Depolarization of rotating spins by random walks on lattices.

Abstract

The depolarization of rotating spins that perform a random walk on a d-dimensional lattice (d=1,2,3) with randomly distributed rotation frequencies is studied by numerical simulations and, especially for d=1, by analytical methods. For a Gaussian frequency distribution an exponential polarization decay is found in all dimensions for large times t, or large step numbers n. The dependence of the decay constant \ensuremath{\lambda} on the width \ensuremath{\sigma} of the frequency distribution is determined by the dimensionality d. In d=1, an approximation that takes the distribution of spans of the random walk into account yields a behavior \ensuremath{\lambda}=const${\ensuremath{\sigma}}^{\ensuremath{\beta}}$ with an exponent \ensuremath{\beta}=(4/3) which is in good agreement with the simulations. The exponent \ensuremath{\beta}\ensuremath{\approxeq}1.77 is numerically obtained for smaller values of \ensuremath{\sigma} in d=2. In d=3 the Gaussian description is appropriate, at least for small \ensuremath{\sigma}: \ensuremath{\lambda}=${a}_{0}$${\ensuremath{\sigma}}^{2}$, where ${a}_{0}$ depends on the structure of the lattice. These results qualitatively agree with the predictions of an effective-medium theory for the decay constant. Other examples of frequency distributions are considered in d=1 to examine the dependence of the polarization decay on the particular choice of the distribution.