Random walks with thermalizing collisions in bounded regions; physical applications valid from the ballistic to diffusive regimes

• Main Authors: Swank, Christopher M, Petukhov, Alexander K, Golub, Robert
• Format: Journal Article
• Language: English
• Published: 30-03-2016
• Subjects: Physics - Statistical Mechanics
• DOI: 10.48550/arxiv.1603.07700

Phys. Rev. A 93, 062703 (2016) The behavior of a spin undergoing Larmor precession in the presence of
fluctuating fields is of interest to workers in many fields. The fluctuating
fields cause frequency shifts and relaxation which are related to their power
spectrum, which can be determined by taking the Fourier transform of the
auto-correlation functions of the field fluctuations. Recently we have shown
how to calculate these correlation functions for all values of mean free path
(ballistic to diffusive motion) in finite bounded regions, using the model of
persistent continuous time random walks (CTRW) for particles subject to
scattering by fixed (frozen) scattering centers so that the speed of the moving
particles is not changed by the collisions. In this work we show how scattering
with energy exchange from an ensemble of scatterers in thermal equilibrium can
be incorporated into the CTRW. We present results for 1,2 and 3 dimensions. The
results agree for all these cases contrary to the previously studied 'frozen'
models. Our results for the velocity autocorrelation function show a long time
tail $\left( \sim t^{-1/2}\right) $, which we also obtain from conventional
diffusion theory, with the same power, independent of dimensionality. Our
results are valid for any Markovian scattering kernel as well as any kernel
based on a scattering cross section $\sim1/v.$