The interval between successive photon detections in a counting experiment

PHYSICS · Subjective · Fermi University Physics · Computational Physics · Monte Carlo and lattice simulation of physical systems · Monte Carlo and lattice simulation of physical systems

Computational Physics, Ch Drawing numbers: random walks and Monte Carlo

Problem

The interval between successive photon detections in a counting experiment, after a dead-time correction, is modelled as P(t)∝f(t)=t e−tP(t)\propto f(t) = t\,e^{-t} for t∈[0,10]t\in[0,10] (time measured in units of the correlation time). It is drawn by rejection sampling. (a) Find the maximum of f(t)f(t) and hence the smallest uniform envelope height MM on [0,10]×[0,M][0,10]\times[0,M] that bounds ff. Using ∫010t e−t dt=1−11e−10≈1.000\int_0^{10}t\,e^{-t}\,\mathrm{d}t = 1 - 11e^{-10} \approx 1.000, compute the acceptance fraction of this uniform envelope. (b) An improved envelope g(t)=C e−t/2g(t) = C\,e^{-t/2}, matched to the slower decay of the tail, is proposed. Find the smallest CC for which g(t)≥f(t)g(t)\ge f(t) for all t≥0t\ge 0, and use ∫010e−t/2 dt=2(1−e−5)\int_0^{10}e^{-t/2}\,\mathrm{d}t = 2(1-e^{-5}) to compute the acceptance fraction of this envelope. (c) State the factor by which the average number of candidate draws needed per accepted sample falls when the matched envelope replaces the uniform one.

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