Nontrivial Exponent for Simple Diffusion

TL;DR

This paper investigates the sign-changing behavior of solutions to the diffusion equation with Gaussian initial conditions, deriving an asymptotic form for the probability of n sign changes and predicting the associated exponents across dimensions.

Contribution

It introduces a simple approximate theory to determine the asymptotic probability of sign changes in diffusion, predicting exponents that match simulations across dimensions.

Findings

Predicted exponents for sign change probabilities in 1D, 2D, 3D.

Asymptotic form for sign change probability involving logarithmic and power-law terms.

Good agreement between theoretical predictions and simulation results.

Abstract

The diffusion equation \partial_t\phi = \nabla^2\phi is considered, with initial condition \phi( _x_ ,0) a gaussian random variable with zero mean. Using a simple approximate theory we show that the probability p_n(t_1,t_2) that \phi( _x_ ,t) [for a given space point _x_ ] changes sign n times between t_1 and t_2 has the asymptotic form p_n(t_1,t_2) \sim [\ln(t_2/t_1)]^n(t_1/t_2)^{-\theta}. The exponent \theta has predicted values 0.1203, 0.1862, 0.2358 in dimensions d=1,2,3, in remarkably good agreement with simulation results.