Stable algorithm for event detection in event-driven particle dynamics: logical states

Abstract

Following the recent development of a stable event-detection algorithm for hard-sphere systems, the implications of more complex interaction models are examined. The relative location of particles leads to ambiguity when it is used to determine the interaction state of a particle in stepped potentials, such as the square-well model. To correctly predict the next event in these systems, the concept of an additional state that is tracked separately from the particle position is introduced and integrated into the stable algorithm for event detection.

Appendix 1: Stable algorithm for square-well molecules

The calculation of the event times in Sect. 3 is expressed in terms of the ball–ball overlap function, $f_\text {BB}$, as given in Eq. 1. This operation is especially sensitive to round-off error in the floating-point representation, therefore two numerically robust subroutines which analyse $f_\text {BB}$ are presented here. Both use the quadratic equation to solve for the roots of $f_\text {BB}$; however, each has different safeguards against numerical errors. The first algorithm, BallBallIntersectionTime (Algorithm 1), calculates the time until two balls begin to intersect. The second, BallBallDisjointTime (Algorithm 2), calculates the time until two balls become disjoint. Both subroutines return $+\infty$ if no event is detected and use the appropriate numerically stable form of the quadratic equation.

The introduction of the logical state and the overlap function requires some changes to the detection of events as outlined in step 2 of the basic algorithm given in Sect. 2. The modified algorithm is presented in SWEventTime (Algorithm 3). The logical state is required as input to this function and must be tracked seperately. For the square-well model, this is a single Boolean value per particle pair indicating whether the particles are captured or not. The specialized routines of Algorithms 1 and 2 are then used to determine the roots of the overlap function. In the case of a captured particle pair, both discontinuities at $\sigma _1$ and $\sigma _2$ are accessible and the minimum of the respective event times has to be selected.