This equation is an early example of a
fluctuation-dissipation relation.[7]
Note that the equation above describes the classical case and should be modified when quantum effects are relevant.
Two frequently used important special forms of the relation are:
Einstein–Smoluchowski equation, for diffusion of
charged particles:[8]
Stokes–Einstein–Sutherland equation, for diffusion of spherical particles through a liquid with low
Reynolds number:
For a particle with
electrical chargeq, its
electrical mobilityμq is related to its generalized mobility μ by the equation μ = μq/q. The parameter μq is the ratio of the particle's terminal
drift velocity to an applied
electric field. Hence, the equation in the case of a charged particle is given as
In the limit of low
Reynolds number, the mobility μ is the inverse of the drag coefficient . A damping constant is frequently used for the inverse momentum relaxation time (time needed for the inertia momentum to become negligible compared to the random momenta) of the diffusive object. For spherical particles of radius r,
Stokes' law gives
where is the
viscosity of the medium. Thus the Einstein–Smoluchowski relation results into the Stokes–Einstein–Sutherland relation
This has been applied for many years to estimating the self-diffusion coefficient in liquids, and a version consistent with isomorph theory has been confirmed by computer simulations of the
Lennard-Jones system.[10]
In the case of
rotational diffusion, the friction is , and the rotational diffusion constant is
This is sometimes referred to as the Stokes–Einstein–Debye relation.
By replacing the diffusivities in the expressions of electric ionic mobilities of the cations and anions from the expressions of the
equivalent conductivity of an electrolyte the Nernst–Einstein equation is derived:
were R is the
gas constant.
Proof of the general case
The proof of the Einstein relation can be found in many references, for example see the work of
Ryogo Kubo.[13]
Suppose some fixed, external
potential energy generates a
conservative force (for example, an electric force) on a particle located at a given position . We assume that the particle would respond by moving with velocity (see
Drag (physics)). Now assume that there are a large number of such particles, with local concentration as a function of the position. After some time, equilibrium will be established: particles will pile up around the areas with lowest potential energy , but still will be spread out to some extent because of
diffusion. At equilibrium, there is no net flow of particles: the tendency of particles to get pulled towards lower , called the drift current, perfectly balances the tendency of particles to spread out due to diffusion, called the diffusion current (see
drift-diffusion equation).
The net flux of particles due to the drift current is
i.e., the number of particles flowing past a given position equals the particle concentration times the average velocity.
The flow of particles due to the diffusion current is, by
Fick's law,
where the minus sign means that particles flow from higher to lower concentration.
Now consider the equilibrium condition. First, there is no net flow, i.e. . Second, for non-interacting point particles, the equilibrium density is solely a function of the local potential energy , i.e. if two locations have the same then they will also have the same (e.g. see
Maxwell-Boltzmann statistics as discussed below.) That means, applying the
chain rule,
Therefore, at equilibrium:
As this expression holds at every position , it implies the general form of the Einstein relation: