A colloidal bead held in an optical trap follows an overdamped Ornstein-Uhlenbeck…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Numerical and computational mathematics

Error, Chance and the Machine, Ch Ordinary differential equations: order, stability and stiffness

Problem

A colloidal bead held in an optical trap follows an overdamped Ornstein-Uhlenbeck process for its position, with relaxation time τ=5 ms\tau=5\,\text{ms} and true stationary variance σ2=5.125×10−15 m2\sigma^2 = 5.125\times10^{-15}\,\text{m}^2. Its position is simulated with the Euler-Maruyama method at fixed step h=1 msh=1\,\text{ms}. ((a)) Predict the stationary variance the simulation will settle to, using σEM2=σ2/(1−h/2τ)\sigma^2_{\rm EM} = \sigma^2/(1-h/2\tau). ((b)) Express the bias as a percentage of the true variance and state whether the simulated bead looks too hot or too cold.

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