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DIFFUSION IN AN EXTERNAL POTENTIAL Here we consider diffusion in an external potential. This is an important issue in physical chemistry and biology. The external potential can be gravity in connection with sedimentation or the electric potential in connection with electrolytes etc. We also derive Einsteins relation in another way. The diffusing particles are subject to an external force fext(x) and a viscous drag force fd. The external force is given by fext(x) = −∇U (x) where U (x) is the external potential. The drag force is given by Stokes law fd = −αvd Here vd is the drift velocity and the friction constant is given by α = 6πηR where η is the viscosity of the medium and R the radius of the sphere. In a steady state the external force and the drag force must balance, i.e., fd + fext = 0 or −αvd + fext = 0 or f 1 vd = ext = − ∇U α α The drift current is given by jd = nvd where n is the density. The diffusion current is given by Fick’s law jdiff = −D∇n The total current is then given by the diffusion current plus the drift current, i.e., jtot = jdiff + jd or inserting 1 jtot = −D∇n − n ∇U α In thermodynamic equilibrium jtot = 0 and the equilibrium density neq(x) is given by the Boltzmann distribution neq(x) ∝ e−U (x)/kT i.e., 1 −D∇neq − neq∇U = 0 α or 1 −U/kT D −U/kT ∇U e − e ∇U = 0 kT α This equation is satisfied provided kT kT D= = α 6πηa which is Einstein’s relation connecting D to microscopics (Einstein 1905)