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 γ. 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 γ). This leads to a notion of generalized quantum mechanics and generalized thermodynamics. When q = 2γ − 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 = γ = 1.2 In the absence of friction (ξ = 0) we get the quantum isothermal Euler equations and in the strong friction limit (ξ → +∞) we get the quantum Smoluchowski equation [58]. This provides an interesting formal connection between quantum mechanics and Brownian theory [55].

The Generalized Stochastic Smoluchowski Equation

“…We hope that one virtue of our paper is to have made a first connection between different communities, generalized thermodynamics [ 108 , 109 ], systems with long-range interactions [ 12 , 13 ] and the physics of simple liquids [ 152 ], by showing that their methods are complementary to each other. On the other hand, many results can be extended to the quantum regime as shown in recent papers (see, e.g., [ 3 ] and references therein).…”

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Synthesis of bioluminescent gold nanoparticle–luciferase hybrid systems for technological applications

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Global dissipative solutions of the defocusing isothermal Euler–Langevin–Korteweg equations

“…This equation first appears in Nassar's paper [20] as a possible way to give a stochastic interpretation of quantum mechanics in the context of Bohmian mechanics. It had a recent renewed interest in the physics community, in particular in quantum mechanics in order to describe the continuous measurement of the position of a quantum particle (see for example [21], [23] or [19]) and in cosmology and statistical mechanics (see [11], [12] or [13]). Note that in its physical interpretation, λ = 2k B τ / corresponds to a quantum friction coefficient, so both positive and negative signs could be of interest (k B and denotes respectively the Boltzmann and the normalized Planck constant, and τ is an effective temperature), unlike the real friction coefficient µ which is taken positive (see [11]).…”