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Mathematically, it takes an infinite amount of time for the transientsolution of a diffusion equation to transition from initial to steady state.Calculating a \textit{finite} transition time, defined as the time required forthe transient solution to transition to within a small prescribed tolerance ofthe steady state solution, is much more useful in practice. In this paper, westudy estimates of finite transition times that avoid explicit calculation ofthe transient solution by using the property that the transition to steadystate defines a cumulative distribution function when time is treated as arandom variable. In total, three approaches are studied: (i) mean action time(ii) mean plus one standard deviation of action time and (iii) a new approachderived by approximating the large time asymptotic behaviour of the cumulativedistribution function. The new approach leads to a simple formula forcalculating the finite transition time that depends on the prescribed tolerance$\delta$ and the $(k-1)$th and $k$th moments ($k \geq 1$) of the distribution.Results comparing exact and approximate finite transition times lead to two keyfindings. Firstly, while the first two approaches are useful at characterisingthe time scale of the transition, they do not provide accurate estimates fordiffusion processes. Secondly, the new approach allows one to calculate finitetransition times accurate to effectively any number of significant digits,using only the moments, with the accuracy increasing as the index $k$ isincreased.

Calculating how long it takes for a diffusion process to effectively reach steady state without computing the transient solution