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Methods of simulating random patterns of non-spherical objects and their application

We consider two mechanisms for simulating spatial patterns of hard-core non-spherical particles, namely the random sequential packing (RSP) and the Markov chain Monte Carlo (MCMC) procedures. The former is described as follows: we put a particle one-by-one into a finite region by sampling its location x and direction θ uniformly at random; if it does not overlap with other particles put before, it is put successfully, otherwise, we discard it and try another uniform sampling of (x, θ); by repeating the above, we can obtain a set of non-overlapping particles. The MCMC procedure is the following: we first give a certain non-overlapping pattern of non-spherical particles prepared in a random or a regular manner; then we select a particle and sample its new trial location x and direction θ at random; if the new sample (x, θ) is accepted, i.e. it does not overlap with other particles, the selected particle is moved to the new ‘position’, otherwise the particle is retained at the old position; by repeating the above, a series of a set of non-overlapping particles is generated.