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Journal of Statistical Physics

, Volume 154, Issue 3, pp 705–722 | Cite as

On the Second Fluctuation–Dissipation Theorem for Nonequilibrium Baths

  • Christian Maes
Article

Abstract

Baths produce friction and random forcing on particles suspended in them. The relation between noise and friction in (generalized) Langevin equations is usually referred to as the second fluctuation–dissipation theorem. We show what is the proper nonequilibrium extension, to be applied when the environment is itself active and driven. In particular we determine the effective Langevin dynamics of a probe from integrating out a steady nonequilibrium environment. The friction kernel picks up a frenetic contribution, i.e., involving the environment’s dynamical activity, responsible for the breaking of the standard Einstein relation.

Keywords

Active medium Fluctuation–dissipation theorem Nonequilibrium reduced dynamics 

Notes

Acknowledgments

We thank Marco Baiesi and Urna Basu for helpful discussions.

References

  1. 1.
    Baerts, P., Basu, U., Maes, C., Safaverdi, S.: The frenetic origin of negative differential response. Phys. Rev. E 88, 052109 (2013)Google Scholar
  2. 2.
    Baiesi, M., Maes, C., Wynants, B.: The modified Sutherland–Einstein relation for diffusive non-equilibria. Proc. R. Soc. A 467, 2792–2809 (2011)ADSCrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Baiesi, M., Maes, C.: An update on nonequilibrium linear response. New J. Phys. 15, 013004 (2013)ADSCrossRefGoogle Scholar
  4. 4.
    Baiesi, M., Maes, C., Wynants, B.: Nonequilibrium linear response for Markov dynamics, I: jump processes and overdamped diffusions. J. Stat. Phys. 137, 1094–1116 (2009)ADSCrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Baiesi, M., Boksenbojm, E., Maes, C., Wynants, B.: Nonequilibrium linear response for Markov dynamics, II: inertial dynamics. J. Stat. Phys. 139, 492–505 (2010)ADSCrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bergman, P.G., Lebowitz, J.L.: New approach to nonequilibrium process. Phys. Rev. 99, 578 (1955)ADSCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bohec, P., Gallet, F., Maes, C., Safaverdi, S., Visco, P., Van Wijland, F.: Probing active forces via a fluctuation–dissipation relation. Eur. Phys. Lett. 102, 50005 (2013)ADSCrossRefGoogle Scholar
  8. 8.
    Cheng, X., Xu, X., Rice, S.A., Dinner, A.R., Cohen, I.: Assembly of vorticity-aligned hard-sphere colloidal strings in a simple shear flow. Proc. Natl. Acad. Sci. 109, 63–67 (2012)ADSCrossRefGoogle Scholar
  9. 9.
    Colangeli, M., Maes, C., Wynants, B.: A meaningful expansion around detailed balance. J. Phys. A 44, 095001 (2011)ADSCrossRefMathSciNetGoogle Scholar
  10. 10.
    Derrida, B.: Non-equilibrium steady states: fluctuations and large deviations of the density and of the current. J. Stat. Mech. 2007, P07023 (2007)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Dzubiella, J., Löwen, H., Likos, C.N.: Depletion forces in non-equilibrium. Phys. Rev. Lett. 91, 248301 (2003)ADSCrossRefGoogle Scholar
  12. 12.
    Einstein, A.: Über die von molekülarkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierter Teilchen. Ann. Physik 17, 549–560 (1905)ADSCrossRefMATHGoogle Scholar
  13. 13.
    Emary, C.: Quantum dynamics in nonequilibrium environments. Phys. Rev. A 78, 032105 (2008)ADSCrossRefMathSciNetGoogle Scholar
  14. 14.
    Eyink, G.L., Lebowitz, J.L., Spohn, H.: Hydrodynamics and fluctuations outside of local equilibrium: driven diffusive systems. J. Stat. Phys. 83, 385–472 (1996)ADSCrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Faxen, H., Angew, Z.: Math. Mech. 7, 79 (1927)MATHGoogle Scholar
  16. 16.
    Fox, R.F., Uhlenbeck, G.E.: Contributions to non-equilibrium thermodynamics. I. Theory of hydro-dynamical fluctuations. Phys. Fluids 13, 1893–1902 (1970)ADSCrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Graham, R.: Covariant formulation of non-equilibrium statistical thermodynamics. Z. Phys. B 26, 397 (1977)ADSCrossRefMathSciNetGoogle Scholar
  18. 18.
    Gupta, S., Rosso, A., Texier, C.: Anomalous dynamics of a tagged monomer of a long polymer chain: The case of harmonic pinning and harmonic absorption. arXiv:1308.0284v1 [cond-mat.stat-mech].Google Scholar
  19. 19.
    Harada, T., Sasa, S.-Y.: Energy dissipation and violation of the fluctuation–response relation in nonequilibrium Langevin systems. Phys. Rev. E 73, 026131 (2006)ADSCrossRefGoogle Scholar
  20. 20.
    Hayashi, K., Sasa, S.: he law of action and reaction for the effective force in a non-equilibrium colloidal system. J. Phys. 18, 2825 (2006)Google Scholar
  21. 21.
    Hernandez, R., Somer Jr, F.L.: Stochastic dynamics in irreversible nonequilibrium environments, 1. The fluctuation–dissipation relation. J. Phys. Chem. B 103, 1064–1069 (1999)CrossRefGoogle Scholar
