Journal of Statistical Physics

, Volume 131, Issue 3, pp 543–558 | Cite as

Response Theory for Equilibrium and Non-Equilibrium Statistical Mechanics: Causality and Generalized Kramers-Kronig Relations



We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.


Axiom A dynamical systems Non-equilibrium steady states Kubo response theory Ruelle response theory SRB measure Chaotic hypothesis Kramers-Kronig relations Harmonic generation Climate 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kubo, R.: Statistical-mechanical theory of irreversible processes. I. J. Phys. Soc. Jpn. 12, 570–586 (1957) CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Zubarev, D.N.: Nonequilibrium Statistical Thermodynamics. Consultant Bureau, New York (1974) Google Scholar
  3. 3.
    Ruelle, D.: General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium. Phys. Lett. A 245, 220–224 (1998) MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Ruelle, D.: Nonequilibrium statistical mechanics near equilibrium: computing higher order terms. Nonlinearity 11, 5–18 (1998) MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–655 (1985) CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Ruelle, D.: Chaotic Evolution and Strange Attractors. Cambridge University Press, Cambridge (1989) MATHGoogle Scholar
  7. 7.
    Ruelle, D.: Differentiation of SRB states. Commun. Math. Phys. 187, 227–241 (1997) MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Ruelle, D.: Differentiation of SRB states: correction and complements. Commun. Math. Phys. 234, 185–190 (2003) MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Dolgopyat, D.: On differentiability of SRB states for partially hyperbolic systems. Invent. Math. 155, 389–449 (2004) MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Butterley, O., Liverani, C.: Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1, 301–322 (2007) MATHMathSciNetGoogle Scholar
  11. 11.
    Jiang, M., de la Llave, R.: Linear response function for coupled hyperbolic attractors. Commun. Math. Phys. 261, 379–404 (2006) MATHCrossRefADSGoogle Scholar
  12. 12.
    Ruelle, D.: Application of hyperbolic dynamics to physics: some problems and conjectures. Bull. Am. Math. Soc. 41, 275–278 (2004) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ruelle, D.: Differentiating the a.c.i.m. of an interval map with respect to f. Commun. Math. Phys. 258, 445–453 (2005) MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Baladi, V.: On the susceptibility function of piecewise expanding interval maps. Commun. Math. Phys. 275, 839–859 (2007) CrossRefADSMathSciNetMATHGoogle Scholar
  15. 15.
    Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970 (1995) MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Gallavotti, G.: Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem. J. Stat. Phys. 84, 899–926 (1996) MATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Nussenzveig, H.M.: Causality and Dispersion Relations. Academic Press, New York (1972) Google Scholar
  18. 18.
    Peiponen, K.-E., Vartiainen, E.M., Asakura, T.: Dispersion, Complex Analysis and Optical Spectroscopy. Springer, Heidelberg (1999) Google Scholar
  19. 19.
    Weber, J.: Fluctuation dissipation theorem. Phys. Rev. 101, 1620–1626 (1956) MATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Kubo, R.: The fluctuation dissipation theorem. Rep. Prog. Phys. 29, 255–284 (1966) CrossRefADSGoogle Scholar
  21. 21.
    Lorenz, E.N.: Forced and free variations of weather and climate. J. Atmos. Sci. 36, 1367–1376 (1979) CrossRefADSGoogle Scholar
  22. 22.
    Lucarini, V., Bassani, F., Saarinen, J.J., Peiponen, K.-E.: Dispersion theory and sum rules in linear and nonlinear optics. Rivista Nuovo Cimento 26, 1–120 (2003) ADSGoogle Scholar
  23. 23.
    Lucarini, V., Saarinen, J.J., Peiponen, K.-E., Vartiainen, E.: Kramers-Kronig Relations in Optical Materials Research. Springer, Heidelberg (2005) Google Scholar
  24. 24.
    Peiponen, K.-E.: Sum rules for the nonlinear susceptibilities in the case of sum frequency generation. Phys. Rev. B 35, 4116–4117 (1987) CrossRefADSGoogle Scholar
  25. 25.
