Rate description of Fokker-Planck processes with time-dependent parameters

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 69, Issue 4 2, 2004, Pages

  Talkner, Peter ^a   Łuczka, Jerzy ^b  

^a UNIVERSITY OF AUGSBURG   (Germany)

^b UNIVERSITY OF SILESIA   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC STABILITY; ATOMIC FORCE MICROSCOPY; BIFURCATION (MATHEMATICS); BROWNIAN MOVEMENT; DEGREES OF FREEDOM (MECHANICS); HYSTERESIS; REACTION KINETICS;

BROWNIAN OSCILLATOR; CONSTRAINED EQUILIBRIUM; FOKKER-PLANCK PROCESSES; SEMI-ADIABATIC LIMIT;

MARKOV PROCESSES;

EID: 42749107024     PISSN: 15393755     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (41)

1. P. A. M. Dirac, Nature (London) 192, 325 (1937); J. K. Webb et al., Phys. Rev. Lett. 87, 091301 (2001).
2. P. A. M. Dirac, Nature (London) 192, 325 (1937); J. K. Webb et al., Phys. Rev. Lett. 87, 091301 (2001).
3. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978); A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); N. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics (Princeton University Press, Princeton, 1947); H. Haken, Synergetics, An Introduction (Springer-Verlag, Berlin, 1983).
4. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978); A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); N. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics (Princeton University Press, Princeton, 1947); H. Haken, Synergetics, An Introduction (Springer-Verlag, Berlin, 1983).
5. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978); A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); N. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics (Princeton University Press, Princeton, 1947); H. Haken, Synergetics, An Introduction (Springer-Verlag, Berlin, 1983).
6. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978); A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); N. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics (Princeton University Press, Princeton, 1947); H. Haken, Synergetics, An Introduction (Springer-Verlag, Berlin, 1983).
7. P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990).
8. P. Talkner, New J. Phys. 1, 4 (1999) (http://www.njp.org/).
9. P. Jung, Phys. Rep. 234, 175 (1993).
10. L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
11. P. Reimann, Phys. Rep. 361, 57 (2002); R. D. Astumian and P. Hänggi, Phys. Today 55(11), 33 (2002).
12. P. Reimann, Phys. Rep. 361, 57 (2002); R. D. Astumian and P. Hänggi, Phys. Today 55(11), 33 (2002).
13. E. Evans, K. Ritchie, Biophys. J. 72, 1541 (1997); R. Merkel, Phys. Rep. 346, 343 (2001).
14. E. Evans, K. Ritchie, Biophys. J. 72, 1541 (1997); R. Merkel, Phys. Rep. 346, 343 (2001).
15. J. Lehmann, P. Reimann, and P. Hänggi, Phys. Status Solidi B 237, 53 (2003); J. Lehmann, P. Reimann, and P. Hänggi, Phys. Rev. E 62, 6282 (2000); J. Lehmann, P. Reimann, and P. Hänggi, Phys. Rev. Lett. 84, 1639 (2000).
16. J. Lehmann, P. Reimann, and P. Hänggi, Phys. Status Solidi B 237, 53 (2003); J. Lehmann, P. Reimann, and P. Hänggi, Phys. Rev. E 62, 6282 (2000); J. Lehmann, P. Reimann, and P. Hänggi, Phys. Rev. Lett. 84, 1639 (2000).
17. J. Lehmann, P. Reimann, and P. Hänggi, Phys. Status Solidi B 237, 53 (2003); J. Lehmann, P. Reimann, and P. Hänggi, Phys. Rev. E 62, 6282 (2000); J. Lehmann, P. Reimann, and P. Hänggi, Phys. Rev. Lett. 84, 1639 (2000).
18. D. Ryter and H. Meyer, Physica A 142, 122 (1987).
19. G. Moro, J. Chem. Phys. 103, 7514 (1995).
20. M. Bier and R. D. Astumian, Phys. Lett. A 247, 385 (1998); I. Derenyi and R. D. Astumian, Phys. Rev. Lett. 82, 2623 (1999); M. Bier, I. Derenyi, M. Kostur, and R. D. Astumian, Phys. Rev. E 59, 6422 (1999).
21. M. Bier and R. D. Astumian, Phys. Lett. A 247, 385 (1998); I. Derenyi and R. D. Astumian, Phys. Rev. Lett. 82, 2623 (1999); M. Bier, I. Derenyi, M. Kostur, and R. D. Astumian, Phys. Rev. E 59, 6422 (1999).
22. M. Bier and R. D. Astumian, Phys. Lett. A 247, 385 (1998); I. Derenyi and R. D. Astumian, Phys. Rev. Lett. 82, 2623 (1999); M. Bier, I. Derenyi, M. Kostur, and R. D. Astumian, Phys. Rev. E 59, 6422 (1999).
23. B. McNamara and K. Wiesenfeld, Phys. Rev. A 39, 4854 (1989).
24. H. Risken, The Fokker-Planck Equation (Springer-Verlag, Berlin, 1989).
