Advertisement

Journal of Statistical Physics

, Volume 135, Issue 1, pp 57–75 | Cite as

Computation of Current Cumulants for Small Nonequilibrium Systems

  • Marco Baiesi
  • Christian Maes
  • Karel Netočný
Article

Abstract

We analyze a systematic algorithm for the exact computation of the current cumulants in stochastic nonequilibrium systems, recently discussed in the framework of full counting statistics for mesoscopic systems. This method is based on identifying the current cumulants from a Rayleigh-Schrödinger perturbation expansion for the generating function. Here it is derived from a simple path-distribution identity and extended to the joint statistics of multiple currents. For a possible thermodynamical interpretation, we compare this approach to a generalized Onsager-Machlup formalism. We present calculations for a boundary driven Kawasaki dynamics on a one-dimensional chain, both for attractive and repulsive particle interactions.

Keywords

Current fluctuations Nonequilibrium Cumulant expansion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bustamante, C., Liphardt, J., Ritort, F.: Thermodynamics of small systems. Phys. Today 43–48 (2005) Google Scholar
  2. 2.
    Rubi, J.M.: Non-equilibrium thermodynamics of small-scale systems. Energy 32, 297–300 (2007) CrossRefGoogle Scholar
  3. 3.
    Saraniti, M., Ravaioli, U.: Nonequilibrium carrier dynamics in semiconductors. In: Proceedings of the 14th International Conference (2005), Chicago, USA. Springer Proceedings in Physics. Springer, Berlin (2005) Google Scholar
  4. 4.
    Blanter, Y.M., Büttiker, M.: Shot noise in mesoscopic conductors. Phys. Rep. 336(1), 1–166 (2000) CrossRefADSGoogle Scholar
  5. 5.
    Nazarov, Y.V. (ed.): Quantum Noise in Mesoscopic Systems. Kluwer, Dordrecht (2003) Google Scholar
  6. 6.
    Jauho, A.P., Flindt, C., Novotny, T., Donarini, A.: Current and current fluctuations in quantum shuttles. Phys. Fluids 17, 10 (2005) CrossRefGoogle Scholar
  7. 7.
    Flindt, C., Novotny, T., Jauho, A.P.: Full counting statistics of nano-electromechanical systems. Europhys. Lett. 69, 475–481 (2005) CrossRefADSGoogle Scholar
  8. 8.
    Lee, H., Levitov, L.S., Yakovets, A.Y.: Universal statistics of transport in disordered conductors. Phys. Rev. B 51, 4079–4083 (1995) CrossRefADSGoogle Scholar
  9. 9.
    Heikkilä, T.T., Ojanen, T.: Quantum detectors for the third cumulant of current fluctuations. Phys. Rev. B 75, 035335 (2007) CrossRefADSGoogle Scholar
  10. 10.
    Fujisawa, T., Hayashi, T., Tomita, R., Hirayama, Y.: Bidirectional counting of singe electrons. Science 312, 1634 (2006) CrossRefADSGoogle Scholar
  11. 11.
    Kurzynski, M.: The Thermodynamic Machinery of Life. Springer, Berlin (2005) Google Scholar
  12. 12.
    Haw, M.: The industry of life. Phys. World 20, 25–30 (2007) Google Scholar
  13. 13.
    Dellago, C., Geissler, P.L. (eds.): Monte Carlo sampling in path space: Calculating time correlation functions by transforming ensembles of trajectories. In: Proceedings of “The Monte Carlo Method in the Physical Sciences: Celebrating the 50th Anniversary of the Metropolis Algorithm”. AIP Conference Proceedings, vol. 690 (2003) Google Scholar
  14. 14.
    Dellago, C. Posch, H.A.: Realizing Boltzmann’s dream: computer simulations in modern statistical mechanics. In: Gallavotti, G., Reiter, W.L., Yngason, J. (eds.) ESI Lectures in Mathematics and Physics “Boltzmann’s Legacy”. European Mathematical Society, Zurich (2008) Google Scholar
  15. 15.
    Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690–2693 (1997) CrossRefADSGoogle Scholar
  16. 16.
    Crooks, G.E.: Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems. J. Stat. Phys. 90, 1481–1487 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Evans, D.J., Cohen, E.G.D., Morriss, G.P.: Probability of second law violations in shearing steady flows. Phys. Rev. Lett. 71, 2401–2404 (1993) zbMATHCrossRefADSGoogle Scholar
