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arXiv:quant-ph/9809061 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Microscopic foundation of nonextensive statistics

Marek Czachor, Jan Naudts

Published Mar 1, 1999• 21 citations
arXivHTMLPDF

Abstract

A combination of the Lie-Poisson equation with the $q$-averaged energy ${U}_{q}=〈H{〉}_{q}$ leads to a microscopic framework for nonextensive $q$ thermodynamics. The resulting von Neumann equation is nonlinear: $i\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}=[H,{\ensuremath{\rho}}^{q}].$ In spite of its nonlinearity the dynamics is consistent with linear quantum mechanics of pure states. The free energy ${F}_{q}{=U}_{q}\ensuremath{-}{\mathrm{TS}}_{q}$ is a stability function for the dynamics. This implies that $q$-equilibrium states are dynamically stable. The (microscopic) evolution of \ensuremath{\rho} is reversible for any $q,$ but for $q\ensuremath{\ne}1$ the corresponding macroscopic dynamics is irreversible.

DOI
10.1103/physreve.59.r2497

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We treat the problem of the approach to thermal equilibrium by only resorting to quantum dynamics of an isolated macroscopic system. Inspired by the two important works in 2009 and in 1929, we have noted that a condition we call "thermodynamic normality" for a macroscopic observable guarantees the approach to equilibrium (in the sense that a measurement of the observable at time ttt almost certainly yields a result close to the corresponding microcanonical average for a sufficiently long and typical ttt). A crucial point is that we make no assumptions on the initial state of the system, except that its energy is distributed close to a certain macroscopic value. We also present three (rather artificial) models in which the thermodynamic normality can be established, thus providing concrete examples in which the approach to equilibrium is rigorously justified. Note that this kind of results which hold for ANY initial state are never possible in classical systems. We are thus dealing with a mechanism which is peculiar to quantum systems. The present note is written in a self-contained (and hopefully readable) manner. It only requires basic knowledge in quantum physics and equilibrium statistical mechanics.
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A Darboux-type method of solving the nonlinear von Neumann equation idotrho=[H,f(rho)]i\\dot \\rho=[H,f(\\rho)]idotrho=[H,f(rho)], with functions f(rho)f(\\rho)f(rho) commuting with rho\\rhorho, is developed. The technique is based on a representation of the nonlinear equation by a compatibility condition for an overdetermined linear system. von Neumann equations with various nonlinearities f(rho)f(\\rho)f(rho) are found to possess the so-called self-scattering solutions. To illustrate the result we consider the Hamiltonian HHH of a one-dimensional harmonic oscillator and f(rho)=rhoq−2rhoq−1f(\\rho)=\\rho^q-2\\rho^{q-1}f(rho)=rhoq−2rhoq−1 with arbitary real qqq. It is shown that self-scattering solutions possess the same asymptotics for all qqq and that different nonlinearities may lead to effectively indistinguishable evolutions. The result may have implications for nonextensive statistics and experimental tests of linearity of quantum mechanics.
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