25. B. J. Matkowsky and Z. Schuss, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 42, 822 (1982).
26. P. Talkner, Z. Phys. B: Condens. Matter 68, 201 (1987).
27. P. Talkner, Chem. Phys. 180, 199 (1994).
28. M. Mangel, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 36, 544 (1979); D. Ryter, J. Stat. Phys. 49, 122 (1987).
29. M. Mangel, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 36, 544 (1979); D. Ryter, J. Stat. Phys. 49, 122 (1987).
30. a the contribution of each saddle point to the matrix element in Eq. (15) can be determined separately. More complicated cases need an extra treatment, see Refs. [3,23].
31. P. Talkner, New Trends in Kramers' Reaction Rate Theory, edited by P. Talkner and P. Hänggi (Kluwer Academic, Dordrecht, 1995), p. 47.
32. R. D. Astumian and B. Robertson, J. Chem. Phys. 91, 4891 (1989); U. Zürcher and C. R. Doering, Phys. Rev. E 47, 3862 (1993); I. A. Goychuk, E. G. Petrov, and V. May, J. Chem. Phys. 103, 4937 (1995); P. Talkner, Ann. Phys. (Leipzig) 9, 741 (2000); R. Rozenfeld, J. A. Freund, A. Neiman, and L. Schimansky-Geier, Phys. Rev. E 64, 051107 (2001); R. D. Astumian, Appl. Phys. A: Mater. Sci. Process. 75, 193 (2002); P. Talkner, Physica A 325, 124 (2003).
33. R. D. Astumian and B. Robertson, J. Chem. Phys. 91, 4891 (1989); U. Zürcher and C. R. Doering, Phys. Rev. E 47, 3862 (1993); I. A. Goychuk, E. G. Petrov, and V. May, J. Chem. Phys. 103, 4937 (1995); P. Talkner, Ann. Phys. (Leipzig) 9, 741 (2000); R. Rozenfeld, J. A. Freund, A. Neiman, and L. Schimansky-Geier, Phys. Rev. E 64, 051107 (2001); R. D. Astumian, Appl. Phys. A: Mater. Sci. Process. 75, 193 (2002); P. Talkner, Physica A 325, 124 (2003).
34. R. D. Astumian and B. Robertson, J. Chem. Phys. 91, 4891 (1989); U. Zürcher and C. R. Doering, Phys. Rev. E 47, 3862 (1993); I. A. Goychuk, E. G. Petrov, and V. May, J. Chem. Phys. 103, 4937 (1995); P. Talkner, Ann. Phys. (Leipzig) 9, 741 (2000); R. Rozenfeld, J. A. Freund, A. Neiman, and L. Schimansky-Geier, Phys. Rev. E 64, 051107 (2001); R. D. Astumian, Appl. Phys. A: Mater. Sci. Process. 75, 193 (2002); P. Talkner, Physica A 325, 124 (2003).
35. R. D. Astumian and B. Robertson, J. Chem. Phys. 91, 4891 (1989); U. Zürcher and C. R. Doering, Phys. Rev. E 47, 3862 (1993); I. A. Goychuk, E. G. Petrov, and V. May, J. Chem. Phys. 103, 4937 (1995); P. Talkner, Ann. Phys. (Leipzig) 9, 741 (2000); R. Rozenfeld, J. A. Freund, A. Neiman, and L. Schimansky-Geier, Phys. Rev. E 64, 051107 (2001); R. D. Astumian, Appl. Phys. A: Mater. Sci. Process. 75, 193 (2002); P. Talkner, Physica A 325, 124 (2003).
36. R. D. Astumian and B. Robertson, J. Chem. Phys. 91, 4891 (1989); U. Zürcher and C. R. Doering, Phys. Rev. E 47, 3862 (1993); I. A. Goychuk, E. G. Petrov, and V. May, J. Chem. Phys. 103, 4937 (1995); P. Talkner, Ann. Phys. (Leipzig) 9, 741 (2000); R. Rozenfeld, J. A. Freund, A. Neiman, and L. Schimansky-Geier, Phys. Rev. E 64, 051107 (2001); R. D. Astumian, Appl. Phys. A: Mater. Sci. Process. 75, 193 (2002); P. Talkner, Physica A 325, 124 (2003).
37. R. D. Astumian and B. Robertson, J. Chem. Phys. 91, 4891 (1989); U. Zürcher and C. R. Doering, Phys. Rev. E 47, 3862 (1993); I. A. Goychuk, E. G. Petrov, and V. May, J. Chem. Phys. 103, 4937 (1995); P. Talkner, Ann. Phys. (Leipzig) 9, 741 (2000); R. Rozenfeld, J. A. Freund, A. Neiman, and L. Schimansky-Geier, Phys. Rev. E 64, 051107 (2001); R. D. Astumian, Appl. Phys. A: Mater. Sci. Process. 75, 193 (2002); P. Talkner, Physica A 325, 124 (2003).
38. R. D. Astumian and B. Robertson, J. Chem. Phys. 91, 4891 (1989); U. Zürcher and C. R. Doering, Phys. Rev. E 47, 3862 (1993); I. A. Goychuk, E. G. Petrov, and V. May, J. Chem. Phys. 103, 4937 (1995); P. Talkner, Ann. Phys. (Leipzig) 9, 741 (2000); R. Rozenfeld, J. A. Freund, A. Neiman, and L. Schimansky-Geier, Phys. Rev. E 64, 051107 (2001); R. D. Astumian, Appl. Phys. A: Mater. Sci. Process. 75, 193 (2002); P. Talkner, Physica A 325, 124 (2003).
39. R. Graham and T. Tel, J. Stat. Phys. 35, 729 (1984); H. R. Jauslin. Physica A 144, 179 (1987).
40. R. Graham and T. Tel, J. Stat. Phys. 35, 729 (1984); H. R. Jauslin. Physica A 144, 179 (1987).
41. A. M. Berezhkovskii, P. Talkner, J. Emmerich, and V. Y. Zitserman, J. Chem. Phys. 105, 10890 (1996).