New Variable Time Step Algorithm

In random flights simulations the single most important parameter is the time step bt, which if not properly converged leads to incorrect results. If the drift remains effectively constant throughout the time step, then the separation between the pair behaves as a one dimensional Wiener process with constant drift. The relative drift 

In order to select a time step, consider the relationship between the stochastic differential equation for df(X) and dX, where f(X) is the same function of X, given by the ltd transformation formula  

This is the standard method for transforming a stochastic differential equation. 

There is a complication in applying this formula to the function f(X) = fj, because when the ions are oppositely charged, there is always a distance where fi = 0 and so di lix cannot be controlled in this region. However, both terms contributing to are of the form ax, and so if both these terms are controlled separately, then the absolute variation in their sum will be acceptable. A and B can then defined as 

Using the definition of the drift term presented earlier (Eq. 4.15) and recognising a = (with D being the diffusion coefficient), Eq. (4.22) can then be expressed as 

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