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Physical Review A

Nonlinear Schrödinger equation for particles obeying an exclusion-inclusion principle

G. Kaniadakis

Published Feb 1, 1997• 13 citations
Publisher

Abstract

By using an appropriate dynamic stochastic quantization method, we quantize a Marcoffian process obeying a generalized Pauli exclusion-inclusion principle. The dynamics of the so obtained quantum system is governed by a nonlinear Schr\"odinger equation. If the exclusion-inclusion principle holds in the momentum space, the ground states of the system are described by statistical distributions interpolating between boson and fermion ones.

DOI
10.1103/physreva.55.941

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By using an appropriate dynamic stochastic quantization method, we quantize a Marcoffian process obeying a generalized Pauli exclusion-inclusion principle. The dynamics of the so obtained quantum system is governed by a nonlinear Schr"odinger equation. If the exclusion-inclusion principle holds in the momentum space, the ground states of the system are described by statistical distributions interpolating between boson and fermion ones.
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 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.
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.