arXiv:1009.4511 · Physical Review E

Revisiting Kawasaki dynamics in one dimension

Marcelo D. Grynberg

Published Nov 16, 2010• 7 citations

Abstract

Critical exponents of the Kawasaki dynamics in the Ising chain are re-examined numerically through the spectrum gap of evolution operators constructed both in spin and domain-wall representations. At low-temperature regimes the latter provides a rapid finite-size convergence to these exponents, which tend to z ≃ 3.11 for instant quenches under ferromagnetic couplings, while approaching to z ≃ 2 in the antiferro case. The spin representation complements the evaluation of dynamic exponents at higher temperature scales, where the kinetics still remains slow.

DOI: 10.1103/physreve.82.051121

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arXiv:1009.4511 · Physical Review E

Revisiting Kawasaki dynamics in one dimension

Marcelo D. Grynberg

Published Nov 16, 2010• 7 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreve.82.051121

Related papers

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Physical review. B, Condensed matterAug 1, 19819 citations

Kawasaki dynamics in the Ising chain: a renormalisation group treatment

J M Nunes da Silva, Eduardo Lage

Journal of Physics C Solid State PhysicsApr 20, 19882 citations

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.

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.