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Fluctuations

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Nonequilibrium and Irreversibility

Part of the book series: Theoretical and Mathematical Physics ((TMP))

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Abstract

Intuition about the SRB distributions, hence about the statistics of chaotic evolutions, requires an understanding of their nature. The physical meaning discussed in Sect. 3.8 is not sufficient because the key notion of SRB potentials is still missing.

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Notes

  1. 1.

    This means fixing for each \(q\) two strings \(\mathbf{a}(\mathbf{q})=\{a_0=q,a_1,\ldots , a_{m-1},a_m=\overline{q}\}\) and \(\mathbf {b}(\mathbf{q})=\{ b_0=\overline{q},b_1,\ldots ,b_{m-1},b_m=q\}\) such that \(\prod _{i=0}^{m-1} M_{a_i,a_{i+1}}=\prod _{i=0}^{m-1} M_{b_i,b_{i+1}}=1\).

  2. 2.

    The continuation has to be done choosing arbitrarily but once and for all a compatible sequence starting with \(\overline{q}\): the simplest is to repeat indefinitely a string of length \(m\) beginning and ending with \(\overline{q}\) to the right and to the left.

  3. 3.

    Because of transitivity of the compatibility matrix.

  4. 4.

    This means that given \(\varepsilon , n>0\) there is \(k\) large enough so that the Gibbs’ distribution \(\mu _{SRB}\) with potential \(\varPhi \) and the Markov process \(\mu _k\) with potential \(\varPhi ^{[\le k]}\), truncation of \(\varPhi \) at range \(k\), will be such that \(\sum _{\mathbf{q}=(q_0,\ldots ,q_n)}\) \(|\mu _{SRB}(E(\mathbf{q}))-\mu _k(E(\mathbf{q}))|<\varepsilon \) and the Kolmogorov-Sinai’s entropy (i.e. \(s( \mu )\,{\mathop {=}\limits ^{def}}\,\) \(\mathop {\lim }\limits _{n\rightarrow \infty } -\frac{1}{n}\sum _{\mathbf{q}=(q_0,\ldots q_{n})} \mu (E(\mathbf{q}))\log \mu (E(\mathbf{q}))\)) of \(\mu _{SRB}\) and \(\mu _k\) are close within \(\varepsilon \).

  5. 5.

    Hence \(x_{ii}=Ix_{ii}\) if \(E_i\cap I E_i\) is a non empty rectangle. Such a fixed point exists because the rectangles are homeomorphic to a ball. The fixed point theorem can be avoided by associating to \(E_i\cap I E_i\) two centers \(x^1_{ii}\) and \(x^2_{ii}=I x^1_{ii}\): we do not do so to simplify the formulae.

  6. 6.

    The transformation \(\varPhi \) of a perturbed Anosov map into the unperturbed one is in general not a smooth change of coordinates but just Hölder continuous. So that the image of \(I\) in the new coordinates might be hard to use.

  7. 7.

    Because the version of Eq. (2.8.6) for evolutions in continuous time is \(\lim _{\tau \rightarrow \infty }\) \(\frac{1}{\tau }\log \det (\partial S_\tau (x))=\sum _{i=0}^{2D-1}\lambda _i\).

  8. 8.

    Since the model is defined in continuous time the matrix \((\partial S_t)^*\partial S_t\) will always have a trivial eigenvalue with average \(0\).

  9. 9.

    This does not preclude the possibility that the attractor has a fractal dimension (smoothness of the closure of an attractor has nothing to do with its fractal dimensionality, see [13–15]). The motion on this lower dimensional surface (whose dimension is smaller than that of phase space by an amount equal to the number of paired negative exponents) will still have an attractor (see p. 39) with dimension lower than the dimension of the surface itself, as suggested by the Kaplan–Yorke formula, [13].

  10. 10.

    As discussed below, it requires a proof and therefore it should not be confused with several identities to which, for reasons that I fail to understand, the same name has been given, [22] and Appendix L.

