-thesis
7. Lebowitz J L and Spohn H 1999 J. Stat. Phys. 95 333
8. Janssen H K 1976 Z. Phys. B 23 377
9. DeDominicis C 1976 J. Physique Coll. 37 C247
10. Bausch R, Janssen H K and Wagner H 1976 Z. Phys. B 24 113
11. DeDominicis C and Peliti L 1978 Phys. Rev. B 18 353
12. Janssen H K 1979 Dynamical Critical Phenomena and Related Topics (Springer Lecture Notes in Physics vol 104) ed C P Enz (Heidelberg: Springer)
13. Janssen H K 1992 From Phase Transitions to Chaos ed G Györgyi, I Kondor, L Sasvri and T Tél (Singapore: World Scientific)
14. Hill T L 1966 J. Theor. Biol. 10 399
15. Schnakenberg J 1976 Rev. Mod. Phys. 48 571
16. Haken H 1983 Synergetics, An Introduction 3rd edn (Berlin: Springer)
17. Kirchhoff G 1847 Poggendorff's Ann. Phys. Chem. 72 497
18. See, e.g. Roberts F S 1984 Applied Combinatorics (Englewood Cliffs, NJ: Prentice-Hall)
19. MacDonald C, Gibbs J and Pipkin A 1968 Biopolymers 6 1
20. MacDonald C and Gibbs J 1969 Biopolymers 7 707
21. Derrida B, Domany E and Mukamel D 1992 J. Stat. Phys. 69 667
22. Derrida B, Evans M R, Hakim V and Pasquier V 1993 J. Phys. A: Math. Gen. 26 1493
23. Schütz G M and Domany E 1993 J. Stat. Phys. 72 277
24. For a review, see Schütz G M 2000 Phase Transitions and Critical Phenomena vol 19 ed C Domb and JL Lebowitz (New York: Academic)
25. Garrido P L and Marro J 1987 Physica A 144 585
26. Blöte H W J, Heringa J R, Hoogland A and Zia R K P 1990 J. Phys. A: Math. Gen. 23 3799
27. Blöte H W J, Heringa J R, Hoogland A and Zia R K P 1991 Int. J. Mod. Phys. B 5 685
28. Rcz Z and Zia R K P 1994 Phys. Rev. E 49 139
29. Schmüser F and Schmittmann B 2002 J. Phys. A: Math. Gen. 35 2569
30. Schmittmann B and Schmüser F 2002 Phys. Rev. E 66 046130
31. Swendsen R H and Wang J-S 1987 Phys. Rev. Lett. 58 86
32. Wolff U 1989 Phys. Rev. Lett. 62 361
33. Niedermayer F 1997 Cluster algorithms available from http://arxiv.org/ PS_cache/hep-lat/pdf/9704/9704009.pdf
34. Bergmann P G and Lebowitz J L 1955 Phys. Rev. 99 578
35. Glansdorff P and Prigogine I 1971 Thermodynamic Theory of Structure, Stability, and Fluctuations (London: Wiley)
36. de Groot S R and Mazur P 1969 Nonequilibrium Thermodynamics (Amsterdam: North-Holland)
37. Maes C and Redig F 2000 J. Stat. Phys. 101 3 and [7]
38. Seifert U 2005 Phys. Rev. Lett. 95 040602
39. Spitzer F 1970 Adv. Math. 5 246
40. For a recent review, see Evans M R and Hanney T 2005 J. Phys. A: Math. Gen. 38 R195
41. Evans M R, Majumdar S N and Zia R K P 2004 J. Phys. A: Math. Gen. 37 L275
42. Evans M R, Majumdar S N and Zia R K P 2006 J. Phys. A: Math. Gen. 39 4859
43. Grosskinsky S, Schütz G M and Spohn H 2003 J. Stat. Phys. 113 389
44. Godrèche C and Luck M 2005 J. Phys. A: Math. Gen. 38 7215
45. Glauber R J 1963 J. Math. Phys. 4 294
46. Kawasaki K 1966 Phys. Rev. 148 375
