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

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