  18. 18.
    Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694–2697 (1995) CrossRefADSGoogle Scholar
  19. 19.
    Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970 (1995) zbMATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Kurchan, J.: Fluctuation theorem for stochastic dynamics. J. Phys. A 31, 3719–3729 (1998) zbMATHCrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Lebowitz, J., Spohn, H.: A Gallavotti–Cohen type symmetry in large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333–365 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Maes, C.: On the origin and the use of fluctuation relations for the entropy. Sémin. Poincaré 2, 29–62 (2003) Google Scholar
  23. 23.
    Maes, C., Netočný, K.: Time-reversal and entropy. J. Stat. Phys. 110, 269–310 (2003) zbMATHCrossRefGoogle Scholar
  24. 24.
    Keldysh, L.V.: Ionization in the field of a strong electromagnetic wave. Sov. Phys. JETP 20, 1307–1314 (1965) MathSciNetGoogle Scholar
  25. 25.
    Craig, R.A.: Perturbation expansion for real-time Green’s functions. J. Math. Phys. 9, 605 (1968) CrossRefADSGoogle Scholar
  26. 26.
    Flindt, C., Novotny, T., Braggio, A., Sassetti, M., Jauho, A.: Counting statistics of non-Markovian quantum stochastic processes. Phys. Rev. Lett. 100, 150601 (2008) CrossRefADSGoogle Scholar
  27. 27.
    Pilgram, S., Jordan, A.N., Sukhorukov, E.V., Büttiker, M.: Stochastic path integral formulation of full counting statistics. Phys. Rev. Lett. 90, 206801 (2003) CrossRefADSGoogle Scholar
  28. 28.
    Kindermann, M., Trauzettel, B.: Current fluctuations of an interacting quantum dot. Phys. Rev. Lett. 94, 166803 (2005) CrossRefADSGoogle Scholar
  29. 29.
    Gershon, G., Bomze, Y., Sukhorukov, E.V., Reznikov, M.: Detection of non-Gaussian fluctuations in a quantum point contact. Phys. Rev. Lett. 101, 016803 (2008) CrossRefADSGoogle Scholar
  30. 30.
    Schmidt, T.L., Komnik, A., Gogolin, A.O.: Full counting statistics of spin transfer through ultrasmall quantum dots. Phys. Rev. B 76, 241307(R) (2007) ADSGoogle Scholar
  31. 31.
    Roche, P., Derrida, B., Doucot, B.: Mesoscopic full counting statistics and exclusion models. Eur. Phys. J. B 43, 529–541 (2005) CrossRefADSGoogle Scholar
  32. 32.
    Bodineau, T., Derrida, B.: Current fluctuations in nonequilibrium diffusive systems: An additivity principle. Phys. Rev. Lett. 92, 180601 (2004) CrossRefADSGoogle Scholar
  33. 33.
    Bodineau, T., Derrida, B.: Distribution of current in nonequilibrium diffusive systems and phase transitions. Phys. Rev. E 72, 066110 (2005) CrossRefADSMathSciNetGoogle Scholar
  34. 34.
    Bertini, L., Sole, A.D., Jona-Lasinio, D.G.G., Landim, C.: Current fluctuations in stochastic lattice gases. Phys. Rev. Lett. 94, 030601 (2005) CrossRefADSGoogle Scholar
  35. 35.
    Bertini, L., Sole, A.D., Jona-Lasinio, D.G.G., Landim, C.: Non equilibrium current fluctuations in stochastic lattice gases. J. Stat. Phys. 123, 237–276 (2006) zbMATHCrossRefADSMathSciNetGoogle Scholar
  36. 36.
    Bodineau, T., Derrida, B.: Cumulants and large deviations of the current through non-equilibrium steady states. C. R. Phys. 8, 540–555 (2007) CrossRefADSGoogle Scholar
  37. 37.
    Appert-Rolland, C., Derrida, B., Lecomte, V., Van Wijland, F.: Universal cumulants of the current in diffusive systems on a ring (2008). arXiv:0804.2590v1
  38. 38.
    Mehl, J., Speck, T., Seifert, U.: Large deviation function for entropy production in driven one-dimensional systems (2008). arXiv:0804.0346v1
  39. 39.
    Giardinà, C., Kurchan, J., Peliti, L.: Direct evaluation of large-deviation functions. Phys. Rev. Lett. 96, 120603 (2006) CrossRefADSGoogle Scholar
  40. 40.
