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

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.

💡 Research Summary

The paper addresses the persistence problem for the linear diffusion equation φ_t(x,t)=∇²φ(x,t) with a random Gaussian initial condition of zero mean and short‑range correlations ⟨φ(x,0)φ(x′,0)⟩=k δ(x−x′). Persistence, in this context, is the probability P(t) that the field at a fixed spatial point x₀ remains positive for the entire time interval