Smoluchowski Equation for Potentials: Extremum Principle and Spectral Expansion

In this section we will consider the general properties of the solution p(r, t) of the Smoluchowski equation in the case that the force field is derived from a potential, i.e., F (r) = −∇U(r) and that the potential is finite everywhere in the diffusion doamain Ω. We will demonstrate first that the solutions, in case of reaction-free boundary conditions, obey an extremum principle, namely, that during the time evolution of p(r, t) the total free energy decreases until it reaches a minimum value corresponding to the Boltzmann distribution. We will then characterize the time-evolution through a so-called spectral expansion, i.e., an expansion in terms of eigenfunctions of the Smoluchowski operator L(r). Since this operator is not self-adjoint, expressed through the fact that, except for free diffusion, the adjoint operator L†(r) as given by (9.22) or (9.38) is not equal to L(r), the existence of appropriate eigenvalues and eigenfunctions is not evident. However, in the present case [F (r) = −∇U(r)] the operators L(r) and L†(r) are similar to a self-adjoint operator for which a complete set of orthonormal eigenfunctions exist. These functions and their associated eigenvalues can be transferred to L(r) and L†(r) and a spectral expansion can be constructed. The expansion will be formulated in terms of projection operators and the so-called propagator, which corresponds to the solutions p(r, t|r0, t0), will be stated in a general form. A pointed out, we consider in this chapter specifically solutions of the Smoluchowski equation