Identical nuclei are present at

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Non-equilibrium statistical mechanics and stochastic thermodynamics

Error, Chance and the Machine, Ch Random processes in continuous time

Problem

N0=1000N_0 = 1000 identical nuclei are present at t=0t=0. Each decays independently at rate λ\lambda, giving a half-life T1/2=5.00 minT_{1/2}=5.00\,\mathrm{min}. Let N(t)N(t) be the number not yet decayed, with distribution P(n,t)=Pr⁡(N(t)=n)P(n,t)=\Pr(N(t)=n). ((a)) Write the master equation for P(n,t)P(n,t) for this pure death process, in which state nn leaves at rate nλn\lambda into state n−1n-1, together with the initial condition. ((b)) Define the generating function G(s,t)=∑nP(n,t)snG(s,t)=\sum_n P(n,t)s^n and show it obeys ∂G∂t=λ(1−s)∂G∂s\dfrac{\partial G}{\partial t}=\lambda(1-s)\dfrac{\partial G}{\partial s}. ((c)) Along a curve s(t)s(t) the chain rule gives dG/dt=∂tG+(ds/dt) ∂sG\mathrm{d}G/\mathrm{d}t=\partial_tG+(\mathrm{d}s/\mathrm{d}t)\,\partial_sG. Show that GG is constant along every curve with ds/dt=−λ(1−s)\mathrm{d}s/\mathrm{d}t=-\lambda(1-s), solve for these curves, and follow each back to t=0t=0, where G(s,0)=sN0G(s,0)=s^{N_0}, to show G(s,t)=[1−(1−s)e−λt]N0,G(s,t)=\left[1-(1-s)e^{-\lambda t}\right]^{N_0}, the generating function of a binomial distribution with parameter p(t)=e−λtp(t)=e^{-\lambda t}. ((d)) Hence give ⟨N(t)⟩\langle N(t)\rangle and var(N(t))\mathrm{var}(N(t)) as functions of tt, and evaluate both at t=8.00 mint=8.00\,\mathrm{min}.

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