  11. 11.

    Actual computation of \(\zeta (p)\) is a task possible in the \(N=1\) case considered in [24] but essentially beyond our capabilities in slightly more general systems in the non linear regime.

  12. 12.

    That is, not infinitesimally close to \(0\) as in the classical theory of nonequilibrium thermodynamics, [30].

  13. 13.

    Colorfully: A waterfall will go up, as likely as we see it going down, in a world in which for some reason, or by the deed of a Daemon, the entropy production rate has changed sign during a long enough time [29, p. 476].

  14. 14.

    Suggested by P. Garrido from the data in the simulation in [12].

  15. 15.

    It says that essentially \(\langle e^{I_E}\rangle _{\mu _E}\equiv 1\) or more precisely it is not too far from \(1\) as \(\tau \rightarrow \infty \) so that Eq. (4.8.4) holds.

  16. 16.

    For a time long enough for being able to consider it as a moving container: for instance while its motion can be considered described by a linear transformation and at the same time long enough to be able to make meaningful observations.

  17. 17.

    Writing the paper [49] I was unaware of these works: I thank Dr. M. Campisi for recently pointing this reference out.

  18. 18.

    Defined, for instance, as the limit of the distributions obtained by evolving in time the Eq. (4.11.6).

References

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

    Article  ADS  MathSciNet  Google Scholar 

  2. Ruelle, D.: Measures describing a turbulent flow. Ann. N. Y. Acad. Sci. 357, 1–9 (1980)

    Article  ADS  Google Scholar 

  3. Ruelle, D.: A measure associated with axiom a attractors. Am. J. Math. 98, 619–654 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  4. Gallavotti, G., Bonetto, F., Gentile, G.: Aspects of the ergodic, qualitative and statistical theory of motion. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  5. Kernigham, B.W., Ritchie, D.M.: The C Programming Language. Prentice Hall Software Series. Prentice Hall, Engelwood Cliffs (1988)

    Google Scholar 

  6. Ornstein, D.: Ergodic theory, randomness and dynamical Systems. Yale Mathematical Monographs, vol. 5. Yale University Press, New Haven (1974)

    Google Scholar 

  7. Gallavotti, G.: Ising model and Bernoulli shifts. Commun. Math. Phys. 32, 183–190 (1973)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Ledrappier, F.: Mesure d’equilibre sur un reseau. Commun. Math. Phys. 33, 119–128 (1973)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Bonetto, F., Gallavotti, G.: Reversibility, coarse graining and the chaoticity principle. Commun. Math. Phys. 189, 263–276 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Dettman, C., Morriss, G.: Proof of conjugate pairing for an isokinetic thermostat. Phys. Rev. E 53, 5545–5549 (1996)

    Article  ADS  Google Scholar 

  12. Bonetto, F., Gallavotti, G., Garrido, P.: Chaotic principle: an experimental test. Physica D 105, 226–252 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  15. Gallavotti, G.: Topics in chaotic dynamics. In: Garrido, P., Marro, J. (ed.) Lecture Notes in Physics, vol. 448, pp. 271–311. Springer, Berlin (1995)

    Google Scholar 

  16. Sinai, Y.G.: Gibbs measures in ergodic theory. Russ. Math. Surv. 27, 21–69 (1972)

    Google Scholar 

  17. Sinai, Y.G.: Lectures in ergodic theory. Lecture notes in Mathematics. Princeton University Press, Princeton (1977)

    Google Scholar 

  18. Sinai, Y.G.: Topics in ergodic theory. Princeton Mathematical Series, vol. 44. Princeton University Press, Princeton (1994)

