Figure 6.

Probability distributions of capture times from the perfect-mixing approximation and Monte Carlo calculations. (a) Linear plot for the two-dimensional case. (b) Logarithmic plot for the two-dimensional case. Here g_CAPT is a histogram of Monte Carlo capture times for a single target and no traps, and g_{CAPT_HIER} (Eq. 19) combines g_CAPT with g_HIER to give the distribution for one set of traps and a target, assuming a fixed escape time from the traps. The final prediction, g_{CAPT_HIER_ESC} (Eq. 21), also takes into account the distribution of escape times from traps. The Monte Carlo result for one set of traps and target is given by g_MC. The distributions g_{CAPT_HIER} and g_{CAPT_HIER_ESC} are of significant magnitude well beyond 3000 time steps; the cumulative distribution functions for the Monte Carlo data at 3000 time steps are 0.568 in two dimensions and 0.654 in three. (c) Ratio of the Monte Carlo distribution to the calculated distribution log g_MC/log g_{CAPT_HIER_ESC} for the two-dimensional case, the three-dimensional case, and the complete graph. For the complete graph the ratio of the log coefficients was 0.9999 ± 0.0070 for 2500 time points. In all these calculations, to get smooth histograms large runs were used (10⁴ trap configurations and 10⁴ tracers per trap configuration).