Skip to main content
SpringerLink
Log in

Fluctuation Relations and Nonequilibrium Response for Chaotic Dissipative Dynamics

  • Chapter
  • First Online:
Applications of Chaos and Nonlinear Dynamics in Science and Engineering - Vol. 3

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

In a recent paper [Colangeli and Rondoni, Physica D 241:681, 2011] it was argued that the Fluctuation Relation for the phase space contraction rate Λ could suitably be extended to non-reversible dissipative systems. We review here those arguments, by discussing the properties of a simple irreversible nonequilibrium baker model. We also consider the problem of the extension of the Fluctuation-Dissipation Theorem to dissipative deterministic dynamical systems, which enjoy a nonvanishing average phase space contraction rate. As noted by Ruelle, the statistical features of the perturbation and, in particular, of the relaxation, cannot be understood solely in terms of the unperturbed dynamics on the attractor. Nevertheless, we show that the singular character of the steady state does not constitute a serious limitation in the case of systems with many degrees of freedom. The reason is that one typically deals with projected dynamics, and these are associated with regular probability distributions in the corresponding lower dimensional spaces.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Castiglione, P., Falcioni, M., Lesne, A., Vulpiani, A.: Chaos and Coarse Graining in Statistical Mechanics. Cambridge University Press, London (2008)

    Book  MATH  Google Scholar 

  2. Evans, D.J., Searles, D.J.: The fluctuation theorem. Adv. Phys. 52, 1529 (2002)

    Article  ADS  Google Scholar 

  3. Ruelle, D.: Smooth dynamics, new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 95, 393 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Rondoni, L., Mejia-Monasterio, C.: Fluctuations in nonequilibrium statistical mechanics: models, mathematical theory, physical mechanisms. Nonlinearity 20, R1–R37 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Klages, R.: Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics. World Scientific, Singapore (2007)

    MATH  Google Scholar 

  6. Marconi, U.M.B., Puglisi, A., Rondoni, L., Vulpiani, A.: Fluctuation-dissipation: response theory in statistical physics. Phys. Rep. 461, 111 (2008)

    Article  ADS  Google Scholar 

  7. Jaksic, V., Pillet, C.-A., Rey-Bellet, L.: Entropic fluctuations in statistical mechanics I. Classical dynamical systems. Nonlinearity 24, 699 (2011)

    Google Scholar 

  8. Evans, D.J., Cohen, E.G.D., Morriss, G.P.: Probability of second law violations in nonequilibrium steady states. Phys. Rev. Lett. 71, 2401 (1993)

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Evans, D.J., Searles, D.J.: Equilibrium microstates which generate second law violating steady states. Phys. Rev. E 50, 1645 (1994)

    Article  ADS  Google Scholar 

  12. Bonetto, F., Gallavotti, G.: Reversibility, coarse graining and the chaoticity principle. Comm. Math. Phys. 189, 263 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Lebowitz, J.L., Spohn, H.: A Gallavotti-Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Chetrite, R., Gawedzki, K.: Fluctuation relations for diffusion processes. Comm. Math. Phys. 282(2), 469 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Searles, D.J., Rondoni, L., Evans, D.J.: The steady state fluctuation relation for the dissipation function. J. Stat. Phys. 128, 1337 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Chechkin, A.V., Klages, R.: Fluctuation relations for anomalous dynamics. J. Stat. Mech. (2009) L03002

    Google Scholar 

  17. Jepps, O.G., Rondoni, L.: Deterministic thermostats, theories of nonequilibrium systems and parallels with the ergodic condition. J. Phys. A: Math. Theor. 43, 1 (2010)

    Article  MathSciNet  Google Scholar 

  18. Gonnella, G., Pelizzola, A., Rondoni, L., Saracco, G.P.: Nonequilibrium work fluctuations in a driven Ising model. Physica A 388, 2815 (2009)