47. Katz S, Lebowitz J L and Spohn H 1983 Phys. Rev. B 28 1655
48. Katz S, Lebowitz J L and Spohn H 1984 J. Stat. Phys. 34 497
49. Mobilia M, Zia R K P and Schmittmann B 2004 J. Phys. A: Math. Gen. 37 L407
50. Hilhorst H J 2002 private communication
51. Schmittmann B and Zia R K P 2007 at press
52. Lax M 1966 Rev. Mod. Phys. 38 541
53. For a recent comment within the context of non-equilibrium steady states, see Zia R K P and Schmittmann B 1995 Z. Phys. B 97 327
54. See, e.g., Graham R 1980 Z. Phys. B 40 149 and [40]
55. Risken H 1989 The Fokker-Planck Equation: Methods of Solution and Applications (Berlin: Springer)
56. For recent studies, see, e.g. Ao P 2005 Phys. Life Rev. 2 117
57. Qian H 2007 Ann. Rev. Phys. Chem. 58 113
58. Zia R K P, Shaw L B and Schmittmann B 2000 Physica A 279 60
59. Levine E, Kafri Y and Mukamel D 2001 Phys. Rev. E 64 026105
60. Janssen H K and Schmittmann B 1986 Z. Phys. B 64 503
61. Leung K-t and Cardy J L 1986 J. Stat. Phys. 44 567
62. Leung K-t and Cardy J L 1986 J. Stat. Phys. 45 1087 (erratum)
63. Schmittmann B and Zia R K P 1991 Phys. Rev. Lett. 66 357
64. Schmittmann B 1993 Europhys. Lett. 24 109
65. Evans D J and Searles D J 1994 Phys. Rev. E 50 1645
66. Gallavotti G and Cohen E G D 1995 J. Stat. Phys. 80 931
67. Kurchan J 1998 J. Phys. A: Math. Gen. 31 3719
68. Bertini L, De Sole A, Gabrielli D, Jona-Lasinio G and Landim C 2002 J. Stat. Phys. 107 635
69. Andrieux D and Gaspard P 2007 J. Stat. Phys. 127 107 and [7,23]
70. For a recent review, see Evans D J and Searles D J 2002 Adv. Phys. 51 1529
71. Note that there is another, quite different class of 'non-equilibrium fluctuation dissipation relations' associated especially with glassy systems, e.g., Parisi G 1998 Phil. Mag. B 77 257
72. For work energy relations, see, e.g. Jarzynski C 1997 Phys. Rev. Lett. 78 2690
73. Crooks G E 1998 J. Stat. Phys. 90 1481
74. Crooks G E 1999 Phys. Rev. E 60 2721
75. Crooks G E 2000 Phys. Rev. E 61 2361
76. Evans D J 2003 Mol. Phys. 101 1551
77. Adib A B 2005 Phys. Rev. E 71 056128 and [7,23]
78. Online 'web-seminars' on these topics may be found at http://www.newton.cam.ac.uk/webseminars/pg+ws/2006/pds/0309/cohen/ and http://www.newton.cam.ac.uk/webseminars/pg+ws/2006/pds/pdsw02/0328/evans/
79. See, for example Liphardt J, Dumont S, Smith S B, Tinoco I and Bustamante C 2002 Science 296 1832
80. Bustamante C, Liphardt J and Ritort F 2005 Phys. Today 58 43
81. Garnier N and Ciliberto S 2005 Phys. Rev. E 71 060101(R)
82. Wang G M, Reid J C, Carberry D M, Williams D R M, Sevick E M and Evans D J 2005 Phys. Rev. E 71 046142
83. Blickle V, Speck T, Helden L, Seifert U and Bechinger C 2006 Phys. Rev. Lett. 96 070603
84. Graham R 1977 Z. Phys. B 26 397
85. Timm C 2006 private communication
86. Timm C 2006 Preprint cond-mat/0612593
87. Wu F Y 1982 Rev. Mod. Phys. 54 235