Skip to main content
SpringerLink
Log in

Fluctuation-Dissipation and Fluctuation Relations: From Equilibrium to Nonequilibrium and Back

  • Chapter
  • First Online:
Large Deviations in Physics

Part of the book series: Lecture Notes in Physics ((LNP,volume 885))

Abstract

The fluctuation-dissipation relation is a most remarkable classical result of statistical physics, which allows us to understand nonequilibrium properties of thermodynamic systems from observations of equilibrium phenomena. The modern transient fluctuation relations do the opposite: they allow us to understand equilibrium properties from nonequilibrium experiments. Under proper conditions, the transient relations turn into statements about nonequilibrium steady states, even far from equilibrium. The steady state relations, in turn, generalize the fluctuation-dissipation relations, as they reduce to them when approaching equilibrium. We will review the progress made since Einstein’s work on the Brownian motion, which gradually evolved from the theory of equilibrium macroscopic systems towards an ever deeper understanding of nonequilibrium phenomena, and is now shedding light on the physics of mesoscopic systems. In this evolution, the focus also shifted from small to large fluctuations, which nowadays constitute a unifying factor for different theories. We will conclude illustrating the recently introduced t-mixing property and discussing a fully general and simple response formula, which applies to deterministic dynamics and naturally extends the Green-Kubo theory.

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 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Notes

  1. 1.

    We recall a few of them, with the intent to show how prolific this research line has been for over a century, and how it has gradually shifted from equilibrium to nonequilibrium problems. Recent reviews on the subject include [3, 4].

  2. 2.

    An exception is provided by gravitational waves detectors, whose resolution is so high that their thermal fluctuations are revealed [46].

  3. 3.

    So that f R ∘ S t x = f R (S t x), with S t x the position of a Brownian particle initially in x, after a time t.

  4. 4.

    There are various points of view on the significance of the random term f R . Some view Nature as intrinsically deterministic and randomness merely as the result of incomplete information. Others take the laws of Nature as ultimately stochastic, and determinism just as a convenient idealization in the description of certain phenomena. We adopt a pragmatic standpoint: mathematical models, whether stochastic or deterministic, are meant to describe non-exhaustively complementary features of natural phenomena. Depending on the feature of interest, one kind of model may be more convenient or may bear deeper insight than the other.

  5. 5.

    At that time, the numerical value of N A was unknown.

  6. 6.

    The terminology originates in the context of electrical circuits, where the power spectrum represents the electrical power dissipated at various frequencies.

  7. 7.

    Except a negligible set of phase space points.

  8. 8.

    In case of Hamiltonian dynamics, and more generally in the case of the so-called adiabatically incompressible systems, probabilities flow like incompressible fluids.

  9. 9.

    Global solution means that particles do no cease to exist after a while; Uniqueness implies that the same particles do not exist at once along distinct trajectories. If these properties are violated, one typically concludes that the equations of motion do not suit the problem at hand.

  10. 10.

    Concerning certain smooth, uniformly hyperbolic dynamical systems.

  11. 11.

    The set E −δ Γ is defined by \(\{\varGamma \in \mathcal{M}: \varGamma +\delta \varGamma \in E\}\).

  12. 12.

    Onsager and Machlup observe that (cf. Footnote 2 of Ref. [14]): “This statement is, of course, charged with meaning, and requires elaborate precautions about ergodicity, etc. It may be said to hold for systems which ‘forget’ their initial states in a ‘reasonably’ short time. It is, however, precisely the choice of time scale that matters. In a sufficiently long time, all physical systems ‘forget’.”

  13. 13.

    An argument similar to that outlined in Sect. 4.4.1 applies if the dynamics is mixing with respect to other invariant densities.

  14. 14.

    Something similar happens when the equilibrium thermodynamic entropy of a physical object is expressed by the equilibrium average of the logarithm of the equilibrium density, which is the Gibbs entropy.

  15. 15.

    The analogy concerning the identification of the thermodynamic entropy with the Gibbs entropy continues: if the steady state is singular, the Gibbs entropy does not represent any thermodynamic entropy at all.

References

  1. A. Einstein, The motion of elements suspended in static liquids as claimed in the molecular kinetic theory of heat. Ann. Phys. 17, 549 (1905)

    Article  MATH  Google Scholar 

  2. A. Einstein, Theory of opalescence of homogeneous liquids and mixtures of liquids in the vicinity of the critical state. Ann. Phys. 33, 1275 (1910)

    Article  MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

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

    Google Scholar 

  5. H. Nyquist, Thermal agitation of electric charge in conductors. Phys. Rev. 32, 110 (1928)

    Article  ADS  Google Scholar 

  6. L. Onsager, Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405 (1931)

    Article  ADS  Google Scholar 

  7. L. Onsager, Reciprocal relations in irreversible processes. II. Phys. Rev. 38, 2265 (1931)

    Article  ADS  MATH  Google Scholar 

  8. H.B. Callen, T.A. Welton, Irreversibility and generalized noise. Phys. Rev. 83, 34 (1951)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. H.B. Callen, R.F. Greene, On a theorem of irreversible thermodynamics. Phys. Rev. 86, 702 (1952)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. M.S. Green, Brownian motion in a gas of noninteracting molecules. J. Chem. Phys. 19, 1036 (1951)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

