Nontrivial Exponent for Simple Diffusion
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) [(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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