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Published June 2016 | Submitted + Published
Journal Article Open

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

Abstract

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 by 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 (∼t^(−1/2)), 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 for any kernel based on a scattering cross section ∼1/v.

Additional Information

© 2016 American Physical Society. (Received 31 March 2016; published 8 June 2016) We are grateful for fruitful discussion with Efim Katz on the long-time-tail problem and with Bart McGuyer concerning scattering kernels. We are grateful for the referee calling our attention to the external force problem in Ref. [13]. This work was supported in part by the US Department of Energy under Grant No. DE-FG02-97ER41042 and the US National Science Foundation under Grant No. 1506459.

Attached Files

Published - PhysRevA.93.062703.pdf

Submitted - 1603.07700v3.pdf

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