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Unorthodox properties of critical clusters

We look at the properties of clusters of order parameter Φ at critical points in thermal systems and consider their significance to statistical–mechanical ground rules. These properties have been previously obtained through the saddle-point approximation in a coarse-grained partition function. We examine both static and dynamical aspects of a single large cluster and indicate that these properties fall outside the canonical Boltzmann–Gibbs (BG) scheme. Specifically, (1) the faster than exponential growth with cluster size of the space-integrated Φ suggests non-extensivity of the BG entropy but extensivity of a q-entropy expression. (2) The finding that the time evolution of Φ is described by the dynamics of an intermittent nonlinear map implies an atypical sensitivity to initial conditions compatible with q-statistics and displays an ‘aging’ scaling property. (3) Both the approach to criticality and the infinite-size cluster limit at criticality manifest through a crossover from canonical to q-statistics and we discuss the non-uniform convergence associated with these features.