The growth of order in Ising models with nonconserved order parameter is considered for quenches to final temperatures T f =0 and T f =T c . The results of numerical simulations in spatial dimension d=2 are presented. In all cases a scaling regime is entered for sufficiently long times, where the characteristic length scale is the 'domain size' L(t) approximately t 1 2/, for T f =0, and the 'nonequilibrium correlation length', xi (t) approximately t 1 z/, for T f =T c . The equal-time correlation function has the expected scaling forms f(r/L(t)) and r -(d-2+ eta ) f c (r/ xi (t)) for T f =0 and T c respectively. The scaling function f c (x) has interesting short-distance behaviour which is elucidated using scaling arguments and by in - and 1/n-expansions. The T=0 scaling function f(x) depends on whether the spin correlations present in the initial conditions are of long or short range, as does the exponent which describes the decay of the autocorrelation function, A(t)=((S i (t))S i (O)) approximately L(t) - . Results for a quench from the equilibrium critical state to T f =0 are consistent with theoretical predictions.
We have investigated the nonequilibrium critical dynamics of a two-dimensional, spin-exchange (i.e., conserved magnetization), kinetic Ising model following quenches to
Tc from random initial states. As in the case of subcritical quenches, we observe dynamical scaling of the equal time order parameter correlation function G(r,t,t), but here the nonequilibrium correlation length grows as L(t)\ensuremath{\sim}
t1/z, where z=4/15 is the usual dynamic critical exponent for this system. Moreover, we find dynamical scaling for correlations with the initial condition G(r,0,t). The temporal decay of autocorrelations at
Tc yields a critical exponent ${\ensuremath{\lambda}}_{\mathit{c}}$=1.96\ifmmode\pm\else\textpm\fi{}0.05 for the nonequilibrium dynamics: G(r,0,t)\ensuremath{\approxeq}${\mathit{L}}_{\mathit{c}}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}$(t)G\ifmmode \tilde{}\else \~{}\fi{}(r/L(t)).
We extend standard finite-size scaling methods to study the dynamical evolution of an unstable state far from equilibrium as the system undergoes a first order phase transition. We suggest that the nonequilibrium structure factor S(q, t, L), at late times and for large enough lattice sizes, scales as S(q, t, L)=LdF(qL, t1/x/L). L is the linear dimension of the system and 1/x is the domain growth exponent. We obtain x=2 in the case of the kinetic Ising model with a nonconserved order parameter. For a critical quench in a system with conserved order parameter, scaling of the peak of the structure factor gives 1/x≈0.27. Higher wavenumbers, however, are more consistent with x=3.
The nonequilibrium dynamics of classical random Ising spin chains with nonconserved magnetization are studied using an asymptotically exact real space renormalization group (RSRG). We focus on random field Ising model (RFIM) spin chains with and without a uniform applied field, as well as on Ising spin glass chains in an applied field. For the RFIM we consider a universal regime where the random field and the temperature are both much smaller than the exchange coupling. In this regime, the Imry-Ma length that sets the scale of the equilibrium correlations is large and the coarsening of domains from random initial conditions (e.g., a quench from high temperature) occurs over a wide range of length scales. The two types of domain walls that occur diffuse in opposite random potentials, of the form studied by Sinai, and domain walls annihilate when they meet. Using the RSRG we compute many universal asymptotic properties of both the nonequilibrium dynamics and the equilibrium limit. We find that the configurations of the domain walls converge rapidly toward a set of system-specific time-dependent positions that are independent of the initial conditions. Thus the behavior of this nonequilibrium system is pseudodeterministic at long times because of the broad distributions of barriers that occur on the long length scales involved. Specifically, we obtain the time dependence of the energy, the magnetization, and the distribution of domain sizes (found to be statistically independent). The equilibrium limits agree with known exact results. We obtain the exact scaling form of the two-point equal time correlation function <S0(t)Sx(t)> and the two-time autocorrelations <S0(t('))S0(t)>. We also compute the persistence properties of a single spin, of local magnetization, and of domains. The analogous quantities for the +/-J Ising spin glass in an applied field are obtained from the RFIM via a gauge transformation. In addition to these we compute the two-point two-time correlation function <S0(t)Sx(t)><S0(t('))Sx(t('))> which can in principle be measured by experiments on spin-glass-like systems. The thermal fluctuations are studied and found to be dominated by rare events; in particular all moments of truncated equal time correlations are computed. Physical properties which are typically measured in aging experiments are also studied, focusing on the response to a small magnetic field which is applied after waiting for the system to equilibrate for a time t(w). The nonequilibrium fluctuation-dissipation ratio X(t,t(w)) is computed. We find that for (t-t(w)) approximately t(alpha)(w) with alpha;<1, the ratio equal to its equilibrium value X=1, although time translational invariance does not hold in this regime. For t-t(w) approximately t(w) the ratio exhibits an aging regime with a nontrivial X=X(t/t(w)) not equal to 1, but the behavior is markedly different from mean field theory. Finally the distribution of the total magnetization and of the number of domains is computed for large finite size systems. General issues about convergence toward equilibrium and the possibilities of weakly history-dependent evolution in other random systems are discussed.
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
TI, that is suddenly placed in contact with a heat bath at temperature
TF, with
TF\ensuremath{\gg}
TI. We present detailed results for the case of
TF=\ensuremath{\infty}, for which we have succeeded in calculating the exact form of S(q,t). For finite
TF, 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
TI 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
TF, \ensuremath{\chi}(q,
TF), in that S(q,t)/\ensuremath{\chi}(q,
TF) 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
TI. Analogous to the behavior for antiferromagnetic interactions, away from the center of the BZ, we find that S(q,t)/\ensuremath{\chi}(q,
TF) is independent of q and is a function of t/
tw, very slowly approaching unity. The characteristic ``waiting time''
tw is anomalously long, proportional to ${\ensuremath{\xi}}^{2}$, where \ensuremath{\xi} is the equilibrium correlation length at temperature
TI. This behavior of
tw can be related to the random walk of domain boundaries. \textcopyright{} 1996 The American Physical Society.