From dynamics to thermodynamics: Linear response and statistical mechanics

Bianucci, Marco; Mannella, R.; West, Bruce J.; Grigolini, Paolo

doi:10.1103/physreve.51.3002

Cited by 62 publications

(137 citation statements)

References 35 publications

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“…In this paper, although the importance of using a very large number of degrees of freedom for the foundation of ordinary statistical mechanics is not ruled out, deterministic chaos of classical trajectories is suggested to be the main ingredient for the foundation of statistical mechanics. The authors of this paper [19] derive indeed for temperature an expression more general than that proposed by Boltzmann, and recover the Boltzmann principle in the limiting case of infinite degrees of freedom.…”

Section: First Consequence: Foundation Of Statistical Mechanicsmentioning

confidence: 99%

“…As an example of the former view we quote Ref. [19]. In this paper, although the importance of using a very large number of degrees of freedom for the foundation of ordinary statistical mechanics is not ruled out, deterministic chaos of classical trajectories is suggested to be the main ingredient for the foundation of statistical mechanics.…”

Section: First Consequence: Foundation Of Statistical Mechanicsmentioning

confidence: 99%

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Strange kinetics: conflict between density and trajectory description

2002

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We study a process of anomalous diffusion, based on intermittent velocity fluctuations, and we show that its scaling depends on whether we observe the motion of many independent trajectories or that of a Liouville-like equation driven density. The reason for this discrepancy seems to be that the Liouville-like equation is unable to reproduce the multi-scaling properties emerging from trajectory dynamics. We argue that this conflict between density and trajectory might help us to define the uncertain border between dynamics and thermodynamics, and that between quantum and classical physics as well.

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Section: First Consequence: Foundation Of Statistical Mechanicsmentioning

confidence: 99%

Section: First Consequence: Foundation Of Statistical Mechanicsmentioning

confidence: 99%

Strange kinetics: conflict between density and trajectory description

2002

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“…Almost a decade ago, Bianucci et al (BMWG henceforth) set out to provide a "derivation," and generalization, of the Boltzmann principle 2 [Eq. (3)] with particular emphasis on systems far away from the thermodynamic limit [5]. Such generalization, which will be concisely reviewed in the next section, is based upon the derivation of an approximate equation of motion for the reduced probability distribution of the degrees of freedom of interest, which is then compared with a postulated form of the Fokker-Planck equation [Eq.…”

Section: Introductionmentioning

confidence: 99%

“…II), it will be shown that the temperature expression found in Ref. [5] is an approximation to the usual one defined through the equipartition theorem, and hence that the expression derived therein as a generalization of the Boltzmann principle is but a particular approximation to the volume entropy (Sec. III).…”

Section: Introductionmentioning

confidence: 99%

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Does the Boltzmann Principle Need a Dynamical Correction?

Adib

2004

Journal of Statistical Physics

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In an attempt to derive thermodynamics from classical mechanics, an approximate expression for the equilibrium temperature of a finite system has been derived [M. Bianucci, R. Mannella, B. J. West, and P. Grigolini, Phys. Rev. E 51, 3002 (1995)] which differs from the one that follows from the Boltzmann principle S = k ln Ω(E) via the thermodynamic relation 1/T = ∂S/∂E by additional terms of "dynamical" character, which are argued to correct and generalize the Boltzmann principle for small systems (here Ω(E) is the area of the constant-energy surface). In the present work, the underlying definition of temperature in the Fokker-Planck formalism of Bianucci et al. is investigated and shown to coincide with an approximate form of the equipartition temperature. Its exact form, however, is strictly related to the "volume" entropy S = k ln Φ(E) via the thermodynamic relation above for systems of any number of degrees of freedom (Φ(E) is the phase space volume enclosed by the constant-energy surface). This observation explains and clarifies the numerical results of Bianucci et al. and shows that a dynamical correction for either the temperature or the entropy is unnecessary, at least within the class of systems considered by those authors. Explicit analytical and numerical results for a particle coupled to a small chain (N ∼ 10) of quartic oscillators are also provided to further illustrate these facts.

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About the Optimal FPE for Non-linear 1d-SDE with Gaussian Noise: The Pitfall of the Perturbative Approach

Bianucci,

Bologna,

Mannella

2024

J Stat Phys

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This paper deals with the problem of finding the Fokker Planck Equation (FPE) for the single-time probability density function (PDF) that optimally approximates the single-time PDF of a 1-D Stochastic Differential Equation (SDE) with Gaussian correlated noise. In this context, we tackle two main tasks. First, we consider the case of weak noise and in this framework we give a formal ground to the effective correction, introduced elsewhere (Bianucci and Mannella in J Phys Commun 4(10):105019, 2020, https://doi.org/10.1088/2399-6528/abc54e), to the Best Fokker Planck Equation (a standard “Born-Oppenheimer” result), also covering the more general cases of multiplicative SDE. Second, we consider the FPE obtained by using the Local Linearization Approach (LLA), and we show that a generalized cumulant approach allows an understanding of why the LLA FPE performs so well, even for noises with long (but finite) time scales and large intensities.

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From dynamics to thermodynamics: Linear response and statistical mechanics

Cited by 62 publications

References 35 publications

Strange kinetics: conflict between density and trajectory description

Strange kinetics: conflict between density and trajectory description

Does the Boltzmann Principle Need a Dynamical Correction?

About the Optimal FPE for Non-linear 1d-SDE with Gaussian Noise: The Pitfall of the Perturbative Approach

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