A Langevin-dynamics simulation of a Brownian particle records its velocity every

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement

Scientific Computing for Physicists, Ch The uncertainty of a simulated number: inputs, sensitivity and correlated output

Problem

\Delta t = 0.02\,\mathrm{s}

, for

N = 65536

samples. The sample variance of the series is

\sigma^2 = 1.5\,\mathrm{m^2\,s^{-2}}

. A one-sided Welch power spectral density estimate of the series (periodogram method, segments averaged as in the spectral-estimation chapter) gives a flat low-frequency plateau

S(0) = 1.14\,\mathrm{m^2\,s^{-2}\,s}

. Separately, a grid-convergence study of the integrator (Richardson extrapolation from runs at two time steps, assuming the integrator's formal order

p = 2

, following the solution-verification method) gives a numerical uncertainty on the mean velocity of

u_{\rm num} = 0.0080\,\mathrm{m\,s^{-1}}

. No input parameter of the simulation carries a stated distribution here. ((a)) Using

s = S(0)/(2\sigma^2\Delta t)

, find the statistical inefficiency

s

of the velocity series. ((b)) Find the effective sample size

N_{\rm eff}

and the statistical uncertainty

u_{\rm stat}

of the mean velocity. ((c)) Assemble the combined uncertainty

u_{\rm sim}

of the simulated mean velocity from its numerical and statistical contributions, treating them as independent.

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