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, 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 $\sim 1/v.$

arXiv comment43 pages, 12 figures

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• Cryogenic magnetic coil and superconducting magnetic shield for neutron electric dipole moment searches
• Spin Dressed Relaxation and Frequency Shifts from Field Imperfections
• A magic magnetic field to measure the neutron electric dipole moment
• Mapping of the magnetic field to correct systematic effects in a neutron electric dipole moment experiment
• The neutron electric dipole moment experiment at the Spallation Neutron Source
• Optimizing Spin Dressing Sensitivity for the nEDMSF Experiment
• Quantum Control of Two Critically Dressed Spin 1/2 Species in Magnetic Fluctuations