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Physical Review Letters

Critical Dynamics of Nonconserved Ising-Like Systems

Kevin E. Bassler, B. Schmittmann

Published Dec 19, 1994• 50 citations
Publisher

Abstract

We show that the dynamical fixed point of Ising-like models, characterized by a single scalar, nonconserved ordering field, is stable near four dimensions with respect to all dynamic perturbations, including those of a nonequilibrium nature.
DOI
10.1103/physrevlett.73.3343

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Stochastic dynamics of a nonconserved scalar order parameter near its critical point, subject to random stirring and mixing, is studied using the field theoretic renormalization group. The stirring and mixing are modelled by a random external Gaussian noise with the correlation function ∝δ(t−t′)k4−d−y\proptoδ(t-t') k^{4-d-y}∝δ(t−t′)k4−d−y and the divergence-free (due to incompressibility) velocity field, governed by the stochastic Navier--Stokes equation with a random Gaussian force with the correlation function ∝δ(t−t′)k4−d−y′\proptoδ(t-t') k^{4-d-y'}∝δ(t−t′)k4−d−y′. Depending on the relations between the exponents yyy and y′y'y′ and the space dimensionality ddd, the model reveals several types of scaling regimes. Some of them are well known (model A of equilibrium critical dynamics and linear passive scalar field advected by a random turbulent flow), but there are three new nonequilibrium regimes (universality classes) associated with new nontrivial fixed points of the renormalization group equations. The corresponding critical dimensions are calculated in the two-loop approximation (second order of the triple expansion in yyy, y′y'y′ and ε=4−dε=4-dε=4−d).
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We numerically investigated the critical behavior for the dilute four-dimensional Ising model at different dilution values. For strong dilution, p<0.5p<0.5p<0.5, we found a significant deviation from the pure system, indicating a fixed point.