From sub-aging to hyper-aging in structural glasses

We demonstrate non-equilibrium scaling laws for the aging dynamics in glass formers that emerge from combining a recent application of Onsager's theory of irreversible processes with the equilibrium scaling laws of glassy dynamics. Different scaling regimes are predicted for the evolution of the system's structural relaxation time $\tau$ with age (waiting time $t_w$), depending on the depth of the quench from the liquid into the glass: \emph{simple aging} ($\tau\sim t_w$) applies for quenches close to the critical point of mode-coupling theory (MCT) and implies \emph{sub-aging} ($\tau\approx\ t_w^\delta$ with $\delta<1$) as a broad cross-over for quenches to nearly-arrested equilibrium states; \emph{hyper-aging} (or \emph{super-aging}, $\tau\sim t_w^{\delta'}$ with $\delta'>1$) emerges for quenches deep into the glass. The latter is cut off by non-mean-field fluctuations that we account for within a recent extension of MCT, the stochastic $\beta$-relaxation theory (SBR). We exemplify the scaling laws by a schematic model that allows to quantitatively fit recent simulation results for density-quenched hard-sphere-like particles.

The response of a viscous fluid to a sudden change in control parameters reveals a rich phenomenology as the system adapts to this change. If the time scale of structural relaxation in the fluid, τ , is large, a slow evolution of both static and dynamic properties of the fluid with system age (i.e., the waiting time t w after the quench) is observed. For kinetically arrested states such as glasses, this aging dynamics implies that the properties of the material depend on the protocol of its fabrication, as a clear signature of the non-equilibrium process [1][2][3]. Hence, the understanding of the relevant non-trivial time scales in aging is of fundamental interest for theoretical physics and materials science alike [4][5][6].
Here we establish non-equilibrium scaling laws of aging and show how they emerge from the equilibrium scaling laws for structural relaxation near the glass transition, specifically near the critical point of mode-coupling theory (MCT). In principle this directly relates non-equilibrium aging exponents to the equilibrium structure of a glass-forming material. The predictions arise from combining two recent theoretical approaches to describe the dynamics of glass-forming fluids: the nonequilibrium self-consistent generalized Langevin equation theory (NE-SCGLE) provides a starting point linking the waiting-time evolution of static properties to the relaxation dynamics of the system, while a recent extension of MCT, the stochastic β-relaxation theory (SBR) provides scaling laws for this relaxation dynamics that also include the effect of non-mean-field fluctuations in the ideal glass of MCT. We demonstrate the scaling laws by a quantitative comparison to recent computer-simulation results for the evolution following density-quenches in hard-spherelike systems [28,29] that elucidate three different scaling regimes predicted by the theory.
MCT is a microscopic theory [30,31] that very successfully describes the liquid-state dynamics close to the glass transition. In its original form it is restricted to the equilibrium ensemble, although recent extensions allow to treat nonlinear response to various external fields [32][33][34][35][36]. Its application to aging dynamics has been proposed 20 years ago by Latz [37,38], but the complexity of that theory has so far only allowed to obtain some results linked to the seminal work by Cugliandolo et al. [4,39] on the pspin model [40]. The complexity stems from the fact that in absence of the equilibrium fluctuation-dissipation theorem (FDT), correlation and response functions are not straightforwardly connected, and are described by coupled integral equations that are not readily evaluated.
To cut this Gordian knot, the NE-SCGLE [41][42][43] invokes an assumption of "local stationarity" for the relaxation process, reducing the complexity of the full problem considerably. Essentially, it partially decouples the arXiv:2202.13384v1 [cond-mat.soft] 27 Feb 2022 evolution of the correlation functions from that of the underlying static response functions. The resulting theory tests favorably against both simulation [28,29,44] and experimental data [45][46][47].
NE-SCGLE in fact refers to two separate ingredients: an evolution equation for the static obervables, and an underlying kinetic theory for the mobility of rearrangements, the SCGLE [48]. The latter is, for the present purposes, structurally identical to MCT. In particular, it provides the same asymptotic scaling laws for the equilibrium structural relaxation [49]. We will use those well-established scaling laws to describe the asymptotic waiting-time dependence after a quench.
