arXiv:1501.05178

Persistence in Ferromagnetic Ordering: Dependence upon initial configuration

Saikat Chakraborty, Subir K. Das

Published Jan 21, 2015• 0 citations

Abstract

We study the dynamics of ordering in ferromagnets via Monte Carlo simulations of the Ising model, employing the Glauber spin-flip mechanism, in space dimensions $d=2$ and $3$. Results for the persistence probability and the domain growth are discussed for quenches to various temperatures ($T_f$) below the critical one ($T_{c}$), from different initial temperatures $T_{i} \geq T_{c}$. In long time limit, for $T_{i} > T_{c}$, the persistence probability exhibits power-law decay with exponents $θ\simeq 0.22$ and $\simeq 0.18$ in $d=2$ and $3$, respectively. For finite $T_i$, the early time behavior is a different power-law whose life-time diverges and exponent decreases as $T_{i} \rightarrow T_{c}$. The crossover length between the two steps diverges as the equilibrium correlation length. $T_i=T_c$ is expected to provide a {\it{new universality class}} for which we obtain $θ\simeq 0.035$ in $d=2$ and $\simeq 0.10$ in $d=3$. The time dependence of the average domain size $\ell$, however, is observed to be rather insensitive to the choice of $T_i$.

DOI: 10.48550/arxiv.1501.05178

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Fractality in persistence decay and domain growth during ferromagnetic ordering: Dependence upon initial correlation

Saikat Chakraborty, Subir K. Das

arXiv:1509.07590Physical review. EMar 22, 201614 citations

Minority-spin dynamics in the nonhomogeneous Ising model: Diverging time scales and exponents

Pratik Mullick, Parongama Sen

arXiv:1608.03399Physical review. EMay 6, 20164 citations

Short-time critical dynamics of damage spreading in the two-dimensional Ising model

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Dynamic crossover in the global persistence at criticality

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Physical review. B, Condensed matterJul 1, 1988824 citations

arXiv:1501.05178

Persistence in Ferromagnetic Ordering: Dependence upon initial configuration

Saikat Chakraborty, Subir K. Das

Published Jan 21, 2015• 0 citations

arXiv HTML PDF

Abstract

We study the dynamics of ordering in ferromagnets via Monte Carlo simulations of the Ising model, employing the Glauber spin-flip mechanism, in space dimensions

d=2

and

3

. Results for the persistence probability and the domain growth are discussed for quenches to various temperatures (

T_f

) below the critical one (

T_{c}

), from different initial temperatures

T_{i} \geq T_{c}

. In long time limit, for

T_{i} > T_{c}

, the persistence probability exhibits power-law decay with exponents

θ\simeq 0.22

and

\simeq 0.18

d=2

and

3

, respectively. For finite

T_i

, the early time behavior is a different power-law whose life-time diverges and exponent decreases as

T_{i} \rightarrow T_{c}

. The crossover length between the two steps diverges as the equilibrium correlation length.

T_i=T_c

is expected to provide a {\it{new universality class}} for which we obtain

θ\simeq 0.035

d=2

and

\simeq 0.10

d=3

. The time dependence of the average domain size

\ell

, however, is observed to be rather insensitive to the choice of

T_i

DOI: 10.48550/arxiv.1501.05178

Related papers

Fractality in persistence decay and domain growth during ferromagnetic ordering: Dependence upon initial correlation

