From nolinear statistical mechanics to nonlinear quantum mechanics - Concepts and applications

TL;DR

This paper explores the transition from linear to nonlinear physics through a generalized statistical mechanics framework, highlighting implications for complex systems, gravitation, and nonlinear quantum equations with various applications.

Contribution

It introduces a perspective linking nonlinear statistical mechanics with nonlinear quantum equations and discusses their applications and implications in physics and complex systems.

Findings

Nonlinear physics characterized by q ≠ 1 is rich and related to complexity.

Nonlinear generalizations of quantum equations have been proposed and tested.

Applications span low- and high-energy physics, including black hole entropy.

Abstract

We briefly review a perspective along which the Boltzmann-Gibbs statistical mechanics, the strongly chaotic dynamical systems, and the Schroedinger, Klein-Gordon and Dirac partial differential equations are seen as linear physics, and are characterized by an index q = 1. We exhibit in what sense q {\neq} 1 yields nonlinear physics, which turn out to be quite rich and directly related to what is nowadays referred to as complexity, or complex systems. We first discuss a few central points like the distinction between additivity and extensivity, and the Central Limit Theorem as well as the large-deviation theory. Then we comment the case of gravitation (which within the present context corresponds to q {\neq} 1, and to similar nonlinear approaches), with special focus onto the entropy of black holes. Finally we briefly focus on recent nonlinear generalizations of the Schroedinger, Klein-Gordon and Dirac equations, and mention various illustrative predictions, verifications and applications within physics (in both low- and high-energy regimes) as well as out of it.