    Lecomte, V., Tailleur, J.: A numerical approach to large deviations in continuous time. J. Stat. Mech. P03004 (2007) Google Scholar
  41. 41.
    Ràkos, A., Harris, R.J.: On the range of validity of the fluctuation theorem for stochastic Markovian dynamics. J. Stat. Mech. P05005 (2008) Google Scholar
  42. 42.
    Meyer, C.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000) zbMATHGoogle Scholar
  43. 43.
    Maes, C., Netočný, K.: The canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states. Europhys. Lett. 82, 30003 (2008) CrossRefADSGoogle Scholar
  44. 44.
    Maes, C., Netočný, K., Wynants, B.: Steady state statistics of driven diffusions. Physica A 387, 2675–2689 (2008) ADSGoogle Scholar
  45. 45.
    Maes, C., Netočný, K., Wynants, B.: On and beyond entropy production: the case of Markov jump processes (2008). Markov Proc. Relat. Fields 14, 445–464 (2008) zbMATHGoogle Scholar
  46. 46.
    Lecomte, V., Appert-Rolland, C., Van Wijland, F.: Thermodynamic formalism and large deviation functions in continuous time Markov dynamics. C. R. Phys. 8, 609–619 (2007) CrossRefADSGoogle Scholar
  47. 47.
    Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Springer, Berlin (1998) Google Scholar
  48. 48.
    Derrida, B.D., Roche, P.E.: Current fluctuations in the one-dimensional symmetric exclusion process with open boundaries. J. Stat. Phys. 115, 717–748 (2004) zbMATHCrossRefADSMathSciNetGoogle Scholar
  49. 49.
    Bodineau, T., Derrida, B.: Current large deviations for asymmetric exclusion processes with open boundaries. J. Stat. Phys. 123, 277–300 (2006) zbMATHCrossRefADSMathSciNetGoogle Scholar
  50. 50.
    Harris, R.J., Rákos, A., Schütz, G.M.: Current fluctuations in the zero-range process with open boundaries. J. Stat. Mech. P08003 (2005) Google Scholar
  51. 51.
    van Wijland, F., Rácz, Z.: Large deviations in weakly interacting boundary driven lattice gases. J. Stat. Phys. 118, 27–54 (2005) zbMATHCrossRefADSMathSciNetGoogle Scholar
  52. 52.
    Derrida, B., Lebowitz, J.L., Speer, E.R.: Free energy functional for nonequilibrium systems: An exactly soluble case. Phys. Rev. Lett. 87, 150601 (2001) CrossRefADSMathSciNetGoogle Scholar
  53. 53.
    Bertini, L., Sole, A.D., Jona-Lasinio, D.G.G., Landim, C.: Large deviations for the boundary driven symmetric simple exclusion process. Math. Phys., Anal. Geom. 6, 231–267 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Onsager, L., Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. Lett. 91, 1505–1512 (1953) zbMATHADSMathSciNetGoogle Scholar
  55. 55.
    Donsker, M.D., Varadhan, S.R.: Asymptotic evaluation of certain Markov process expectations for large time, I. Commun. Pure Appl. Math. 28, 1–47 (1975) zbMATHMathSciNetGoogle Scholar
  56. 56.
    Meyer, C.D.: The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev. 17, 443–464 (1975) zbMATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Sonin, I., Thornton, J.: Recursive algorithm for the fundamental/group inverse matrix of a Markov chain from an explicit formula. SIAM J. Matrix Anal. Appl. 23, 209–224 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Heyman, D.P.: Accurate computation of the fundamental matrix of a Markov chain. SIAM J. Matrix Anal. Appl. 16, 954–963 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1995) zbMATHGoogle Scholar
  60. 60.
    Rayleigh, J.W.S.: Theory of Sound, 2nd edn., vol. I. Macmillan, London (1894), pp. 115–118 zbMATHGoogle Scholar
  61. 61.
    Schrödinger, E.: Annalen der Physik. Vierte Folge 80, 437 (1926) Google Scholar
  62. 62.
    Landau, L.D., Lifschitz, E.M.: Quantum Mechanics: Non-relativistic Theory, 3rd edn. Butterworth-Heinemann, London (1981) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Instituut voor Theoretische FysicaK. U. LeuvenLeuvenBelgium
  2. 2.Institute of Physics AS CRPragueCzech Republic

Personalised recommendations