Dissipative effects in nonlinear Klein-Gordon dynamics

A. R. Plastino; C. Tsallis

doi:10.1209/0295-5075/113/50005

All issues

Volume 113 / No 5 (March 2016)

EPL, 113 5 (2016) 50005

Abstract

Issue		EPL Volume 113, Number 5, March 2016


Article Number		50005
Number of page(s)		5
Section		General
DOI		https://doi.org/10.1209/0295-5075/113/50005
Published online		01 April 2016

EPL, 113 (2016) 50005

Dissipative effects in nonlinear Klein-Gordon dynamics

A. R. Plastino¹^,2 and C. Tsallis²^,3

¹ CeBio y Secretaría de Investigación, Universidad Nacional Buenos Aires- Noreoeste (UNNOBA) and Conicet Roque Saenz Peña 456, Junin, Argentina
² Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro - RJ, Brazil
³ Santa Fe Institute - 1399 Hyde Park Road, Santa Fe, NM 87501, USA

Received: 10 January 2016
Accepted: 22 March 2016

Abstract

We consider dissipation in a recently proposed nonlinear Klein-Gordon dynamics that admits exact time-dependent solutions of the power-law form $e_q^{i(kx-wt)}$ , involving the q-exponential function naturally arising within the nonextensive thermostatistics ( $e_q^z \equiv [1+(1-q)z]^{1/(1-q)}$ , with $e_1^z=e^z$ ). These basic solutions behave like free particles, complying, for all values of q, with the de Broglie-Einstein relations $p=\hbar k$ , $E=\hbar \omega$ and satisfying a dispersion law corresponding to the relativistic energy-momentum relation $E^2 = c^2p^2 + m^2c^4$ . The dissipative effects explored here are described by an evolution equation that can be regarded as a nonlinear generalization of the celebrated telegraph equation, unifying within one single theoretical framework the nonlinear Klein-Gordon equation, a nonlinear Schrödinger equation, and the power-law diffusion (porous-media) equation. The associated dynamics exhibits physically appealing traveling solutions of the q-plane wave form with a complex frequency ω and a q-Gaussian square modulus profile.

PACS: 05.90.+m – Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems (restricted to new topics in section 05) / 02.30.Jr – Partial differential equations / 03.50.-z – Classical field theories

© EPLA, 2016

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