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

TL;DR

This paper derives an exact formula for the persistence exponent of the diffusion equation in any integer dimension, revealing a simple relation for low dimensions and a constant value for higher dimensions, challenging previous estimates.

Contribution

It provides the first exact calculation of the persistence exponent for the diffusion equation in all integer dimensions, using selective averaging.

Findings

Persistence exponent $ heta_o = d/4$ for $d \\leq 4$

Persistence exponent $ heta_o = 1$ for $d > 4$

Results differ from previously accepted numerical estimates.

Abstract

The persistence exponent $\theta_o$ for the simple diffusion equation ${\phi}_t({\it x},t) = \triangle \phi (x,t)$ , 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 $\phi$ 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 $d\leq 4$ 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.