Scrutiny of the Critical Exponent Paradigm, as Exemplified by Gelation

The conflict between classical and modern theories of criticality is resolved by recognising that both theories work with approximate models, whose relative merits can be assessed by their response to efforts to refine them. Contrary to many claims, critical exponents are not tools fitted to discriminate between different models. True, Riemannian theory which characterises analytic functions in terms of their singularities, can be mapped into experimentally measured functions, that have finite ranges, finite errors, and replace singularities by rounded corners. Under the mapping, the uniqueness of series expansions is not preserved. Weierstrassian theory shows that a single infinite series is mapped into a non-denumerable set of series, each with a finite number of terms, which are equivalent in fitting experimental functions uniformly within any c however small, and whose leading exponents x range over -°°<x<+°°. From the statistical viewpoint this means that tests for or against a theory, by a null-hypothesis based on the critical exponent, fail because this parameter is not identifiable, or (in other variants of the test) inconsistently estimated, r lacking in robustness. We illustrate these principles by reference to gelation data in the literature, and exemplify the refinement processes envisaged by classical and modern theories. Modern theories aim to discover universal features, working outward from the critical point by addition of successively larger corrections to the Hamiltonian or free energy. Classical theory works inward towards the critical point by adding successively smaller corrections, and with due regard to system-specific features. The classical gelation theory of Flory and Stockmayer has long since been refined in this way in respect of cycle formation and substituent effects and fits some good data almost within experimental error. The parameters of the refined theories, used in such fittings, are available from measurements independent of gelation.