Random initial conditions and nonlinear relaxation

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Abstract
The author studies the effect of randomness in the initial conditions on the deterministic diffusion equation with nonlinear terms. Physically, this describes, among other things, the time development of a system quenched from a high temperature to the vicinity of the critical point, in the approximation where the effects of thermal noise are neglected. He considers the case of a non-conserved order parameter with O(n) symmetry, and shows that the nonlinearities are irrelevant for the large time behaviour for dimension d)2. The model is investigated for d(2 using the renormalization group and in -expansion. It is found, to all orders in in , that the local fluctuations in the order parameter scale like t-12/, and have a universal distribution. The time dependence of the response function, describing the dependence on the initial condition, is characterized by another exponent which is computed to O( in 2). These results are checked in the exactly soluble cases of n to infinity and d=0.