Physical review. B, Condensed matter

Physics of the dynamical critical exponent in one dimension

Robert Cordery, Sanjoy Kumar Sarker, Jan Tobochnik

Published Nov 1, 1981• 83 citations

Abstract

The value of the dynamical critical exponent $z$ is calculated for various one-dimensional kinetic Ising models using simple physical arguments. The values agree with all known exact results and with the lower bounds obtained by Haake and Thol.

DOI: 10.1103/physrevb.24.5402

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The dynamical critical exponent, z , of the two-dimensional kinetic Ising model is estimated by the Monte Carlo method using finite-size scaling. The Monte Carlo simulalion is performed on a special-purpose computer system, m -TIS, designed and constructed by the present authors. The obtained value of z is 2.132±0.008. This result supports the conjecture by Racz and Collins that z is equal to 17/8 in the two-dimensional kinetic Ising model (Racz and Collins; Rhys. Rev. B 13 (1976) 3074).

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Physical review. B, Condensed matter

Physics of the dynamical critical exponent in one dimension

Robert Cordery, Sanjoy Kumar Sarker, Jan Tobochnik

Published Nov 1, 1981• 83 citations

Publisher

Abstract

The value of the dynamical critical exponent

z

is calculated for various one-dimensional kinetic Ising models using simple physical arguments. The values agree with all known exact results and with the lower bounds obtained by Haake and Thol.

DOI: 10.1103/physrevb.24.5402

Related papers

Finite-size calculations for the kinetic Ising model

Rahul Anil Pandit, Gabor Forgács, Pál Ruján

Using a Hamiltonian formulation of the master equation, we carry out finite-size calculations for the dynamical critical exponent $z$ of the kinetic Ising model in one and two dimensions. In one dimension, different appropriately normalized transition probabilities give different values for $z$ ; this result is based on both exact and finite-size calculations. We show that this "nonuniversal" behavior is due to the fact that the critical temperature for the one-dimensional Ising model is zero. In two dimensions we combine our finite-size calculations with finite-size scaling theory to calculate $z$ . Even in this case we find that $z$ depends on the form of the transition probability---a result that contradicts the universality hypothesis for dynamical critical behavior. We discuss the possible reasons for this contradiction.Using a Hamiltonian formulation of the master equation, we carry out finite-size calculations for the dynamical critical exponent $z$ of the kinetic Ising model in one and two dimensions. In one dimension, different appropriately normalized transition probabilities give different values for $z$; this result is based on both exact and finite-size calculations. We show that this "nonuniversal" behavior is due to the fact that the critical temperature for the one-dimensional Ising model is zero. In two dimensions we combine our finite-size calculations with finite-size scaling theory to calculate $z$. Even in this case we find that $z$ depends on the form of the transition probability---a result that contradicts the universality hypothesis for dynamical critical behavior. We discuss the possible reasons for this contradiction.

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A novel dynamic Monte Carlo renormalization-group method is devised. Application to the kinetic Ising model in one dimension yields a value of $z$ , the dynamic critical exponent, which is in excellent agreement with the exact result. In two dimensions we find $z=2.22\ifmmode\pm\else\textpm\fi{}0.13$ with the most probable value $z=2.17$ .A novel dynamic Monte Carlo renormalization-group method is devised. Application to the kinetic Ising model in one dimension yields a value of $z$, the dynamic critical exponent, which is in excellent agreement with the exact result. In two dimensions we find $z=2.22\ifmmode\pm\else\textpm\fi{}0.13$ with the most probable value $z=2.17$.

