A polystyrene bead of radius and density is suspended in water at

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Non-equilibrium statistical mechanics and stochastic thermodynamics · Non-equilibrium statistical mechanics and stochastic thermodynamics

Error, Chance and the Machine, Ch Random processes in continuous time

Problem

A polystyrene bead of radius a=0.3 μma = 0.3\,\mu\mathrm{m} and density ρ=1050 kg m−3\rho = 1050\,\mathrm{kg\,m^{-3}} is suspended in water at T=298 KT = 298\,\mathrm{K}, viscosity η=0.89×10−3 Pa s\eta = 0.89\times10^{-3}\,\mathrm{Pa\,s}. In the Langevin model (which neglects the inertia of the surrounding fluid), the drag coefficient is γ=6πηa\gamma = 6\pi\eta a and the diffusion coefficient is D=kBT/γD = k_{\rm B}T/\gamma. ((a)) Compute the drag coefficient γ\gamma. ((b)) Compute the diffusion coefficient DD. ((c)) Compute the velocity relaxation time τv=m/γ\tau_v = m/\gamma, taking the bead's mass from its volume and density, and state how this compares to a typical laboratory video frame time of 1 ms1\,\mathrm{ms}.

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