Generalized Euler, Smoluchowski and Schr\"odinger equations admitting self-similar solutions with a Tsallis invariant profile

Abstract

The damped isothermal Euler equations, the Smoluchowski equation and the damped logarithmic Schr\"odinger 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 Weizs\"acker 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 Schr\"odinger equation involving a generalized kinetic term characterized by an index $q$ and a power-law nonlinearity characterized by an index $\gamma$. We derive an $H$-theorem satisfied by a generalized free energy functional involving a generalized von Weizs\"acker functional (associated with $q$) and a Tsallis functional (associated with $\gamma$). This leads to a notion of generalized quantum mechanics and generalized thermodynamics. When $q=2\gamma-1$, our nonlinear Schr\"odinger equation admits an exact self-similar solution with a Tsallis invariant profile. Standard quantum mechanics (Schr\"odinger) and standard thermodynamics (Boltzmann) are recovered for $q=\gamma=1$.