Derivation of a generalized Schr\"odinger equation from the theory of scale relativity

TL;DR

This paper derives a generalized Schr"odinger equation from scale relativity theory, incorporating environmental interactions, leading to irreversible evolution towards equilibrium, with potential applications in modeling dark matter halos.

Contribution

It introduces a new generalized Schr"odinger equation derived from scale relativity, including damping and thermal effects, and links it to a fluctuation-dissipation framework.

Findings

Derivation of a generalized Schr"odinger equation with damping and thermal terms.

Establishment of an $H$-theorem for the quantum Boltzmann free energy.

Potential application to dark matter halos as self-gravitating Bose-Einstein condensates.

Abstract

Using Nottale's theory of scale relativity relying on a fractal space-time, we derive a generalized Schr\"odinger equation taking into account the interaction of the system with the external environment. This equation describes the irreversible evolution of the system towards a static quantum state. We first interpret the scale-covariant equation of dynamics stemming from Nottale's theory as a hydrodynamic viscous Burgers equation for a potential flow involving a complex velocity field and an imaginary viscosity. We show that the Schr\"odinger equation can be directly obtained from this equation by performing a Cole-Hopf transformation equivalent to the WKB transformation. We then introduce a friction force proportional and opposite to the complex velocity in the scale-covariant equation of dynamics in a way that preserves the local conservation of the normalization condition. We find that the resulting generalized Schr\"odinger equation, or the corresponding fluid equations obtained from the Madelung transformation, involve not only a damping term but also an effective thermal term. The friction coefficient and the temperature are related to the real and imaginary parts of the complex friction coefficient in the scale-covariant equation of dynamics. This may be viewed as a form of fluctuation-dissipation theorem. We show that our generalized Schr\"odinger equation satisfies an $H$-theorem for the quantum Boltzmann free energy. As a result, the probability distribution relaxes towards an equilibrium state which can be viewed as a Boltzmann distribution including a quantum potential. We propose to apply this generalized Schr\"odinger equation to dark matter halos in the Universe, possibly made of self-gravitating Bose-Einstein condensates.