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 〈[Xₗ(t)-Xₗ(0)]²〉~tᶜ and of the correlation function 〈(1√N)×Σl^[Xₗ(t)-Xₗ(0)]²〉~tᵈ. Invoking the dynamic scaling hypothesis, it is shown that d=2c=2/b, and the bounds 1/2<~1/b<~1 are derived for the Hamiltonian case. The relativistic ideal-kink-gas picture leads to 1/b=1. For a chain evolving according to a Langevin equation with large damping 1/b=1/2 is obtained.