    Peiponen, K.-E.: Nonlinear susceptibilities as a function of several complex angular-frequency variables. Phys. Rev. B 37, 6463–6467 (1988) CrossRefADSGoogle Scholar
  26. 26.
    Bassani, F., Lucarini, V.: General properties of optical harmonic generation from a simple oscillator model. Il Nuovo Cimento D 20, 1117–1125 (1998) ADSCrossRefGoogle Scholar
  27. 27.
    Young, L.S.: What are SRB measures, and which dynamical systems have them? J. Stat. Phys. 108(5), 733–754 (2002) MATHCrossRefGoogle Scholar
  28. 28.
    Cessac, B., Sepulchre, J.-A.: Linear response, susceptibility and resonances in chaotic toy models. Physica D 225, 13–28 (2007) MATHCrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Bassani, F., Scandolo, S.: Dispersion relations and sum rules in nonlinear optics. Phys. Rev. B 44, 8446–8453 (1991) CrossRefADSGoogle Scholar
  30. 30.
    Peiponen, K.-E., Saarinen, J.J., Svirko, Y.: Derivation of general dispersion relations and sum rules for meromorphic nonlinear optical spectroscopy. Phys. Rev. A 69, 043818 (2004) CrossRefADSGoogle Scholar
  31. 31.
    Reick, C.H.: Linear response of the Lorenz system. Phys. Rev. E 66, 036103 (2002) CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Bassani, F., Lucarini, V.: Asymptotic behaviour and general properties of harmonic generation susceptibilities. Eur. Phys. J. B 17, 567–573 (2000) CrossRefADSGoogle Scholar
  33. 33.
    Frye, G., Warnock, R.L.: Analysis of partial-wave dispersion relations. Phys. Rev. 130, 478–494 (1963) CrossRefADSMathSciNetGoogle Scholar
  34. 34.
    Lucarini, V., Peiponen, K.-E.: Verification of generalized Kramers-Kronig relations and sum rules on experimental data of third harmonic generation susceptibility on polymer. J. Phys. Chem. 119, 620–627 (2003) CrossRefGoogle Scholar
  35. 35.
    Bassani, F., Altarelli, M.: Interaction of radiation with condensed matter. In: Koch, E.E. (ed.) Handbook on Synchrotron Radiation. North–Holland, Amsterdam (1983) Google Scholar
  36. 36.
    Aspnes, D.E.: The accurate determination of optical properties by ellipsometry. In: Palik, E.D. (ed.) Handbook of Optical Constants of Solids, pp. 89–112. Academic Press, New York (1985) Google Scholar
  37. 37.
    Peiponen, K.-E., Vartiainen, E.M.: Kramers-Kronig relations in optical data inversion. Phys. Rev. B 44, 8301–8303 (1991) CrossRefADSGoogle Scholar
  38. 38.
    King, F.W.: Efficient numerical approach to the evaluation of Kramers-Kronig transforms. J. Opt. Soc. Am. B 19, 2427–2436 (2002) CrossRefADSGoogle Scholar
  39. 39.
    Palmer, K.F., Williams, M.Z., Budde, B.A.: Multiply subtractive Kramers-Kronig analysis of optical data. Appl. Opt. 37, 2660–2673 (1998) ADSCrossRefGoogle Scholar
  40. 40.
    Lucarini, V., Saarinen, J.J., Peiponen, K.-E.: Multiply subtractive generalized Kramers-Kronig relations: application on third-harmonic generation susceptibility on polysilane. J. Chem. Phys. 119, 11095–11098 (2003) CrossRefADSGoogle Scholar
  41. 41.
    Leith, C.E.: Climate response and fluctuation dissipation. J. Atmos. Sci. 32, 2022–2026 (1975) CrossRefADSGoogle Scholar
  42. 42.
    Lindenberg, K., West, B.J.: Fluctuation and dissipation in a barotropic flow field. J. Atmos. Sci. 41, 3021–3031 (1984) CrossRefADSGoogle Scholar
  43. 43.
    Corti, S., Molteni, F., Palmer, T.N.: Signature of recent climate change in frequencies of natural atmospheric circulation regimes, Nature 398, 799–802 (1999) Google Scholar
  44. 44.
    Lucarini, V., Speranza, A., Vitolo, R.: Parametric smoothness and self-scaling of the statistical properties of a minimal climate model: what beyond the mean field theories? Physica D 234, 105–123 (2007) MATHCrossRefADSMathSciNetGoogle Scholar
  45. 45.
    Lucarini, V.: Towards a definition of climate science. Int. J. Environ. Pollut. 18, 409–414 (2002) CrossRefGoogle Scholar
  46. 46.
    Speranza, A., Lucarini, V.: Environmental science: physical principles and applications. In: Bassani, F., Liedl, J., Wyder, P. (eds.) Encyclopedia of Condensed Matter Physics. Elsevier, Amsterdam (2005) Google Scholar
  47. 47.
    Intergovernmental Panel on Climate Change, Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge (2007) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of BolognaBolognaItaly

Personalised recommendations