Equilibrium Wigner Function Research Articles

Correction: Camiola, V.D., et al. Equilibrium Wigner Function for Fermions and Bosons in the Case of a General Energy Dispersion Relation. Entropy 2020, 22, 1023

In Section 5 of Equilibrium Wigner Function for Fermions and Bosons in the Case of a General Energy Dispersion Relation [...].

Equilibrium Wigner Function for Fermions and Bosons in the Case of a General Energy Dispersion Relation.

The approach based on the Wigner function is considered as a viable model of quantum transport which allows, in analogy with the semiclassical Boltzmann equation, to restore a description in the phase-space. A crucial point is the determination of the Wigner function at the equilibrium which stems from the equilibrium density function. The latter is obtained by a constrained maximization of the entropy whose formulation in a quantum context is a controversial issue. The standard expression due to Von Neumann, although it looks a natural generalization of the classical Boltzmann one, presents two important drawbacks: it is conserved under unitary evolution time operators, and therefore cannot take into account irreversibility; it does not include neither the Bose nor the Fermi statistics. Recently a diagonal form of the quantum entropy, which incorporates also the correct statistics, has been proposed in Snoke et al. (2012) and Polkovnikov (2011). Here, by adopting such a form of entropy, with an approach based on the Bloch equation, the general condition that must be satisfied by the equilibrium Wigner function is obtained for general energy dispersion relations, both for fermions and bosons. Exact solutions are found in particular cases. They represent a modulation of the solution in the non degenerate situation.

Quantum corrected hydrodynamic models for charge transport in graphene

A quantum hydrodynamic model for charge transport in graphene is derived from a moment expansion of the Wigner–Boltzmann equation. The needed closure relations are obtained by adding quantum corrections based on the equilibrium Wigner function to the semiclassical model formulated in Camiola and Romano (2014) and Luca and Romano (2018) [2,3] by exploiting the Maximum Entropy Principle. The expression of the equilibrium Wigner function which takes into account the energy band of graphene has been obtained by solving the corresponding Bloch equation.

Rate of convergence of Feynman approximations of semigroups generated by the oscillator Hamiltonian

We determine the rate with which finitely multiple approximations in the Feynman formula converge to the exact expression for the equilibrium density operator of a harmonic oscillator in the linear τ-quantization. We obtain an explicit analytic expression for a finitely multiple approximation of the equilibrium density operator and the related Wigner function. We show that in the class of τ-quantizations, the equilibrium Wigner function of a harmonic oscillator is positive definite only in the case of the Weyl quantization.

Quantum Brownian motion and generalizations of Hermite polynomials

We introduce new families of orthogonal polynomials H D , n , motivated by the non-equilibrium evolution of a quantum Brownian particle (qBp). The H D , n ’s generalize non-trivially the standard Hermite polynomials, employed for classical Brownian motion. We treat several models (labelled by D ) for a non-equilibrium qBp, by means of the Wigner function W , in the presence of a “heat bath” at thermal equilibrium, with and without ab initio friction. For long times (for a suitable class of initial conditions), the non-equilibrium Wigner function W should approach, in some sense, the (time-independent) equilibrium Wigner function W e q , D , which describes the thermal equilibrium of the qBp with the “heat bath” and plays a central role. W e q , D is chosen to be the weight function which orthogonalizes the H D , n ’s. New results on W e q , D and on the H D , n ’s are reported. We justify the key role of the H D , n ’s as follows. Using the H D , n ’s, moments W e q , D , n and W n are introduced for W e q , D and W , respectively. At equilibrium, all moments W e q , D , n except the lowest one ( W e q , D , 0 ) vanish identically. Off-equilibrium, one expects that, for long times (for suitable initial conditions): (i) all non-equilibrium moments W n (except the lowest moment W 0 ), will approach zero, while (ii) the lowest non-equilibrium moment W 0 will tend to W e q , D , 0 ( ≠ 0 ) . To complete the justification, we outline how the approximate long-time non-equilibrium theories determined by W 0 for the different models ( D ) yield Smoluchowski equations and irreversible evolutions of the qBp towards thermal equilibrium.

