Spectral Decomposition of a Fokker–Planck Equation at Criticality

The mean field for a complex network consisting of a large but finite number of random two-state elements, $$M$$M, has been shown to satisfy a nonlinear Langevin equation. The noise intensity is inversely proportional to $$\sqrt{M} $$M. In the limiting case $$M = \infty $$M=∞, the solution to the Langevin equation exhibits a transition from exponential to inverse power law relaxation as criticality is approached from above or below the critical point. When $$M < \infty $$M<∞, the inverse power law is truncated by an exponential decay with rate $$\varGamma $$Γ, the evaluation of which is the main purpose of this article. An analytic/numeric approach is used to obtain the lowest-order eigenvalues in the spectral decomposition of the solution to the corresponding Fokker–Planck equation and its equivalent Schrödinger equation representation.