Relaxation dynamics of a particle in the presence of an external potential: exact solution in terms of matrix continued fractions

Exact expressions for the Laplace transform of the after effect function arising from the solution of the Langevin or underlying Fokker-Planck equation for Brownian motion in an external potential are obtained as a sum of products of infinite matrix continued fractions. This is accomplished by reducing the scalar multiterm recurrence relations associated with the Fokker-Planck equation to matrix three term recurrence relations. The solution is illustrated by considering the problem of dielectric relaxation of a single axis rotator subjected to both constant and crystalline anisotropy fields.