Physical review. B, Condensed matter

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

Stephen J. Cornell, Michel Droz

Published Dec 1, 1992• 0 citations

Abstract

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.

DOI: 10.1103/physrevb.46.14261

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Physical review. B, Condensed matter

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

Stephen J. Cornell, Michel Droz

Published Dec 1, 1992• 0 citations

Publisher

Abstract

DOI: 10.1103/physrevb.46.14261

Related papers

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.

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

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.

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.