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Microscopic Theory of a Fluctuation-Induced Dynamical Crossover in Supercooled Liquids

Corentin C. L. Laudicina, Liesbeth M. C. Janssen, Grzegorz Szamel

Abstract

Mean-field theories of the glass transition predict a phase transition to a dynamically arrested state, yet no such transition is observed in experiments or simulations of finite-dimensional systems. We resolve this long-standing discrepancy by incorporating critical dynamical fluctuations into a microscopic mode-coupling framework. We show that these fluctuations round off the mean-field singularity and restore ergodicity at all finite densities (or temperatures) without invoking activated dynamics or facilitation. The resulting effective theory describes the order parameter as a stochastic process with self-induced, annealed disorder, determined self-consistently at the mean-field level. In the $β$-relaxation regime it reduces to stochastic beta-relaxation theory, thereby unifying mode-coupling and replica-based approaches beyond mean-field. All parameters of the stochastic $β$-relaxation theory are fixed by the static structure, enabling parameter-free predictions that extend mean-field theory into finite dimensions.

Microscopic Theory of a Fluctuation-Induced Dynamical Crossover in Supercooled Liquids

Abstract

Mean-field theories of the glass transition predict a phase transition to a dynamically arrested state, yet no such transition is observed in experiments or simulations of finite-dimensional systems. We resolve this long-standing discrepancy by incorporating critical dynamical fluctuations into a microscopic mode-coupling framework. We show that these fluctuations round off the mean-field singularity and restore ergodicity at all finite densities (or temperatures) without invoking activated dynamics or facilitation. The resulting effective theory describes the order parameter as a stochastic process with self-induced, annealed disorder, determined self-consistently at the mean-field level. In the -relaxation regime it reduces to stochastic beta-relaxation theory, thereby unifying mode-coupling and replica-based approaches beyond mean-field. All parameters of the stochastic -relaxation theory are fixed by the static structure, enabling parameter-free predictions that extend mean-field theory into finite dimensions.

Paper Structure

This paper contains 14 equations, 1 figure.

Figures (1)

  • Figure 1: (a) Diagrammatic structure of Eq. \ref{['eq:bethe_salpeter_3_vertex']} for $\chi_{\boldsymbol{q}}^-(\boldsymbol{k})$, here represented by diagram. The $\chi_{\boldsymbol{q}}^+(\boldsymbol{k})$ susceptibility is obtained by using a right-handed vertex instead. Bonds represent the mode-coupling propagator $F_{\mathrm{MCT}}(k)$. (b) Diagrammatic structure of Eq. \ref{['eq:generic_overarching']}. The dark and light grey dashed box with two external legs represents $F^{(2n)}(k)$ and $A^{(2n)}(k)$ respectively. (c) All possible divergent diagrams with two connected rainbow insertions. (d) Second order effective sources $A_u^{(2)}(\boldsymbol{k}_+, \boldsymbol{k}_-)$ generated by the perturbative solution to Eqs. \ref{['eq:postulate_microscopic_stochastic']}–\ref{['eq:inhomogeneous_Mirr']}. Note that $= \sum_{\nu=\pm}\chi_{\boldsymbol{q}}^{(\nu)}(\boldsymbol{k})u^{(\nu)}_{\boldsymbol{q}}$, where theline represents a sum over the random fields $u^{\pm}_{\boldsymbol{q}}$.