The role of the number of degrees of freedom and chaos in macroscopic irreversibility

Cerino, Luca; Cecconi, Fabio; Cencini, Massimo; Vulpiani, Angelo

doi:10.1016/j.physa.2015.09.036

Cited by 18 publications

(24 citation statements)

References 31 publications

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“…Roughly, this means that "typically" the ensemble members enjoy the same behavior, i.e., those that do not constitute a set of vanishing Lebesgue measure. 1 While the notion of ergodicity based on invariant measures led to many insights of physical interest [4][5][6][7][8][9][10], the idea of typicality that it entails is problematic in the case of dissipative dynamics, such as those of nonequilibrium molecular dynamics (NEMD) models. Indeed the invariant measures of such models are singular with respect to the Lebesgue measure, as they attribute probability 1 to sets of vanishing volume.…”

Section: Introductionmentioning

confidence: 99%

On Typicality in Nonequilibrium Steady States

et al. 2016

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From the statistical mechanical viewpoint, relaxation of macroscopic systems and response theory rest on a notion of typicality, according to which the behavior of single macroscopic objects is given by appropriate ensembles: ensemble averages of observable quantities represent the measurements performed on single objects, because "almost all" objects share the same fate. In the case of non-dissipative dynamics and relaxation toward equilibrium states, "almost all" is referred to invariant probability distributions that are absolutely continuous with respect to the Lebesgue measure. In other words, the collection of initial micro-states (single systems) that do not follow the ensemble is supposed to constitute a set of vanishing, phase space volume. This approach is problematic in the case of dissipative dynamics and relaxation to nonequilibrium steady states, because the relevant invariant distributions attribute probability 1 to sets of zero volume, while evolution commonly begins in equilibrium states, i.e., in sets of full phase space volume. We consider the relaxation of classical, thermostatted particle systems to nonequilibrium steady states. We show that the dynamical condition known as T-mixing is necessary and sufficient for relaxation of ensemble averages to steady state values. Moreover, we find that the condition known as weak T-mixing applied to smooth observables is sufficient for ensemble relaxation to be independent of the B Lamberto Rondoni initial ensemble. Lastly, we show that weak T-mixing provides a notion of typicality for dissipative dynamics that is based on the (non-invariant) Lebesgue measure, and that we call physical ergodicity.

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Section: Introductionmentioning

confidence: 99%

On Typicality in Nonequilibrium Steady States

et al. 2016

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“…In practice, one starts by identifying a space of states in which the system under consideration can find itself and one assigns a prior distribution to it (maximizing the Shannon entropy, given the information available at the initial time), which is then updated when new information becomes available. 6 Note that probabilistic statements, understood subjectively, are forms of reasoning, although not deductive ones. Therefore, one cannot check them empirically, because reasonings, whether they are inductive or deductive, are either correct or not, but that depends on the nature of the reasoning not on any facts.…”

Section: Two Notions Of Probabilitymentioning

confidence: 99%

“…Laplace emphasized that human intelligence will forever remain 'infinitely distant' from the one of his demon. 6 For an introduction to Bayesian updating, see e.g. [14,15].…”

Section: Two Notions Of Probabilitymentioning

confidence: 99%

Probabilistic Explanations and the Derivation of Macroscopic Laws

Bricmont

2020

Statistical Mechanics and Scientific Explanation

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We will discuss the link between scientific explanations and probabilities, specially in relationship with statistical mechanics and the derivation of macroscopic laws from microscopic ones. 1 18 See for example figure 4 where i = −1 and i+1 = +1. 19There is a small abuse here, because it seems that we change the laws of motion by changing the orientation (from clockwise to counterclockwise). But one can attach another discrete "velocity" parameter to the particles, having the same value for all of them, and indicating the orientation, clockwise or counterclockwise, of their motion. Then, the motion is truly reversible, and we have simply to assume that the analogue here of the operation I of (9) changes also that extra velocity parameter.20 This is analogous to the Poincaré cycles in mechanics, except that, here, the length of the cycle is the same for all configurations (there is no reason for this feature to hold in general mechanical systems).

