Gibbs, Einstein and the Foundations of Statistical Mechanics

Navarro, Luis

doi:10.1007/s004070050025

Cited by 18 publications

(20 citation statements)

References 29 publications

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“…As discussed later on in this paper, many financial signals can be classified as random scaling fractal signals (with a fractal dimension D ∈ [1,2]). This is the basis for the fractal market hypothesis in mathematical economics and hence, the applications of fractional calculus.…”

Section: Fractional Integrals and Differentialsmentioning

confidence: 99%

“…A simple derivation of this result can be obtained by noting that (2). The non-asymptotic Lévy distribution for arbitrary values of γ can easily be evaluated numerically through application of a discrete Fourier transform.…”

Section: Lévy and Associated Distributionsmentioning

confidence: 99%

“…In addition to the work of Josiah Gibbs, the evolution equation that Einstein derived is one the foundations of statistical mechanics [2,3]. The approach can, for example, be applied equally well to modelling the diffusion of light propagating through a complex of scatterers.…”

Section: Introductionmentioning

confidence: 99%

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Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction

et al. 2019

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This paper examines a range of results that can be derived from Einstein’s evolution equation focusing on the effect of introducing a Lévy distribution into the evolution equation. In this context, we examine the derivation (derived exclusively from the evolution equation) of the classical and fractional diffusion equations, the classical and generalised Kolmogorov–Feller equations, the evolution of self-affine stochastic fields through the fractional diffusion equation, the fractional Poisson equation (for the time independent case), and, a derivation of the Lyapunov exponent and volatility. In this way, we provide a collection of results (which includes the derivation of certain fractional partial differential equations) that are fundamental to the stochastic modelling associated with elastic scattering problems obtained under a unifying theme, i.e., Einstein’s evolution equation. This includes an analysis of stochastic fields governed by a symmetric (zero-mean) Gaussian distribution, a Lévy distribution characterised by the Lévy index γ ∈ [ 0 , 2 ] and the derivation of two impulse response functions for each case. The relationship between non-Gaussian distributions and fractional calculus is examined and applications to financial forecasting under the fractal market hypothesis considered, the reader being provided with example software functions (written in MATLAB) so that the results presented may be reproduced and/or further investigated.

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Section: Fractional Integrals and Differentialsmentioning

confidence: 99%

Section: Lévy and Associated Distributionsmentioning

confidence: 99%

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Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction

et al. 2019

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“…Incidentally, Albert Einstein was unaware of Gibbs's contributions in that field when, between 1902 and 1904, he wrote three papers on statistical mechanics. After reading Gibbs's textbook, in the 1905 German translation by Ernst Zermelo, Einstein declared that Gibbs's treatment was superior to his own and explained that he would not have written those papers if he had known Gibbs's work [6].…”

Section: Gibbs (1902)mentioning

confidence: 99%

“…This of course abets vagueness, confusion, misuse, and abuse, and the propagation of "urban legends" about it that occasionally penetrate even responsible scientific quarters (see the Shannon entry in Section 4). 6. The myth that entropy, whether physical or information-theoretical, is something that you measure.…”

Section: Introductionmentioning

confidence: 99%

Entropy? Honest!

Toffoli

2016

Entropy

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Here we deconstruct, and then in a reasoned way reconstruct, the concept of "entropy of a system", paying particular attention to where the randomness may be coming from. We start with the core concept of entropy as a count associated with a description; this count (traditionally expressed in logarithmic form for a number of good reasons) is in essence the number of possibilities-specific instances or "scenarios"-that match that description. Very natural (and virtually inescapable) generalizations of the idea of description are the probability distribution and its quantum mechanical counterpart, the density operator. We track the process of dynamically updating entropy as a system evolves. Three factors may cause entropy to change: (1) the system's internal dynamics; (2) unsolicited external influences on it; and (3) the approximations one has to make when one tries to predict the system's future state. The latter task is usually hampered by hard-to-quantify aspects of the original description, limited data storage and processing resource, and possibly algorithmic inadequacy. Factors 2 and 3 introduce randomness-often huge amounts of it-into one's predictions and accordingly degrade them. When forecasting, as long as the entropy bookkeping is conducted in an honest fashion, this degradation will always lead to an entropy increase. To clarify the above point we introduce the notion of honest entropy, which coalesces much of what is of course already done, often tacitly, in responsible entropy-bookkeping practice. This notion-we believe-will help to fill an expressivity gap in scientific discourse. With its help, we shall prove that any dynamical system-not just our physical universe-strictly obeys Clausius's original formulation of the second law of thermodynamics if and only if it is invertible. Thus this law is a tautological property of invertible systems!

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Statistical Thermodynamics

Oono

2003

Encyclopedia of Polymer Science and Technology

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The objective and mode of thinking of statistical thermodynamics are explained with key references. The problems of statistical thermodynamics fall into two categories: studies of universal structures and studies of materials constants. The modern statistical thermodynamics emphasizes the former category. Consequently, recognizing a universality class described by a well‐defined phenomenological framework and constructing a minimal model to elucidate it statistically are the key problems. From this modern point of view equilibrium and nonequilibrium statistical thermodynamics are outlined, with applications to polymer solutions.

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Gibbs, Einstein and the Foundations of Statistical Mechanics

Cited by 18 publications

References 29 publications

Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction

Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction

Entropy? Honest!

Statistical Thermodynamics

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