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First passage time (FPT) is the time a particle, subject to some stochasticprocess, hits or crosses a closed surface for the very first time.$\tau$-leaping methods are a class of stochastic algorithms in which, insteadof simulating every single reaction, many reactions are ``leaped" over in orderto shorten the computing time. In this paper we developed a $\tau$-leapingmethod for computing the FPT and position in arbitrary volumes for a Brownianparticle governed by the Langevin equation. The $\tau$-leaping method proposedhere works as follows. A sphere is inscribed within the volume of interest(VOI) centered at the initial particle's location. On this sphere, the FPT issampled, as well as the position, which becomes the new initial position. Then,another sphere, centered at this new location, is inscribed. This processcontinues until the sphere becomes smaller than some minimal radius$R_{\text{min}}$. When this occurs, the $\tau$-leaping switches to theconventional Monte Carlo, which runs until the particle either crosses thesurface of the VOI or finds its way to a position where a sphere of radius$>R_{\text{min}}$ can be inscribed. The switching between $\tau$-leaping and MCcontinues until the particle crosses the surface of the VOI. The size of thisradius depends on the system parameters and on one's notion of accuracy: thelarger this radius the more accurate the $\tau$-leaping method, but also lessefficient. This trade off between accuracy and efficiency is discussed. For twoVOI, the $\tau$-leaping method is shown to be accurate and more efficient thanMC by at least a factor of 10 and up to a factor of about 110. However, whileMC becomes exponentially slower with increasing VOI, the efficiency of the$\tau$-leaping method remains relatively unchanged. Thus, the $\tau$-leapingmethod can potentially be many orders of magnitude more efficient than MC.

A tau-leaping method for computing joint probability distributions of the first-passage time and position of a Brownian particle