Theory of the $β$-Relaxation Beyond Mode-Coupling Theory: A Microscopic Treatment

TL;DR

Starting from a microscopic diagrammatic formulation of MCT, the paper identifies the dominant fluctuation-driven diagrams that destabilize the mean-field transition below the upper critical dimension $d_c=8$. By mapping the divergent series onto a stochastic process for the density-order parameter, it achieves a controlled resummation and derives a fully microscopic stochastic beta-relaxation (SBR) theory, with coupling constants fixed by the static structure factor. The analysis shows ergodicity is restored and the putative glass transition becomes a smooth crossover, while preserving the MCT exponents in a fluctuating, beyond-mean-field framework. This work unifies static and dynamical perspectives on glassy relaxation, providing parameter-free predictions for structural relaxation beyond mean-field, including explicit results for the hard-sphere system.

Abstract

We develop a systematic extension of mode-coupling theory (MCT) that incorporates critical dynamical fluctuations. Starting from a microscopic diagrammatic theory, we identify dominant classes of divergent diagrams near the mode-coupling transition and show that the corresponding asymptotic series dominates the mean-field below an upper critical dimension $d_c=8$. To resum these divergences, we construct a mapping to a stochastic dynamical process in which the order parameter evolves under random spatiotemporal fields. This reformulation provides a controlled, fully dynamical derivation of an effective theory for the $β$-relaxation which remarkably coincides with stochastic beta-relaxation theory [T. Rizzo, EPL 106, 56003 (2014)]. All coupling constants of the latter theory are expressed microscopically in terms of the liquid static structure factor and are computed for the paradigmatic hard-sphere system. The analysis demonstrates that fluctuations alone restore ergodicity and replace the putative mean-field transition by a smooth crossover. Our results establish a predictive framework for structural relaxation beyond mean-field.

Figures (13)

• Figure 1: Diagrammatic rules of the kinetic theory to lowest order in a cluster expansion.
• Figure 2: Diagrammatic expansion of the normalized correlation function $F(k;t)$ in terms of the bare propagator $F_0(k;t)$.
• Figure 3: Illustration of the diagrammatic regularization procedure for the vertices of the diagrammatic theory. Regularization of the (a) $\mathcal{V}_{21}$, (b) $\mathcal{V}_{12}$ and (c) $\mathcal{V}_{22}$ vertices. In each, the grey bubble corresponds to the memory matrix defined in Eq. \ref{['eq:memory_matrix']}. The time labels refer to the time-slice of the renormalized vertices.
• Figure 4: Diagrammatic form of the Dyson equation for the intermediate scattering function $F(k,t)$ using the mode-coupling approximation for the self-energy. The thick line represents the full propagator $F(k,t)$, and the thinner one denotes the bare propagator $F_0(k,t)$. The second row shows the first few bare diagrams that make up the MCT.
• Figure 5: Diagrammatic representation of the resummation of rainbow diagrams yielding the dynamical susceptibilities $\chi_{\boldsymbol{q}}^+(\boldsymbol{k}; t ; t')$ [panel (a)] and $\chi_{\boldsymbol{q}}^-(\boldsymbol{k}; t ; t')$ [panel (b)]. Dangling ends at time $t'$ indicate points at which a propagator carrying momentum $\boldsymbol{q}$ can be attached; solid lines represent full propagators $F(k;t)$, and square vertices correspond to regularized mode-coupling vertices $\mathcal{v}^{\mathrm{reg}}_{12},\ \mathcal{v}^{\mathrm{reg}}_{21}$.