An overdamped particle diffusing in a one-dimensional harmonic potential has drift…

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

An overdamped particle diffusing in a one-dimensional harmonic potential U(x)=12kx2U(x)=\tfrac12 kx^2 has drift A(x)=−U′(x)/γ=−kx/γA(x)=-U'(x)/\gamma = -kx/\gamma and constant diffusion coefficient D=kBT/γD=k_{\rm B}T/\gamma, obeying the Fokker-Planck equation ∂tP=−∂x[A(x)P]+D ∂x2P\partial_t P = -\partial_x[A(x)P]+D\,\partial_x^2P. ((a)) From the stationary, zero-current condition J(x)=A(x)P(x)−D dP/dx=0J(x)=A(x)P(x)-D\,\mathrm{d}P/\mathrm{d}x = 0, find P(x)P(x) up to a normalisation constant, show that it is the Boltzmann distribution P(x)∝e−U(x)/kBTP(x)\propto e^{-U(x)/k_{\rm B}T}, a Gaussian of variance σ2=kBT/k\sigma^2 = k_{\rm B}T/k, and note that γ\gamma cancels. ((b)) A bead is released at the trap centre at t=0t=0. Use the moment equation d⟨x2⟩/dt=2⟨xA⟩+2D\mathrm{d}\langle x^2\rangle/\mathrm{d}t=2\langle xA\rangle+2D of the Fokker-Planck equation to find ⟨x2⟩(t)\langle x^2\rangle(t). Show that at short times it grows as 2Dt2Dt, as for a free bead, and that it approaches σ2\sigma^2 with a time constant you should identify. Explain why γ\gamma appears here but not in (a). ((c)) The silica sphere of this chapter, with γ=9.44×10−9 kg s−1\gamma=9.44\times10^{-9}\,\mathrm{kg\,s^{-1}} in water at T=293 KT=293\,\mathrm{K}, is held in an optical trap of spring constant k=5.0×10−6 N m−1k=5.0\times10^{-6}\,\mathrm{N\,m^{-1}}. Compute the root-mean-square displacement σ\sigma and the time constant of (b).

Browse the problem library