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“…In the present paper we do not discuss irreversibility in terms of entropy, for two main reasons. First, the word entropy can be source of confusion: for instance the entropy S G , defined in terms of the probability distribution function in the Γ-space, has a completely different behavior from S B , i.e the entropy obtained from the probability density of a single particle (µ-space); for a discussion on this point see [4,8,14]. Second, at a practical, as well as at a conceptual level, for understanding of irreversibility it is enough to observe that, if the system starts from a typical far-from-equilibrium initial state, the macroscopic observables stay close to their mean values during the evolution, and therefore they approach their equilibrium values.…”

Section: Remarks About Ensembles Entropies and Typicalitymentioning

confidence: 99%

Irreversibility and typicality: A simple analytical result for the Ehrenfest model

Baldovin

Caprini

Vulpiani

2019

Physica A: Statistical Mechanics and its Applications

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With the aid of simple analytical computations for the Ehrenfest model, we clarify some basic features of macroscopic irreversibility. The stochastic character of the model allows us to give a nonambiguous interpretation of the general idea that irreversibility is a typical property: for the vast majority of the realizations of the stochastic process, a single trajectory of a macroscopic observable behaves irreversibly, remaining "very close" to the deterministic evolution of its ensemble average, which can be computed using probability theory. The validity of the above scenario is checked through simple numerical simulations and a rigorous proof of the typicality is provided in the thermodynamic limit. I. INTRODUCTIONUnderstanding the irreversibility from first principles is an old and noble problem of Physics. The technical reason of its difficulty is rather clear: on the one hand, the microscopic world is ruled by laws (Hamilton equations) which are invariant under the transformation of time reversal (t → −t , q → q , p → −p, being q and p the positions and momenta of the system); the macroscopic world, on the other hand, is described by irreversible equations, e.g. the Fick equation for the diffusion [1][2][3][4].How is it possible to conciliate the two above facts? On this topic there is an aged debate which started with the celebrated Boltzmann's H theorem and the well known criticisms by Loschmidt (about reversibility) and Zermelo (about recurrency). Of course we cannot enter in a detailed discussion about this fascinating chapter of statistical mechanics [2,5]. Already Boltzmann and Smoluchowski understood that the criticism by Zermelo is not a real serious problem as long as macroscopic systems are considered: basically, because of the Kac's lemma, in macroscopic systems the recurrence time is so large that it cannot be observed [2,6,7]. We can summarise the conclusions of Boltzmann by saying that the irreversibility describes an empirical regularity of macroscopic objects which is valid for a "vast majority" of the possible initial conditions. Often such validity for the "vast majority" of initial conditions is called typicality. According to Lebowitz [8] (as well as many others) a certain behavior is typical if the set of microscopic states for which it occurs comprises a region whose volume fraction goes to one as the number of molecules N grows. We can state that irreversibility is an emergent property [5,6,9] which appears as the number of degrees of freedom becomes (sufficiently) large; in such a limit, a single observation of the system is enough to determine its macroscopic properties.Several mathematical results, as well as detailed numerical simulations, support the coherence of the scenario proposed by Boltzmann [2]. Among the others, Lanford's work about the rarified gases is particularly important: he was able to prove, in a rigorous fashion, the validity of the Boltzmann equation for short times (of the order of the collision time) in the so-called Boltzmann-Grad limit [10].In spite of the ...

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The role of the number of degrees of freedom and chaos in macroscopic irreversibility

Cited by 18 publications

References 31 publications

On Typicality in Nonequilibrium Steady States

On Typicality in Nonequilibrium Steady States

Probabilistic Explanations and the Derivation of Macroscopic Laws

Irreversibility and typicality: A simple analytical result for the Ehrenfest model

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