Saikat Chakraborty, Subir K. Das

The dynamics of ordering in the Ising model, following quench to zero temperature, has been studied via Glauber spin-flip Monte Carlo simulations in space dimensions d=2 and 3. One of the primary objectives has been to understand phenomena associated with the persistent spins, viz., time decay in the number of unaffected spins, growth of the corresponding pattern, and its fractal dimensionality for varying correlation length in the initial configurations, prepared at different temperatures, at and above the critical value. It is observed that the fractal dimensionality and the exponent describing the power-law decay of persistence probability are strongly dependent upon the relative values of nonequilibrium domain size and the initial equilibrium correlation length. Via appropriate scaling analyses, these quantities have been estimated for quenches from infinite and critical temperatures. The above-mentioned dependence is observed to be less pronounced in the higher dimension. In addition to these findings for the local persistence, we present results for the global persistence as well. Furthermore, important observations on the standard domain growth problem are reported. For the latter, a controversy in d=3, related to the value of the exponent for the power-law growth of the average domain size with time, has been resolved.The dynamics of ordering in the Ising model, following quench to zero temperature, has been studied via Glauber spin-flip Monte Carlo simulations in space dimensions d=2 and 3. One of the primary objectives has been to understand phenomena associated with the persistent spins, viz., time decay in the number of unaffected spins, growth of the corresponding pattern, and its fractal dimensionality for varying correlation length in the initial configurations, prepared at different temperatures, at and above the critical value. It is observed that the fractal dimensionality and the exponent describing the power-law decay of persistence probability are strongly dependent upon the relative values of nonequilibrium domain size and the initial equilibrium correlation length. Via appropriate scaling analyses, these quantities have been estimated for quenches from infinite and critical temperatures. The above-mentioned dependence is observed to be less pronounced in the higher dimension. In addition to these findings for the local persistence, we present results for the global persistence as well. Furthermore, important observations on the standard domain growth problem are reported. For the latter, a controversy in d=3, related to the value of the exponent for the power-law growth of the average domain size with time, has been resolved.

arXiv:1509.07590Physical review. EMar 22, 201614 citations

Minority-spin dynamics in the nonhomogeneous Ising model: Diverging time scales and exponents

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We investigate the dynamical behavior of the Ising model under a zero-temperature quench with the initial fraction of up spins $0\ensuremath{\le}x\ensuremath{\le}1$. In one dimension, the known results for persistence probability are verified; it shows algebraic decay for both up and down spins asymptotically with different exponents. It is found that the conventional finite-size scaling is valid here. In two dimensions, however, the persistence probabilities are no longer algebraic; in particular for $x\ensuremath{\le}0.5$, persistence for the up (minority) spins shows the behavior ${P}_{\text{min}}(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\gamma}}exp[\ensuremath{-}{(t/\ensuremath{\tau})}^{\ensuremath{\delta}}]$ with time $t$ , while for the down (majority) spins, ${P}_{\text{maj}}(t)$ approaches a finite value. We find that the timescale $\ensuremath{\tau}$ diverges as ${({x}_{c}\ensuremath{-}x)}^{\ensuremath{-}\ensuremath{\lambda}}$, where ${x}_{c}=0.5$ and $\ensuremath{\lambda}\ensuremath{\simeq}2.31$. The exponent $\ensuremath{\gamma}$ varies as ${\ensuremath{\theta}}_{2d}+{c}_{0}{({x}_{c}\ensuremath{-}x)}^{\ensuremath{\beta}}$ where ${\ensuremath{\theta}}_{2d}\ensuremath{\simeq}0.215$ is very close to the persistence exponent in two dimensions; $\ensuremath{\beta}\ensuremath{\simeq}1$. The results in two dimensions can be understood qualitatively by studying the exit probability, which for different system size is found to have the form $E(x)=f\left[(\frac{x\ensuremath{-}{x}_{c}}{{x}_{c}}){L}^{1/\ensuremath{\nu}}\right]$, with $\ensuremath{\nu}\ensuremath{\approx}1.47$. This result suggests that $\ensuremath{\tau}\ensuremath{\sim}{L}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}=\frac{\ensuremath{\lambda}}{\ensuremath{\nu}}=1.57\ifmmode\pm\else\textpm\fi{}0.11$ is an exponent not explored earlier.We investigate the dynamical behavior of the Ising model under a zero-temperature quench with the initial fraction of up spins $0\ensuremath{\le}x\ensuremath{\le}1$. In one dimension, the known results for persistence probability are verified; it shows algebraic decay for both up and down spins asymptotically with different exponents. It is found that the conventional finite-size scaling is valid here. In two dimensions, however, the persistence probabilities are no longer algebraic; in particular for $x\ensuremath{\le}0.5$, persistence for the up (minority) spins shows the behavior ${P}_{\text{min}}(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\gamma}}exp[\ensuremath{-}{(t/\ensuremath{\tau})}^{\ensuremath{\delta}}]$ with time $t$, while for the down (majority) spins, ${P}_{\text{maj}}(t)$ approaches a finite value. We find that the timescale $\ensuremath{\tau}$ diverges as ${({x}_{c}\ensuremath{-}x)}^{\ensuremath{-}\ensuremath{\lambda}}$, where ${x}_{c}=0.5$ and $\ensuremath{\lambda}\ensuremath{\simeq}2.31$. The exponent $\ensuremath{\gamma}$ varies as ${\ensuremath{\theta}}_{2d}+{c}_{0}{({x}_{c}\ensuremath{-}x)}^{\ensuremath{\beta}}$ where ${\ensuremath{\theta}}_{2d}\ensuremath{\simeq}0.215$ is very close to the persistence exponent in two dimensions; $\ensuremath{\beta}\ensuremath{\simeq}1$. The results in two dimensions can be understood qualitatively by studying the exit probability, which for different system size is found to have the form $E(x)=f\left[(\frac{x\ensuremath{-}{x}_{c}}{{x}_{c}}){L}^{1/\ensuremath{\nu}}\right]$, with $\ensuremath{\nu}\ensuremath{\approx}1.47$. This result suggests that $\ensuremath{\tau}\ensuremath{\sim}{L}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}=\frac{\ensuremath{\lambda}}{\ensuremath{\nu}}=1.57\ifmmode\pm\else\textpm\fi{}0.11$ is an exponent not explored earlier.

