Persistence Exponent for the Simple Diffusion Equation: The Exact Solution for any Integer Dimension
The persistence exponent $θ_o$ for the simple diffusion equation $ϕ_t({\it x},t) = \triangle ϕ(x,t)$ , 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 $ϕ$ at a specified spatial coordinate remains positive throughout for a certain time $t$ behaves as $t^{-θ_o}$ for asymptotically large time $t$. The value of $θ_o$, calculated here for any integer dimension $d$, is $θ_o = \frac{d}{4}$ 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 $ θ_o = 0.12, 0.18,0.23$ for $d=1,2,3$ respectively.
cond-mat.stat-mech↗