Depolarization of rotating spins by random walks on lattices

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 λ on the width σ 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 λ=const${\sigma }^{\beta }$ with an exponent β=(4/3) which is in good agreement with the simulations. The exponent β≊1.77 is numerically obtained for smaller values of σ in d=2. In d=3 the Gaussian description is appropriate, at least for small σ: λ=$a_0$ ${\sigma }^2$, 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.