Light scattered by particles much smaller than its wavelength (Rayleigh scattering)…

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

Light scattered by particles much smaller than its wavelength (Rayleigh scattering) leaves at an angle θ\theta, measured from the incident direction, with probability density P(θ)=38(1+cos⁡2θ)sin⁡θP(\theta) = \tfrac{3}{8}(1+\cos^2\theta)\sin\theta for θ∈[0,π]\theta\in[0,\pi]. Inverting this cumulative distribution means solving a cubic in cos⁡θ\cos\theta for every draw, so θ\theta is drawn instead by rejection sampling on the unnormalised density f(θ)=(1+cos⁡2θ)sin⁡θf(\theta) = (1+\cos^2\theta)\sin\theta, using a candidate θ∼U(0,π)\theta\sim U(0,\pi) and a test value y∼U(0,M)y\sim U(0,M), accepting whenever y≤f(θ)y \le f(\theta). (a) Show that f(θ)f(\theta) has its maximum where cos⁡2θ=1/3\cos^2\theta = 1/3, and evaluate that maximum. Confirm that M=1.1M = 1.1 is a safe envelope height. (b) Derive the acceptance fraction (∫0πf dθ)/(πM)\left(\int_0^\pi f\,\mathrm{d}\theta\right)/(\pi M) and evaluate it numerically, using ∫0π(1+cos⁡2θ)sin⁡θ dθ=8/3\int_0^\pi(1+\cos^2\theta)\sin\theta\,\mathrm{d}\theta = 8/3. (c) If the program draws N=50 000N = 50\,000 candidate pairs (θ,y)(\theta,y), estimate the number of accepted samples and its standard error, treating acceptance as an independent event of fixed probability for each candidate.

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