  14. L. Onsager, S. Machlup, Fluctuations and irreversible processes. Phys. Rev. 91, 1505 (1953)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. S. Machlup, L. Onsager, Fluctuations and irreversible process. II. Systems with kinetic energy. Phys. Rev. 91, 1512 (1953)

    MATH  MathSciNet  Google Scholar 

  16. B.J. Alder, T.E. Wainwright, Velocity autocorrelations for hard spheres. Phys. Rev. Lett. 18, 988 (1967)

    Article  ADS  Google Scholar 

  17. L.P. Kadanoff, J. Swift, Transport coefficients near the liquid-gas critical point. Phys. Rev. 166, 89 (1968)

    Article  ADS  Google Scholar 

  18. I. Procaccia, D. Ronis, I. Oppenheim, Light scattering from nonequilibrium stationary states: the implication of broken time-reversal symmetry. Phys. Rev. Lett. 42, 287 (1979)

    Article  ADS  Google Scholar 

  19. T.R. Kirkpatrick, E.G.D. Cohen, J.R. Dorfman, Kinetic theory of light scattering from a fluid not in equilibrium. Phys. Rev. Lett. 42, 862 (1979)

    Article  ADS  Google Scholar 

  20. H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, Heidelberg, 1991)

    Book  MATH  Google Scholar 

  21. P. Hänggi, H. Thomas, Linear response and fluctuation theorems for nonstationary stochastic-processes. Z. Phys. B 22, 295 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  22. P. Hänggi, Stochastic-processes.2. Response theory and fluctuation theorems. Helv. Phys. Acta 51, 202 (1978)

    Google Scholar 

  23. W.M. Visscher, Transport processes in solids and linear-response theory. Phys. Rev. A 10, 2461 (1974)

    Article  ADS  Google Scholar 

  24. J.W. Dufty, M.J. Lindenfeld, Non-linear transport in the Boltzmann limit. J. Stat. Phys. 20, 259 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  25. E.G.D. Cohen, Kinetic-theory of non-equilibrium fluids. Physica A 118, 17 (1983)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. G.P. Morriss, D.J. Evans, Application of transient correlation functions to shear flow far from equilibrium. Phys. Rev. A 35, 792 (1987)

    Article  ADS  Google Scholar 

  27. M. Falcioni, S. Isola, A. Vulpiani, Correlation functions, relaxation properties in chaotic dynamics, statistical mechanics. Phys. Lett. A 144, 341 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  28. G. Boffetta, G. Lacorata, S. Musacchio, A. Vulpiani, Relaxation of finite perturbations: beyond the fluctuation-response relation. Chaos 13, 806 (2003)

    Article  ADS  Google Scholar 

  29. D. Ruelle, Differentiation of SRB states. Commun. Math. Phys. 187, 227 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  30. F. Corberi, E. Lippiello, M. Zannetti, Fluctuation dissipation relations far from equilibrium. J. Stat. Mech. 2007, P07002 (2007)

    Article  Google Scholar 

  31. M. Baiesi, C. Maes, An update on nonequilibrium linear response. New J. Phys. 15, 013004 (2013)

    Article  ADS  Google Scholar 

  32. T. Speck, U. Seifert, Restoring a fluctuation-dissipation theorem in a nonequilibrium steady state. Europhys. Lett. 74, 391 (2006)

    Article  ADS  Google Scholar 

  33. T. Speck, U. Seifert, Extended fluctuation-dissipation theorem for soft matter in stationary flow. Phys. Rev. E 79, 040102 (2009)

    Article  ADS  Google Scholar 

  34. R. Chetrite, K. Gawedzki, Fluctuation relations for diffusion processes. Commun. Math. Phys. 282, 469 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. D.J. Evans, E.G.D. Cohen, G.P. Morriss, Probability of second law violations in shearing steady flows. Phys. Rev. Lett. 71, 2401 (1993)

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  37. D.J. Evans, D.J. Searles, Steady sates, invariant measures, response theory. Phys. Rev. E 52, 5839 (1995)

    Article  ADS  Google Scholar 

  38. G. Gallavotti, E.G.D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 94, 2694 (1995)

    Article  ADS  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  44. C. Jarzynski, Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690 (1997)

    Article  ADS  Google Scholar 

  45. C. Bustamante, J. Liphardt, F. Ritort, The nonequilibrium thermodynamics of small systems. Phys. Today 58, 43 (2005)

    Article  Google Scholar 

  46. M. Bonaldi et al., Nonequilibrium steady-state fluctuations in actively cooled resonators. Phys. Rev. Lett. 103, 010601 (2009)

    Article  ADS  Google Scholar 

  47. R. Brown, A brief account of microscopical observations made in the months of June, July, and August, 1827 on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Philos. Mag. 4, 161 (1828)

