Quantum Maximum Entropy Principle and Quantum Statistics in Extended Thermodynamics
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Quantum Maximum Entropy Principle and Quantum Statistics in Extended Thermodynamics Massimo Trovato
Received: 20 January 2014 / Accepted: 2 April 2014 © Springer Science+Business Media Dordrecht 2014
Abstract By using a functional of the reduced density matrix, a formulation of quantum maximum entropy principle (QMEP) for the fractional exclusion statistics (FES) is proposed. In this way, compatibly with the uncertainty principle, we include a nonlocal description for the anyonic systems satisfying FES. By considering the Wigner formalism, we present a general scheme to develop a closed quantum hydrodynamic models in the framework of Extended Thermodynamics. Accordingly, the QMEP including FES is here asserted as the most advanced formulation of the fundamental principle of quantum statistical mechanics. Keywords Quantum maximum entropy principle · Extended thermodynamics · Fractional statistics
1 Introduction In thermodynamics and statistical mechanics entropy is the fundamental physical quantity to describe the evolution of a statistical ensemble. Its microscopic definition was provided by Boltzmann through the celebrated expression S = kB ln Γ
(1)
where kB is the Boltzmann constant and Γ is the number of microstates exploiting the given macroscopic properties. In this context, it is well known that in classical mechanics the entropy: (i) allows the violation of the uncertainty principle [1]; (ii) can be considered as a special case of the so-called Boltzmann-Gibbs-Shannon entropy that enables one to apply results of information theory to physics [1, 2]. In particular, maximum entropy principle (MEP) allows one to derive [2–4] the nonequilibrium distribution function associated with particles, and to determine the microstate corresponding to the given macroscopic quantity.
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M. Trovato ( ) Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria, 95125 Catania, Italy e-mail: [email protected]
M. Trovato
The MEP can be exploited in the completely nonlinear case, without any assumption on the nonequilibrium processes. Alternatively, an approximate distribution function is usually derived through a formal expansion around a local equilibrium configuration and so Extended Thermodynamics (ET) theories [3, 5, 6] of N moments and degree α (ET αN models) were obtained. In this way, it was found possible to derive rigorous hydrodynamic (HD) models based on the moments of the distribution function to different orders of a power expansion and including appropriate closure conditions [3–13]. Accordingly, making use of the Lagrange multipliers technique, it was found possible to construct the set of evolution equations for the macro-variables of interest. Apart from some partial attempts [2, 14–17] this is no longer the case in quantum mechanics. Here, the main difficulties concern with: (i) the definition of a proper quantum entropy for the explicit incorporation of statistics into problems involving a system of identical particles; (ii) the formulation of a global quantum MEP (QMEP) that
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