Skip to main content
SpringerLink
Log in

Thermostats, Chaos and Onsager Reciprocity

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

Finite thermostats are studied in the context of nonequilibrium statistical mechanics. Entropy production rate has been identified with the mechanical quantity expressed by the phase space contraction rate and the currents have been linked to its derivatives with respect to the parameters measuring the forcing intensities. In some instances Green–Kubo formulae, hence Onsager reciprocity, have been related to the fluctuation theorem. However, mainly when dissipation takes place at the boundary (as in gases or liquids in contact with thermostats), phase space contraction may be independent on some of the forcing parameters or, even in absence of forcing, phase space contraction may not vanish: then the relation with the fluctuation theorem does not seem to apply. On the other hand phase space contraction can be altered by changing the metric on phase space: here this ambiguity is discussed and employed to show that the relation between the fluctuation theorem and Green–Kubo formulae can be extended and is, by far, more general.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Nosé, S.: A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys. 81, 511–519 (1984)

    Article  ADS  Google Scholar 

  2. Hoover, W.: Canonical equilibrium phase-space distributions. Phys. Rev. A 31, 1695–1697 (1985)

    Article  ADS  Google Scholar 

  3. Evans, D.J., Morriss, G.P.: Statistical Mechanics of Nonequilibrium Fluids. Academic Press, New York (1990)

    Google Scholar 

  4. Gallavotti, G.: Heat and fluctuations from order to chaos. Eur. Phys. J. B 61, 1–24 (2008)

    ADS  Google Scholar 

  5. Gallavotti, G.: Fluctuation relation, fluctuation theorem, thermostats and entropy creation in non equilibrium statistical physics. C.R. Phys. 8, 486–494 (2007). doi:10.1016/j.crhy.2007.04.011

    Article  ADS  Google Scholar 

  6. Bonetto, F., Gallavotti, G., Giuliani, A., Zamponi, F.: Chaotic hypothesis, fluctuation theorem and singularities. J. Stat. Phys. 123, 39–54 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Becker, R.: Electromagnetic Fields and Interactions. Blaisdell, New York (1964)

    Google Scholar 

  8. Chernov, N.I., Eyink, G.L., Lebowitz, J.L., Sinai, Ya.G.: Steady state electric conductivity in the periodic Lorentz gas. Commun. Math. Phys. 154, 569–601 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Garrido, P., Gallavotti, G.: Boundary dissipation in a driven hard disk system. J. Stat. Phys. 126, 1201–1207 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. 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)

    Article  MATH  ADS  Google Scholar 

  11. Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694–2697 (1995)

    Article  ADS  Google Scholar 

  12. Gallavotti, G.: Statistical Mechanics. A Short Treatise. Springer, Berlin (2000)

    Google Scholar 

  13. Gallavotti, G.: Chaotic hypothesis: Onsager reciprocity and fluctuation–dissipation theorem. J. Stat. Phys. 84, 899–926 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Jarzynski, C.: Hamiltonian derivation of a detailed fluctuation theorem. J. Stat. Phys. 98, 77–102 (1999)

    Article  MathSciNet  Google Scholar 

  15. Bochkov, G.N., Kuzovlev, Yu.E.: Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem. Physica A 106, 443–479 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  16. Gallavotti, G.: New methods in nonequilibrium gases and fluids. Open Syst. Inf. Dyn. 6, 101–136 (1999). chao-dyn/9610018

    Article  MATH  MathSciNet  Google Scholar 

  17. Ruelle, D.: What are the measures describing turbulence. Prog. Theor. Phys. Suppl. 64, 339–345 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  18. Ruelle, D.: Natural nonequilibrium states in quantum statistical mechanics. J. Stat. Phys. 98, 57–75 (1998)

    Article  MathSciNet  Google Scholar 

  19. Ruelle, D.: Ergodic theory. In: Cohen, E.G.D., Thirring, W. (eds.) The Boltzmann Equation. Acta Physica Austriaca, vol. Suppl X. Springer, New York (1973)

