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Physical Review A

Rate processes and decay of metastable states inside an anharmonic well driven by a random zero-point field

Michele Battezzati

Published May 13, 2009• 0 citations
Publisher

Abstract

A procedure previously developed by us for the evaluation of the diffusion coefficient and drift velocity of a charged Newtonian particle in a zero-point field is applied here to a sextic potential-one-dimensional system, for which an exact solution to the related Hamilton-Jacobi-Yasue-Riccati equation is available. This potential energy function represents either a stiff single well or a triple well. For this model, the averaged diffusion coefficient has been evaluated by assuming that the particle under study is confined for a long time inside one anharmonic well, that is, in the limit of infinite relaxation time, and the results are compared with quantum-mechanical predictions. A double-well model potential is subsequently examined.

DOI
10.1103/physreva.79.052112

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•Oct 22, 2015•5 citations
A procedure previously developed by us for the evaluation of the diffusion coefficient and drift velocity of a charged Newtonian particle in a zero-point field is applied here to a sextic potential-one-dimensional system, for which an exact solution to the related Hamilton-Jacobi-Yasue-Riccati equation is available. This potential energy function represents either a stiff single well or a triple well. For this model, the averaged diffusion coefficient has been evaluated by assuming that the particle under study is confined for a long time inside one anharmonic well, that is, in the limit of infinite relaxation time, and the results are compared with quantum-mechanical predictions. A double-well model potential is subsequently examined.
A method previously proposed by the same author, of solving the equations of motion in the presence of friction and an external stochastic force, is applied to the nonrelativistic Dirac equation for a charged bound pointlike particle in a black-body radiation field. It separates the particle velocity field, under the assumtion of stationarity, into a position-dependent component and a randomly fluctuating component depending mainly upon the time. This description of the possible stationary states of the bound particle in the random force field is taken as a starting point for establishing a diffusion equation in configuration space. Since we identify the first component with the drift velocity, we show that it must be the solution to a modified Hamilton–Jacobi equation by using a term that is the counterpart of the quantum modification to the same equation, which has as a first approximation exactly the same form. This term gives results proportional to the diffusion coefficient and, thereby, to the spectral density of the external random force. The diffusion equation that is obtained has complex coefficients and therefore it defines a probability density for the complex values of the variable coordinate. We propose an interpretation of this probability density, based on a specific example.
We study the dynamics of an interacting classical gas trapped in a double-well potential at finite temperature. Two model potentials are considered: a cubic box with a square barrier in the middle, and a harmonic trap with a Gaussian barrier along one direction. The study is performed using the Boltzmann equation, solved numerically via the test-particle method. We introduce and discuss a simple analytical model that allows one to provide estimates of the relaxation time, which are compared with numerical results. Finally, we use our findings to make numerical and analytical predictions for the case of a fermionic mixture in the normal-fluid phase in a realistic double-well potential relevant for experiments with cold atoms.
The following calculations provide expressions for the drift velocity and the average diffusion coefficient, applying to diffusion in configuration space of a nonrelativistic charged particle interacting with the zero-point field of stochastic electrodynamics. The particle is assumed to evolve according to a widely popular reduced form of the Braffort–Marshall equation, which is free from runaway solutions, and supposed to be found in stationary conditions, with regard to the average values of physical quantities. The results show an interesting similarity with the equations of quantum mechanics.
The diffusion Monte Carlo method is applied to study the ground-state properties of charged bosons in one dimension confined in a harmonic double-well trap. The particles interact repulsively through a Coulombic 1/ r potential. Numerical results show that the well separation has significant influence on the ground-state properties of the system. When the interaction of the system is weak, ground-state energy decreases with the increasing well separation and has a minimal value. If the well separation increases continually, the ground-state energy increases and approaches to a constant gradually. This effect will be abatable in the strong interacting system. In addition, by calculating the density of the systems for different interaction strengths with various well separations, we find that the density increases abnormally when the well separation is large at the centre of the system.
Wallstrom's criticism of existing formulations of stochastic mechanics is that they fail to derive the empirical predictions of orthodox quantum mechanics because they require an ad hoc quantization condition on the postulated velocity potential, \emph{S}, in order to derive Schr\odinger wave functions. We propose an answer to this criticism by modifying the Nelson-Yasue formulation of non-relativistic stochastic mechanics for a spinless particle with the following hypothesis: a spinless Nelson-Yasue particle of rest mass mmm continuously undergoes a driven steady-state oscillation of `zitterbewegung' (\emph{zbw}) frequency, ωc=(1/ℏ)mc2\omega_{c}=\left(1/\hbar\right)mc^{2}ωc​=(1/ℏ)mc2, in its instantaneous mean forward (and backward) translational rest frame. With this hypothesis we show that, in the lab frame, \emph{S} arises from imposing the constraint of conservative diffusions on the time-symmetrized steady-state phase of the \emph{zbw} particle, satisfies the required quantization condition, and evolves in time by the Hamilton-Jacobi-Madelung equations (when generalized to describe a statistical ensemble of \emph{zbw} particles). From the mathematical equivalence of Schr\odinger's equation with the Hamilton-Jacobi-Madelung equations plus the quantization condition, Schr\odinger wave functions for a spinless particle (in and excluding external fields) are thereby recovered. We also apply this `zitterbewegung stochastic mechanics' (ZSM) to the case of a central potential and show that it predicts angular momentum quantization. This paper sets the foundation for Part II, which will (primarily) work out the many-particle version of ZSM.