Exact nonequilibrium work generating function for a small classical system

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

Volumn 82, Issue 2, 2010, Pages

  Morgado, W A M ^a   Soares Pinto, D O ^b  

^a PONTIFICAL CATHOLIC UNIVERSITY OF RIO DE JANEIRO   (Brazil)

^b UNIVERSITY OF SÃO PAULO   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

BROWNIAN PARTICLES; CLASSICAL SYSTEMS; EXTERNAL WORK; FINITE TIME INTERVALS; GENERATING FUNCTIONS; JARZYNSKI EQUALITY; NONEQUILIBRIUM WORK; SMALL SYSTEMS;

FUNCTION EVALUATION;

EID: 77955593698     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.82.021112     Document Type: Article

References (41)

1. E. H. Trepagnier, C. Jarzynski, F. Ritort, G. E. Crooks, C. J. Bustamante, and J. Liphardt, Proc. Natl. Acad. Sci. U.S.A. 101, 15038 (2004). 10.1073/pnas.0406405101
2. C. Bustamante, J. Liphardt, and F. Ritort, Phys. Today 58 (7), 43 (2005). 10.1063/1.2012462
3. D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco, and C. Bustamante, Nature (London) 437, 231 (2005). 10.1038/nature04061
4. J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, Science 296, 1832 (2002). 10.1126/science.1071152
5. A. Imparato and L. Peliti, Phys. Biol. 6, 025011 (2009). 10.1088/1478-3975/6/2/025011
6. C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997). 10.1103/PhysRevLett.78. 2690
7. G. E. Crooks, Phys. Rev. E 61, 2361 (2000). 10.1103/PhysRevE.61.2361
8. C. Jarzynski, C. R. Phys. 8, 495 (2007). 10.1016/j.crhy.2007.04.010
9. J. Horowitz and C. Jarzynski, J. Stat. Mech.: Theory Exp. (2007), P11002. 10.1088/1742-5468/2007/11/P11002
10. C. Lanczos, The Variational Principles of Mechanics (Dover Publications, Mineola, NY, 1970).
11. An example of such a potential is that of an instantaneously switched-on constant force f (t): for t < 0, there is no potential energy, while for t > 0 a potential energy - f 0 x arises. For instance, this can be thought as the energy of a very small bare electric charge placed in the inside of charged capacitor plates, with plates orthogonal to the x axis. The charging of the capacitors happens almost instantly at t = 0, and we assume that the plates were grounded at t < 0. In fact, the external work done by the force, described by the equation above, corresponds to the work done by the batteries in order to initially charge the capacitors up to a fixed voltage (part of that work shifts the particles potential energy and the remaining is used to create the electric field between the plates), producing a constant force upon the particle. It has the effect of shifting the initial potential energy of the charged particle by an amount - f 0 x 0, where x 0 is the initial position of the charged particle (if we include the batteries within the system, then the potential energy they provide is taken to be internal). That external work is then given by consistent with references. The important point to be consistent with is that the external work is the one changing the energy landscape for the phase-space point. In contrast, if we keep the batteries as an external agent for all t, then the total external work is given by (x, t) = - f (t) xi - f 0 Θ (t) x. W ext, 0 = - f 0 x 0 = - t 1 < 0 t 2 > 0 d t f 0 Θ (t) x t, W ext = - f 0 (x t 2 - x 0) + W ext, 0 = - f 0 x t 2. The definition of what is the internal Hamiltonian, and what is external, is crucial as to which Work Fluctuation relation can be derived.
12. G. N. Bochkov and Y. E. Kuzovlev, Zh. Eksp. Teor. Fiz. 72, 238 (1977).
13. G. N. Bochkov and Y. E. Kuzovlev, Sov. Phys. JETP 45, 125 (1977).
14. G. N. Bochkov and Y. E. Kuzovlev, Physica 106, 443 (1981). 10.1016/0378-4371(81)90122-9
15. G. N. Bochkov and Y. E. Kuzovlev, Physica 106, 480 (1981). 10.1016/0378-4371(81)90123-0
16. J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett. 100, 020601 (2008). 10.1103/PhysRevLett.100.020601
17. J. Horowitz and C. Jarzynski, Phys. Rev. Lett. 101, 098901 (2008). 10.1103/PhysRevLett.101.098901
18. J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett. 101, 098902 (2008). 10.1103/PhysRevLett.101.098902
19. L. Peliti, Phys. Rev. Lett. 101, 098903 (2008). 10.1103/PhysRevLett.101. 098903
20. J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett. 101, 098904 (2008). 10.1103/PhysRevLett.101.098904
21. B. Palmieri and D. Ronis, Phys. Rev. E 75, 011133 (2007). 10.1103/PhysRevE.75.011133
22. K. Huang, Statistical Mechanics (Wiley, New York, 1987).
23. O. Mazonka and C. Jarzynski, e-print arXiv:cond-mat/9912121.
24. A. Imparato and L. Peliti, J. Stat. Mech.: Theory Exp. (2006), P03005. 10.1088/1742-5468/2006/03/P03005
25. A. Imparato and L. Peliti, EPL 69, 643 (2005). 10.1209/epl/i2004-10390-3
26. N. C. Harris, Y. Song, and C. H. Kiang, Phys. Rev. Lett. 99, 068101 (2007). 10.1103/PhysRevLett.99.068101
27. A. Imparato, S. Luccioli, and A. Torcini, Phys. Rev. Lett. 99, 168101 (2007). 10.1103/PhysRevLett.99.168101
28. A. Imparato, A. Pelizzola, and M. Zamparo, J. Chem. Phys. 127, 145105 (2007). 10.1063/1.2776271
29. P. Maragakis, F. Ritort, C. Bustamante, M. Karplus, and G. E. Crooks, J. Chem. Phys. 129, 024102 (2008). 10.1063/1.2937892
30. A. Mossa, S. de Lorenzo, J. M. Huguet, and F. Ritort, J. Chem. Phys. 130, 234116 (2009). 10.1063/1.3155084
31. T. Sagawa and M. Ueda, Phys. Rev. Lett. 104, 090602 (2010). 10.1103/PhysRevLett.104.090602
32. W. A. M. Morgado and D. O. Soares-Pinto, Phys. Rev. E 79, 051116 (2009). 10.1103/PhysRevE.79.051116
33. D. O. Soares-Pinto and W. A. M. Morgado, Phys. Rev. E 77, 011103 (2008). 10.1103/PhysRevE.77.011103
34. N. C. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992).
35. A. Imparato and L. Peliti, Phys. Rev. E 72, 046114 (2005). 10.1103/PhysRevE.72.046114
36. A. Imparato and L. Peliti, EPL 70, 740 (2005). 10.1209/epl/i2005-10067-5
37. D. O. Soares-Pinto and W. A. M. Morgado, Physica A 365, 289 (2006). 10.1016/j.physa.2006.01.063
38. T. Speck and U. Seifert, Eur. Phys. J. B 43, 521 (2005). 10.1140/epjb/e2005-00086-6
39. A. Imparato, L. Peliti, G. Pesce, G. Rusciano, and A. Sasso, Phys. Rev. E 76, 050101 (2007). 10.1103/PhysRevE.76.050101
40. T. Taniguchi and E. G. D. Cohen, J. Stat. Phys. 126, 1 (2007). 10.1007/s10955-006-9252-2
41. P. Hänggi and H. Thomas, Phys. Rep. 88, 207 (1982). 10.1016/0370-1573(82)90045-X