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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Wigner equation for particles obeying an exclusion-inclusion principle

G. Kaniadakis, A. Lavagno, P. Quarati

Published Feb 1, 1998• 2 citations
Publisher

Abstract

We consider a nonlinear Schr\"odinger equation for particles obeying an exclusion-inclusion principle recently proposed by us and derive, within the Wigner formalism, a generalized quantum phase-space transport equation. We calculate the $\ensuremath{\Elzxh}$ expansion of this equation and study the quantum corrections, up to the ${\ensuremath{\Elzxh}}^{2}$ order, of the finite temperature Thomas-Fermi phase-space distribution. The effects and the quantitative relevance of the exclusion principle on the distribution function are considered for a system of fermionic particles in a harmonic confining potential.

DOI
10.1103/physreve.57.1395

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We consider a nonlinear Schr"odinger equation for particles obeying an exclusion-inclusion principle recently proposed by us and derive, within the Wigner formalism, a generalized quantum phase-space transport equation. We calculate the \ensuremath\Elzxh\ensuremath{\Elzxh}\ensuremath\Elzxh expansion of this equation and study the quantum corrections, up to the \ensuremath\Elzxh2{\ensuremath{\Elzxh}}^{2}\ensuremath\Elzxh2 order, of the finite temperature Thomas-Fermi phase-space distribution. The effects and the quantitative relevance of the exclusion principle on the distribution function are considered for a system of fermionic particles in a harmonic confining potential.
An analysis of the Wigner function for identical particles is presented. Four situations have been considered. i) A scattering process between two indistinguishable electrons described by a minimum uncertainty wave packets showing the exchange and correlation hole in Wigner phase space. ii) An equilibrium ensemble of N electrons in a one-dimensional box and in a one-dimensional harmonic potential showing that the reduced single particle Wigner function as a function of the energy defined in the Wigner phase-space tends to a Fermi distribution. iii) The reduced one-particle transport-equation for the Wigner function in the case of interacting electrons showing the need for the two-particle reduced Wigner function within the BBGKY hierarchy scheme. iv) The electron-phonon interaction in the two-particle case showing co-participation of two electrons in the interaction with the phonon bath.
Recently [G. Kaniadakis, Phys. Rev. A 55, 941 (1997)], we introduced a Schr\"odinger equation containing a complex nonlinearity $W(\ensuremath{\rho},\mathbf{j})+i\mathcal{W}(\ensuremath{\rho},\mathbf{j})$ which describes the collective interaction introduced by an exclusion-inclusion principle (EIP). The EIP does not affect $W(\ensuremath{\rho},\mathbf{j})$ and determines $\mathcal{W}(\ensuremath{\rho},\mathbf{j})$ univocally. In the above reference $W(\ensuremath{\rho},\mathbf{j})$ was deduced by means of a stochastic quantization approach, in this way obtaining a noncanonical quantum system. In this work we introduce a family of nonlinearities $W(\ensuremath{\rho},\mathbf{j})$ generating a family of nonlinear canonical quantum systems, and derive their Lagrangian and the Hamiltonian functions and the evolution equations of the fields. We derive also the Ehrenfest relations and study the soliton properties. The shape of the soliton, propagating in the system obeying the EIP, can be obtained by solving a first-order ordinary differential equation. We show that, in the case of soliton solutions, by means of a unitary transformation, the EIP potential is equivalent to a real algebraic nonlinear potential proportional to $\ensuremath{\kappa}{\ensuremath{\rho}}^{2}/(1+\ensuremath{\kappa}\ensuremath{\rho}).$
We express the response of a normal Fermi system to a weak external charge in terms of a one-particle Wigner distribution function which obeys a generalized transport equation valid at all temperatures. The derivation of this transport equation is closely related to the method developed by Prigogine and co-workers in nonequilibrium situations, both classical and quantum; it is based on a reclassification of the perturbation expansion of the autocorrelation formula expressing the response. The proof is greatly facilitated by the use of a suitable diagram technique which describes both the dynamical processes in the system as well as the effect of the interactions on equilibrium properties; full advantage is taken of the contraction theorem for averages over the unperturbed equilibrium ensemble.
In a previous work [Phys. Rev. E 48, 4263 (1993)] we have derived a nonlinear one-dimensional kinetic equation for the distribution function of particles obeying an exclusion principle. In the present work, on the same grounds, we extend this kinetics to D-dimensional continuous or discrete space, in order to study the distribution function of particles obeying a generalized exclusion-inclusion Pauli principle (EIP). This exclusion or inclusion principle is introduced into the classical transition rates by means of an inhibition or an enhancement factor, which contains a parameter \ensuremath{\kappa}, whose values range between -1 and +1 and can balance the effect of the full or partial validity of EIP. After deriving the kinetic equation we obtain a general expression of the stationary distribution function, depending on the value we give to the parameter \ensuremath{\kappa}. When we limit ourselves to Brownian particles, we derive exactly for \ensuremath{\kappa}=-1 the Fermi-Dirac (FD) distribution, for \ensuremath{\kappa}=0 the Maxwell-Boltzmann distribution, and for \ensuremath{\kappa}=1 the Bose-Einstein (BE) distribution. When \ensuremath{\kappa} assumes an intermediate value, except zero, between the extreme values -1 and +1, we obtain statistical distributions different from the FD and BE ones. We attribute to the parameter \ensuremath{\kappa} the meaning of the degree of indistinguishability of identical particles, the degree of antisymmetrization, or the symmetrization of the wave function of the particle system.
In this work is studied a many body system obeying to a generalized Exclusion-Inclusion Principle (EIP) originated by collective effect, the dynamics, in mean field approximation, being ruled by a nonlinear Schroedinger equation. The EIP is introduced by a judicious generalization of the particle current. By means of variational principle is obtained a canonical nonlinear Schroedinger equation. We study the Lagrangian and Hamiltonian formulation. We study the symmetries of the nonlinear Schroedinger equation obeying to EIP and by means of Noether theorem we obtain and discuss conserved quantities. Successively, EIP-Schroedinger equation is coupled in a minimal way to an abelian gauge field which dynamics is described by the Maxwell-Chern-Simons Lagrangian. We show that the anyonic statistic behavior ascribed to the system by the Chern-Simons Lagrangian is not destroyed by the presence of EIP potential. Finally, we study special solutions of the system. Applications on the Bose-Einstein condensation and vortex-like solution in the picture of Chern-Simons interaction are considered.