arXiv:1608.03399Physical review. EMay 6, 20164 citations

Short-time critical dynamics of damage spreading in the two-dimensional Ising model

M. Leticia Rubio Puzzo, Ezequiel Vicente Albano

The short-time critical dynamics of propagation of damage in the Ising ferromagnet in two dimensions is studied by means of Monte Carlo simulations. Starting with equilibrium configurations at $T=\ensuremath{\infty}$ and magnetization $M=0$ , an initial damage is created by flipping a small amount of spins in one of the two replicas studied. In this way, the initial damage is proportional to the initial magnetization ${M}_{0}$ in one of the configurations upon quenching the system at ${T}_{C}$ , the Onsager critical temperature of the ferromagnetic-paramagnetic transition. It is found that, at short times, the damage increases with an exponent ${\ensuremath{\theta}}_{D}=1.915(3)$, which is much larger than the exponent $\ensuremath{\theta}=0.197$ characteristic of the initial increase of the magnetization $M(t)$ . Also, an epidemic study was performed. It is found that the average distance from the origin of the epidemic $(⟨{R}^{2}(t)⟩)$ grows with an exponent ${z}^{\ensuremath{\ast}}\ensuremath{\approx}\ensuremath{\eta}\ensuremath{\approx}1.9$, which is the same, within error bars, as the exponent ${\ensuremath{\theta}}_{D}$. However, the survival probability of the epidemics reaches a plateau so that $\ensuremath{\delta}=0$. On the other hand, by quenching the system to lower temperatures one observes the critical spreading of the damage at ${T}_{D}\ensuremath{\simeq}0.51{T}_{C}$, where all the measured observables exhibit power laws with exponents ${\ensuremath{\theta}}_{D}=1.026(3)$, $\ensuremath{\delta}=0.133(1)$, and ${z}^{\ensuremath{\ast}}=1.74(3)$.The short-time critical dynamics of propagation of damage in the Ising ferromagnet in two dimensions is studied by means of Monte Carlo simulations. Starting with equilibrium configurations at $T=\ensuremath{\infty}$ and magnetization $M=0$, an initial damage is created by flipping a small amount of spins in one of the two replicas studied. In this way, the initial damage is proportional to the initial magnetization ${M}_{0}$ in one of the configurations upon quenching the system at ${T}_{C}$, the Onsager critical temperature of the ferromagnetic-paramagnetic transition. It is found that, at short times, the damage increases with an exponent ${\ensuremath{\theta}}_{D}=1.915(3)$, which is much larger than the exponent $\ensuremath{\theta}=0.197$ characteristic of the initial increase of the magnetization $M(t)$. Also, an epidemic study was performed. It is found that the average distance from the origin of the epidemic $(⟨{R}^{2}(t)⟩)$ grows with an exponent ${z}^{\ensuremath{\ast}}\ensuremath{\approx}\ensuremath{\eta}\ensuremath{\approx}1.9$, which is the same, within error bars, as the exponent ${\ensuremath{\theta}}_{D}$. However, the survival probability of the epidemics reaches a plateau so that $\ensuremath{\delta}=0$. On the other hand, by quenching the system to lower temperatures one observes the critical spreading of the damage at ${T}_{D}\ensuremath{\simeq}0.51{T}_{C}$, where all the measured observables exhibit power laws with exponents ${\ensuremath{\theta}}_{D}=1.026(3)$, $\ensuremath{\delta}=0.133(1)$, and ${z}^{\ensuremath{\ast}}=1.74(3)$.