The non-equilibrium extension of the SCGLE is usually derived by referencing Onsager's laws of linear irreversible thermodynamics and the corresponding stochastic theory of thermal fluctuations (see Refs. [50,51]). Under certain assumptions, it leads to an innocuous looking relaxation equation for the waiting-time evolution of the non-equilibrium static structure factor S(k; t w ). We demonstrate that this equation can also be rationalized in a spirit closer to MCT employing the integrationthrough transients (ITT) formalism [32]: writing the evolution equation of the non-equilibrium distribution function p(t) of a system as ∂ t p(t) = Ω(t)p(t), with some linear differential operator Ω(t), a formal solution is where k are the microscopic number-density fluctuations and (f, g) is the usual L 2 scalar product in Hilbert space. For a sudden quench, Ω(t) = Ω i for t < 0 and Ω(t) = Ω f for t > 0, we can make use of the relation Ω(t )p(t w ) = ∂ tw p(t w ) for all t ≥ t w > 0, which avoids the need to formulate the effect of the quench in the time-evolution operator explicitly. Projecting onto density-pair modes as the relevant variables, P 2 = k − k p(t w ))N −1 ( − k k (suitably normalized), and neglecting memory effects, we obtain S(k; t) − S(k; t w ) ≈ t tw dt C 4 (k; t, t ) ∂ tw S(k; t w ) with some four-point density correlation function C 4 (k; t, t ), and thus for t → ∞, where µ(k; t w ) is a mobility factor that is slaved to the structural relaxation dynamics [7,52]. The initial state before the quench is S(k; 0) = S i (k), and S f (k) characterizes the quenched-to final state. Equation (1) essentially is a formalized extension of the empirical Tool-Narayanaswamy model of physical aging [53]. Equation (1) already predicts universal scaling laws for the aging dynamics to be encoded in the equilibrium dynamics: since the glass transition is a dynamical phenomenon, in its vicinity the static structure functions remain regular, and we can linearize S(k; t w ) for small control-parameter distances ε(t w ) to the transition. The temporal evolution is thus asymptotically governed by the evolution of the distance parameter along the relevant direction in k-space (MCT's critical eigenvector [30,54]), Now enter the scaling laws for µ(ε): close to the critical point of MCT, µ(ε) ∼ 1/τ (ε) ∼ (−ε) γ for liquid states (ε < 0), and µ(ε) = 0 in the ideal-glass state (ε ≥ 0). The non-trivial exponent γ is related to the equilibrium structure of the system at its glass transition through the MCT exponent parameter λ [30,49]. The fact that µ approaches zero, allows for non-equilibrium stationary solutions of Eq. (2), where the relaxation towards equilibrium gets "stuck". We immediately get two important scaling laws from Eq. (2): (i ) for quenches close to the glass-transition point (|ε f | |ε i |), there exists a growing window in t w , where ∂ tw ε ∼ |ε| γ+1 , which results in |ε| ∼ t −1/γ w and, thus, simple or full aging, τ ∼ t w as t w → ∞.
(ii ) for a deep quench into the ideal glass, ε f |ε(t w )| holds in the limit of t w → ∞, because the relaxation gets stuck around values close to zero. Then, ∂ tw ε ∼ |ε| γ , resulting in the asymptotic law τ ∼ t γ/(γ−1) w . Since γ > 1, the exponent δ = γ/(γ − 1) is also larger than unity, and we find hyper -aging or super-aging, τ ∼ t δ w for t w → ∞. These scaling laws describe the idealized indefinite aging of a system that is quenched to a state with infinite relaxation time. In reality, the ultimate MCT-like divergence of the relaxation time is not observed; this one can attribute to long-wavelength fluctuations that cause deviations from the mean-field like scenario [55][56][57]. It will provide a cut-off for the scaling laws, rendering them transient rather than truly infinite-waiting-time asymptotes, as we shall discuss below.
For quenches to liquid states close to the glass transition, ε f < 0, the mobility always remains positive, and the corresponding long-time asymptote is then (iii ) τ ∼ const. for t w → ∞. For the typical slow evolution of the structural relaxation time, this implies a broad cross-over where τ grows sublinearly with t w , and hence sub-aging. Although not a rigorous asymptote, an empirical power law, τ ≈ t δ w with δ < 1, typically fits well in this regime [40].
To elucidate the emergence of the three regimessimple, sub-and hyper -aging -we devise a schematic model of aging. Qualitatively, the mobility is the inverse of an integrated friction memory kernel; in the spirit of MCT schematic models, we assume that the slow dynamics of all such microscopic correlation functions is governed by a single-mode (density) correlation function φ(t; t w ), 10 −1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 1 The latter obeys a Mori-Zwanzig type integral equation, (3b) In Eq. (3b) we anticipate that t w only enters parametrically in determining the coupling coefficients of the memory kernel m(t; t w ). This encodes the assumption of local stationarity, and is in the spirit of the ITT framework [33] that relates non-equilibrium transport coefficients to such "transient" correlation functions.
We complete the schematic model by the closure with two coupling parameters v 1 and v 2 that describe the current t w -dependent state of the system. For fixed t w , the model specified by Eqs. (3b) and (3c) is the widely studied schematic F 12 model of MCT. It has a line of glass transitions (v c 1 , v c 2 ) where ε = 0. Equations (3) define our schematic model. Together with the (mean-field) assumption τ (t w ) ∝ 1/µ(t w ), and v 1 = v c 1 , v 2 (t w ) = v c 2 (1 + ε(t w )) to define the distance to the glass transition point, it allows to fit available computer-simulation data for τ (t w ) after mapping ε i = ε(0) and ε f to the simulation's control parameters.
