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

Devashish Sanyal
Arxiv ID: 1005.0120Last updated: 8/11/2021
The persistence exponent θ_o for the simple diffusion equation ϕ_t( x,t) = ϕ (x,t) , 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 t^-θ_o for asymptotically large time t. The value of θ_o, calculated here for any integer dimension d, is θ_o = d/4 for d≤ 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.

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