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Universal scaling function for domain growth in the Glauber-Ising chain

Zhifang Lin, Xiaomin Wang, Ruibao Tao

The equal-time correlation function for an Ising chain with alternating ferromagnetic coupling constants ( ${\mathit{J}}_{1}$ > ${\mathit{J}}_{2}$ ) and Glauber dynamics has been calculated. Averaging over random initial conditions shows that the correlation function is independent of the probability distribution for the initial conditions, but depends on the nontrivial combination of different coupling constants, which casts some doubt as to the validity of the scaling form C(r,t)=f( ${\mathit{r}}^{2}$ /t). Furthermore, by calculating the spin-spin correlations for different times, we find that the power-law decay of the autocorrelation function has the same scaling form as in the simple uniform case C(0,t,t')\ensuremath{\sim}(2/\ensuremath{\pi})(2t/t' ${)}^{1/2}$ , for t'\ensuremath{\gg}t. This may suggest that a different exponent, instead of the dynamic critical exponent z=1+ ${\mathit{J}}_{1}$ / ${\mathit{J}}_{2}$ , is needed to characterize the scaling of the autocorrelation function.The equal-time correlation function for an Ising chain with alternating ferromagnetic coupling constants (${\mathit{J}}_{1}$>${\mathit{J}}_{2}$) and Glauber dynamics has been calculated. Averaging over random initial conditions shows that the correlation function is independent of the probability distribution for the initial conditions, but depends on the nontrivial combination of different coupling constants, which casts some doubt as to the validity of the scaling form C(r,t)=f(${\mathit{r}}^{2}$/t). Furthermore, by calculating the spin-spin correlations for different times, we find that the power-law decay of the autocorrelation function has the same scaling form as in the simple uniform case C(0,t,t')\ensuremath{\sim}(2/\ensuremath{\pi})(2t/t'${)}^{1/2}$, for t'\ensuremath{\gg}t. This may suggest that a different exponent, instead of the dynamic critical exponent z=1+${\mathit{J}}_{1}$/${\mathit{J}}_{2}$, is needed to characterize the scaling of the autocorrelation function.

Physical review. B, Condensed matterApr 1, 19923 citations

Dynamic crossover in the global persistence at criticality

Raja Paul, Andrea Gambassi, Grégory Schehr

We investigate the global persistence properties of critical systems relaxing from an initial state with non-vanishing value of the order parameter ( e.g. , the magnetization in the Ising model). The persistence probability of the global order parameter displays two consecutive regimes in which it decays algebraically in time with two distinct universal exponents. The associated crossover is controlled by the initial value m 0 of the order parameter and the typical time at which it occurs diverges as m 0 vanishes. Monte Carlo simulations of the two-dimensional Ising model with Glauber dynamics display clearly this crossover. The measured exponent of the ultimate algebraic decay is in rather good agreement with our theoretical predictions for the Ising universality class.We investigate the global persistence properties of critical systems relaxing from an initial state with non-vanishing value of the order parameter ( e.g. , the magnetization in the Ising model). The persistence probability of the global order parameter displays two consecutive regimes in which it decays algebraically in time with two distinct universal exponents. The associated crossover is controlled by the initial value m 0 of the order parameter and the typical time at which it occurs diverges as m 0 vanishes. Monte Carlo simulations of the two-dimensional Ising model with Glauber dynamics display clearly this crossover. The measured exponent of the ultimate algebraic decay is in rather good agreement with our theoretical predictions for the Ising universality class.

