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Physical review. B, Condensed matter

Dynamical scaling and decay of correlations for spinodal decomposition at Tc

Francis J. Alexander, David A. Huse, Steven A. Janowsky

Published Jul 1, 1994• 16 citations
Publisher

Abstract

We have investigated the nonequilibrium critical dynamics of a two-dimensional, spin-exchange (i.e., conserved magnetization), kinetic Ising model following quenches to ${\mathit{T}}_{\mathit{c}}$ 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}${\mathit{t}}^{1/\mathit{z}}$, 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 ${\mathit{T}}_{\mathit{c}}$ 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)).

DOI
10.1103/physrevb.50.663

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We have investigated the nonequilibrium critical dynamics of a two-dimensional, spin-exchange (i.e., conserved magnetization), kinetic Ising model following quenches to Tc{\mathit{T}}_{\mathit{c}}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{\mathit{t}}^{1/\mathit{z}}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{\mathit{T}}_{\mathit{c}}Tc​ yields a critical exponent \ensuremathλc{\ensuremath{\lambda}}_{\mathit{c}}\ensuremathλc​=1.96\ifmmode\pm\else\textpm\fi{}0.05 for the nonequilibrium dynamics: G(r,0,t)\ensuremath{\approxeq}Lc\ensuremath−\ensuremathλ{\mathit{L}}_{\mathit{c}}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}Lc\ensuremath−\ensuremathλ​(t)G\ifmmode \tilde{}\else ~{}\fi{}(r/L(t)).
A quench of a d-dimensional spin system from a random initial configuration, {Si{\mathrm{S}}_{i}Si​(0)}, to a critical point is considered. The decay with time t of the autocorrelation with the initial condition is q0{q}_{0}q0​(t)==〈Si{\mathrm{S}}_{i}Si​(0)\ensuremath{\cdot}Si{\mathrm{S}}_{i}Si​(t)〉\ensuremath{\sim}${t}_{\mathit{c}}^{\ensuremath{-}\ensuremath{\lambda}}$/z, where z is the usual dynamic critical exponent. Naively, ${\ensuremath{\lambda}}_{c}$=d, but I find ${\ensuremath{\lambda}}_{c}$simulations of pure Ising models in d=2, 3 and the \ifmmode\pm\else\textpm\fi{}J Ising spin glass in d=3. This suggests that ${\ensuremath{\lambda}}_{c}$ is a new critical exponent for nonequilibrium dynamics. For a spin glass the decay of q0{q}_{0}q0​(t) is the same as that of the remanent magnetization; the exponent ${\ensuremath{\lambda}}_{c}$/z observed in the spin-glass simulation is in good agreement with a recent experimental measurement by Granberg et al.
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 study the autocorrelation function of a conserved spin system following a quench at the critical temperature. Defining the correlation length $L(t)\ensuremath{\sim}{t}^{1/z}$, we find that for times ${t}^{\ensuremath{'}}$ and ttt satisfying $L({t}^{\ensuremath{'}})\ensuremath{\ll}L(t)\ensuremath{\ll}L({t}^{\ensuremath{'}}{)}^{\ensuremath{\phi}}$ well inside the scaling regime, the spin autocorrelation function behaves like $\ensuremath{\langle}s(t)s({t}^{\ensuremath{'}})\ensuremath{\rangle}\ensuremath{\sim}L({t}^{\ensuremath{'}}{)}^{\ensuremath{-}(d\ensuremath{-}2+\ensuremath{\eta})}[L({t}^{\ensuremath{'}})/L(t){]}^{{\ensuremath{\lambda}}_{c}^{\ensuremath{'}}}$. For the O(n)O(n)O(n) model in the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, we show that ${\ensuremath{\lambda}}_{c}^{\ensuremath{'}}=d+2$ and $\ensuremath{\phi}=z/2$. We give a heuristic argument suggesting that this result is, in fact, valid for any dimension ddd and spin vector dimension nnn. We present numerical simulations for the conserved Ising model in d=1d=1d=1 and d=2d=2d=2, which are fully consistent with the present theory.
We study the nonequilibrium dynamics of the q-state Potts model following a quench from the high-temperature disordered phase to zero temperature. The time-dependent two-point correlation functions of the order parameter field satisfy dynamic scaling with a length scale L(t)\ensuremath{\sim}t1/2{\mathit{t}}^{1/2}t1/2. In particular, the autocorrelation function decays as [L(t)${]}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}(\mathit{q})}$. We illustrate these properties by solving exactly the kinetic Potts model in d=1. We then analyze a Langevin equation of an appropriate field theory to compute these correlation functions for general q and d. We establish a correspondence between the two-point correlations of the q-state Potts model and those of a kinetic Ising model evolving with a fixed magnetization (2/q-1). The dynamics of this Ising model is solved exactly in the large q limit and in the limit of a large number of components n for the order parameter. For general q and in any dimension, we introduce a Gaussian closure approximation and calculate within this approximation the scaling functions and the exponent \ensuremath{\lambda}(q). These are in good agreement with the direct numerical simulations of the Potts model as well as the kinetic Ising model with fixed magnetization. We also discuss the existing and possible experimental realizations of these models.
The dynamics of spin fluctuations in nonequilibrium and equilibrium states is studied in the one-phase region above Tc{\mathit{T}}_{\mathit{c}}Tc​. The equal-time structure factor is computed after quenching the system from an infinite temperature to T>Tc{\mathit{T}}_{\mathit{c}}Tc​. The result shows growing spatial patterns which can be scaled with temperature-dependent scaling functions. The different-time (dynamic) structure factor is computed by taking time averages while the system is fluctuating in equilibrium states. The equilibrium autocorrelation decays following a diffusive exponential term and a slower but transient term. The relaxation rate due to the diffusion supports the dynamic scaling.
We study the nonequilibrium dynamics of the qqq-state Potts model following a quench from the high-temperature disordered phase to zero temperature. We calculate, within a Gaussian closure approximation, the time-dependent two-point correlation functions for general qqq. These correlation functions obey dynamic scaling with a length scale $L(t)\ensuremath{\sim}{t}^{1/2}$, while the autocorrelation function decays as $L(t{)}^{\ensuremath{-}\ensuremath{\lambda}(q)}$. We also establish a correspondence of this model to the Ising model evolving with a fixed magnetization $(2/q\ensuremath{-}1)$. Extensive numerical simulations of the Potts model in two dimensions show good agreement with our theory.