    Article  ADS  Google Scholar 

  19. Bonaldi, M., Conti, L., De Gregorio, P., Rondoni, L., Vedovato, G., Vinante, A., Bignotto, M., Cerdonio, M., Falferi, P., Liguori, N., Longo, S., Mezzena, R., Ortolan, A., Prodi, G.A., Salemi, F., Taffarello, L., Vitale, S., Zendri, J.P.: Nonequilibrium steady-state fluctuations in actively cooled resonators. Phys. Rev. Lett. 103, 010601 (2009)

    Article  ADS  Google Scholar 

  20. Rondoni, L., Morriss, G.P.: Large fluctuations and axiom-C structures in deterministically thermostatted systems. Open Syst. Inform. Dynam. 10, 105 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mejia-Monasterio, C., Rondoni, L.: On the fluctuation relation for Nos-Hoover boundary thermostated systems. J. Stat. Phys. 133, 617 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Colangeli, M., Maes, C., Wynants, B.: A meaningful expansion around detailed balance. J. Phys. A: Math. Theor. 44, 095001 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  23. Lucarini, V., Colangeli, M.: Beyond the fluctuation-dissipation theorem: the role of causality. J. Stat. Mech.: Theor. Exp. (2012) P05013

    Google Scholar 

  24. Zwanzig, R.: Nonequilibrium Statistical Mechanics. Oxford University Press, London (2001)

    Google Scholar 

  25. Colangeli, M., Karlin, I.V., Kröger, M.: Hyperbolicity of exact hydrodynamics for three-dimensional Grad equations. Phys. Rev. E 76, 022201 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  26. Colangeli, M., Kröger, M., Öttinger, H.C.: Boltzmann equation and hydrodynamic fluctuations. Phys. Rev. E 80, 051202 (2009)

    Article  ADS  Google Scholar 

  27. Boffetta, G., Lacorata, G., Musacchio, S., Vulpiani, A.: Relaxation of finite perturbations: beyond the fluctuation dissipation relation. Chaos 13, 3 (2003)

    Article  Google Scholar 

  28. Evans, D., Rondoni, L.: Comments on the entropy of nonequilibrium steady states. J. Stat. Phys. 109, 895 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Bonetto, F., Kupiainen, A., Lebowitz, J.L.: Absolute continuity of projected SRB measures of coupled Arnold cat map lattices. Ergod. Theor. Dynam. Syst. 25, 59 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Colangeli, M., De Gregorio, P.M., Klages, R., Rondoni, L.: Steady state fluctuation relation with discontinuous “time reversibility” and invariant measures. J. Stat. Mech. (2011) P04021

    Google Scholar 

  31. Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics. Cambridge University Press, London (1985)

    Google Scholar 

  32. Roberts, J.A.G., Quispel, G.R.W.: Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems. Phys. Rep. 216, 63 (1992)

    Google Scholar 

  33. Harris, R.J., Schütz, G.M.: Fluctuation theorems for stochastic dynamics. J. Stat. Mech. (2007) P07020

    Google Scholar 

  34. Morriss, G.P., Rondoni, L.: Equivalence of “nonequilibrium” ensembles for simple maps. Physica A 233, 767 (1996)

    Article  ADS  Google Scholar 

  35. Tolman, R.C.: The Principles of Statistical Mechanics. Oxford University Press, London (1938)

    Google Scholar 

  36. Falcioni, M., Palatella, L., Pigolotti, S., Rondoni, L., Vulpiani, A.: Initial growth of Boltzmann entropy and chaos in a large assembly of weakly interacting systems. Physica A 385, 170–184 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  37. Onsager, L.: Reciprocal relations in irreversible processes I. Phys. Rev. 37, 405 (1931); Reciprocal relations in irreversible processes II. Phys. Rev. 38, 2265 (1931)

    Google Scholar 

  38. Ellis, R.S.: An overview of the theory of large deviations and applications to statistical mechanics. Scand. Actuarial J. 1, 97 (1995)

    Article  Google Scholar 

  39. Rondoni, L., Morriss, G.P.: Applications of periodic orbit theory to N-particle systems. J. Stat. Phys. 86, 991 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Evans, D.J., Searles, S.J., Rondoni, L.: On the application of the Gallavotti-Cohen fluctuation relation to thermostatted steady states near equilibrium. Phys. Rev. E 71, 056120 (2005)

