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Ergodicity: How Can It Be Broken?

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Large Deviations in Physics

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

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Abstract

The introduction of the ergodic hypothesis can be traced back to the contributions by Boltzmann to the foundations of Statistical Mechanics. The formulation of this hypothesis was at the origin of a long standing debate between supporters and opponents of the Boltzmann mechanistic formulation of thermodynamics. The great intuition of the Austrian physicist nevertheless inspired the following contributions that aimed at establishing rigorous mathematical basis for ergodicity. The first part of this chapter will be devoted to reconstructing the evolution of the concept of ergodicity, going through the basic contributions by Birkhoff, Khinchin, Kolmogorov, Sinai etc. The second part will be focused on more recent case studies, associated with the phenomenon known as “ergodicity breaking” and its relations with physical systems. In particular, we describe how it can be related to the presence of exceedingly large relaxation time scales that emerge in nonlinear systems (e.g., the Fermi-Pasta-Ulam model and the Discrete Nonlinear Schrödinger Equation) and to the coexistence of more than one equilibrium phase in disordered systems (spins and structural glasses).

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Notes

  1. 1.

    It is worth pointing out that the second law of thermodynamics allows to reach the conclusion that in the absence of external forcing a thermodynamic system evolves spontaneously to its equilibrium state, that corresponds to an extremal point (maximum) of the state function entropy.

  2. 2.

    This is the basic consideration that inspired even the modern formulation of the so-called fluctuation theorem [3].

  3. 3.

    Variants that we shall not considered here include the discrete case \(t \in \mathbb{Z}\) and the non-invertible cases \(t \in {\mathbb{R}}^{+}\) and \(t \in \mathbb{N}\).

  4. 4.

    Where the gradient is smaller, nearby surfaces of constant energy are, so to speak, more separated; correspondingly, the measure associated to the same area of Γ E is larger. The measure μ L keeps the Boltzmann original attitude of looking at the 2n-dimensional volume in a thin layer between nearby surfaces of constant energy, just turning it into a precise mathematical language.

  5. 5.

    One might also consider the whole Γ in place of Γ E and the volume in Γ in place of μ L : an easier point of view, which however does not lead to any interesting result because the dynamics never mixes different constant energy surfaces.

  6. 6.

    Usually reported as: any measurable constant of motion (i.e. f(ϕ t(x)) = f(x) for any x) is trivial, i.e. almost everywhere constant in M; the restriction to ρ positive and normalized is irrelevant.

  7. 7.

    E4 is known as metrical indecomposability of M: no decomposition of M into A and M ∖ A can be invariant, unless it is trivial.

  8. 8.

    This happens of course also if we change t in − t, i.e. there is no preferred direction of time. A deep discussion could be opened here. Take a common gas and consider a low entropy initial condition, for example with all molecules initially confined in a corner of the container. For most initial states satisfying such condition, both in the forward and in the backward evolution of the system, with perfect symmetry, the gas evolves towards equilibrium and entropy increases. This is not always told to students, when discussing about the “arrow of time”.

  9. 9.

    One point bounces elastically inside a bounded region. The ergodic properties depend in a non trivial way on the shape of the boundary. The first example, due to Sinai [5, 6], represented an important breakthrough in ergodic theory.

  10. 10.

    This means, essentially, that the system behaves nicely, as if it were ergodic, if one does not look at the behavior of microscopic degrees of freedom, but restricts the observation to macroscopic observables, possibly to a selected subset of them. Such a point of view was proposed by Khinchin [7]. It looks physically deep, but never gave rise to a well-posed mathematical theory.

  11. 11.

    Mathematical Analyzer, Numerical Integrator (or perhaps Numerator, Integrator) and Computer.

  12. 12.

    Such a paper contains a mathematical theorem which suitably generalizes a well known result by Poincaré [9], followed by an heuristic application to the ergodic problem. This latter part is open to criticism (a crucial regularity that Fermi assumes to be generic, is not).

  13. 13.

    Quite interestingly, the paper remained for several years an internal report of the Los Alamos Laboratories [10], and only in 1965 it got published in Fermi’s Collected Papers [11].

  14. 14.

    MANIAC I is referred to as capable of 104 “operations” per second. This means a possible ratio 105, compared to common nowadays cpu’s; a cluster of 100 cpu’s—a quite common object—rises the ratio to 107. This means that what is easily done, nowadays, in 1 h, would have required about 400, 000 years of computation on MANIAC I.

  15. 15.

