Dynamic structure factor in a random diffusion model

Abstract

Let {Xt:≥0} denote random walk in the random waiting time model, i.e., simple random walk with jump ratew−1(Xt), where {w(x):x∈ℤd} is an i.i.d. random field. We show that (under some mild conditions) theintermediate scattering functionF(q,t)=E0\(e^{iqX_l } \) (q∈ℝd) is completely monotonic int (E0 denotes double expectation w.r.t. walk and field). We also show that thedynamic structure factorS(q, w)=2∫0∞ cos(ωt)F(q, t) exists for ω≠0 and is strictly positive. Ind=1, 2 it diverges as 1/|ω|1/2, resp. −ln(|ω|), in the limit ω→0; ind≥3 its limit value is strictly larger than expected from hydrodynamics. This and further results support the conclusion that the hydrodynamic region is limited to smallq and small ω such that |ω|≫D |q|2, whereD is the diffusion constant.