Generalized Euler, Smoluchowski and Schrödinger equations admitting self-similar solutions with a Tsallis invariant profile

Abstract.

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 $\gamma$ . 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 $\gamma$ . This leads to a notion of generalized quantum mechanics and generalized thermodynamics. When $q=2\gamma-1$ , 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 $q=\gamma=1$ .