The role of excitations in supercooled liquids: Density, geometry, and relaxation dynamics

Significance

Understanding the mechanisms controlling the glass transition is a central challenge of condensed matter. Here, we introduced an algorithm that extracts the elementary excitations of glasses numerically up to an energy scale never reached before. We measure excitations systematically to characterize their density, geometry, and location and find that these quantities are remarkably predictive of structural relaxation. The geometry of excitations reveals the presence of a dynamical transition, supporting that the latter governs structural relaxation even at low temperatures.

Abstract

Low-energy excitations play a key role in all condensed-matter systems, yet there is limited understanding of their nature in glasses, where they correspond to local rearrangements of groups of particles. Here, we introduce an algorithm to systematically uncover these excitations up to the activation energy scale relevant to structural relaxation. We use it in a model system to measure the density of states on a scale never achieved before, confirming that this quantity shifts to higher energy under cooling, precisely as the activation energy does. Second, we show that the excitations’ energetic and spatial features allow one to predict with great accuracy the dynamic propensity, i.e., the location of future relaxation dynamics. Finally, we find that excitations have a primary field whose properties, including the displacement of the most mobile particle, scale as a power-law of their activation energy and are independent of temperature. Additionally, they exhibit an outer deformation field that depends on the material’s stability and, therefore, on temperature. We build a scaling description of these findings. Overall, our analysis supports that excitations play a crucial role in regulating relaxation dynamics near the glass transition, effectively suppressing the transition to dynamical arrest predicted by mean-field theories while also being strongly influenced by it.

In low-temperature glasses, elementary excitations are two-level systems, groups of particles that can tunnel between two states (1–6). At much higher temperatures near the glass transition T_g, structural relaxation occurs on a time scale $\tau ={\tau }_{0}exp (E_a/T)$, where τ₀ is microscopic time and E_a is an activation energy that grows on cooling in fragile liquids (7). Thermally activated elementary rearrangements of a few particles are important in this temperature range as well, as they contribute to structural relaxation (8–11). Such rearrangements thus play a vital role in theories of the glass transition (12–15). In Kinetically Constrained Models, local rearrangements are defects that can diffuse and interact to relax the system (16, 17). In mean field theories, local rearrangements correspond to the string-like (18–21) “hopping processes” through which finite-dimensional systems relax below the dynamical or mode coupling temperature T_c, where the dynamics would halt in infinite dimensions where these processes are absent (22, 23). Finally, in elastic models of the glass transition (24–28), the energy of local excitations directly determines the activation energy E_a. Theoretically, distinct ideas have been proposed to understand the geometry of local rearrangements, including entropic considerations (29) or the existence of defects around a hexatic phase in two dimensions (17). Alternatively, building on the notion of a length scale diverging (30–32) at the dynamical transition T_c, ref. 33 derived relationships between geometric and energetic properties of the excitations with the minimal energy.

Differentiating between different scenarios for the geometry of these excitations requires measuring their density of states $N(E)$ and geometry across a broad energy spectrum, a challenge that remains unresolved. Indeed, potential energy landscape studies (34, 35) access the consecutive excitations activated in a liquid during its relations but cannot provide the density of states of excitations. Studies conducted on the few lowest-energy excitations (6, 36–39), are pertinent to the plastic and quantum properties of glasses and not directly relevant to the glass transition, which typically involves much higher energy rearrangements. We recently developed SEER (40), an algorithm based on thermal exploration that allowed us to measure $N(E)$. However, SEER only accesses excitations with energy notably smaller than the activation energy $E_a=Tlog (\tau /{\tau }_0)$ and hence does not inform on many excitations that have a significant probability to be triggered dynamically and contribute to structural relaxation (34). The excitations’ density of state and geometric properties in such a broad energy range have yet to be studied, as well as the connection between these observables and the dynamics.

Indeed, the hypothesis that excitations govern where future structural relaxation will occur remains to be tested. A key quantity to predict is the particle propensity (41), which is the squared particle displacement averaged over an isoconfigurational ensemble of trajectories. These trajectories share the same initial configuration but differ in their momenta, which are randomly drawn from the expected equilibrium distribution. If excitations regulate the relaxation dynamics, they should enable the prediction of propensity at later times, with high-energy excitations becoming increasingly relevant for these predictions over time.

In this work, we introduce an algorithm to identify excitations via mechanical perturbations. This algorithm offers fast computation and enables access to a wide range of energies, surpassing the activation energy of the model under consideration. Our findings reveal three key insights: i) We observe that the density of states $N(E)$ approximately follows $N(E)\propto {(E-E_g(T))}^a$ with $a\approx 2.7$ up to the activation energy. Additionally, we confirm that the variation in $E_g(T)$ predicts the change in activation energy (40). ii) We introduce an excitation-based propensity predictor that successfully correlates with the propensity and demonstrate that high-energy excitations contribute to long-term relaxation. iii) Remarkably, we show that certain excitations’ geometric properties, such as the displacement δ of the most-moving particle or the probability of displaying string-like motion, only depend on their energy E. However, other properties, like the volume of an excitation, depend on both energy and temperature. We reconcile these observations by considering that excitations possess a primary field solely governed by their energy, which then influences the surrounding medium on a scale determined by material stability and, consequently, temperature. Overall, our results suggest that the hopping processes that suppress the mean-field dynamical transition are very much affected by it.

