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A Nonlocal Formulation of Quantum Maximum Entropy Principle Including Fractional Exclusion Statistics

Massimo Trovato, Lino Reggiani, Viale A. Doria

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

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, 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], [2]; 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], [3]. In particular, by introducing the principle of maximum entropy (MEP) it was found possible to derive rigorous hydrodynamic (HD) models based on the moments of the distribution function to all orders of a power expansion and including appropriate closure conditions [4], [5]. 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 [3], [6], this is no longer the case in quantum mechanics. Here, the main difficulties concern with: i) the definition of a proper quantum entropy that includes particle indistinguishability; ii) the formulation of a global quantum MEP (QMEP) that allows one to obtain a quantum distribution function both for thermodynamic equilibrium and nonequilibrium configurations. From one hand, in the framework of a nonlocal quantum theory, the generalization of the corresponding Lagrange multipliers is also an open problem. Fron another hand, a rigorous formulation of quantum HD (QHD) closed models is a demanding issue for many kinds of problems in quantum systems like, interacting fermionic and bosonic gases, quantum turbulence, quantum fluids, quantized vortices, nuclear physics, confined carrier transport in semiconductor heterostrucures, phonon and electron transport in nanostructures, nanowires and thin layers. The aim of this talk is to address the above drawbacks by defining a quantum entropy in terms of the reduced density matrix, thus explicitly incorporating quantum statistics into problems involving a system of identical particles. As a further step, with respect to the uncertainty principle, starting from the Wigner representation we formulate a quantum maximum entropy principle which, in the framework of information theory, allows us to obtain a non-local theory for the system under study. As a final step, we determine a closed quantum hydrodynamic model for the macroscopic variables used as constraints in the QMEP approach.

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Quantum Maximum Entropy Principle for Fractional Exclusion Statistics

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Using the Wigner representation, compatibly with the uncertainty principle, we formulate a quantum maximum entropy principle for the fractional exclusion statistics. By considering anyonic systems satisfying fractional exclusion statistic, all the results available in the literature are generalized in terms of both the kind of statistics and a nonlocal description for excluson gases. Gradient quantum corrections are explicitly given at different levels of degeneracy and classical results are recovered when ℏ→0.

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Entropy is in Flux

Leo P. Kadanoff

The science of thermodynamics was put together in the Nineteenth Century to describe large systems in equilibrium. One part of thermodynamics defines entropy for equilibrium systems and demands an ever-increasing entropy for non-equilibrium ones. However, starting with the work of Ludwig Boltzmann in 1872, and continuing to the present day, various models of non-equilibrium behavior have been put together with the specific aim of generalizing the concept of entropy to non-equilibrium situations. This kind of entropy has been termed {\em kinetic entropy} to distinguish it from the thermodynamic variety. Knowledge of kinetic entropy started from Boltzmann's insight about his equation for the time dependence of gaseous systems. In this paper, his result is stated as a definition of kinetic entropy in terms of a local equation for the entropy density. This definition is then applied to Landau's theory of the Fermi liquid thereby giving the kinetic entropy within that theory. Entropy has been defined and used for a wide variety of situations in which a condensed matter system has been allowed to relax for a sufficient period so that the very most rapid fluctuations have been ironed out. One of the broadest applications of non-equilibrium analysis considers quantum degenerate systems using Martin-Schwinger Green's functions\cite{MS} as generalized of Wigner functions, $g^$. This paper describes once again these how the quantum kinetic equations for these functions give locally defined conservation laws for mass momentum and energy. In local thermodynamic equilibrium, this kinetic theory enables a reasonable local definition of entropy density. However, when the system is outside of local equilibrium, this definition fails. It is speculated that quantum entanglement is the source of this failure.

