PHYS6562 F5 Daily Quaetion

Loaded dice, and sticky spheres

For discrete system, the entropy \[S_{discrete} = -k_B \sum p_i\log p_i\] \[S_{discrete} = \frac{3}{2}k_B \log 2\]

Entropy for being hit by a comet with \(\rho_c(\theta,\phi) = \frac{1}{4\pi}\): \[S_c = -k_B \iint \rho_s \log (\rho_s) d\Omega = k_B \log (4\pi)\] Entropy for being hit by an asteroid with \(\rho_a(\theta,\phi) = \frac{cos(\theta)}{\pi^2}\): \[S_a = -k_B \iint \rho_a \log (\rho_a) d\Omega = -k_B \int \left( \frac{2\cos^2\theta}{\pi}\log(\cos\theta) - \frac{4\cos^2\theta}{\pi}\log\pi \right)d\theta\] \[S_a = -k_B \left(\frac{1}{2}(1-\log 4) - 2\log(\pi) \right) = k_B \log \left(\frac{2\pi^2}{\sqrt{e}} \right)\]

The entropy difference \[\Delta S = S_c - S_a = k_B\log\left(\frac{2\sqrt{e}}{\pi} \right)\]

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