(a) A measured time interval carries two independent random errors added together:…

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) A measured time interval carries two independent random errors added together: the first has mean zero and standard deviation 3.0 ms3.0\ \mathrm{ms}, the second has mean zero and standard deviation 4.0 ms4.0\ \mathrm{ms}. Find the mean and the standard deviation of the total error. (b) Estimate ln⁡30!\ln 30! with ln⁡N!≈Nln⁡N−N\ln N! \approx N\ln N - N, and again with the correction 12ln⁡(2πN)\tfrac12\ln(2\pi N) included. (c) A nitrogen molecule in still air travels a mean free path λ=65 nm\lambda = 65\ \mathrm{nm} between collisions, one collision every τ=1.3×10−10 s\tau = 1.3\times10^{-10}\ \mathrm{s}. Model it as a one-dimensional random walk of fixed step λ\lambda taken every τ\tau, the steps independent and equally likely in either direction, so that D=λ2/2τD = \lambda^2/2\tau and ⟨x2⟩=2Dt\langle x^2\rangle = 2Dt. Find DD and the time for the root-mean-square displacement to reach 0.50 m0.50\ \mathrm{m}.

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