Abstract
The persistence exponent 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 at a specified spatial coordinate remains positive throughout for a certain time t behaves as for asymptotically large time t. The value of , calculated here for any integer dimension d, is for and 1 otherwise. This exact theoretical result is being reported possibly for the first time and is not in agreement with the accepted values for respectively.
Highlights
The persistence exponent θo for the simple diffusion equation φt ( x,t ) = ∆φ ( x,t ), with random Gaussian initial condition, has been calculated exactly using a method known as selective averaging
The problem in the present paper is to find the persistence exponent for the simple diffusion equation φt ( x,t ) = ∆φ ( x,t )
The event is that φ at a specified location remains positive throughout the time evolution till a certain time t i.e. the φ at the location does not change sign even once
Summary
The problem in the present paper is to find the persistence exponent for the simple diffusion equation φt ( x,t ) = ∆φ ( x,t ). The diffusion equation is an equation that has no stochasticity. The stochasticity is introduced through the random initial conditions. The event is that φ at a specified location remains positive throughout the time evolution till a certain time t i.e. the φ at the location does not change sign even once. This probability for asymptotically large time is characterised by an exponent θo called the persistence exponent
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