arXiv:1403.6162•Mar 24, 2014•0 citations
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, 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], [2]; 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], [3]. In particular, by introducing the principle of maximum entropy (MEP) it was found possible to derive rigorous hydrodynamic (HD) models based on the moments of the distribution function to all orders of a power expansion and including appropriate closure conditions [4], [5]. 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 [3], [6], this is no longer the case in quantum mechanics. Here, the main difficulties concern with: i) the definition of a proper quantum entropy that includes particle indistinguishability; ii) the formulation of a global quantum MEP (QMEP) that allows one to obtain a quantum distribution function both for thermodynamic equilibrium and nonequilibrium configurations. From one hand, in the framework of a nonlocal quantum theory, the generalization of the corresponding Lagrange multipliers is also an open problem. Fron another hand, a rigorous formulation of quantum HD (QHD) closed models is a demanding issue for many kinds of problems in quantum systems like, interacting fermionic and bosonic gases, quantum turbulence, quantum fluids, quantized vortices, nuclear physics, confined carrier transport in semiconductor heterostrucures, phonon and electron transport in nanostructures, nanowires and thin layers. The aim of this talk is to address the above drawbacks by defining a quantum entropy in terms of the reduced density matrix, thus explicitly incorporating quantum statistics into problems involving a system of identical particles. As a further step, with respect to the uncertainty principle, starting from the Wigner representation we formulate a quantum maximum entropy principle which, in the framework of information theory, allows us to obtain a non-local theory for the system under study. As a final step, we determine a closed quantum hydrodynamic model for the macroscopic variables used as constraints in the QMEP approach.
Using the Wigner representation, compatibly with the uncertainty principle, we formulate a quantum maximum entropy principle for the fractional exclusion statistics. By considering anyonic systems satisfying fractional exclusion statistic, all the results available in the literature are generalized in terms of both the kind of statistics and a nonlocal description for excluson gases. Gradient quantum corrections are explicitly given at different levels of degeneracy and classical results are recovered when ℏ→0.
Entropy in nonequilibrium statistical mechanics is investigated theoretically so as to extend the well-established equilibrium framework to open nonequilibrium systems. We first derive a microscopic expression of nonequilibrium entropy for an assembly of identical bosons/fermions interacting via a two-body potential. This is performed by starting from the Dyson equation on the Keldysh contour and following closely the procedure of Ivanov, Knoll and Voskresensky [Nucl. Phys. A {\bf 672} (2000) 313]. The obtained expression is identical in form with an exact expression of equilibrium entropy and obeys an equation of motion which satisfies the HHH-theorem in a limiting case. Thus, entropy can be defined unambiguously in nonequilibrium systems so as to embrace equilibrium statistical mechanics. This expression, however, differs from the one obtained by Ivanov {\em et al}., and we show explicitly that their ``memory corrections'' are not necessary. Based on our expression of nonequilibrium entropy, we then propose the following principle of maximum entropy for nonequilibrium steady states: ``The state which is realized most probably among possible steady states without time evolution is the one that makes entropy maximum as a function of mechanical variables, such as the total particle number, energy, momentum, energy flux, etc.'' During the course of the study, we also develop a compact real-time perturbation expansion in terms of the matrix Keldysh Green's function.
The interactions and correlations among the constituents of manybody systems are manifested in characteristic physical properties such as ferromagnetism, superconductivity, etc. A list of some that have been studied in the last century is given in the Box (see below). A parallel development in quantum information was initially slow but over the past two decades progress has been very rapid. Fundamentally this is another aspect of quantum correlations in composite systems arising from the twin features of the superposition principle and the tensor product structure of state space. These features are not utilized in the same manner in quantum many-body physics. In the Box, a corresponding parallel list with properties of many-body systems is given because there has been an interplay between the two research efforts and their understanding. The basic quantum mechanical principles apply to both cases except different aspects are utilized because the goals are different in each. Both areas of investigation are based on a probabilistic foundation with a variational underpinning founded on an “entropy” maximization, which may be called Quantum Statistical Mechanics [1]. The specific form of the entropy as a functional of the density matrix will be made explicit presently. See Table I for definitions of density matrix and associated quantities. The equilibrium properties of the many-body systems are then given by a maximization of von Neumann entropy subject to certain constraints such as the average value of the Hamiltonian of the system “energy”. This leads to the familiar exponential probabilities of the Boltzmann-Gibbs (BG) form. Any nonequilibrium properties are studied by a quantum time evolution equation. Non-equilibrium properties such as the anomalous relaxation in time are often analyzed with the quantum version of the Tsallis entropy with an entropic index, q, which leads to powerlaw probabilities in contrast to the BG type [2]. (See the Box in the Introduction of Tsallis and Boon). In quantum information, the maximum entropy scheme is not of use because its origins are elsewhere as will be made clear presently. The evolution is replaced by processing of information
