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 $q=\gamma =1$.

References in corpus (10)

• Ultralight scalars as cosmological dark matter
• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations
• A General Nonlinear Fokker-Planck Equation and its Associated Entropy
• Logotropic distributions
• White dwarf stars in D dimensions
• Kinetic and hydrodynamic models of chemotactic aggregation
• Critical dynamics of self-gravitating Langevin particles and bacterial populations
• Nonlinear mean-field Fokker-Planck equations and their applications in physics, astrophysics and biology
• Phase separation of bacterial colonies in a limit of high degradation. Analogy with Jupiter's great red spot

Cited by in corpus (3)

• Landau equation for self-gravitating classical and quantum particles: Application to dark matter
• Kinetic theory of collisionless relaxation for systems with long-range interactions
• Global dissipative solutions of the defocusing isothermal Euler-Langevin-Korteweg equation