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

Abstract

The persistence exponent o θ for the simple diffusion equation, with random Gaussian initial condition, 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 o t θ − for asymptotically large time t. The value of o θ , calculated here for any integer dimension d, is 4 o d θ = for 4 d ≤ and 1 otherwise. This exact theoretical result is being reported possibly for the first time and is not in agreement with the accepted values 0.12, 0.18, 0.23 o θ = for 1, 2, 3 d = respectively.