Exact exponent λ of the autocorrelation function for a soluble model of coarsening

The exponent λ that describes the decay of the autocorrelation function A(t) in a phase ordering system, A(t)∼${\mathit{L}}^{\mathrm{-}(\mathit{d}\mathrm{-}\lambda )}$, where d is the dimension and L the characteristic length scale at time t, is calculated exactly for the time-dependent Ginzburg-Landau equation in d=1. We find λ=0.3993835.... We also show explicitly that a small bias of positive domains over negative gives a magnetization which grows in time as M(t)∼${\mathit{L}}^{\mathrm{\mu }}$ and prove that for the one-dimensional Ginzburg-Landau equation, μ=λ, exemplifying a general result.