Inequivalence of time and ensemble averages in ergodic systems: Exponential versus power-law relaxation in confinement

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

Volumn 85, Issue 2, 2012, Pages

  Jeon, Jae Hyung ^a   Metzler, Ralf ^a,b  

^a TAMPERE UNIVERSITY OF TECHNOLOGY   (Finland)

^b UNIVERSITY OF POTSDAM   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ANOMALOUS DIFFUSION; BIOLOGICAL CELLS; DIFFUSIVE PROPERTIES; ENSEMBLE AVERAGES; ENSEMBLE-AVERAGED; ERGODICS; FRACTIONAL BROWNIAN MOTION; FRACTIONAL LANGEVIN EQUATIONS; MEASUREMENT TIME; PHYSICAL INTERPRETATION; POWER LAW RELAXATION; RELAXATION TIME SCALE; SINGLE-PARTICLE; SMALL SYSTEMS; STANDARD TOOLS; TIME AVERAGES; TIME-AVERAGED;

DIFFERENTIAL EQUATIONS;

TIME SERIES;

EID: 84857741874     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.85.021147     Document Type: Article

References (44)

1. C. Bräuchle, D. C. Lamb, and J. Michaelis, Single Particle Tracking and Single Molecule Energy Transfer (Wiley-VCH, Weinheim, 2010);
2. X. S. Xie, P. J. Choi, G.-W. Li, N. K. Lee, and G. Lia, Annu. Rev. Biophys. 1936-122X 10.1146/annurev.biophys.37.092607.174640 37, 417 (2008).
3. I. Golding and E. C. Cox, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.96.098102 96, 098102 (2006). (Pubitemid 43385256)
4. S. C. Weber, A. J. Spakowitz, and J. A. Theriot, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.104.238102 104, 238102 (2010).
5. J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sørensen, L. Oddershede, and R. Metzler, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.106.048103 106, 048103 (2011).
6. G. Seisenberger, M. U. Ried, T. Endreß, H. Büning, M. Hallek, and C. Bräuchle, Science SCIEAS 0036-8075 10.1126/science.1064103 294, 1929 (2001). (Pubitemid 33101590)
7. I. Bronstein, Y. Israel, E. Kepten, S. Mai, Y. Shav-Tal, E. Barkai, and Y. Garini, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.103.018102 103, 018102 (2009).
8. A. V. Weigel, B. Simon, M. M. Tamkun, and D. Krapf, Proc. Nat. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.1016325108 108, 6438 (2011).
9. S. C. Kou and X. S. Xie, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.93.180603 93, 180603 (2004);
10. A. Taloni, A. V. Chechkin, and J. Klafter, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.104.160602 104, 160602 (2010), and references therein;
11. L. Lizana, T. Ambjörnsson, A. Taloni, E. Barkai, and M. A. Lomholt, Phys. Rev. E 1539-3755 10.1103/PhysRevE.81.051118 81, 051118 (2010);
12. I. Goychuk, Adv. Chem. Phys. 0065-2385 10.1002/9781118197714.ch5 150, 187 (2012).
13. J. Szymanski and M. Weiss, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.103.038102 103, 038102 (2009).
14. M. Magdziarz, A. Weron, K. Burnecki, and J. Klafter, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.103.180602 103, 180602 (2009).
15. Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.101.058101 101, 058101 (2008).
16. A. Lubelski, I. M. Sokolov, and J. Klafter, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.100.250602 100, 250602 (2008).
17. R. Metzler and J. Klafter, Phys. Rep. PRPLCM 0370-1573 10.1016/S0370-1573(00)00070-3 339, 1 (2000);
18. R. Metzler and J. Klafter, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/37/31/R01 37, R161 (2004). (Pubitemid 39063557)
19. W. Deng and E. Barkai, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.79. 011112 79, 011112 (2009).
20. S. Burov, J.-H. Jeon, R. Metzler, and E. Barkai, Phys. Chem. Chem. Phys. PPCPFQ 1463-9076 10.1039/c0cp01879a 13, 1800 (2011).
21. I. Goychuk, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.80.046125 80, 046125 (2009).
22. E. W. Montroll and G. H. Weiss, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.1704269 6, 167 (1965).
23. H. Scher and E. W. Montroll, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.12.2455 12, 2455 (1975).
24. The related concept of weak ergodicity breaking was developed by Bouchaud and, subsequently, by Barkai and coworkers: J.-P. Bouchaud, J. Phys. I JPGCE8 1155-4304 10.1051/jp1:1992238 2, 1705 (1992);
25. G. Bel and E. Barkai, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.94.240602 94, 240602 (2005); (Pubitemid 41636460)
26. A. Rebenshtok and E. Barkai, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.99.210601 99, 210601 (2007); (Pubitemid 350137198)
27. The consequences for the mixing behavior between boundaries and bulk diffusion were explored by M. A. Lomholt, I. M. Zaid, and R. Metzler, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.98.200603 98, 200603 (2007). (Pubitemid 47139586)
28. S. Burov, R. Metzler, and E. Barkai, Proc. Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.1003693107 107, 13228 (2010).
29. T. Neusius, I. M. Sokolov, and J. C. Smith, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.80.011109 80, 011109 (2009).
30. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
31. A. N. Kolmogorov, Dokl. Acad. Sci. USSR 26, 115 (1940);
32. B. B. Mandelbrot and J. W. van Ness, SIAM Rev. SIREAD 0036-1445 10.1137/1010093 1, 422 (1968).
33. O. Yu. Sliusarenko, V. Yu. Gonchar, A. V. Chechkin, I. M. Sokolov, and R. Metzler, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.81.041119 81, 041119 (2010).
34. E. Lutz, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.64.051106 64, 051106 (2001).
35. J.-H. Jeon and R. Metzler, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.81.021103 81, 021103 (2010).
36. A. Erdélyi, ed., Higher Transcendental Functions, Bateman Manuscript Project Volume (McGraw-Hill, New York, 1954).
37. S. C. Kou, Ann. Appl. Stat. 1932-6157 10.1214/07-AOAS149 2, 501 (2008).
38. α (t) by combined effects of the memory of fractional noise and confinement [S. Burov and E. Barkai, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.100.070601 100, 070601 (2008); (Pubitemid 351393590)
39. S. Burov and E. Barkai, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.78.031112 78, 031112 (2008)].
40. J.-H. Jeon and R. Metzler, J. Phys. A 1751-8113 10.1088/1751-8113/43/25/ 252001 43, 252001 (2010).
41. I. M. Sokolov, E. Heinsalu, P. Hänggi, and I. Goychuk, Europhys. Lett. EULEEJ 0295-5075 10.1209/0295-5075/86/30009 86, 30009 (2009).
42. V. Tejedor, O. Bénichou, R. Voituriez, R. Jungmann, F. Simmel, C. Selhuber-Unkel, L. Oddershede, and R. Metzler, Biophys. J. BIOJAU 0006-3495 10.1016/j.bpj.2009.12.4282 98, 1364 (2010).
43. T. Akimoto, E. Yamamoto, K. Yasuoka, Y. Hirano, and M. Yasui, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.107.178103 107, 178103 (2011).
44. S. Condamin, V. Tejedor, R. Voituriez, O. Bénichou, and J. Klafter, Proc. Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.0712158105 105, 5675 (2008). (Pubitemid 351758430)