A Model connecting Quantum, Diffusion, Soliton, and Periodic Localized States under Brownian motion

TL;DR

This paper introduces new equations of motion derived from Brownian motion theory that unify quantum, diffusion, soliton, and periodic localized states, revealing diverse phenomena including phase transitions and soliton collapse.

Contribution

It presents a novel set of classical equations with additional terms that connect various physical states and phenomena, extending the Schrödinger equation and incorporating quantum potential effects.

Findings

Unified framework for quantum, diffusion, soliton, and periodic states

Revealed dynamics of phase transitions and soliton collapse

Extended nonlinear Schrödinger equation with new phenomena

Abstract

We propose new equations of motion under the theory of the Brownian motion to connect the states of quantum, diffusion, soliton, and periodic localization. The new equations are nothing but the classical equations of motion with two additional terms and the one of them can be regarded as the the quantum potential. By choosing a parameter space, various important states are obtained. Further, the equations contain other interesting phenomena such as general dynamics of diffusion process, collapse of the soliton, the nonlinear extension of the Schr\"dinger equation, and the dynamics of phase transition.