Physical Review LettersMay 25, 198173 citations

Migdal-type renormalization-group calculation for the kinetic Ising model

S. T. Chui, Gabor Forgács, H. L. Frisch

We make use of the calculation of Achiam and Kosterlitz and investigate the generalization of the Migdal-type renormalization-group calculation to critical dynamics, by looking at the one- and two-dimensional kinetic Ising model with no conserved magnetization. In the two-dimensional case the dynamical critical index ${Z}_{M}(\mathrm{magnetic}\mathrm{perturbation})=2.064$ and ${Z}_{E}(\mathrm{energylike}\mathrm{perturbation})=1.819$ for scale factor $\ensuremath{\lambda}=1$. ${Z}_{M}$ involves the static $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ exponent, while ${Z}_{E}$ involves $\frac{1}{\ensuremath{\nu}}$, and therefore, it is not surprising that ${Z}_{M}$ is closer to the high-temperature expansion results, since $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ in the Migdal approximation for statics is much closer to the exact value than $\frac{1}{\ensuremath{\nu}}$. In one dimension we obtain ${Z}_{M}={Z}_{E}=2$ , which is the exact result of Glauber.We make use of the calculation of Achiam and Kosterlitz and investigate the generalization of the Migdal-type renormalization-group calculation to critical dynamics, by looking at the one- and two-dimensional kinetic Ising model with no conserved magnetization. In the two-dimensional case the dynamical critical index ${Z}_{M}(\mathrm{magnetic}\mathrm{perturbation})=2.064$ and ${Z}_{E}(\mathrm{energylike}\mathrm{perturbation})=1.819$ for scale factor $\ensuremath{\lambda}=1$. ${Z}_{M}$ involves the static $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ exponent, while ${Z}_{E}$ involves $\frac{1}{\ensuremath{\nu}}$, and therefore, it is not surprising that ${Z}_{M}$ is closer to the high-temperature expansion results, since $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ in the Migdal approximation for statics is much closer to the exact value than $\frac{1}{\ensuremath{\nu}}$. In one dimension we obtain ${Z}_{M}={Z}_{E}=2$, which is the exact result of Glauber.

Physical review. B, Condensed matterJul 1, 197917 citations

On the critical dynamics of one-dimensional Potts models

Michel Droz, J. Kamphorst Leal da Silva, Anna‐Sapfo Malaspinas +1 more

Journal of Physics A Mathematical and GeneralSep 11, 198611 citations

Accurate Estimation of the Dynamical Critical Exponent of the Two-Dimensional Kinetic Ising Model Based on the Ising Machinem-TIS

Nobuyasu Ito, Makoto Taiji, Masuo Suzuki

Journal of the Physical Society of JapanDec 15, 198724 citations

Dynamics of the one-dimensional Potts model

P. O. Weir, J. Michael Kosterlitz

We consider the q-state Potts model in one dimension for the cases q ${=2}^{m}$ , m=1,2,3,. . . . The dynamical critical exponent is calculated with use of the real-space renormalization-group method. We find z=2 in the 4- and 8-state models, which agrees with the Ising case, and speculate that this generalizes to any m.We consider the q-state Potts model in one dimension for the cases q${=2}^{m}$, m=1,2,3,. . . . The dynamical critical exponent is calculated with use of the real-space renormalization-group method. We find z=2 in the 4- and 8-state models, which agrees with the Ising case, and speculate that this generalizes to any m.

Physical review. B, Condensed matterJan 1, 198612 citations

The value of the dynamical critical exponent

z

is calculated for various one-dimensional kinetic Ising models using simple physical arguments. The values agree with all known exact results and with the lower bounds obtained by Haake and Thol.

Using a Hamiltonian formulation of the master equation, we carry out finite-size calculations for the dynamical critical exponent

z

of the kinetic Ising model in one and two dimensions. In one dimension, different appropriately normalized transition probabilities give different values for

z

; this result is based on both exact and finite-size calculations. We show that this "nonuniversal" behavior is due to the fact that the critical temperature for the one-dimensional Ising model is zero. In two dimensions we combine our finite-size calculations with finite-size scaling theory to calculate

z

. Even in this case we find that

z

depends on the form of the transition probability---a result that contradicts the universality hypothesis for dynamical critical behavior. We discuss the possible reasons for this contradiction.

A novel dynamic Monte Carlo renormalization-group method is devised. Application to the kinetic Ising model in one dimension yields a value of

z

, the dynamic critical exponent, which is in excellent agreement with the exact result. In two dimensions we find $z=2.22\ifmmode\pm\else\textpm\fi{}0.13$ with the most probable value

z=2.17

{Z}_{M}(\mathrm{magnetic}\mathrm{perturbation})=2.064

and

{Z}_{E}(\mathrm{energylike}\mathrm{perturbation})=1.819

for scale factor $\ensuremath{\lambda}=1$.

{Z}_{M}

involves the static $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ exponent, while

{Z}_{E}

involves $\frac{1}{\ensuremath{\nu}}$, and therefore, it is not surprising that

{Z}_{M}

is closer to the high-temperature expansion results, since $\frac{\ensuremath{\beta}}{\ensuremath{\nu}}$ in the Migdal approximation for statics is much closer to the exact value than $\frac{1}{\ensuremath{\nu}}$. In one dimension we obtain

{Z}_{M}={Z}_{E}=2

, which is the exact result of Glauber.

We consider the q-state Potts model in one dimension for the cases q

{=2}^{m}

, m=1,2,3,. . . . The dynamical critical exponent is calculated with use of the real-space renormalization-group method. We find z=2 in the 4- and 8-state models, which agrees with the Ising case, and speculate that this generalizes to any m.