Exact Exponent $λ$ of the Autocorrelation Function for a Soluble Model of Coarsening

The exponent $\lambda$ that describes the decay of the autocorrelation function $A(t)$ in a phase ordering system, $A(t) \sim L^{-(d-\lambda)}$, 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 $\lambda = 0.399\,383\,5\ldots$. We also show explicitly that a small bias of positive domains over negative gives a magnetization which grows in time as $M(t) \sim L^\mu$ and prove that for the $1d$ Ginzburg-Landau equation, $\mu=\lambda$, exemplifying a general result.