Long-time behavior and dynamic scaling properties of displacement fluctuations in the sine-Gordon chain

The long-time behavior of the displacement fluctuations in the sine-Gordon chain is studied in terms of the exponents $c$ and $d$, characterizing the asymptotic time dependence of the meansquare displacement $〈{\left[X_l\mathrm{}\right(t)-X_l\mathrm{}(0\left)\right]}^2〉\sim t^c$ and of the correlation function $〈{\left\{\right(\frac{1}{\sqrt{N}})\times \Sigma l^{}\left[X_l\mathrm{}\right(t)-X_l\mathrm{}(0\left)\right]\}}^2〉\sim t^d$. Invoking the dynamic scaling hypothesis, it is shown that $d=2c=\frac{2}{b}$, and the bounds $\frac{1}{2}<~\frac{1}{b}<~1$ are derived for the Hamiltonian case. The relativistic ideal-kink-gas picture leads to $\frac{1}{b}=1\mathrm{}$. For a chain evolving according to a Langevin equation with large damping $\frac{1}{b}=\frac{1}{2}$ is obtained.