Relaxation in systems with hierarchical organization: Analytical derivation of the relaxation and dispersion functions

We report a successful attempt to derive a closed-form expression for the relaxation function of a complex system by solving a set of coupled kinetic equations connecting the excitation/de-excitation rates to the number of particles (such as electrical charges, dipoles, etc.) in excited states. Our approach, originating from developments in dielectric and mechanical relaxation studies, allows the use of a unified treatment for a wide array of natural processes which often pose challenges to theoretical modeling. We use the notions that (i) a dipole represents any pair formed by a particle in an excited state (such as an energy level in optically excited molecules, or an electrode in dielectric spectroscopy) and its image in the ground state (or reference electrode), that (ii) coupling between such dipoles may be described as particle transfer from one excited state to another with lower energy, and that (iii) the relaxation function for such a system of dipoles is mathematically equivalent to the cumulative distribution function of particles, i.e., the total number of particles that are still in an excited state at a time t following excitation. Taken together, these ideas naturally lead to the identification of two types of relaxation – parallel and serial relaxation – and allow one to tackle systems with either geometrical or physical self-similarity within a unified mathematical scheme.