    Google Scholar 

  20. Boltzmann, L.: Über die mechanische Bedeutung des zweiten Hauptsatzes der Wärmetheorie. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen, vol. 1, p. 9. Chelsea, New York (1968)

    Google Scholar 

  21. Clausius, R.: Ueber die zurückführung des zweites hauptsatzes der mechanischen wärmetheorie aud allgemeine mechanische prinzipien. Ann. Phys. 142, 433–461 (1871)

    Google Scholar 

  22. Helmholtz, H.: Prinzipien der Statistik Monocyklischer Systeme. Wissenschaftliche Abhandlungen, vol. III. Barth, Leipzig (1895)

    Google Scholar 

  23. Mejia-Monasterio, C., Rondoni, L.: On the fluctuation relation for Nose-Hoover locally thermostated systems (2007). arXiv:0710.3673 [cond-mat]

  24. Sinai, Ya.G.: Markov partitions and C-diffeomorphisms. Funct. Anal. Appl. 2(1), 64–89 (1968)

    Article  MathSciNet  Google Scholar 

  25. Bowen, R.: Markov partitions for axiom A diffeomorphisms. Am. J. Math. 92, 725–747 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  26. Bowen, R., Ruelle, D.: The ergodic theory of axiom A flows. Invent. Math. 29, 181–205 (1975)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Gentile, G.: A large deviation theorem for Anosov flows. Forum Math. 10, 89–118 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sinai, Ya.G.: Lectures in Ergodic Theory. Lecture Notes in Mathematics. Princeton University Press, Princeton (1977)

    Google Scholar 

  29. Gallavotti, G., Bonetto, F., Gentile, G.: Aspects of the Ergodic, Qualitative and Statistical Theory of Motion. Springer, Berlin (2004)

    MATH  Google Scholar 

  30. Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970 (1995)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. Gallavotti, G.: Reversible Anosov diffeomorphisms and large deviations. Math. Phys. Electron. J. 1, 1–12 (1995)

    MathSciNet  Google Scholar 

  32. Cohen, E.G.D., Gallavotti, G.: Note on two theorems in nonequilibrium statistical mechanics. J. Stat. Phys. 96, 1343–1349 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  33. Gallavotti, G.: Fluctuation patterns and conditional reversibility in nonequilibrium systems. Ann. Inst. Henri Poincaré 70, 429–443 (1999). chao-dyn/9703007

    MATH  MathSciNet  Google Scholar 

  34. Gallavotti, G.: Foundations of Fluid Dynamics, 2nd edn. Springer, Berlin (2005)

    Google Scholar 

  35. Gallavotti, G., Ruelle, D.: SRB states and nonequilibrium statistical mechanics close to equilibrium. Commun. Math. Phys. 190, 279–285 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. Ruelle, D.: Positivity of entropy production in nonequilibrium statistical mechanics. J. Stat. Phys. 85, 1–25 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  37. Gallavotti, G.: Extension of Onsager’s reciprocity to large fields and the chaotic hypothesis. Phys. Rev. Lett. 77, 4334–4337 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Gomez-Marin, A., Parrondo, J.M.R., Van den Broeck, C.: The footprints of irreversibility. Eur. Phys. Lett. 82, 50002 (2008)

    Article  ADS  Google Scholar 

  39. Chernov, N.I., Eyink, G.L., Lebowitz, J.L., Sinai, Ya.G.: Derivation of Ohm’s law in a deterministic mechanical model. Phys. Rev. Lett. 70, 2209–2212 (1993)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica and INFN, Università di Roma La Sapienza, P. A. Moro 2, 00185, Rome, Italy

    Giovanni Gallavotti

Authors
  1. Giovanni Gallavotti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovanni Gallavotti.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gallavotti, G. Thermostats, Chaos and Onsager Reciprocity. J Stat Phys 134, 1121–1131 (2009). https://doi.org/10.1007/s10955-008-9655-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-008-9655-3

Keywords

Search

Navigation