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Symmetry

Generalized Equations in Quantum Mechanics and Brownian Theory

Pierre-Henri Chavanis

Published Dec 13, 2023• 6 citations
Publisher

Abstract

We discuss formal analogies between a nonlinear Schrödinger equation derived by the author from the theory of scale relativity and the equations of Brownian theory. By using the Madelung transformation, the nonlinear Schrödinger equation takes the form of hydrodynamic equations involving a friction force, an effective thermal pressure, a pressure due to the self-interaction, and a quantum potential. These hydrodynamic equations have a form similar to the damped Euler equations obtained for self-interacting Brownian particles in the theory of simple liquids. In that case, the temperature is due to thermal motion and the pressure arises from spatial correlations between the particles. More generally, the correlations can be accounted for by using the dynamical density functional theory. We determine the excess free energy of Brownian particles that reproduces the standard quantum potential. We then consider a more general form of excess free energy functionals and propose a new class of generalized Schrödinger equations. For a certain form of excess free energy, we recover the generalized Schrödinger equation associated with the Tsallis entropy considered in a previous paper.

DOI
10.3390/sym15122195

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We discuss formal analogies between a nonlinear Schrödinger equation derived by the author from the theory of scale relativity and the equations of Brownian theory. By using the Madelung transformation, the nonlinear Schrödinger equation takes the form of hydrodynamic equations involving a friction force, an effective thermal pressure, a pressure due to the self-interaction, and a quantum potential. These hydrodynamic equations have a form similar to the damped Euler equations obtained for self-interacting Brownian particles in the theory of simple liquids. In that case, the temperature is due to thermal motion and the pressure arises from spatial correlations between the particles. More generally, the correlations can be accounted for by using the dynamical density functional theory. We determine the excess free energy of Brownian particles that reproduces the standard quantum potential. We then consider a more general form of excess free energy functionals and propose a new class of generalized Schrödinger equations. For a certain form of excess free energy, we recover the generalized Schrödinger equation associated with the Tsallis entropy considered in a previous paper.
In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.
Nonlinear friction forces are a problem in classical Brownian motion for a long time [1]. They are described either by Langevin or by Fokker-Planck equations [2-4]. However, the rigorous generalized Langevin equation is linear, which points out that nonlinear friction forces possess a macroscopic hydrodynamic origin [1]. For this reason, they are not present in the modern quantum theory of open systems [5]. The scope of the present paper is to investigate the nonlinear friction effect on quantum mechanics. The analysis is based on a dissipative Madelung quantum hydrodynamics. Thus, the nonlinear friction is a result of nonlinear non-equilibrium thermodynamics in contrast to the traditional mechanical nonlinear friction forces. A quantum particle in vacuum is described by the Schrodinger equation
We propose a generalization of quantum mechanical equations in the hydrodynamic form by introducing, into the Lagrangian density, terms taking into account the diffusion velocity at zero and finite temperatures and the diffusion pressure energy of the warm vacuum. Based on this, for the model of one-dimensional hydrodynamics, we construct a system of equations that are analogous to the Euler equations, but with the inclusion of quantum and thermal effects. They are a generalization of the equations of the Nelson stochastic mechanics. The numerical analysis of the system’s solutions’ behavior determined that this system can be used to describe the process of quantum-thermal fluctuation relaxation.
We propose a generalization of equations of quantum mechanics in the hydrodynamic form by introducing the terms taking into account the diffusion velocity at zero and finite temperatures and the density energy of diffusion pressure of the thermal vacuum into the Lagrangian density. Based on this, for a model of one-dimensional hydrodynamics, we construct a system of equations that are similar to the Euler equations but taking quantum and thermal effects into account. They are a generalization of equations of Nelson's stochastic mechanics and can be used to describe a new matter state, namely, nearly perfect fluidity.
Madelung's hydrodynamical forms of the Schrodinger equation and the Klein-Gordon equation are presented. The physical nature of the quantum potential is explored. It is demonstrated that the geometrical origin of the quantum potential is in the scalar curvature of the of the metric that defines the kinetic energy density for an extended particle and that the quantization of circulation (Bohr-Sommerfeld) is a consequence of associating an SO(2)-reduction of the Lorentz frame bundle with wave motion. The Madelung equations are then cast in a basis-free form in terms of exterior differential forms in such a way that they represent the equations for a timelike solution to the conventional wave equations whose rest mass density satisfies a differential equation of the "Klein-Gordon minus nonlinear term" type. The role of non-zero vorticity is briefly examined.
We first recall that the system of fluid mechanics equations (Euler and continuity) that describes a fluid in irrotational motion subjected to a generalized quantum potential (depending on a constant which may be different from the standard quantum constant ℏ) is equivalent to a generalized Schrödinger equation for a 'wavefunction' whose modulus squared yields the fluid density. Then we show that even in the case of the presence of vorticity, it is also possible to obtain a nonlinear Schrödinger-like equation (now of the vectorial field type) from the continuity and Euler equations including a quantum potential. The same kind of transformation also applies to a classical charged fluid subjected to an electromagnetic field and to an additional potential having the form of a quantum potential. Such a fluid can therefore be described by an equation of the Ginzburg–Landau type, and is expected to show some superconducting-like properties. Moreover, a Schrödinger form can be obtained for a fluctuating rotational motion of a solid. In this case the mass is replaced by the tensor of inertia, and a generalized form of the quantum potential is derived. We finally reconsider the case of a standard diffusion process, and we show that, after a change of variable, the diffusion equation can also be given the form of a continuity and Euler system including an additional potential energy. Since this potential is exactly the opposite of a quantum potential, the quantum behavior may be considered, in this context, as an anti-diffusion.