Nontrivial Exponent for Simple Diffusion

The diffusion equation ${\ensuremath{\partial}}_{t}\ensuremath{\varphi}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{▽}^{2}\ensuremath{\varphi}$ is considered, with initial condition $\ensuremath{\varphi}(\mathbf{x},0)$, a Gaussian random variable with zero mean. Using a simple approximate theory we show that the probability ${p}_{n}({t}_{1}{,t}_{2})$ that $\ensuremath{\varphi}(\mathbf{x},t)$ (for a given space point $\mathbf{x}$) changes sign $n$ times between ${t}_{1}$ and ${t}_{2}$ has the asymptotic form ${p}_{n}({t}_{1}{,t}_{2})\ensuremath{\sim}{c}_{n}[\mathrm{ln}({t}_{2}/{t}_{1}){]}^{n}({t}_{1}/{t}_{2}{)}^{\ensuremath{-}\ensuremath{\theta}}$. The exponent $\ensuremath{\theta}$ has predicted values $0.1203$, $0.1862$, $0.2358$ in dimensions $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,3$, in remarkably good agreement with simulation results.