Generalized Euler, Smoluchowski and Schrödinger equations admitting self-similar solutions with a Tsallis invariant profile
Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, Toulouse, France
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Accepted: 26 April 2019
Published online: 25 July 2019
The damped isothermal Euler equations, the Smoluchowski equation and the damped logarithmic Schrödinger equation with a harmonic potential admit stationary and self-similar solutions with a Gaussian profile. They satisfy an H -theorem for a free energy functional involving the von Weizsäcker functional and the Boltzmann functional. We derive generalized forms of these equations in order to obtain stationary and self-similar solutions with a Tsallis profile. In particular, we introduce a nonlinear Schrödinger equation involving a generalized kinetic term characterized by an index q and a power-law nonlinearity characterized by an index . We derive an H -theorem satisfied by a generalized free energy functional involving a generalized von Weizsäcker functional (associated with q and a Tsallis functional (associated with . This leads to a notion of generalized quantum mechanics and generalized thermodynamics. When , our nonlinear Schrödinger equation admits an exact self-similar solution with a Tsallis invariant profile. Standard quantum mechanics (Schrödinger) and standard thermodynamics (Boltzmann) are recovered for .
© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature, 2019