    Google Scholar 

  48. P. Langevin, Sur la theorie du mouvement brownien. C. R. Acad. Sci. (Paris) 146, 530 (1908). Translated in Am. J. Phys. 65, 1079 (1997)

    Google Scholar 

  49. J. Perrin, Les Atomes (Alcan, Paris, 1913)

    MATH  Google Scholar 

  50. R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics (Springer, Berlin, 1985)

    Book  Google Scholar 

  51. A.I. Khinchin, Mathematical Foundations of Statistical Mechanics (Dover, New York, 1949)

    MATH  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  55. F. Bonetto, A. Kupiainen, J.L. Lebowitz, Absolute continuity of projected SRB measures of coupled Arnold cat map lattices. Ergod. Theory Dyn. Syst. 25, 59 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  57. L. Bertini, A.D. Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim, Macroscopic fluctuation theory for stationary non-equilibrium states. J. Stat. Phys. 107, 635 (2002)

    Article  ADS  MATH  Google Scholar 

  58. A. Gamba, L. Rondoni, Current fluctuations in the nonequilibrium Lorentz gas. Physica A 340, 274 (2004); C. Giberti, L. Rondoni, C. Vernia, Asymmetric fluctuation-relaxation paths in FPU models. Physica A 365, 229 (2006); C. Paneni, D.J. Searles, L. Rondoni, Temporal asymmetry of fluctuations in nonequilibrium states. J. Chem. Phys. 124, 114109 (2006); C. Paneni, D.J. Searles, L. Rondoni, Temporal asymmetry of fluctuations in nonequilibrium steady states: links with correlation functions and nonlinear response. J. Chem. Phys. 128, 164515 (2008)

    Google Scholar 

  59. J.D. Weeks, D. Chandler, H.C. Andersen, Role of repulsive forces in determining the equilibrium structure of simple liquids. J. Chem. Phys. 54, 5237 (1971)

    Article  ADS  Google Scholar 

  60. G. Gallavotti, Reversible Anosov diffeomorphisms, large deviations. Math. Phys. Electron. J. 1, 1 (1995)

    MathSciNet  Google Scholar 

  61. G. Gallavotti, Fluctuation theorem revisited (2004), http://arXiv.org/cond-mat/0402676

  62. J. Kurchan, Fluctuation theorem for stochastic dynamics. J. Phys. A 31, 3719 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  64. C. Maes, The fluctuation theorem as a Gibbs property. J. Stat. Phys. 95, 367 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  65. Ya.G. Sinai, Lectures in Ergodic Theory. Lecture Notes in Mathematics (Princeton University Press, Princeton, New Jersey, USA, 1977)

    Google Scholar 

  66. D.J. Evans, E.G.D. Cohen, D.J. Searles, F. Bonetto, Note on the Kaplan-Yorke dimension and linear transport coefficients. J. Stat. Phys. 101, 17 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  68. D.J. Evans, G.P. Morriss, Statistical Mechanics of Nonequlibrium Liquids (Academic, New York, 1990)

    Google Scholar 

  69. B. Johnston, D.J. Evans, D.J. Searles, L. Rondoni, Time reversibility, correlation decay and the steady state fluctuation relation for dissipation. Entropy 15, 1503 (2013)

    Article  ADS  Google Scholar 

  70. D.J. Searles, D.J. Evans, Ensemble dependence of the transient fluctuation theorem. J. Chem. Phys. 113, 3503 (2000)

    Article  ADS  Google Scholar 

  71. E.G.D. Cohen, D. Mauzerall, A note on the Jarzynski equality. J. Stat. Mech. 2004, P07006 (2004)

    Article  Google Scholar 

Download references

Acknowledgements

This work has received funding from the European Research Council under the EU Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement n. 202680. The EU is not liable for any use made on the information contained herein.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, I-10129, Italy

    Paolo Adamo, Roman Belousov & Lamberto Rondoni

Authors
  1. Paolo Adamo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roman Belousov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lamberto Rondoni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lamberto Rondoni .

Editor information

Editors and Affiliations

  1. Dipartimento di Fisica, Università di Roma Sapienza, Roma, Italy

    Angelo Vulpiani

  2. Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, Roma, Italy

    Fabio Cecconi

  3. Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, Roma, Italy

    Massimo Cencini

  4. Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, Roma, Italy

    Andrea Puglisi

  5. Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, Roma, Italy

    Davide Vergni

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Adamo, P., Belousov, R., Rondoni, L. (2014). Fluctuation-Dissipation and Fluctuation Relations: From Equilibrium to Nonequilibrium and Back. In: Vulpiani, A., Cecconi, F., Cencini, M., Puglisi, A., Vergni, D. (eds) Large Deviations in Physics. Lecture Notes in Physics, vol 885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54251-0_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-54251-0_4

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-54250-3

  • Online ISBN: 978-3-642-54251-0

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

Publish with us

Policies and ethics

Search

Navigation