  22. 22.
    Holzer, L., Bammert, J., Rzehak, R., Zimmermann, W.: Dynamics of a trapped Brownian particle in shear flows. Phys. Rev. E 81, 041124 (2010)ADSCrossRefMathSciNetGoogle Scholar
  23. 23.
    van Kampen, N.G.: Elimination of fast variables. Phys. Rep. 124, 69–160 (1985)ADSCrossRefMathSciNetGoogle Scholar
  24. 24.
    van Kampen, N.G., Oppenheim, I.: Brownian motion as a problem of eliminating fast variables. Physica A 138, 231–248 (1986)ADSCrossRefMathSciNetGoogle Scholar
  25. 25.
    Katz, S., Lebowitz, J.L., Spohn, H.: Stationary nonequilibrium states for stochastic lattice gas models of ionic superconductors. J. Stat. Phys. 34, 497–537 (1984)ADSCrossRefMathSciNetGoogle Scholar
  26. 26.
    Krüger, M., Emig, T., Bimonte, G., Kardar, M.: Non-equilibrium Casimir forces: spheres and sphere-plate. Eur. Phys. Lett. 95, 21002 (2011)ADSCrossRefGoogle Scholar
  27. 27.
    Kubo, R.: The fluctuation–dissipation theorem. Rep. Prog. Phys. 29, 255–284 (1966)ADSCrossRefGoogle Scholar
  28. 28.
    Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics: Nonequilibrium Statistical Mechanics, 2nd edn. Springer, New York (1992)Google Scholar
  29. 29.
    Lebowitz, J.L., Spohn, H.: Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys. 38, 109 (1978)Google Scholar
  30. 30.
    MacKay, R.S.: Langevin equation for slow degrees of freedom of Hamiltonian systems. In: Theil, M., Kurths, J., Romano, M.C., Karolyi, G., Moura, A. (eds.) Nonlinear Dynamics and Chaos, pp. 89–102. Springer, London (2010)Google Scholar
  31. 31.
    Maes, C., Netočný, K.: Time-reversal and entropy. J. Stat. Phys. 110, 269–310 (2003)CrossRefMATHGoogle Scholar
  32. 32.
    Maes, C., Safaverdi, S., Visco, P., van Wijland, F.: Fluctuation–response relations for nonequilibrium diffusions with memory. Phys. Rev. E 87, 022125 (2013)ADSCrossRefGoogle Scholar
  33. 33.
    Maes, C., Thomas, S.R.: From Langevin to generalized Langevin equations for the nonequilibrium Rouse mode. Phys. Rev. E 87, 022145 (2013)ADSCrossRefGoogle Scholar
  34. 34.
    Marchetti, M.C., Joanny, J.F., Ramaswamy, S., Liverpool, T.B., Prost, J.: Madan Rao, and R. Aditi Simha, Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143–1189 (2013)ADSCrossRefGoogle Scholar
  35. 35.
    Mejia-Monasterio, C., Oshanin, G.: Bias- and bath-mediated pairing of particles driven through a quiescent medium. Soft Matter. 7, 993–1000 (2011)ADSCrossRefGoogle Scholar
  36. 36.
    Mizuno, D., Tardin, C., Schmidt, C.F., MacKintosh, F.C.: Nonequilibrium mechanics of active cytoskeletal networks. Science 315, 370–373 (2007)ADSCrossRefGoogle Scholar
  37. 37.
    Mori, H.: Prog. Theor. Phys. 33, 423 (1965)ADSCrossRefMATHGoogle Scholar
  38. 38.
    Perez-Madrid, A., Rubí, J.M., Mazur, P.: Brownian motion in the presence of a temperature gradient. Physica A 212, 231–238 (1994)ADSCrossRefGoogle Scholar
  39. 39.
    Radu, M., Schilling, T.: Hydrodynamic interactions of colloidal spheres under shear flow. arXiv:1203.3441v1 (cond-mat.soft).Google Scholar
  40. 40.
    Rubí, J.M., Bedeaux, D.J.: Brownian-motion in a fluid in elongational flow. J. Stat. Phys. 53, 125 (1988)ADSCrossRefGoogle Scholar
  41. 41.
    Santamaria-Holek, I., Reguera, D., Rubí, J.M.: Diffusion in stationary flow from mesoscopic non-equilibrium thermodynamics. Phys. Rev. E 63, 051106 (2001)ADSCrossRefGoogle Scholar
  42. 42.
    Sasa, S.-I.: Long range spatial correlation between two Brownian particles under external driving. Physica D 205, 233–241 (2005)ADSCrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Shea, J.-E.: Brownian motion in a non-equilibrium bath. MIT Ph.D. thesis (1997).Google Scholar
  44. 44.
    Shea, J.-E., Oppenheim, I.: Fokker–Planck equation and Langevin equation for one Brownian particle in a nonequilibrium bath. J. Phys. Chem. 100, 19035–19042 (1996)CrossRefGoogle Scholar
  45. 45.
    Stratonovich, R.L.: Nonlinear nonequilibrium thermodynamics II: advanced theory. In: Mishra, R.K., Maas, D., Zwierlein, E. (eds.) Springer Series in Synergetics, pp. 345–354. Springer, Berlin (1994)Google Scholar
  46. 46.
    H. Tasaki, Two theorems that relate discrete stochastic processes to microscopic mechanics. arXiv:0706.1032v1 [cond-mat.stat-mech].Google Scholar
  47. 47.
    Tomita, K., Tomita, H.: Prog. Theor. Phys. 51, 1731 (1874)ADSCrossRefGoogle Scholar
  48. 48.
    Zubarev, D.N., Bashkirov, A.G.: Physica 39, 334 (1968)ADSCrossRefGoogle Scholar
  49. 49.
    Zwanzig, R.: Ensemble method in the theory of irreversibility. J. Chem. Phys. 33, 1338 (1960)ADSCrossRefMathSciNetGoogle Scholar
  50. 50.
    Zwanzig, R.: Memory effects in irreversible thermodynamics. Phys. Rev. 124, 983 (1961)ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Instituut voor Theoretische FysicaKU LeuvenLouvainBelgium

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