Semiconductor Simulations Using a Coupled Quantum Drift‐Diffusion Schrödinger–Poisson Model

A coupled quantum drift‐diffusion Schrodinger–Poisson model for stationary resonant tunneling simulations in one space dimension is proposed. In the ballistic quantum zone with the resonant quantum barriers, the Schrodinger equation is solved. Near the contacts, where collisional effects are assumed to be important, the quantum drift‐diffusion model is employed. The quantum drift‐diffusion model was derived by a quantum moment method from a collisional Wigner equation by Degond et al. [J. Statist. Phys., 118 (2005), pp. 625–665]. The derivation yields an $O(\hbar^4)$ approximation of the equilibrium Wigner function which is used as the “alimentation function” in the mixed‐state formula for the electron and current densities at the interface. The coupling of the two models is realized by assuming the continuity of the electron and current densities at the interface points. Current‐voltage characteristics of a one‐dimensional tunneling diode are numerically computed. The results are compared to those from t...

A Critical Examination of the Basis of Macroscopic Quantum Transport Approaches

The density gradient, effective potential and smooth quantum potential approaches are derived from particular approximations of the equilibrium Wigner function, density matrix or the average carrier energy. These approximations in turn are perturbations upon the free carrier solution and are hence questionable near large abrupt barriers or sharp gradients in potential. In this paper we examine these approximations near large barriers and discuss a new form for the equilibrium Wigner function/density matrix that takes cognizance of the presence of a large abrupt barrier.

The Wigner function for thermal equilibrium

We provide a simple mathematical framework which is useful for a precise description of the thermal equilibrium Wigner function and for obtaining approximations to it in both cases of Boltzmann and Fermi Dirac statistics. Regular and singular asymptotic expansions of this Wigner function are derived and their validity shown for respectively smooth and discontinuous potentials. We also generalize some continuation methods, previously suggested for Boltzmann statistics, to the case of Fermi Dirac statistics.

The Wigner function for thermal equilibrium

We provide a simple mathematical framework which is useful for a precise description of the thermal equilibrium Wigner function and for obtaining approximations to it in both cases of Boltzmann and Fermi Dirac statistics. Regular and singular asymptotic expansions of this Wigner function are derived and their validity shown for respectively smooth and discontinuous potentials.

Relativistic quantum kinetic analysis of a pion-nucleon system

A relativistic plasma of nucleons interacting through pions via the usual isospin-invariant Yukawa coupling is analyzed in the framework of the covariant Wigner function technique. The method is manifestly covariant and the temperature effects are considered. The relativistic quantum BBGKY hierarchy for the pion-nucleon system is derived. By generalizing the Bogolioubov analysis of the classical BBGKY hierarchy a non-perturbative renormalizable method is elaborated which allows the solution of the kinetic problem in form of power series of two cluster parameters which measure the importance of correlations. In the lowest order of the cluster expansion (Hartree approximation or zero-order approximation) the quasi-nucleon Fock space is introduced, the fermion Wigner function in the thermodynamic equilibrium is obtained and the vacuum effects are renormalized. In this approximation the plasma behaves as a perfect Fermi gas of nucleons and antinucleons, but there exists an abnormal configuration with a uniform pion condensate which is unstable. In the next approximation (quadratic in the small parameters) the quasi-pion dispersion relation is obtained and the vacuum polarization tensor is renormalized. The quasi-pion rest-mass spectra (“plasma frequency”) and the effective-coupling behaviour as functions of the thermodynamic state are given. By estimating the size of the cluster parameters the self-consistency of the approximation scheme is proved. The quasi-pion Fock space is introduced and the quasi-pion equilibrium Wigner function is obtained. From these results the problem of the higher-order corrections to the Hartree thermodynamics is outlined.

Covariant Wigner function approach to the relativistic quantum electron gas in a strong magnetic field

This paper is concerned with a unified approach to some equilibrium properties of the relativistic quantum electron plasma embedded in a strong external magnetic field. This unified approach rests on the systematic use of a covariant Wigner function. The equilibrium Wigner function of the noninteracting gas is derived and its main properties are studied. In particular, it satisfies equations that are the complete analog of the usual Liouville equation and thus can be termed “relativistic quantum Liouville equation” whose properties are considered. The equations of state are rederived in this formalism and the results obtained earlier by Canuto and Chiu are found anew. Also, the covariant Wigner funetion of the magnetized vacuum is derived: it is needed, in this formalism, in order to obtain, e.g., the vacuum polarization tensor. Since we are also interested in the plasma modes, the fluctuations of one-particle quantities—and their spectrum—(in particular, of the four current) are calculated in view of their use in the fluctuation-dissipation theorem. We also outline a microscopic proof of this theorem, on the basis of a BBGKY hierarchy for the covariant Wigner functions, and point out the existence of an effective plasma frequency.