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

Nonlinear canonical quantum system of collectively interacting particles via an exclusion-inclusion principle

G. Kaniadakis, P. Quarati, Antonio Maria Scarfone

Published Nov 1, 1998• 15 citations
Publisher

Abstract

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 $\ensuremath{\kappa}{\ensuremath{\rho}}^{2}/(1+\ensuremath{\kappa}\ensuremath{\rho}).$

DOI
10.1103/physreve.58.5574

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Recently [G. Kaniadakis, Phys. Rev. A 55, 941 (1997)], we introduced a Schr"odinger equation containing a complex nonlinearity W(\ensuremathρ,j)+iW(\ensuremathρ,j)W(\ensuremath{\rho},\mathbf{j})+i\mathcal{W}(\ensuremath{\rho},\mathbf{j})W(\ensuremathρ,j)+iW(\ensuremathρ,j) which describes the collective interaction introduced by an exclusion-inclusion principle (EIP). The EIP does not affect W(\ensuremathρ,j)W(\ensuremath{\rho},\mathbf{j})W(\ensuremathρ,j) and determines W(\ensuremathρ,j)\mathcal{W}(\ensuremath{\rho},\mathbf{j})W(\ensuremathρ,j) univocally. In the above reference W(\ensuremathρ,j)W(\ensuremath{\rho},\mathbf{j})W(\ensuremathρ,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(\ensuremathρ,j)W(\ensuremath{\rho},\mathbf{j})W(\ensuremathρ,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 \ensuremathκ\ensuremathρ2/(1+\ensuremathκ\ensuremathρ).\ensuremath{\kappa}{\ensuremath{\rho}}^{2}/(1+\ensuremath{\kappa}\ensuremath{\rho}).\ensuremathκ\ensuremathρ2/(1+\ensuremathκ\ensuremathρ).
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.
We study the quantization of a classical system of interacting particles obeying a recently proposed kinetic interaction principle (KIP) [G. Kaniadakis, Physica A 296, 405 (2001)]. The KIP fixes the expression of the Fokker-Planck equation describing the kinetic evolution of the system and imposes the form of its entropy. In the framework of canonical quantization, we introduce a class of nonlinear Schrödinger equations (NSEs) with complex nonlinearities, describing, in the mean-field approximation, a system of collectively interacting particles whose underlying kinetics is governed by the KIP. We derive the Ehrenfest relations and discuss the main constants of motion arising in this model. By means of a nonlinear gauge transformation of the third kind, it is shown that in the case of constant diffusion and linear drift, the class of NSEs obeying the KIP is gauge-equivalent to another class of NSEs containing purely real nonlinearities depending only on the field rho=|psi|(2) .
Starting from a many-body classical system governed by a trace-form entropy we derive, in the stochastic quantization picture, a family of non linear and non-Hermitian Schroedinger equations describing, in the mean filed approximation, a quantum system of interacting particles. The time evolution of the main physical observables is analyzed by means of the Ehrenfest equations showing that, in general, this family of equations takes into account dissipative and damped effects due to the interaction of the system with the background. We explore the presence of steady states by means of solitons, describing conservative solutions. The results are specialized to the case of a system governed by the Boltzmann-Gibbs entropy.
Starting from a many-body classical system governed by a trace-form entropy we derive, in the stochastic quantization picture, a family of non-linear and non-Hermitian Schrödinger equations describing, in the mean field approximation, a quantum system of interacting particles. The time evolution of the main physical observables is analysed by means of the Ehrenfest equations, showing that, in general, this family of equations takes into account dissipative and damped effects due to the interaction of the system with the background. We explore the presence of steady states by means of solitons, describing conservative solutions. The results are specialized to the case of a system governed by the Boltzmann–Gibbs entropy.