Nonlinear Wave Equations Related to Nonextensive Thermostatistics

Abstract

Abstract:We advance two nonlinear wave equations related to the nonextensive thermostatistical formalism based upon the power-law nonadditive S q entropies. Our present contribution is in line with recent developments, where nonlinear extensions inspired on the q-thermostatistical formalism have been proposed for the Schroedinger, Klein-Gordon, and Dirac wave equations. These previously introduced equations share the interesting feature of admitting q-plane wave solutions. In contrast with these recent developments, one of the nonlinear wave equations that we propose exhibits real q-Gaussian solutions, and the other one admits exponential plane wave solutions modulated by a q-Gaussian. These q-Gaussians are q-exponentials whose arguments are quadratic functions of the space and time variables. The q-Gaussians are at the heart of nonextensive thermostatistics.The wave equations that we analyze in this work illustrate new possible dynamical scenarios leading to time-dependent q-Gaussians. One of the nonlinear wave equations considered here is a wave equation endowed with a nonlinear potential term, and can be regarded as a nonlinear Klein-Gordon equation. The other equation we study is a nonlinear Schroedinger-like equation.

Lorentzian solitary wave in a generalised nonlinear Schrödinger equation

“…(12) and (13), for the values which give the Lorentzian solitary waves in eqs. (20) and (21), into the density eq. (25) and integrating over all space we obtain…”

Entropic Characterization of Quantum States with Maximal Evolution under Given Energy Constraints

“…, and constitute useful tools for the analysis of diverse problems both in classical and in quantum physics (see, for example, [20][21][22][23] and references therein). Notice that, in the present application of the S q entropies, the particular value q = 2 is an inevitable consequence of the structure of the inner product in Hilbert space, which provides a natural way to assess the distinguishability between quantum pure states.…”

Revisiting Entanglement within the Bohmian Approach to Quantum Mechanics

“…The de Broglie-Bohm approach to quantum mechanics [ 1 , 2 , 3 , 4 ], also referred to as the pilot-wave theory, or the quantum theory of motion, has been, since the publication of the seminal works by Bohm, a subject of constant interest in the field of the foundations of quantum mechanics. Indeed, there has been a sustained research activity on the de Broglie-Bohm formulation along the years [ 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 ]. In an article reviewing and celebrating the first 100 years of quantum physics, Tegmark and Wheeler included the formulation of Bohmian mechanics within the list of the most significant events in the development of this field of Science [ 30 ].…”