We show that an exponent of a powerlike time domain growth is determined not only by the conservation or nonconservation of the order parameter, but also by the asymmetry of single-particle jumps. Domains that have an anisotropic pattern, such as (2 X 1), have a tendency to grow faster in a certain direction than they do in others. The rate of expansion in different directions depends on the barriers for single-particle jumps. As a result, dynamical behavior of systems which start in the same configurations and eventually reach the same equilibrium states is completely different. We show how differences in microscopic dynamics in a one-dimensional Potts model lead to different rates of domain growth. We observe a similar effect for a two-dimensional (2 X 1) ordering by changing the way in which a barrier for a jump depends on the number of neighboring particles. We show examples of the domain power growth, which are characterized by different exponents.
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