The diffusion equation for a particle in a zero-point field: A free particle in a finite volume

Summary

The stochastic motion in one dimension of a free particle in a finite volume, subjected to a zero-point-field of random forces, is considered. The velocity of the particle is split, as usual, into a stochastic plus a stationary component, depending upon the coordinate only. The modified Hamilton-Jacobi equation for the deterministic component is then solved, and the resulting zeroth-order trajectories are computed. The diffusion equation can be written down in terms of the stationary drift component of the velocity, which turns out to be dependent upon a parameter γ, which characterizes the uncertainty in the velocity polarization, and the resulting flux through the walls. The zeroth-order trajectories can hence be classified following this parameter, which indicates a transition between two different regimes for some value lying between 0 and 1. For each type a diffusion equation in configuration space is obtained, together with an expression for the particle number flux in steady state. It is then briefly examined the effects of introducing a cut-off frequency into the zero-point spectrum, and a comparison with previous treatments of the same problem is made. It is found that the diffusion equations for non-zero values of γ are of a new type which had not been considered before.