Canonical quantization of classical systems with generalized entropies

TL;DR

This paper develops a framework for canonical quantization of classical systems characterized by generalized entropies, resulting in nonlinear Schrödinger equations that describe interacting particle systems with kinetic interactions.

Contribution

It introduces a novel quantization approach linking generalized entropies and nonlinear Schrödinger equations via kinetic interaction principles.

Findings

Derivation of nonlinear Schrödinger equations from classical kinetic principles.

Connection between generalized entropy and quantum nonlinearities.

Framework applicable to systems with collective interactions.

Abstract

We present, in the framework of the canonical quantization, a class of nonlinear Schroedinger equations with a complex nonlinearity describing, in the mean field approximation, systems of collectively interacting particles. The quantum evolution equation is obtained starting from the study of a N-body classical system where the underlined nonlinear kinetics is governed by a kinetic interaction principle (KIP) recently proposed [G. Kaniadakis: Physica A 296 (2001), 405--425]. The KIP, both imposes the form of the generalized entropy associated to the classical system, and determines the Fokker-Planck equation describing the kinetic evolution of the system towards equilibrium. Keywords: Nonlinear Schroedinger equation, Nonlinear kinetics, Generalized entropy.