arXiv:cond-mat/0612656Europhysics Letters (EPL)Mar 13, 200716 citations

Nonequilibrium dynamics of spin glasses

Daniel S. Fisher, David A. Huse

We consider the nonequilibrium behavior of the spin-glass ordered phase within the droplet scaling theory introduced previously. The fundamental long-time nonequilibrium process is assumed to be the thermally activated growth of spin-glass ordered domains. The remanent magnetization, m(t), in zero field is found to decay at long times as m(t)\ensuremath{\sim}${R}_{t}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}$, where ${R}_{t}$ \ensuremath{\sim}(lnt${)}^{1/\ensuremath{\psi}}$ is the linear domain size, \ensuremath{\psi} is the previously introduced barrier exponent describing the growth of activation-barrier heights with length scale, and \ensuremath{\lambda} is a new nonequilibrium dynamic exponent, satisfying the relation \ensuremath{\lambda}\ensuremath{\ge}d/2 for d-dimensional systems. The effects of waiting for partial equilibration before making a measurement are studied in various regimes. The effects of quenching first to one temperature and then to another are also examined. Such experiments can, in principle, be used to obtain information about the relative rate of dynamic evolution as well as the overlap between the equilibrium states at different temperatures. In particular, the length scale ${L}_{\ensuremath{\Delta}T}$, below which equilibrium correlations at temperatures T and T+\ensuremath{\Delta}T are similar, plays an important role. The decay of m(t) and the growth of spin-glass order after a quench are examined in Monte Carlo simulations of the Sherrington-Kirkpatrick model.We consider the nonequilibrium behavior of the spin-glass ordered phase within the droplet scaling theory introduced previously. The fundamental long-time nonequilibrium process is assumed to be the thermally activated growth of spin-glass ordered domains. The remanent magnetization, m(t), in zero field is found to decay at long times as m(t)\ensuremath{\sim}${R}_{t}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}$, where ${R}_{t}$\ensuremath{\sim}(lnt${)}^{1/\ensuremath{\psi}}$ is the linear domain size, \ensuremath{\psi} is the previously introduced barrier exponent describing the growth of activation-barrier heights with length scale, and \ensuremath{\lambda} is a new nonequilibrium dynamic exponent, satisfying the relation \ensuremath{\lambda}\ensuremath{\ge}d/2 for d-dimensional systems. The effects of waiting for partial equilibration before making a measurement are studied in various regimes. The effects of quenching first to one temperature and then to another are also examined. Such experiments can, in principle, be used to obtain information about the relative rate of dynamic evolution as well as the overlap between the equilibrium states at different temperatures. In particular, the length scale ${L}_{\ensuremath{\Delta}T}$, below which equilibrium correlations at temperatures T and T+\ensuremath{\Delta}T are similar, plays an important role. The decay of m(t) and the growth of spin-glass order after a quench are examined in Monte Carlo simulations of the Sherrington-Kirkpatrick model.

Physical review. B, Condensed matterJul 1, 1988824 citations

We study the dynamics of ordering in ferromagnets via Monte Carlo simulations of the Ising model, employing the Glauber spin-flip mechanism, in space dimensions

d=2

and

3

. Results for the persistence probability and the domain growth are discussed for quenches to various temperatures (

T_f

) below the critical one (

T_{c}

), from different initial temperatures

T_{i} \geq T_{c}

. In long time limit, for

T_{i} > T_{c}

, the persistence probability exhibits power-law decay with exponents

θ\simeq 0.22

and

\simeq 0.18

d=2

and

3

, respectively. For finite

T_i

, the early time behavior is a different power-law whose life-time diverges and exponent decreases as

T_{i} \rightarrow T_{c}

. The crossover length between the two steps diverges as the equilibrium correlation length.

T_i=T_c

is expected to provide a {\it{new universality class}} for which we obtain

θ\simeq 0.035

d=2

and

\simeq 0.10

d=3

. The time dependence of the average domain size

\ell

, however, is observed to be rather insensitive to the choice of

T_i

t

, while for the down (majority) spins,

{P}_{\text{maj}}(t)

approaches a finite value. We find that the timescale $\ensuremath{\tau}$ diverges as ${({x}_{c}\ensuremath{-}x)}^{\ensuremath{-}\ensuremath{\lambda}}$, where

