Out-of-equilibrium dynamics of the marginal (Gardner) phase of the infinite-dimensional hard-sphere glass

The exact mean-field (replica) solution of hard spheres in the limit of infinite spatial dimension predicts that deep in the glass phase each metastable basin undergoes the Gardner transition, fracturing into a marginally stable, hierarchically organized (full replica-symmetry-breaking) landscape that contains the jamming transition. The statics of this marginal phase is solved, its critical exponents are confirmed by numerical simulations in dimensions $2$ to $13$, and signatures of the marginal glass phase have been observed in granular, colloidal and molecular experiments. What is not known is the out-of-equilibrium dynamics of this phase: the exact dynamical mean-field-theory equations for infinite-dimensional hard spheres under general protocols have been derived, but their solution inside the marginal phase — the aging laws, the fluctuation-dissipation structure, and the dynamical signatures of marginal stability following a quench or compression into that phase — remains open, as flagged by the source authors themselves. The problem is to solve these equations for specified protocols, with consistency against the solved equilibrium and static limits, and to determine whether the dynamics can enter or remain in the marginal phase.

Background

A glass is an amorphous solid obtained when a liquid is cooled or compressed too fast to crystallize: it falls out of equilibrium and remains confined, on any observable timescale, to a small region of configuration space called a metastable basin. The Random First Order Transition (RFOT) theory describes this at the mean-field level, and its central tool is the replica method, in which $n$ copies of the system are introduced in order to count and weigh the metastable basins. RFOT predicts that below the glass transition the equilibrium measure splits into exponentially many basins, each producing a plateau in the density-correlation function and hence the glassy slowdown.

The theory becomes exact, and fully solvable, for hard spheres in the limit of spatial dimension $d\to\infty$. Deep in the glass phase each basin undergoes the Gardner transition: it fractures into a hierarchy of sub-basins within sub-basins, described by full replica symmetry breaking (full RSB, the Parisi scheme in which this hierarchy is continuous), and becomes marginal — the replicon eigenvalue, the softest mode measuring the stability of a basin, vanishes, so the free-energy landscape is rough and self-similar on all scales. This marginal phase exists at all pressures above the Gardner line and contains the jamming transition, the infinite-pressure point at which the packing becomes mechanically rigid and isostatic (held together by the minimum number of contacts consistent with mechanical equilibrium). Solving the full-RSB equations analytically fixed the critical exponents of jamming — the weak-force exponent $\theta=0.42311$ governing the small-force distribution $P(f)\sim f^{\theta}$, the quasi-contact exponent $\alpha=0.41269$, and the cage exponent $\kappa=1.41574$ governing the decay of the cage size $\Delta\sim p^{-\kappa}$ with pressure $p$ — together with the scaling function $\mathcal{F}(\lambda)$ of the pair-correlation contact peak, all in agreement with numerical simulations in dimensions $2$ to $13$. The whole construction is consolidated in the monograph by Parisi, Urbani and Zamponi.

The dynamical counterpart is dynamical mean-field theory (DMFT), the technique — originally developed for lattice models — in which a many-body dynamics is replaced by a single effective stochastic process with memory and colored noise, determined self-consistently; in $d\to\infty$ the dynamics of a tagged hard sphere reduces exactly to such an effective one-dimensional process. The equilibrium dynamics of the liquid has been solved in this framework, the exact out-of-equilibrium DMFT equations (valid for quenches, compressions, shear, and general aging protocols) have been derived, and the statics of a prepared glass under compression or shear is separately controlled by state following, the replica construction that tracks one selected glass state along such a protocol. What is missing is the solution of the dynamical equations inside the marginal phase. The source paper states that characterizing the off-equilibrium dynamics in the marginal phase 'is at present only poorly understood', and its long companion paper calls this study 'the most important point' that 'remains an open problem'. Aging — the dependence of relaxation on the waiting time $t_w$ elapsed since preparation, with ever-growing relaxation times and violation of the fluctuation-dissipation theorem (FDT, the equilibrium relation between correlation and response, quantified by a non-trivial fluctuation-dissipation ratio and effective temperatures) — is expected to carry specific signatures of marginal stability, such as a diverging four-point susceptibility (the correlation between local relaxation events at different points), but these have so far been conjectured only from statics. Even in the spherical mixed $p$-spin model — a solvable mean-field spin glass with random $p$-body couplings that is the prototypical RFOT system and the standard testbed for aging dynamics — the asymptotic dynamics in the marginal regime is not solved analytically and continues to reveal surprises.

Problem Statement

Derive and solve the out-of-equilibrium (aging) dynamics of the exact infinite-dimensional hard-sphere glass inside the marginal (Gardner, full-RSB) phase, using the already-derived out-of-equilibrium dynamical mean-field-theory (DMFT) equations — the exact equations established for infinite-dimensional hard spheres under arbitrary isotropic protocols by Agoritsas, Maimbourg and Zamponi, extended by them to shear strain, with initial conditions tied to the state-following construction of Rainone, Urbani, Yoshino and Zamponi and of Rainone and Urbani. Take a glass prepared in equilibrium at a pressure below the Gardner line (a preparation for which the static state-following solution is known) and follow a specified protocol that takes it into the marginal phase; state the class of protocols for which the result is expected to hold. For the chosen protocol, determine: (a) the two-time observables: the mean-square displacement, the correlation functions, and the linear response functions, as functions of waiting time $t_w$ and observation time $t$; (b) the aging laws: how the relaxation times (and lengths, via cage growth) grow with age; the structure of the fluctuation-dissipation relation in the aging regime, including the fluctuation-dissipation-ratio / effective-temperature structure and any sub-aging exponents; (c) the dynamical manifestation of marginal stability: whether the trajectory settles on threshold states (free-energy stationary points whose replicon eigenvalue vanishes, at the top of the hierarchy) or on other marginal states, how the hierarchical full-RSB structure emerges dynamically in the self-consistent memory kernels, and the predicted dynamical signatures of marginality, such as the behavior of the four-point susceptibility; (d) consistency with all solved limits: the equilibrium DMFT of the liquid, the state-following solution below the Gardner line, and the static full-RSB solution (including the cage scaling $\Delta\sim p^{-\kappa}$ and the fluctuation-dissipation ratio implied by statics). Determine, within the same framework, whether the mean-field dynamics can neither enter nor remain in the marginal phase — a determination to be made for the class of admissible protocols as a whole, rather than only for a single protocol or limiting regime.

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