Nontrivial Exponent for Simple Diffusion

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Abstract
The diffusion equation ∂tφ=▽2φ is considered, with initial condition φ(x,0), a Gaussian random variable with zero mean. Using a simple approximate theory we show that the probability pn(t1,t2) that φ(x,t) (for a given space point x) changes sign n times between t1 and t2 has the asymptotic form pn(t1,t2)∼cn[ln(t2/t1)]n(t1/t2)−θ. The exponent θ has predicted values 0.1203, 0.1862, 0.2358 in dimensions d=1,2,3, in remarkably good agreement with simulation results.

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