Physical review. B, Condensed matter

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

Zhifang Lin, Xiaomin Wang, Ruibao Tao

Published Apr 1, 1992• 3 citations

Abstract

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.

DOI: 10.1103/physrevb.45.8131

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

Alan J. Bray

The equal-time correlation function is calculated at T=0 for the one-dimensional Ising model with Glauber dynamics. Random initial conditions, appropriate to a sudden quench from non-zero temperature, are imposed. Averaging over initial conditions yields the scaling form C(r, t)=f(r 2 /t). The scaling function is given by f(x)= integral 0 1 (dy/ pi )(y(1-y)) -1/2 exp(-x/4y), and is universal, i.e. independent of the probability distribution for the initial conditions.

Journal of Physics A Mathematical and GeneralJan 21, 199077 citations

Comment on ‘‘Universal scaling function for domain growth in the Glauber-Ising chain’’

Stephen J. Cornell, Michel Droz

We show that the nonuniversal behavior during domain growth found by Lin et al. in the alternating-bond Glauber chain [Phys. Rev. B 45, 8131 (1992)] has a simple physical origin, and that similar dependences arise in more complex and higher-dimensional systems. However, since the universal forms are recovered by rescaling the time, there is no contradiction between the nonuniversality (in one dimension) of the dynamic critical exponent and the universality of the autocorrelation function.

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Persistence in Ferromagnetic Ordering: Dependence upon initial configuration

Saikat Chakraborty, Subir K. Das

arXiv:1501.05178Jan 21, 20150 citations

Nonuniversal critical dynamics of the alternating-bond Ising chain: Relaxational and diffusive kinetics

James H. Luscombe

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Universality in the time correlations of the long-range 1d Ising model

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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

Physical review. B, Condensed matter

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

Zhifang Lin, Xiaomin Wang, Ruibao Tao

Published Apr 1, 1992• 3 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevb.45.8131

Related papers

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

Alan J. Bray

Journal of Physics A Mathematical and GeneralJan 21, 199077 citations

Comment on ‘‘Universal scaling function for domain growth in the Glauber-Ising chain’’

Stephen J. Cornell, Michel Droz

Physical review. B, Condensed matterDec 1, 19920 citations

Persistence in Ferromagnetic Ordering: Dependence upon initial configuration

Saikat Chakraborty, Subir K. Das

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 $T_i$ .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 $T_i$.

arXiv:1501.05178Jan 21, 20150 citations

Nonuniversal critical dynamics of the alternating-bond Ising chain: Relaxational and diffusive kinetics

James H. Luscombe

Nonuniversal dynamic critical exponents are obtained for both the Glauber and Kawasaki dynamics of the Ising chain with alternating near-neighbor interactions, ${J}_{1}$ , ${J}_{2}$ , with the exponents related to the ratio of the two interaction strengths. We expand upon the recent work of Droz et al. [Phys. Lett. 115A, 448 (1986)] for the ferromagnetic system ( ${J}_{1}$ >0, ${J}_{2}$ >0). For Glauber (spin-flip) dynamics we obtain the exact exponent z=1+\ensuremath{\rho} (\ensuremath{\rho}\ensuremath{\equiv}\ensuremath{\Vert} ${J}_{1}$ / ${J}_{2}$ \ensuremath{\Vert}) which generalizes the result of Droz et al. and is valid irrespective of the signs of ${J}_{1}$ and ${J}_{2}$ , where it is assumed that \ensuremath{\Vert} ${J}_{1}$ \ensuremath{\Vert}>\ensuremath{\Vert} ${J}_{2}$ \ensuremath{\Vert} (\ensuremath{\rho}\ensuremath{\ge}1). For Kawasaki (spin-exchange) dynamics, we obtain dynamic critical exponents from conventional theory which provides a rigorous lower bound for the exponent z. For the case of the conserved (ferromagnetic) order parameter, however, we present arguments that the conventional exponent is exact. We obtain z=4+\ensuremath{\rho} in this case. For ${J}_{1}$ >0,${J}_{2}$0 we derive the conventional exponent z=1+\ensuremath{\rho}, whereas for ${J}_{1}$0 we find z=2 irrespective of ${J}_{2}$ . A key aspect of this system is the narrowing of the dynamic critical region as compared with the isotropic system ( ${J}_{1}$ = ${J}_{2}$ ). The extra bond periodicity splits the isotropic order parameter into components such that only comparatively closer to criticality does the order parameter become the dominant slow mode. The nonuniversal critical dynamics is intrinsically linked to the nonuniform bond distribution, being shown to arise from a kinetic coefficient which vanishes with a nonuniversal critical exponent. Nonuniversality of the dynamic exponent for Ising systems with inhomogeneous couplings is argued to be specific to zero-temperature critical points.Nonuniversal dynamic critical exponents are obtained for both the Glauber and Kawasaki dynamics of the Ising chain with alternating near-neighbor interactions, ${J}_{1}$, ${J}_{2}$, with the exponents related to the ratio of the two interaction strengths. We expand upon the recent work of Droz et al. [Phys. Lett. 115A, 448 (1986)] for the ferromagnetic system (${J}_{1}$>0,${J}_{2}$>0). For Glauber (spin-flip) dynamics we obtain the exact exponent z=1+\ensuremath{\rho} (\ensuremath{\rho}\ensuremath{\equiv}\ensuremath{\Vert}${J}_{1}$/${J}_{2}$\ensuremath{\Vert}) which generalizes the result of Droz et al. and is valid irrespective of the signs of ${J}_{1}$ and ${J}_{2}$, where it is assumed that \ensuremath{\Vert}${J}_{1}$\ensuremath{\Vert}>\ensuremath{\Vert}${J}_{2}$\ensuremath{\Vert} (\ensuremath{\rho}\ensuremath{\ge}1). For Kawasaki (spin-exchange) dynamics, we obtain dynamic critical exponents from conventional theory which provides a rigorous lower bound for the exponent z. For the case of the conserved (ferromagnetic) order parameter, however, we present arguments that the conventional exponent is exact. We obtain z=4+\ensuremath{\rho} in this case. For ${J}_{1}$>0,${J}_{2}$0 we derive the conventional exponent z=1+\ensuremath{\rho}, whereas for ${J}_{1}$0 we find z=2 irrespective of ${J}_{2}$. A key aspect of this system is the narrowing of the dynamic critical region as compared with the isotropic system (${J}_{1}$=${J}_{2}$). The extra bond periodicity splits the isotropic order parameter into components such that only comparatively closer to criticality does the order parameter become the dominant slow mode. The nonuniversal critical dynamics is intrinsically linked to the nonuniform bond distribution, being shown to arise from a kinetic coefficient which vanishes with a nonuniversal critical exponent. Nonuniversality of the dynamic exponent for Ising systems with inhomogeneous couplings is argued to be specific to zero-temperature critical points.

