DYNAMIC STRUCTURE FACTOR IN A RANDOM DIFFUSION-MODEL

Let {X(t):t greater-than-or-equal-to 0} denote random walk in the rand om waiting time model, i.e., simple random walk with jump rate w-1(X(t )), where {w(x): x is-an-element-of Z(d)} is an i.i.d. random field. W e show that (under some mild conditions) the intermediate scattering f unction F(q, t) = E0e(iqXt) (q is-an-element-of R(d)) is completely mo notonic in t (E0 denotes double expectation w.r.t. walk and field). We also show that the dynamic structure factor S(q, omega) = 2 integral0 infinity cos(omega t) F(q, t) dt exists for omega not-equal 0 and is s trictly positive. In d = 1,2 it diverges as 1/\omega\1/2, resp. -ln(\o mega\), in the limit omega --> 0; in d greater-than-or-equal-to 3 its limit value is strictly larger than expected from hydrodynamics. This and further results support the conclusion that the hydrodynamic regio n is limited to small q and small omega such that \omega\ much-greater -than D \q\2, where D is the diffusion constant.