Smoluchowski equation

In physics, the diffusion equation with drift term is often called Smoluchowski equation (after Marian von Smoluchowski[1]).

Contents

The equation

Let w(r, t) be a density, D a diffusion constant, ζ a friction coefficient, and U(r, t) a potential. Then the Smoluchowski equation states that the density evolves according to

\frac{\partial w}{\partial t} = D \nabla^2w %2B \zeta^{-1}\nabla\cdot(w\nabla U)

The diffusivity term D \nabla^2 w acts to smoothen out the density, while the drift term \zeta^{-1} \nabla \cdot (w \nabla U) shifts the density towards regions of low potential U. The equation is consistent with each particle moving according to a stochastic differential equation, with a bias term -\zeta^{-1} \nabla U and a diffusivity D. Physically, the drift term originates from a force -\nabla U being balanced by a viscous drag given by \zeta.

The Smoluchowski equation is formally identical to the Fokker–Planck equation, the only difference being the physical meaning of w: a distribution of particles in space for the Smoluchowski equation, a distribution of particle velocities for the Fokker–Planck equation.

Steady-state solution

A time-invariant solution to the Smoluchowski equation is

w(x) = C .\exp( - D^{-1} \zeta^{-1} U(x))

where C is a normalization constant. Thus, in steady-state, concentration highs are found in potential lows, and that the accumulation is more pronounced when diffusivity D and/or the friction \zeta is low. This distribution is formally identical to the Gibbs canonical distribution in statistical physics, and very closely related to the Boltzmann distribution.

Other equations named for Smoluchowski

In the first half of the 20th century, a number of different equations were referred to as Smoluchowski's. In an influential review,[2] Chandrasekhar stated in 1943 that the diffusion equation with drift term "is sometimes called Smoluchowski equation". Since then, this has become the standard nomenclature.

Other equations due to Smoluchowski are now usually given more specific names:

References

  1. ^ M. v. Smoluchowski, Über Brownsche Molekularbewegung unter Einwirkung äußerer Kräfte und den Zusammenhang mit der verallgemeinerten Diffusionsgleichung, Ann. Phys. 353 (4. Folge 48), 1103–1112 (1915) http://matwbn.icm.edu.pl/ksiazki/pms/pms2/pms2132.pdf
  2. ^ S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943), equation (312).