Persistence Exponent for the Simple Diffusion Equation: The Exact Solution for any Integer Dimension

Abstract

The persistence exponent ${\theta }_o$ for the simple diffusion equation ${\varphi }_t(x,t)=△\varphi (x,t)$ , with random Gaussian initial condition {\color{red},} has been calculated exactly using a method known as selective averaging. The probability that the value of the field $\varphi$ at a specified spatial coordinate remains positive throughout for a certain time $t$ behaves as $t^{-{\theta }_o}$ for asymptotically large time $t$. The value of ${\theta }_o$, calculated here for any integer dimension $d$, is ${\theta }_o=\frac{d}{4}$ for $d\le 4$ and $1$ otherwise. This exact theoretical result is being reported possibly for the first time and is not in agreement with the accepted values ${\theta }_o=0.12,0.18,0.23$ for $d=1,2,3$ respectively.

5 pages

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