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arXiv:2308.01437

Can Schrodingerist Wavefunction Physics Explain Brownian Motion? II. The Diffusion Coefficient

W. David Wick

Published Aug 2, 2023• 0 citations
arXivHTMLPDF

Abstract

In the first paper of this series, I investigated whether a wavefunction model of a heavy particle and a collection of light particles might generate "Brownian-Motion-Like" trajectories of the heavy particle. I concluded that it was possible, but left unsettled the second claim in Einstein's classical program: diffusive motion, proportional to the square-root of time, as opposed to ballistic motion, proportional to the time. In this paper, I derive a criterion for diffusive motion, as well as an expression for the diffusion coefficient. Unfortunately, as in paper I, no exact solutions are available for the models, making checking the criterion difficult. But a virtue of the method employed here is that, given adequate information about model eigenvalues and eigenfunctions, diffusion can be definitively ruled in or out.

DOI
10.48550/arxiv.2308.01437

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In the first paper of this series, I investigated whether a wavefunction model of a heavy particle and a collection of light particles might generate "Brownian-Motion-Like" trajectories of the heavy particle. I concluded that it was possible, but left unsettled the second claim in Einstein's classical program: diffusive motion, proportional to the square-root of time, as opposed to ballistic motion, proportional to the time. In this paper, I derive a criterion for diffusive motion, as well as an expression for the diffusion coefficient. Unfortunately, as in paper I, no exact solutions are available for the models, making checking the criterion difficult. But a virtue of the method employed here is that, given adequate information about model eigenvalues and eigenfunctions, diffusion can be definitively ruled in or out.
In two prior papers of this series, it was proposed that a wavefunction model of a heavy particle and a collection of light particles might generate ``Brownian-Motion-Like" trajectories as well as diffusive motion (displacement proportional to the square-root of time) of the heavy particle, but did not exhibit a concrete instance. Here we introduce a one-space-dimensional model which, granted a finite perturbation series, fulfills the criteria for BML trajectories and diffusion. We note that Planck's constant and the molecular mass of light particles make an appearance in the diffusion coefficient, which further differentiates the present theory from the work of Poincar{é} and Einstein in the previous century.
Einstein's 1905 analysis of the Brownian Motion of a pollen grain in a water droplet as due to statistical variations in the collisions of water molecules with the grain, followed up by Perrin's experiments, provided one of the most convincing demonstrations of the reality of atoms. But in 1926 Schroedinger replaced classical particles by wavefunctions, which cannot undergo collisions. Can a Schroedingerist wavefunction physics account for Perrin's observations? As systems confined to a finite box can only generate quasiperiodic signals, this seems impossible, but I argue here that the issue is more subtle. I then introduce several models of the droplet-plus-grain; unfortunately, no explicit solutions are available (related is the remarkable fact that the harmonics of a general right triangle are still unknown). But from generic features of the models I conclude that: (a) wavefunction models may generate trajectories resembling those of a stochastic process; (b) diffusive behavior may appear for a restricted time interval; and (c) additional ``Wave Function Energy", by restricting ``cat" formation, can render the observations more ``classical". But completing the Einstein program of linking diffusion to viscosity and temperature in wavefunction models is still challenging.
The connection between the Schrödinger equation and Einstein's diffusion theory based on the Brownian motion of independent particles is well known. However, in contrast to diffusion theory, quantum mechanics theory has suffered controversial interpretations due to the counterintuitive concept of wavefunction. Here, while we confirm there is no difference in the mathematical form of these two equations, we derive the imaginary version of displacement. Using the diffusion theory of particles in a medium, as simple as it is, we describe that quantum mechanics is just an elegant and subtle equation to describe the probability of all the trajectories that a particle can take to propagate in time by a predictive wavefunction. Therefore, information on the position of particles through time in quantum theory is embedded in the wavefunction, which predicts the evolution of an ensemble of individual Brownian particles.
We examine the hypothesis that every particle of mass mmm is subject to a Brownian motion with diffusion coefficient $\frac{\ensuremath{\hbar}}{2m}$ and no friction. The influence of an external field is expressed by means of Newton's law F=ma\mathbf{F}=m\mathbf{a}F=ma, as in the Ornstein-Uhlenbeck theory of macroscopic Brownian motion with friction. The hypothesis leads in a natural way to the Schr\"odinger equation, but the physical interpretation is entirely classical. Particles have continuous trajectories and the wave function is not a complete description of the state. Despite this opposition to quantum mechanics, an examination of the measurement process suggests that, within a limited framework, the two theories are equivalent.
By modelling quantum systems as emerging from a (classical) sub-quantum thermodynamics, the quantum mechanical "decay of the wave packet" is shown to simply result from sub-quantum diffusion with a specific diffusion coefficient varying in time due to a particle's changing thermal environment. It is thereby proven that free quantum motion strictly equals ballistic diffusion. The exact quantum mechanical trajectory distributions and the velocity field of the Gaussian wave packet are thus derived solely from classical physics. Moreover, also quantum motion in a linear (e.g., gravitational) potential is shown to equal said ballistic diffusion. Quantitative statements on the trajectories' characteristic behaviours are obtained which provide a detailed "micro-causal" explanation in full accordance with momentum conservation.