Modern Physics Letters B

FINITE-SIZE SCALING ANALYSIS OF THE DECAY OF AN UNSTABLE STATE DURING A FIRST ORDER PHASE TRANSITION

Jorge Viñals, David Jasnow

Published Apr 1, 1988• 4 citations

Abstract

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.

DOI: 10.1142/s0217984988000126

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Modern Physics Letters B

FINITE-SIZE SCALING ANALYSIS OF THE DECAY OF AN UNSTABLE STATE DURING A FIRST ORDER PHASE TRANSITION

Jorge Viñals, David Jasnow

Published Apr 1, 1988• 4 citations

Publisher

Abstract

DOI: 10.1142/s0217984988000126

Related papers

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The effect of random impurities (quenched disorder) on the growth of correlations is studied for model A and model B after a sudden quench to ${\mathit{T}}_{\mathit{c}}$ from the high-temperature phase (i.e., random initial conditions). Exponents and scaling functions of the nonequilibrium dynamic response function ${\mathit{G}}_{\mathit{k}}$ (t)=[〈\ensuremath{\partial}${\mathrm{\ensuremath{\varphi}}}_{\mathbf{k}}$(t)/\ensuremath{\partial}${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{-}}\mathbf{k}}$(0)〉] and the structure factor ${\mathit{S}}_{\mathit{k}}$ (t)=[〈${\mathrm{\ensuremath{\varphi}}}_{\mathbf{k}}$(t)${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{-}}\mathbf{k}}$(t)〉] are calculated to first order in \ensuremath{\epsilon} (\ensuremath{\epsilon}=4-d) for the O(n) model. For a nonconserved order parameter, the scaling form ${\mathit{G}}_{\mathit{k}}$ (t)=${\mathit{t}}^{\ensuremath{\lambda}/\mathit{z}}$f( ${\mathit{k}}^{\mathit{z}}$ t) is obtained, with f(0)=const and \ensuremath{\lambda}=\ensuremath{\epsilon}/4+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$), for 1n4. For n>4, random impurities are irrelevant, and \ensuremath{\lambda}=[(n+2)/2(n+8)]\ensuremath{\epsilon}+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$), in agreement with calculations on the pure system. For a conserved order parameter \ensuremath{\lambda}=0, but the scaling function f(x) is nontrivial. For both conserved and nonconserved order parameter, disorder gives rise to algebraically decaying scaling functions.The effect of random impurities (quenched disorder) on the growth of correlations is studied for model A and model B after a sudden quench to ${\mathit{T}}_{\mathit{c}}$ from the high-temperature phase (i.e., random initial conditions). Exponents and scaling functions of the nonequilibrium dynamic response function ${\mathit{G}}_{\mathit{k}}$(t)=[〈\ensuremath{\partial}${\mathrm{\ensuremath{\varphi}}}_{\mathbf{k}}$(t)/\ensuremath{\partial}${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{-}}\mathbf{k}}$(0)〉] and the structure factor ${\mathit{S}}_{\mathit{k}}$(t)=[〈${\mathrm{\ensuremath{\varphi}}}_{\mathbf{k}}$(t)${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{-}}\mathbf{k}}$(t)〉] are calculated to first order in \ensuremath{\epsilon} (\ensuremath{\epsilon}=4-d) for the O(n) model. For a nonconserved order parameter, the scaling form ${\mathit{G}}_{\mathit{k}}$(t)=${\mathit{t}}^{\ensuremath{\lambda}/\mathit{z}}$f(${\mathit{k}}^{\mathit{z}}$t) is obtained, with f(0)=const and \ensuremath{\lambda}=\ensuremath{\epsilon}/4+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$), for 1n4. For n>4, random impurities are irrelevant, and \ensuremath{\lambda}=[(n+2)/2(n+8)]\ensuremath{\epsilon}+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$), in agreement with calculations on the pure system. For a conserved order parameter \ensuremath{\lambda}=0, but the scaling function f(x) is nontrivial. For both conserved and nonconserved order parameter, disorder gives rise to algebraically decaying scaling functions.

Physical review. B, Condensed matterAug 1, 199228 citations

The dynamics of spin fluctuations in nonequilibrium and equilibrium states is studied in the one-phase region above

{\mathit{T}}_{\mathit{c}}

. The equal-time structure factor is computed after quenching the system from an infinite temperature to T>

{\mathit{T}}_{\mathit{c}}

. 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 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)).

The effect of random impurities (quenched disorder) on the growth of correlations is studied for model A and model B after a sudden quench to

{\mathit{T}}_{\mathit{c}}

from the high-temperature phase (i.e., random initial conditions). Exponents and scaling functions of the nonequilibrium dynamic response function

{\mathit{G}}_{\mathit{k}}

(t)=[〈\ensuremath{\partial}${\mathrm{\ensuremath{\varphi}}}_{\mathbf{k}}$(t)/\ensuremath{\partial}${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{-}}\mathbf{k}}$(0)〉] and the structure factor

{\mathit{S}}_{\mathit{k}}

(t)=[〈${\mathrm{\ensuremath{\varphi}}}_{\mathbf{k}}$(t)${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{-}}\mathbf{k}}$(t)〉] are calculated to first order in \ensuremath{\epsilon} (\ensuremath{\epsilon}=4-d) for the O(n) model. For a nonconserved order parameter, the scaling form

{\mathit{G}}_{\mathit{k}}

(t)=${\mathit{t}}^{\ensuremath{\lambda}/\mathit{z}}$f(

{\mathit{k}}^{\mathit{z}}

t) is obtained, with f(0)=const and \ensuremath{\lambda}=\ensuremath{\epsilon}/4+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$), for 1n4. For n>4, random impurities are irrelevant, and \ensuremath{\lambda}=[(n+2)/2(n+8)]\ensuremath{\epsilon}+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$), in agreement with calculations on the pure system. For a conserved order parameter \ensuremath{\lambda}=0, but the scaling function f(x) is nontrivial. For both conserved and nonconserved order parameter, disorder gives rise to algebraically decaying scaling functions.