Order Variables and Their Correlations, and the Mean-Field Approximation
For a phase transition in a crystal the order parameter η is a thermodynamical variable signifying the ordered phase at temperatures below the critical region. Originating from microscopic variables σ m attached to ions or molecules that are active at sites m for a structural change, their ensemble average is considered to represent the macroscopic-order parameter η. Such a variable σ m is a function of space—time coordinates at the site m, so that the time variation as well as the spatial distribution are significant for the averaging of distributed variables. In the disordered phase above T c, those variables σ m are usually in fast random motion so that the time average <σ m> t vanishes at each lattice point, hence independent of the site m. In contrast, at temperatures below T c, they are correlated in slow motion, so that the ordered phase is dominated by their spatial distribution. In a binary system, as illustrated schematically in Fig. 2.1, the “ordered” phase below T c are topologically inhomogeneous, comprising either domains or sublattices . The ensemble average in such a state is meaningful, only if calculated for one of these subsystems instead of the whole crystal.
KeywordsIsing Model Order Variable Collective Motion Binary State Unequal Pair
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