Results for τ (t w ) from the schematic model for quenches to various final states close to the MCT transition give a consistent description of computer-simulation data for density-quenched quasi-hard spheres (Fig. 1). For the fit, we have allowed to adjust a global time scale and the proportionality factor between µ and 1/τ , and we have chosen a transition point (v c 1 , v c 2 ) such that the exponent parameter of MCT matches a value usually found for hard-sphere like systems, λ = 0.735. This determines the exponent γ = 1/2a , and thus the exponent δ ≈ 1.684.
The schematic model elucidates the three aging regimes of the ideal-glass theory: empirical sub-aging is found as a cross-over for quenches to final states in the liquid, ε f < 0, while hyper-aging emerges from the model as the asymptote for quenches to the glass, ε f > 0. A growing intermediate-t w window that extends to t w → ∞ at the critical point of MCT, ε f = 0, displays simple aging.
The evolution of τ after the quench relates to the wellknown problem of determining a diverging relaxation time at fixed waiting time t w (corresponding to a typical experiment duration or probing time scale): approaching the transition, the power-law divergence of τ as a function of quenched-to state ε f that is predicted by the idealized theory, is cut off at any finite t w , and replaced by a cross-over to a slower growth (Fig. 2). In our model, we obtain τ ∼ |ε| δ , with a prefactor that diverges with increasing t w (dash-dotted lines in Fig. 2).
Deviations from the ideal theory are noted in the simulation data for quenches to the highest final densities and at large t w . We attribute this to the avoidance of the ideal MCT transition, that also causes the hyper-aging regime to be interrupted.
To understand this, we turn to the SBR [55,56], a recent extension of MCT that includes fluctuations in the local glassiness, viewing σ ∼ ε as a dynamical fluctuating order parameter. SBR predicts scaling laws that re- place the divergent power law with a cross-over between a power law on the liquid side and exponential growth on the glassy side of the transition. Specifically [58], for the structural relaxation time and for the mobility where we have identified σ = ε. Here, ∆σ is a material parameter that quantifies the strength of longwavelength order-parameter fluctuations. Using Eqs. (4) to evaluate µ in Eq. (2) and to calculate τ , we obtain an improved asymptotic description of the τ -vs-t w curves (colored dashed lines in Fig. 1) that account for the crossover from hyper-aging to a constant τ as the system finally equilibrates even in the ideal-MCT glass. Interestingly, the hyper-aging law predicted by the ideal theory still survives as a transient. In the simulation data, this is best seen as a non-monotonic variation of the ratio τ /t w as a function of t w that is present for all quenches to ϕ f > ϕ c (Fig. 3). This transient hyperaging signature fits well the corresponding SBR prediction (solid lines in Fig. 3).
In conclusion, we present scaling laws for the evolution of the structural relaxation time τ as a function of system age t w after the quench of a glass-forming fluid to states close to the ideal glass-transition point of MCT. Based on the NE-SCGLE to describe the evolution of static quantities after such quenches, the scaling laws delineate regimes of simple and transient hyper-and subaging.
The results link the hyper-aging exponent δ to the exponent characterizing the equilibrium relaxation time. Hence, they link a non-equilibrium dynamical exponent of the system to a non-trivial equilibrium exponent, and through this to the equilibrium static structure of the system. Sub-aging on the other hand, emerges only as an effective cross-over, i.e., as a finite-t w deviation from the mathematically rigorous simple-aging asymptote.
Interrupted hyper-aging versus sub-aging emerges as a clear indicator of the separation between ideal-glass like dynamics, and the dynamics that arises from the avoidance of the ideal glass transition. It could in principle be used to determine more precisely the position of the ideal glass transition.
This separation leads us to speculate that models with a non-avoided MCT-like glass transition might show clear hyper-aging asymptotes. High-dimensional systems of hard spheres, approaching the expected mean-field-like behavior in d = ∞ [59,60], might be suitable candidates. On the other hand, in the context of spin glasses with MCT transitions, e.g., the spherical p-spin model, numerical solutions so far favor sub-and normal aging [4,40,61]. But the analytical determination of the scaling laws is still a critical open issue [40,61]. Hyper-aging in a trapped phase has been discussed very recently in the context of decision-making models that incorporate reinforcement by memory effects [62]. Our Eq. (2) predicts weak ergodicity breaking and aging that gets stuck at the MCT-cricial point; it will be interesting to explore the connection to the strong ergodicity breaking discussed in spin glasses [63] and the loss of ultrametricity connected with the hyper-aging asymptote in suitably enhanced models.
We thank L. Berthier, M. Fuchs, and G. Szamel for their valuable comments, and A. Meyer and M. Medina-Noyola for continued support. Part of this work has benefited from discussions at the CECAM Flagship Workshop "Memory Effects in Dynamical Processes" of the Erwin-Schrödinger Institut (ESI) in Vienna. Th.V. also thanks the Glass & Time group at Roskilde University and specifically Jeppe Dyre for their kind hospitality during a research visit where this manuscript was finalized.