Passage-time statistics for the decay of unstable equilibrium states

The decay of unstable equilibrium states is accompanied by large-scale fluctuations. The statistical properties of such processes can be characterized by using the time at which a representative observable first passes through a fixed threshold value. We present an asymptotic probability distribution for that passage time which is valid when the threshold is set sufficiently far from the initial state. For the simplest example of linear isotropic amplification of an $n$-component vector we calculate both the exact first-passage time distribution and our asymptotic distribution. We verify that the asymptotic distribution coincides with the exact one in the appropriate limit. We then evaluate our asymptotic distribution for a number of more complicated systems including one in which an $n$-component vector field in $d$ spatial dimensions departs from an unstable equilibrium state. The resulting expression has a considerable degree of universality. Its form is independent of $d$ and of details of the field dynamics. It is insensitive, in particular, to whether the underlying field considered is conserved or not. Our procedure is applicable to a wide variety of problems in which an order parameter departs spontaneously from an unstable initial value.