Model a soot particle as a sphere of radius and density suspended in air 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

Model a soot particle as a sphere of radius a=1.0 μma=1.0\,\mu\mathrm{m} and density ρ=1800 kg m−3\rho=1800\,\mathrm{kg\,m^{-3}} suspended in air at T=293 KT=293\,\mathrm{K}, with air's viscosity η=1.81×10−5 Pa s\eta=1.81\times10^{-5}\,\mathrm{Pa\,s} giving Stokes drag γ=6πηa\gamma=6\pi\eta a. Stokes's law treats the air as a continuum, which needs the particle to be large against the mean free path of the air molecules, about 65 nm65\,\mathrm{nm}; at this radius the true drag is about 88 per cent below the Stokes value, and we neglect that. In the Langevin model one horizontal component of the velocity obeys m dv=−γv dt+b dWm\,\mathrm{d}v = -\gamma v\,\mathrm{d}t + b\,\mathrm{d}W, with bb the noise strength and dW\mathrm{d}W a Wiener increment of variance dt\mathrm{d}t. The stationary variance of vv is b2/2mγb^2/2m\gamma. ((a)) Using equipartition, ⟨v2⟩=kBT/m\langle v^2\rangle = k_{\rm B}T/m at equilibrium, derive the noise strength bb in terms of γ\gamma, kBk_{\rm B} and TT, and state explicitly how bb depends on the drag coefficient γ\gamma. ((b)) Compute γ\gamma, the particle's mass mm (treating it as a uniform sphere), the velocity relaxation time τv=m/γ\tau_v = m/\gamma, and the noise strength bb for the stated soot particle in air. ((c)) Compute the diffusion coefficient D=kBT/γD=k_{\rm B}T/\gamma and the root-mean-square displacement along a horizontal axis after t=1.0 st=1.0\,\mathrm{s}. Verify that t=1.0 st=1.0\,\mathrm{s} lies deep in the diffusive regime by comparing it with τv\tau_v.

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