Physical review. B, Condensed matterJul 1, 198729 citations

Universality in the time correlations of the long-range 1d Ising model

Federico Corberi, Eugenio Lippiello, Paolo Politi

Abstract The equilibrium and nonequilibrium properties of ferromagnetic systems may be affected by the long-range nature of the coupling interaction. Here we study the phase separation process of a one-dimensional Ising model in the presence of a power-law decaying coupling, with , and we focus on the two-time autocorrelation function . We find that it obeys the scaling form , where is the typical domain size at time t , and where can only be of two types. For , when domain walls diffuse freely, falls in the nearest-neighbour (nn) universality class. Conversely, for , when domain walls dynamics is driven, displays a new universal behavior. In particular, the so-called Fisher-Huse exponent, which characterizes the asymptotic behavior of for , is in the nn universality class ( ) and for .Abstract The equilibrium and nonequilibrium properties of ferromagnetic systems may be affected by the long-range nature of the coupling interaction. Here we study the phase separation process of a one-dimensional Ising model in the presence of a power-law decaying coupling, with , and we focus on the two-time autocorrelation function . We find that it obeys the scaling form , where is the typical domain size at time t , and where can only be of two types. For , when domain walls diffuse freely, falls in the nearest-neighbour (nn) universality class. Conversely, for , when domain walls dynamics is driven, displays a new universal behavior. In particular, the so-called Fisher-Huse exponent, which characterizes the asymptotic behavior of for , is in the nn universality class ( ) and for .

arXiv:1904.05595Journal of Statistical Mechanics Theory and ExperimentJul 1, 201925 citations

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

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

{\mathit{J}}_{1}

{\mathit{J}}_{2}

{\mathit{r}}^{2}

{)}^{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.

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

Nonuniversal dynamic critical exponents are obtained for both the Glauber and Kawasaki dynamics of the Ising chain with alternating near-neighbor interactions,

{J}_{1}

{J}_{2}

, with the exponents related to the ratio of the two interaction strengths. We expand upon the recent work of Droz et al. [Phys. Lett. 115A, 448 (1986)] for the ferromagnetic system (

{J}_{1}

>0,

{J}_{2}

>0). For Glauber (spin-flip) dynamics we obtain the exact exponent z=1+\ensuremath{\rho} (\ensuremath{\rho}\ensuremath{\equiv}\ensuremath{\Vert}

{J}_{1}

{J}_{2}

\ensuremath{\Vert}) which generalizes the result of Droz et al. and is valid irrespective of the signs of

{J}_{1}

and

{J}_{2}

, where it is assumed that \ensuremath{\Vert}

{J}_{1}

\ensuremath{\Vert}>\ensuremath{\Vert}

{J}_{2}

\ensuremath{\Vert} (\ensuremath{\rho}\ensuremath{\ge}1). For Kawasaki (spin-exchange) dynamics, we obtain dynamic critical exponents from conventional theory which provides a rigorous lower bound for the exponent z. For the case of the conserved (ferromagnetic) order parameter, however, we present arguments that the conventional exponent is exact. We obtain z=4+\ensuremath{\rho} in this case. For

{J}_{1}

>0,${J}_{2}$0 we derive the conventional exponent z=1+\ensuremath{\rho}, whereas for ${J}_{1}$0 we find z=2 irrespective of

{J}_{2}

. A key aspect of this system is the narrowing of the dynamic critical region as compared with the isotropic system (

{J}_{1}

{J}_{2}

). The extra bond periodicity splits the isotropic order parameter into components such that only comparatively closer to criticality does the order parameter become the dominant slow mode. The nonuniversal critical dynamics is intrinsically linked to the nonuniform bond distribution, being shown to arise from a kinetic coefficient which vanishes with a nonuniversal critical exponent. Nonuniversality of the dynamic exponent for Ising systems with inhomogeneous couplings is argued to be specific to zero-temperature critical points.