Rules of motion as a Wiener process
The main results of this post about the stochastic motion of single particles can be obtained by considering such motion - for large times - as a Brownian motion with drift,
where W(t) is a Wiener process with zero mean and unit variance. The momentum propensity V here plays the role of the drift, while the variance is given by the diffusivity coefficient D = 1-e, where e is the energy propensity (1+V2)/2.
For instance, we can derive the probability density from the Fokker-Planck equation of the Brownian motion,
whose solution is clearly the Gaussian distribution shown in the aforementioned post,
with the initial condition w = δ(x-x0).
Or, we can use the properties of stochastic processes to find the probability density of the first return time to the source, i.e., the time at which the particle comes back for the first time at x = x0. For some reason, I could not find a formula for that in the case of a Wiener process with drift. For this reason, I have performed my own calculations starting from the original, discrete pmf of position, and found that
where ⟨τ⟩ is the expected first return time and e, as usual, the energy propensity. The derivation is detailed in this document, appendix A.1. This expression looks rather complicated, but it is well approximated by the simpler relation ⟨τ⟩ = 1/e, at least for large energy propensities. It is interesting to note that this relation, once expressed as f = e, where f is the frequency of returns, is formally identical to the Planck-Einstein relation for the photon,
where E is the photon's energy and ν its frequency, once the latter is put into lattice units.
 |
| Illustrative depiction of a particle's random walk as a matter wave |
Although this matter wave relation does not play any special role in the model development, it is interesting that it emerges rather naturally from the simple rules of motion.
Comments
Post a Comment