Droplet model for autocorrelation functions in an Ising ferromagnet

The autocorrelation function, C(t)=〈$S_i$(0)$S_i$(t)〉-〈$S_i^2$ (0)〉, of Ising spins in an ordered phase (T<$T_c$) is studied via a droplet model. Only noninteracting spherical droplets are considered. The Langevin equation for droplet fluctuations is studied in detail. The relaxation-rate spectra for the corresponding Fokker-Planck equation are found to be (1) continuous from zero for dimension d=2, (2) continuous with a finite gap for d=3, and (3) discrete for d≥4. These spectra are different from the gapless form assumed by Takano, Nakanishi, and Miyashita for the kinetic Ising model. The asymptotic form of C(t) is found to be exponential for d≥3 and stretched exponential with the exponent β=1/2 for d=2. Our results for C(t) are consistent with the scaling argument of Huse and Fisher, but not with Ogielski’s Monte Carlo simulations.