A crystal surface offers adsorption sites

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles

Thermal Physics, Ch Probability, random walks and the statistics of large numbers

Problem

A crystal surface offers N=1200N = 1200 adsorption sites. Each site is, independently of the others, occupied by an atom with probability p=0.250p = 0.250 and empty with probability q=0.750q = 0.750, so the number nn of occupied sites has the binomial distribution P(n)=(Nn)pnqN−nP(n) = \binom{N}{n}p^{n}q^{N-n}. The text kept pp general when it derived the Gaussian limit; carry that derivation through yourself for this pp, and then test it against the exact binomial. (a) Write ln⁡P(n)\ln P(n), apply Stirling's approximation in the form ln⁡M!≈Mln⁡M−M\ln M! \approx M\ln M - M to each factorial, and show that dln⁡Pdn=ln⁡ ⁣[(N−n)pnq]\dfrac{\mathrm{d}\ln P}{\mathrm{d}n} = \ln\!\left[\dfrac{(N-n)p}{nq}\right], treating nn as continuous. (b) Show that this derivative vanishes at n0=Npn_0 = Np and that the second derivative there is −1/(Npq)-1/(Npq). (c) Hence write the second-order Taylor expansion of ln⁡P\ln P about the peak, exponentiate, and fix the prefactor by normalisation, ∑nP(n)≈∫P(n) dn=1\sum_n P(n) \approx \int P(n)\,\mathrm{d}n = 1. You may use ∫−∞∞e−au2du=π/a\displaystyle\int_{-\infty}^{\infty}e^{-au^{2}}\mathrm{d}u = \sqrt{\pi/a}. (d) Evaluate σn\sigma_n, P(300)P(300) and P(330)P(330) for the numbers given, and compare the two probabilities with the exact binomial values 0.026590.02659 and 0.0036390.003639.

Browse the problem library