    Reference [13] was published only in 1972, but the results certainly circulated since 1961, see Ulam’s introduction to FPU in [11].

  16. 16.

    We shall not define integrability here. In the very essence, an integrable system is a system having many constants of motion—as many as the number of degrees of freedom—and all motions are quasi periodic. Ergodicity is far away, and statistical approaches like using a microcanonical measure are meaningless.

  17. 17.

    How many degrees of freedom do already represent the thermodynamic limit, is a delicate question discussed rather widely in [24]. Basically, the more \(\varepsilon\) is small, the larger N needs to be to reasonably approach the thermodynamic limit. Taking instead the limit \(\varepsilon \rightarrow 0\) at fixed large N (no matter how large) is not appropriate: the limits are in the reverse order. The exchange of the limits, although very spontaneous having finite computational resources, might lead to a wrong picture of the thermodynamic limit behavior.

  18. 18.

    With a minor change of the definition of \(\overline{E}_{k}\), namely

    $$\displaystyle{\overline{E}_{k}(t) = \frac{1} {t/3}\,\int _{(2/3)t}^{t}E_{ k}(Q_{k}(t^{\prime}),P_{k}(t^{\prime}))\,\mathrm{d}t^{\prime};}$$

    similar averages in a running window (of amplitude proportional to t) are in principle equivalent to the usual time averages from t = 0, but are less “lazy” to change, and better show the evolution of the time averages on the appropriate time-scale.

  19. 19.

    If instead N is kept fixed, even if large, then \(T(N,\varepsilon )\), for small \(\varepsilon\) below a certain \(\varepsilon _{N}\), abandons the power law to follow a stretched exponential \(T(N,\varepsilon ) \sim {e}^{1{/\varepsilon }^{\gamma } }\), \(\gamma = 1/8\). This is different from the thermodynamic limit. Performing the limits in the correct way is numerically painful but necessary.

  20. 20.

    Explicit calculations of gran-partition function of the DNLSE were carried out by transfer integral techniques.

  21. 21.

    In classical statistical mechanics it is an Abelian algebra with identity while it would be a non-Abelian algebra in the quantum case.

  22. 22.

    The experienced reader may ask why we have introduced the extra length R that may seems to be unnecessary. One would be tempted to assume that the volume average in the box of side Λ k of the observable should become k independent for large k. This simpler formulation does not have problems as far the expectation values of the observables are space independent. In the most general case where no (even approximate) translational invariance is present we have better to stick to the formulation we use in the main text.

  23. 23.

    The limit may not exist also in non-random systems: the simplest example is a two dimensional Ising model that is ferromagnetic in the y direction (with periodic boundary conditions) and antiferromagnetic in the x direction with fixed boundary conditions (positive at the left and negative on the right). It is possible to check that at low temperatures two different states are obtained in the infinite L limit, depending on the parity of L.

  24. 24.

    It is very easy to arrive to contradictions if one does not make a clear distinction between these two different concepts.

  25. 25.

    In the Ising case the configuration space contains 2N points.

  26. 26.

    The precise definition of small at large distance in a finite volume system can be phrased in many different ways. For example we can introduce a function g(x) which goes to zero when x goes to infinity and require that the connected correlation functions evaluated in a given phase are smaller than g(x). Of course the function g(x) should be carefully chosen in order to avoid to give trivial results; one should also prove the independence of the results from the choice of g in a given class of functions.

  27. 27.

    The probability distribution in a finite volume pure state is not the Gibbs one: the DLR relations, which tell us that the probability distribution is locally a Gibbs distribution, are violated, but the violation should go to zero in the large volume limit.

  28. 28.

    This refinement is not crucial in the infinite volume limit.

  29. 29.

    It may be possible that also for non-random systems we have a similar description where the average over the number of degrees of freedom plays the same role of the average over the disorder.

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Authors and Affiliations

  1. Dipartmento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, Padova, I-35121, Italy

    Giancarlo Benettin

  2. Dipartimento di Fisica, Università di Firenze, Via Sansone 1, Sesto Fiorentino, I-50019, Italy

    Roberto Livi

  3. Dipartimento di Fisica, Università di Roma “Sapienza”, Piazzale Aldo Moro 5, Rome, I-00185, Italy

    Giorgio Parisi

Authors
  1. Giancarlo Benettin
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  2. Roberto Livi
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  3. Giorgio Parisi
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Correspondence to Roberto Livi .

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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

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Benettin, G., Livi, R., Parisi, G. (2014). Ergodicity: How Can It Be Broken?. 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_2

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