    Google Scholar 

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

    Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  22. Gallavotti, G., Cohen, E.G.D.: Note on nonequilibrium stationary states and entropy. Phys. Rev. E 69, 035104 (+4) (2004)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  25. Bowen, R., Ruelle, D.: The ergodic theory of axiom a flows. Inventiones Mathematicae 29, 181–205 (1975)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. Gentile, G.: A large deviation theorem for anosov flows. Forum Mathematicum 10, 89–118 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Gallavotti, G.: Fluctuation patterns and conditional reversibility in nonequilibrium systems. Annales de l’ Institut H. Poincaré 70, 429–443 (1999) (chao-dyn/9703007)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  30. de Groot, S., Mazur, P.: Nonequilibrium Thermodynamics. Dover, Mineola (1984)

    Google Scholar 

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

    Google Scholar 

  32. Gomez-Marin, A., Parondo, J.M.R., Van den Broeck, C.: The footprints of irreversibility. European. Phys. Lett. 82, 5002(+4) (2008)

    Google Scholar 

  33. Hurtado, P., Péres-Espigares, C., Pozo, J., Garrido, P.: Symmetries in fluctuations far from equilibrium. Proc. Nat. Acad. Sci. 108, 7704G7709 (2011)

    Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. Bonetto, F.: Entropy theorem. Private communication, see (9.10.4) in [Ga00] (1997)

    Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  38. Ruelle, D.: Differentiation of srb states. Commun. Math. Phys. 187, 227–241 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. Gallavotti, G.: A local fluctuation theorem. Physica A 263, 39–50 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  40. Pesin, Y.B., Sinai, Y.G.: Space-time chaos in chains of weakly inteacting hyperbolic mappimgs. Adv. Sov. Math. 3, 165–198 (1991)

    MathSciNet  Google Scholar 

  41. Bricmont, J., Kupiainen, A.: High temperature expansions and dynamical systems. Commun. Math. Phys. 178, 703–732 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  42. Jiang, M., Pesin, Y.B.: Equilibrium measures for coupled map lattices: existence, uniqueness and finite-dimensional approximations. Commun. Math. Phys. 193, 675–711 (1998)

    Google Scholar 

  43. Gallavotti, G.: Equivalence of dynamical ensembles and Navier Stokes equations. Phys. Lett. A 223, 91–95 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  44. Bonetto, F., Gallavotti, G., Gentile, G.: A fluctuation theorem in a random environment. Ergodic Theory Dyn. Syst. 28, 21–47 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  45. Olla, S.: Large deviations for gibbs random fields. Probab. Theory Relat. Fields 77, 343–357 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  46. Gallavotti, G., Lebowitz, J.L., Mastropietro, V.: Large deviations in rarefied quantum gases. J. Stat. Phys. 108, 831–861 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  47. Gallavotti, G., Perroni, F.: An experimental test of the local fluctuation theorem in chains of weakly interacting anosov systems (unpublished, draft). http://ipparcoroma1.infn.it (1999)

  48. Mauri, F., Car, R., Tosatti, E.: Canonical statistical averages of coupled quantum-classical systems. Europhys. Lett. 24, 431–436 (1993)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  50. Alonso, J.L., Castro, A., Clemente-Gallardo, J., Cuchi, J.C., Echenique, P., Falceto, F.: Statistics and nosé formalism for Ehrenfest dynamics. J. Phys. A 44, 395004 (2011)

    Article  MathSciNet  Google Scholar 

  51. Gallavotti, G.: Entropy, thermostats and chaotic hypothesis. Chaos 16, 043114 (+6) (2006)

    Google Scholar 

  52. Gallavotti, G.: The Elements of Mechanics, 2nd edn. http://ipparco.roma1.infn.it, Roma (2008) (I edition was Springer 1984)

  53. Strocchi, F.: Complex coordinates and quantum mechanics. Rev. Mod. Phys. 38, 36–40 (1966)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  54. Campisi, M.: Quantum Fluctuation Relations for Ensembles of Wave Functions. arxiv:1306.5557, pp. 1–12 (2013)

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  1. Department of Physics, University of Rome “La Sapienza”, Rome, Italy

    Giovanni Gallavotti

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Gallavotti, G. (2014). Fluctuations. In: Nonequilibrium and Irreversibility. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-06758-2_4

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