Statistical mechanics reveals that the properties of a macroscopic physical system emerge as an average over an ensemble of statistically independent microscopic subsystems, each occupying a specific microstate. In the study of quantum systems, these microstates can be chosen to correspond to the pure state wavefunctions of individual quantum systems. However, the physical principles that govern the distribution of a pure state wavefunction ensemble, even under conditions of thermal equilibrium, are not well established. For instance, the canonical Boltzmann distribution cannot be applied to wavefunctions because they lack a definite energy. In this manuscript, we present a maximum entropy principle for the quantum wavefunction ensemble at thermal equilibrium, the so-called Scrooge ensemble. We highlight that a constraint on the energy expectation value, or even the shape of the associated eigenstate distribution, fails to yield a valid equilibrium state. We find that in addition to these constraints, one must also constrain the measurement entropy to be equal to the Rényi divergence of the ensemble with respect to the Gibbs state, indicating that the Rényi divergence may have uninvestigated physical importance to thermal equilibrium in quantum systems.
The science of thermodynamics was put together in the Nineteenth Century to describe large systems in equilibrium. One part of thermodynamics defines entropy for equilibrium systems and demands an ever-increasing entropy for non-equilibrium ones. However, starting with the work of Ludwig Boltzmann in 1872, and continuing to the present day, various models of non-equilibrium behavior have been put together with the specific aim of generalizing the concept of entropy to non-equilibrium situations. This kind of entropy has been termed {\em kinetic entropy} to distinguish it from the thermodynamic variety. Knowledge of kinetic entropy started from Boltzmann's insight about his equation for the time dependence of gaseous systems. In this paper, his result is stated as a definition of kinetic entropy in terms of a local equation for the entropy density. This definition is then applied to Landau's theory of the Fermi liquid thereby giving the kinetic entropy within that theory. Entropy has been defined and used for a wide variety of situations in which a condensed matter system has been allowed to relax for a sufficient period so that the very most rapid fluctuations have been ironed out. One of the broadest applications of non-equilibrium analysis considers quantum degenerate systems using Martin-Schwinger Green's functions\cite{MS} as generalized of Wigner functions, $g^ $. This paper describes once again these how the quantum kinetic equations for these functions give locally defined conservation laws for mass momentum and energy. In local thermodynamic equilibrium, this kinetic theory enables a reasonable local definition of entropy density. However, when the system is outside of local equilibrium, this definition fails. It is speculated that quantum entanglement is the source of this failure.
The science of thermodynamics was put together in the Nineteenth Century to describe large systems in equilibrium. One part of thermodynamics defines entropy for equilibrium systems and demands an ever-increasing entropy for non-equilibrium ones. However, starting with the work of Ludwig Boltzmann in 1872, and continuing to the present day, various models of non-equilibrium behavior have been put together with the specific aim of generalizing the concept of entropy to non-equilibrium situations. This kind of entropy has been termed {\em kinetic entropy} to distinguish it from the thermodynamic variety. Knowledge of kinetic entropy started from Boltzmann's insight about his equation for the time dependence of gaseous systems. In this paper, his result is stated as a definition of kinetic entropy in terms of a local equation for the entropy density. This definition is then applied to Landau's theory of the Fermi liquid thereby giving the kinetic entropy within that theory. Entropy has been defined and used for a wide variety of situations in which a condensed matter system has been allowed to relax for a sufficient period so that the very most rapid fluctuations have been ironed out. One of the broadest applications of non-equilibrium analysis considers quantum degenerate systems using Martin-Schwinger Green's functions\cite{MS} as generalized of Wigner functions, $g^$. This paper describes once again these how the quantum kinetic equations for these functions give locally defined conservation laws for mass momentum and energy. In local thermodynamic equilibrium, this kinetic theory enables a reasonable local definition of entropy density. However, when the system is outside of local equilibrium, this definition fails. It is speculated that quantum entanglement is the source of this failure.