Mori–Zwanzig–Daubechies decomposition of Ising-model Monte Carlo dynamics

Abstract

Monte Carlo dynamics of one- and two-dimensional Ising lattices were studied by computer simulation. A comparison of decompositions made with Haar and Daubechies wavelets finds that wavelet–wavelet time correlation functions 〈cn(t)cn(t+τ)〉 and their long-time decay constants Γn are virtually independent of the choice of wavelet basis. An intermediate-temperature scaling relation between Γn and 〈cn(t)cn(t+τ)〉 fails at low temperature. The temperature at which failure occurs decreases with increasing wavelet decimation level n. Mori–Zwanzig memory kernels φ(τ) are extracted from 〈cn(t)cn(t+τ)〉 without resort to Laplace transforms. Numerically, φ(τ) computed from the random force autocorrelation function 〈 fi(t)fi(t+τ)〉 is in good agreement with φ(τ) computed from the 〈cn(t)cn(t+τ)〉. Even for a system as simple as the two-dimensional periodic Ising lattice with nearest-neighbor interactions, φ(τ) is nonexponential; our results are consistent with a power-law decay of φ(τ) at large τ. © 1995 American Institute of Physics.