Nonequilibrium structure factor for conserved spin dynamics: Abrupt temperature increase.

Abstract

We consider the nonequilibrium, elastic-scattering structure factor S(q,t) (q denotes the wave vector, t the time), for the Kawasaki spin-conserving kinetic Ising model of a one-dimensional system with nearest-neighbor interactions, initially in equilibrium at temperature ${\mathit{T}}_{\mathit{I}}$, that is suddenly placed in contact with a heat bath at temperature ${\mathit{T}}_{\mathit{F}}$, with ${\mathit{T}}_{\mathit{F}}$\ensuremath{\gg}${\mathit{T}}_{\mathit{I}}$. We present detailed results for the case of ${\mathit{T}}_{\mathit{F}}$=\ensuremath{\infty}, for which we have succeeded in calculating the exact form of S(q,t). For finite ${\mathit{T}}_{\mathit{F}}$, we present an approximation scheme for the higher-order nonequilibrium correlation functions that leads to closure of the hierarchy of equations of motion. The merits of this approximation are that (i) S(q,t) is guaranteed to satisfy an exact sum rule over the Brillouin zone (BZ) of wave vectors q, and (ii) S(q,t) evolves to the correct value in the long-time limit. For antiferromagnetic coupling, the structure factor, initially dominated by the Bragg peak associated with ${\mathit{T}}_{\mathit{I}}$ at the edge of the BZ, decays exponentially with time, ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathit{t}/{\mathrm{\ensuremath{\tau}}}_{\mathit{q}}}$ while approximately preserving its shape in q space, since the lifetime ${\mathrm{\ensuremath{\tau}}}_{\mathit{q}}$ is nearly independent of q. Except near the center of the BZ, after the Bragg peak has decayed sufficiently, the dependence of S(q,t) on q can be characterized as though the spins rapidly quasiequilibrate to the equilibrium structure factor associated with ${\mathit{T}}_{\mathit{F}}$, \ensuremath{\chi}(q,${\mathit{T}}_{\mathit{F}}$), in that S(q,t)/\ensuremath{\chi}(q,${\mathit{T}}_{\mathit{F}}$) is independent of q, but is time dependent, slowly approaching unity as ${\mathit{t}}^{\mathrm{\ensuremath{-}}1/2}$ for large t. For q\ensuremath{\approxeq}0 the initial form of S remains in effect until the value of t is of order ${\mathit{q}}^{\mathrm{\ensuremath{-}}2}$. For ferromagnetic coupling, the initial Bragg peak for q\ensuremath{\approxeq}0 does not preserve its shape while decaying exponentially, since the lifetime ${\mathrm{\ensuremath{\tau}}}_{\mathit{q}}$ strongly depends on the wave-vector q, diverging as ${\mathit{q}}^{\mathrm{\ensuremath{-}}2}$ for q\ensuremath{\rightarrow}0, and, in particular, it is as though the spins for q\ensuremath{\approxeq}0 remain ``frozen'' at ${\mathit{T}}_{\mathit{I}}$. Analogous to the behavior for antiferromagnetic interactions, away from the center of the BZ, we find that S(q,t)/\ensuremath{\chi}(q,${\mathit{T}}_{\mathit{F}}$) is independent of q and is a function of t/${\mathit{t}}_{\mathit{w}}$, very slowly approaching unity. The characteristic ``waiting time'' ${\mathit{t}}_{\mathit{w}}$ is anomalously long, proportional to ${\ensuremath{\xi}}^{2}$, where \ensuremath{\xi} is the equilibrium correlation length at temperature ${\mathit{T}}_{\mathit{I}}$. This behavior of ${\mathit{t}}_{\mathit{w}}$ can be related to the random walk of domain boundaries. \textcopyright{} 1996 The American Physical Society.