Relaxation at critical points: Deterministic and stochastic theory

A generalized critical point can be characterized by non-linear dynamics. We formulate the deterministic and stochastic theory of relaxation at such a point. Canonical problems are used to motivate the general solutions. In the deterministic theory, we show that at the critical point certain modes have polynomial (rather than exponential) growth or decay. The stochastic relaxation rates can be calculated in terms of various incomplete special functions. Three examples are considered. First, a substrate inhibited reaction (marginal type dynamical system) is treated. Second, the relaxation of a mean field ferromagnet is considered. We obtain a result that generalizes the work of Griffiths et al. Third, we study the relaxation of a critical harmonic oscillator.