(a) A barometer reading fluctuates: one reading is a random variable of mean and…

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Thermal, Statistical & Complex Systems · Uncertainty in measurement

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

Problem

(a) A barometer reading fluctuates: one reading is a random variable of mean 101.30 kPa101.30\ \mathrm{kPa} and standard deviation 0.60 kPa0.60\ \mathrm{kPa}, and successive readings are independent. Nine readings are taken and their arithmetic average is formed. Find the mean and the standard deviation of that average. (b) Estimate ln⁡60!\ln 60! using ln⁡N!≈Nln⁡N−N+12ln⁡(2πN)\ln N! \approx N\ln N - N + \tfrac12\ln(2\pi N). (c) A tagged water molecule in liquid water is modelled as a one-dimensional random walk of fixed step length ℓ=0.30 nm\ell = 0.30\ \mathrm{nm}, one independent step every τ=5.0 ps\tau = 5.0\ \mathrm{ps}, each step equally likely in either direction. Find the diffusion coefficient D=ℓ2/2τD = \ell^2/2\tau and the root-mean-square displacement after 1.0 s1.0\ \mathrm{s}.

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