The mode-coupling theory of the glass transition treats the dynamics of supercooled liquids in terms of two-point density correlation functions. Here we consider a generalized, hierarchical formulation of schematic mode-coupling equations in which the full basis of multipoint density correlations is taken into account. By varying the parameters that control the effective contributions of higher-order correlations, we show that infinite hierarchies can give rise to both sharp and avoided glass transitions. Moreover, small changes in the form of the coefficients result in different scaling behaviors of the structural relaxation time, providing a means to tune the fragility in glass-forming materials. This demonstrates that the infinite-order construct of generalized mode-coupling theory constitutes a powerful and unifying framework for kinetic theories of the glass transition.
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