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Rondoni, L., Cohen, E.G.D.: Orbital measures in non-equilibrium statistical mechanics: the Onsager relations. Nonlinearity 11, 1395 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. Rondoni, L.: Deterministic thermostats and fluctuation relations. In: Dynamics of Dissipation. Lecture Notes in Physics, vol. 597, p. 35. Springer, Berlin (2002)

    Google Scholar 

  44. Rondoni, L., Cohen, E.G.D.: Gibbs entropy and irreversible thermodynamics. Nonlinearity 13, 1905 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. Rondoni, L., Tel, T., Vollmer, J.: Fluctuation theorems for entropy production in open systems. Phys. Rev. E 61, R4679–R4682 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  46. Cohen, E.G.D., Rondoni, L.: Particles, maps and irreversible thermodynamics. Physica A 306, 117 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. Gallavotti, G.: Breakdown and regeneration of time reversal symmetry in nonequilibrium statistical mechanics. Physica D 112, 250–257 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. Rondoni, L., Cohen, E.G.D.: On some derivations of irreversible thermodynamics from dynamical systems theory. Physica D 168–169, 341 (2002)

    Article  MathSciNet  Google Scholar 

  49. Green, M.S.: Markoff random processes and the statistical mechanics of time-dependent phenomena. J. Chem. Phys. 20, 1281 (1952)

    Article  MathSciNet  ADS  Google Scholar 

  50. Green, M.S.: Markoff random processes and the statistical mechanics of time-dependent phenomena: II. Irreversible processes in fluids. J. Chem. Phys. 22, 398 (1954)

    Google Scholar 

  51. Kubo, R.: Statistical-mechanical theory of irreversible processes: I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn. 12, 570 (1957)

    Google Scholar 

  52. Cencini, M., Cecconi, F., Vulpiani, A.: Chaos: From Simple Models to Complex Systems. World Scientific, Singapore (2009)

    Google Scholar 

  53. Ruelle, D.: General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium. Phys. Lett. A 245, 220 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  54. Cessac, B., Sepulchre, J.-A.: Linear response, susceptibility and resonances in chaotic toy models. Physica D 225, 13 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. Colangeli, M., Rondoni, L., Vulpiani, A.: Fluctuation-dissipation relation for chaotic non-Hamiltonian systems. J. Stat. Mech. (2012) L04002

    Google Scholar 

  56. Colangeli, M., Rondoni, L.: Equilibrium, fluctuation relations and transport for irreversibile deterministic dynamics. Physica D 241, 681 (2011)

    Article  ADS  Google Scholar 

  57. Gabrielli, D., Jona-Lasinio, G., Landim, C.: Onsager reciprocity relations without microscopic reversibility. Phys. Rev. Lett. 77, 1202–1205 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. Gabrielli, D., Jona-Lasinio, G., Landim, C.: Onsager Symmetry from Microscopic TP Invariance. J. Stat. Phys. 96, 639 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  59. Cessac, B.: Does the complex susceptibility of the Hénon map have a pole in the upper-half plane? A numerical investigation. Nonlinearity 20, 2883 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Matteo Colangeli & Lamberto Rondoni

Authors
  1. Matteo Colangeli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lamberto Rondoni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matteo Colangeli .

Editor information

Editors and Affiliations

  1. , Institute for Mathematical Research, University Putra Malaysia, Hak Cipta Terpelihara UPM 2007, Serdang, 43400, Malaysia

    Santo Banerjee

  2. , Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, Torino, 10129, Italy

    Lamberto Rondoni

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Colangeli, M., Rondoni, L. (2013). Fluctuation Relations and Nonequilibrium Response for Chaotic Dissipative Dynamics. In: Banerjee, S., Rondoni, L. (eds) Applications of Chaos and Nonlinear Dynamics in Science and Engineering - Vol. 3. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34017-8_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-34017-8_1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-34016-1

  • Online ISBN: 978-3-642-34017-8

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation