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Journal of Physics A Mathematical and General

On the generalised Glauber kinetics for a one-dimensional Potts chain

Magdalena A. Załuska–Kotur, Łukasz A. Turski

Published Feb 21, 1989• 4 citations
Publisher

Abstract

The one-dimensional Potts chain dynamics is analysed by means of the direct space renormalisation technique. In contrast to previous work one needs no restriction on the value of the Potts dimensionality q. They use the generalised Glauber kinetics for the chain and find that the value of the dynamical critical exponent z is not universal, i.e. it depends on both Potts dimensionality and the transition rates asymmetry coefficient lambda .

DOI
10.1088/0305-4470/22/4/010

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The one-dimensional Potts chain dynamics is analysed by means of the direct space renormalisation technique. In contrast to previous work one needs no restriction on the value of the Potts dimensionality q. They use the generalised Glauber kinetics for the chain and find that the value of the dynamical critical exponent z is not universal, i.e. it depends on both Potts dimensionality and the transition rates asymmetry coefficient lambda .
The one-dimensional kinetic Potts models are studied for both Glauber and Kawasaki dynamics with different transition rates. Rigorous lower bounds for the dynamical exponent (z) are obtained from the initial response rate and the scaling hypothesis for the relaxation time. It is shown that the z exponent is non-universal for several periodic models with a basic unit cell (J 1 ,J 2 ,. . .,JM) containing different coupling constants. The critical dynamics of the alternating bond model is also studied by a simple argument about the movement of domain walls.
Recent results concerning non-universality in the dynamical critical behavior of the one-dimensional Ising model, are generalised to several formulations of the Potts model, using arguments based on domain wall motion. By applying real-space renormalisation group methods, the authors also study the dynamical Ising model, with single flips, both for a period distribution of nearest-neighbor exchanges and for the random chain (with nonzero couplings). They show that the critical dynamical exponent is z=1+J M /J m where J M (J m ) is the greatest (smallest) of the exchanges.
A plausible argument using the concepts of dynamic renormalisation group is used to derive the dynamical exponent z of the one-dimensional q=2, 3 and 4 Potts and four-state clock models. The exact result (z=2) for the Ising model with Glauber dynamics is recovered and in addition the authors find that the q=3, 4 Potts and four-state clock models belong to the same universality class. Moreover they find that z=3.05+or-0.1 for the (q=2) Potts model with Kawasaki dynamics in agreement with the z=4- eta prediction of Halperin, Hohenberg and Ma (1974). A simple qualitative derivation is also given.