Theory of the relaxation and fluctuation from unstable and metastable states

An extension of the usual system-size expansion method is given to study the relaxation and fluctuation from unstable and metastable states. The short-time and the long-time behavior of the moments y(t) and Mn(t) and of the distribution function P(x, t) of a macroscopic variable x are investigated by using the generating function G(α, β; t) = δ(α − y(t))Π∞n = 2δ(βn − Mn(t)), where y(t) = ∫ xP(x, t) dx and Mn(t) ≡ ∫ (x − y(t))nP(x, t) dx. The dominant part of the time evolution of G(α, β; t) is extracted by introducing a scale transformation of β. The theory is applied to the one-dimensional laser model.