The one-dimensional kinetic Ising models with order-parameter conservation are studied for different transition rates. The effect on critical dynamics of global and local conservation laws is considered. 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 for several dynamical models that z\ensuremath{\ge}5 even if the order-parameter conservation occurs only globally. For models in which the maximal separation between two exchanged spins is proportional to the equilibrium correlation length, the inequality z\ensuremath{\ge}3 is obtained. The d-dimensional case is discussed briefly.
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 .
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.
Nonuniversal dynamic critical exponents are obtained for both the Glauber and Kawasaki dynamics of the Ising chain with alternating near-neighbor interactions,
J1,
J2, 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 (
J1>0,
J2>0). For Glauber (spin-flip) dynamics we obtain the exact exponent z=1+\ensuremath{\rho} (\ensuremath{\rho}\ensuremath{\equiv}\ensuremath{\Vert}
J1/
J2\ensuremath{\Vert}) which generalizes the result of Droz et al. and is valid irrespective of the signs of
J1 and
J2, where it is assumed that \ensuremath{\Vert}
J1\ensuremath{\Vert}>\ensuremath{\Vert}
J2\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
J1>0,${J}_{2}$0 we derive the conventional exponent z=1+\ensuremath{\rho}, whereas for ${J}_{1}$0 we find z=2 irrespective of
J2. A key aspect of this system is the narrowing of the dynamic critical region as compared with the isotropic system (
J1=
J2). 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.
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.
The critical dynamics of a disordered Ising ferromagnetic chain with two coupling constants (J 1 >or=J 2 >0) is studied for Glauber dynamics. Using a domain wall argument the dynamical critical exponent z is found to be non-universal but independent of the disorder, namely z=1+J 1 /J 2 . The problem is formulated in terms of diffusion in a random medium. The diffusion is shown to be normal. Relationships with apparently very different diffusion problems, like the diffusion in hierarchically structured media, are established.