Activated processes and the four-point dynamical susceptibility of supercooled liquids: rounding the mean-field divergence of $\chi_4$ and its correlation length at the mode-coupling transition

When a supercooled liquid approaches the mode-coupling (dynamical) transition temperature $T_c$, mean-field theory predicts that the four-point dynamical susceptibility $\chi_4$ — the standard measure of dynamical heterogeneity — diverges as $|T-T_c|^{-1/2}$, with an associated dynamical correlation length diverging as $|T-T_c|^{-1/4}$. Franz and Parisi established this in the spherical $p$-spin model and in a replicated liquid theory, but had to leave open what happens in real, finite-dimensional liquids, where activated (hopping) processes restore ergodicity below $T_c$ and are expected to round the divergence. The problem asks for a quantitative and self-consistent — not merely phenomenological — theoretical treatment of activated barrier crossing and its effect on $\chi_4$ and on the dynamical correlation length through the crossover into the deeply supercooled regime. The controlled confirmation of the mean-field scaling exponents close to $T_c$ and the explicit relation between the overlap-based $\chi_4$ and a true spatial correlation length are treated as settled and lie outside this deliverable; the open question is the quantitative microscopic theory of how activated processes modify $\chi_4$ and its length scale.

Background

A supercooled liquid is a liquid cooled below its freezing point without crystallizing; its structural (alpha) relaxation time grows by many orders of magnitude upon cooling until the liquid falls out of equilibrium and forms a glass. A key discovery of the last twenty-five years is that this slowing-down is dynamically heterogeneous: at any instant, particle mobility is spatially clustered, and the clusters grow on cooling. The standard quantitative diagnostic is the four-point dynamical susceptibility $\chi_4$, defined from the covariance of a two-point-like overlap order parameter: one introduces a weight function $w$ (e.g. a window of width $a$ in the relative displacement of a particle between times 0 and $t$), defines the local overlap $q_i(t)$ of particle $i$, and integrates the spatial correlation $g_4$ of the fluctuations $\delta q_i(t)$ over distance to obtain $\chi_4(t)=N\langle(\delta q)^2\rangle$, whose maximum quantifies the number of particles relaxing cooperatively. In the mean-field theory of the glass transition this quantity becomes sharp and computable. Two mean-field frameworks are relevant. First, the spherical $p$-spin model: $N$ continuous spins confined to a sphere, coupled by quenched random $p$-body Gaussian interactions; it has a dynamical (mode-coupling-like) transition at a temperature $T_c$ below which the Langevin dynamics arrests into a metastable glassy state with a nonzero Edwards-Anderson-like order parameter (the frozen long-time value of the self-overlap of the configuration with its initial state), and below which mode-coupling theory (MCT) — the microscopic theory of caging (the trapping of a particle by the cage of its neighbors) in liquids, which predicts a power-law divergence of the relaxation time at $T_c$ — applies. Second, the Franz-Parisi constrained (replicated) potential: one considers two copies of the system, constrains their mutual overlap $q$ (computed with the same weight function $w$ as for $\chi_4$), and studies the free energy $V(q)$, which for $T<T_c$ develops a secondary minimum signaling the existence of long-lived amorphous (glassy) states. Franz and Parisi showed that in this framework $\chi_4$ is the inverse curvature of $V(q)$ at its minimum, that $\chi_4$ grows as $|T-T_c|^{-\gamma}$ with the mean-field exponent $\gamma=1/2$ on approaching $T_c$ from below, and independently computed $\chi_4(t)$ from dynamical (MCT-like) equations for the $p=3$ spherical $p$-spin with a persistent coupling to the initial condition, fitting $\gamma=0.52\pm0.02$ and $\alpha=1.1\pm0.1$ from numerical solutions that reached only $T=0.7$, compared with the transition temperature $T_c=0.612$. For realistic liquids they combined this replica construction with the hypernetted-chain (HNC) approximation — a standard integral-equation closure of liquid-state theory used to evaluate the replicated free energy, here for hard spheres. The theory they built is the dynamical counterpart of the static random first-order transition (RFOT) theory of the glass transition (the framework that develops the mean-field picture of exponentially many metastable amorphous states into a theory of finite-dimensional glasses). Crucially, all of these mean-field schemes miss one physical ingredient that is present in any finite-dimensional system: activated processes (also called hopping), i.e. thermally activated barrier-crossing events that restore ergodicity below $T_c$ and are expected to round off the predicted divergence. In the source paper these could only be included phenomenologically, 'by hand'. Hopping has since been generated self-consistently within extended mode-coupling schemes for the two-point dynamics, but those schemes do not compute the four-point correlators that define $\chi_4$. The paper is otherwise the seed of the modern theory and measurement of dynamical heterogeneity: it fixed the definition of $\chi_4$ via the overlap with weight $w$ (their eq. (9)), which became the standard field definition, and it launched the experimental nonlinear-susceptibility program that followed.

Problem Statement

Provide a quantitative theory of the four-point dynamical susceptibility $\chi_4$ of supercooled liquids and of its associated dynamical correlation length $\xi$ across the glass crossover, in the presence of activated processes. The mean-field theory (spherical $p$-spin dynamics, mode-coupling theory, and the Franz-Parisi constrained potential) predicts $\chi_4\simeq|T-T_c|^{-1/2}$ at the dynamical transition temperature $T_c$, and inhomogeneous mode-coupling theory (the spatially resolved version of MCT) gives $\xi\sim|T-T_c|^{-1/4}$; in real finite-dimensional systems, activated (hopping) processes restore ergodicity below $T_c$ and are expected to round this divergence. Two further issues raised by the source are now established mean-field results and form part of the background to, rather than deliverables of, this problem: the confirmation of the mean-field scaling exponents $\gamma=1/2$ for $\chi_4$ and $\xi\sim|T-T_c|^{-1/4}$ close to $T_c$ (the source's numerics reached only $T=0.7$ against $T_c=0.612$), and the explicit relation between the overlap-based $\chi_4$ (the four-point function defined with the weight $w$, the source's eq. (9)) and a true spatial correlation length, the original $p$-spin model having no spatial structure. The open problem comprises one coherent objective with the following jointly required components. (i) Compute the quantitative effect of activated processes on $\chi_4(T)$ and on $\xi(T)$: how the mean-field divergence is rounded, what the maximum of $\chi_4$ becomes, and how both quantities behave through the crossover into the deeply supercooled regime. (ii) Construct a self-consistent theoretical treatment of activated barrier crossing and its effect on the dynamical susceptibility, replacing the phenomenological, 'by hand' inclusion of hopping: the activated rates should be determined by, and feed back on, the structure of the quasi-arrested state that the dynamics itself produces. Component (ii) is the stronger requirement and subsumes (i): a self-consistent construction whose activated rates feed back on the quasi-arrested state already yields the quantitative rounding of $\chi_4(T)$ and $\xi(T)$ that (i) requests, so solving (ii) resolves both, whereas (i) alone, without the self-consistent construction, does not resolve the objective. (iii) Determine which mechanism — activated processes, critical dynamical fluctuations (the fluctuations of the collective dynamics that below the upper critical dimension $d_c=8$ also round the mean-field singularity), or both — quantitatively controls the rounding, with predictions testable against simulation and experiment. Components (i)-(iii) are jointly required: (i) and (ii) must report the self-consistent activated contribution to $\chi_4(T)$ and $\xi(T)$ — the deliverable the source left open — and (iii) is a genuine determination, with activated control expected but not assumed, of how much of the rounding that contribution, rather than critical dynamical fluctuations, accounts for.

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