{x}_{c}=0.5

and $\ensuremath{\lambda}\ensuremath{\simeq}2.31$. The exponent $\ensuremath{\gamma}$ varies as ${\ensuremath{\theta}}_{2d}+{c}_{0}{({x}_{c}\ensuremath{-}x)}^{\ensuremath{\beta}}$ where ${\ensuremath{\theta}}_{2d}\ensuremath{\simeq}0.215$ is very close to the persistence exponent in two dimensions; $\ensuremath{\beta}\ensuremath{\simeq}1$. The results in two dimensions can be understood qualitatively by studying the exit probability, which for different system size is found to have the form $E(x)=f\left[(\frac{x\ensuremath{-}{x}_{c}}{{x}_{c}}){L}^{1/\ensuremath{\nu}}\right]$, with $\ensuremath{\nu}\ensuremath{\approx}1.47$. This result suggests that $\ensuremath{\tau}\ensuremath{\sim}{L}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}=\frac{\ensuremath{\lambda}}{\ensuremath{\nu}}=1.57\ifmmode\pm\else\textpm\fi{}0.11$ is an exponent not explored earlier.

M=0

, an initial damage is created by flipping a small amount of spins in one of the two replicas studied. In this way, the initial damage is proportional to the initial magnetization

{M}_{0}

in one of the configurations upon quenching the system at

{T}_{C}

, the Onsager critical temperature of the ferromagnetic-paramagnetic transition. It is found that, at short times, the damage increases with an exponent ${\ensuremath{\theta}}_{D}=1.915(3)$, which is much larger than the exponent $\ensuremath{\theta}=0.197$ characteristic of the initial increase of the magnetization

M(t)

. Also, an epidemic study was performed. It is found that the average distance from the origin of the epidemic

(⟨{R}^{2}(t)⟩)

grows with an exponent ${z}^{\ensuremath{\ast}}\ensuremath{\approx}\ensuremath{\eta}\ensuremath{\approx}1.9$, which is the same, within error bars, as the exponent ${\ensuremath{\theta}}_{D}$. However, the survival probability of the epidemics reaches a plateau so that $\ensuremath{\delta}=0$. On the other hand, by quenching the system to lower temperatures one observes the critical spreading of the damage at ${T}_{D}\ensuremath{\simeq}0.51{T}_{C}$, where all the measured observables exhibit power laws with exponents ${\ensuremath{\theta}}_{D}=1.026(3)$, $\ensuremath{\delta}=0.133(1)$, and ${z}^{\ensuremath{\ast}}=1.74(3)$.

The equal-time correlation function for an Ising chain with alternating ferromagnetic coupling constants (

{\mathit{J}}_{1}

{\mathit{J}}_{2}

) and Glauber dynamics has been calculated. Averaging over random initial conditions shows that the correlation function is independent of the probability distribution for the initial conditions, but depends on the nontrivial combination of different coupling constants, which casts some doubt as to the validity of the scaling form C(r,t)=f(

{\mathit{r}}^{2}

/t). Furthermore, by calculating the spin-spin correlations for different times, we find that the power-law decay of the autocorrelation function has the same scaling form as in the simple uniform case C(0,t,t')\ensuremath{\sim}(2/\ensuremath{\pi})(2t/t'

{)}^{1/2}

, for t'\ensuremath{\gg}t. This may suggest that a different exponent, instead of the dynamic critical exponent z=1+

{\mathit{J}}_{1}

{\mathit{J}}_{2}

, is needed to characterize the scaling of the autocorrelation function.

{R}_{t}

\ensuremath{\sim}(lnt${)}^{1/\ensuremath{\psi}}$ is the linear domain size, \ensuremath{\psi} is the previously introduced barrier exponent describing the growth of activation-barrier heights with length scale, and \ensuremath{\lambda} is a new nonequilibrium dynamic exponent, satisfying the relation \ensuremath{\lambda}\ensuremath{\ge}d/2 for d-dimensional systems. The effects of waiting for partial equilibration before making a measurement are studied in various regimes. The effects of quenching first to one temperature and then to another are also examined. Such experiments can, in principle, be used to obtain information about the relative rate of dynamic evolution as well as the overlap between the equilibrium states at different temperatures. In particular, the length scale ${L}_{\ensuremath{\Delta}T}$, below which equilibrium correlations at temperatures T and T+\ensuremath{\Delta}T are similar, plays an important role. The decay of m(t) and the growth of spin-glass order after a quench are examined in Monte Carlo simulations of the Sherrington-Kirkpatrick model.