# Computational Physics

## Effective N-particle method for simulating growth processes driven by stochastic particle motion

Brown++ is an N-particle code for simulating the growth of fractal particles embedded in a dilute gas driven by Brownian motion. Under such circumstances, the mean particle distance exceeds the typical aggregate diameter by far, and a collision is a very rare event. The N-particle method employs an adaptive time step scheme respecting the individual dynamic states of the clusters. The basic idea is to perform on average constant „length steps“, instead of using constant time steps. This is achieved by an „event calendar technique“ similar to MD methods simulating hard sphere fluids. The numerical cost of the algorithm scales with the particle number better than N∙log N.

The basic idea of the algorithm is shown below:

A 2-dimensional example of three time steps as performed by the event-driven algorithm. The arrows show the predicted trajectories within the cells, while the italic numbers are the predicted instants of cell boundary crossings. For each time step the trajectory of the bold-printed particle is simulated.

Evolution of the event tree of the particle ensemble illustrated above (the node colours correspond with the particle colours used in this plot). The upper part of a node shows the particle index and the predicted time of the particle‘s cell boundary crossing. The lower left number is the index of a linked later event, while the lower right number is the time of a linked earlier event. The bold arrows mark the search path to the earliest event, while the broken arrows identify the search path for sorting in a new event.

## References:

- S. Kempf & S. Pfalzner, An effective algorithm for simulating diffusion-driven aggregation,
*Comput. Phys. Com.*, 2001,**137**, 225-235 - S. Kempf, S. Pfalzner, & Th. Henning,N-body calculations of cluster growth in proto-planetary disks,
*Molecular Dynamics on Parallel Computers*, 2000, World Scientific. - S. Kempf, S. Pfalzner, & Th. Henning, N-particle-simulations of dust growth: I. Growth driven by Brownian motion,
*Icarus*, 1999,**141**, 388-398