Simplified approach for calculating moments of action for linear reaction-diffusion equations

摘要

The mean action time is the mean of a probability density function that can be interpreted as a critical time,nwhich is a finite estimate of the time taken for the transient solution of a reaction-diffusion equation to effectivelynreach steady state. For high-variance distributions, the mean action time underapproximates the critical timensince it neglects to account for the spread about the mean. We can improve our estimate of the critical time byncalculating the higher moments of the probability density function, called the moments of action, which providenadditional information regarding the spread about the mean. Existing methods for calculating the nth moment ofnaction require the solution of n nonhomogeneous boundary value problems which can be difficult and tediousnto solve exactly. Here we present a simplified approach using Laplace transforms which allows us to calculatenthe nth moment of action without solving this family of boundary value problems and also without solving fornthe transient solution of the underlying reaction-diffusion problem.We demonstrate the generality of our methodnby calculating exact expressions for the moments of action for three problems from the biophysics literature.nWhile the first problem we consider can be solved using existing methods, the second problem, which is readilynsolved using our approach, is intractable using previous techniques. The third problem illustrates how the Laplacentransform approach can be used to study coupled linear reaction-diffusion equations.