ASEER: Athermal Systematic Excitation ExtRaction

Thermal cycling is the most straightforward strategy to uncover the excitations or rearrangements associated with a reference inherent structure (IS₀). In this approach, one repeatedly evolves the configuration, each time with a diverse momenta initialization, until an excitation occurs. Practically, given that the probability of observing an excitation is exponential in its activation energy, this approach can only expose the very low-activation energy barriers. We recently mitigated this issue by introducing the Systematic Excitation ExtRaction algorithm (40). This algorithm still relies on thermal cycles. However, after discovering an excitation, it modifies the energy functional to suppress it, ensuring that successive cycles discover novel excitations. SEER proved able to expose many low-energy excitations. Yet, it can still not expose excitations with a high activation energy relevant to the dynamics.

Here, we tackle this issue by introducing the Athermal Systematic Excitation ExtRaction or “ASEER.” This protocol builds on the idea that changing the topology of the Voronoi neighbors induces an excitation, in analogy with the triggering of T1 transition in two spatial dimensions; see, e.g., refs. 27, 28, and 42. To uncover an excitation, we increase the separation between two adjacent (à la Voronoi) particles i and j by modifying the energy functional via the addition of an elastic spring, $E_{\mathrm{spring}}(\mathrm{\Delta }r)=k{[(r_{\mathit{ij}}^0+\mathrm{\Delta }r)-r_{\mathit{ij}}]}^2$, $r_{\mathit{ij}}^0$ being the distance between the particles in IS₀. We stress that the algorithm constrains one degree of freedom, the distance between the particles, as the added spring can rotate. We have checked that the value of k is not critical and empirically fix it so that the spring is never compressed by more than $5\%$. We slowly increase $\mathrm{\Delta }r$ while continuously minimizing the energy to keep the system in a minimum of the expanded energy functional $U(\{{\mathbf{r}}_{\mathbf{i}}\})+E_{\mathrm{spring}}(\mathrm{\Delta }r)$. The $\mathrm{\Delta }r$ dependence of the spring energy (or of the total one) comprises smooth elastic branches punctuated by sudden drops corresponding to irreversible rearrangements. We focus on the first plastic event identified via a standard thresholding approach. When this event occurs, we remove the spring and minimize the energy again, potentially leading the system to a new IS^∗. This approach is illustrated in Fig. 1. We provide our implementation of ASEER in SI Appendix. This implementation has a computational cost scaling as $N^2$, as it involves the simulation of an N particle system to measure the energy as a function of the distance of each pair of Voronoi neighbors ($\propto N$).

Fig. 1.

Illustration of the ASEER algorithm. We detect excitations by enforcing a dipolar force through a spring connecting adjacent particles of energy-minimized configurations. We gradually increase the spring rest length until an instability occurs. We then remove the spring and minimize the energy, bringing the system to a novel energy minimum. The minimum energy pathway going from one state to the other is analyzed in the absence of the spring.

We investigate the minimum energy path connecting IS₀, and each uncovered IS^∗ via the nudge-elastic-band (NEB) method (43) to estimate the energy $E_{\mathrm{Saddle}}$ of the saddle point separating the considered ISs. If not stated otherwise, when the minimum energy path traverses additional ISs (≃20% of cases in the considered system), we redefine IS^∗ as the first encountered IS and repeat the NEB analysis. This results in a catalog of unique (we ensure each IS^∗ appears once) excitations, each characterized by its energy barrier $E=E_{\mathrm{Saddle}}-E_{{\mathrm{IS}}_0}$ and displacement field $\mathbf{dr}={\mathbf{r}}^{\ast }-{\mathbf{r}}^0$, with ${\mathbf{r}}^{\ast }$ and ${\mathbf{r}}^0$ the positions in the two ISs. This algorithm leads to large catalogs of unique excitations, as it uncovers from $1.2$ to $2$ excitations per particle on cooling. Notice that we uncover less than $NZ/2$ excitations, with Z the coordination number, as ASEER may induce duplicated excitations by inducing plastic events localized on distant particles rather than the coupled ones.

While excitations require particle rearrangements similar to those triggered by ASEER, finite-temperature excitations may, in principle, differ. We follow two approaches to show that ASEER uncovers excitations relevant to the relaxation dynamics. First, we compare the uncovered excitations with those extracted by the thermal SEER algorithm (40). This comparison is limited to excitations with low activation energy, as these are the only ones accessible to thermal cycling algorithms. Second, we investigate whether ASEER’s excitations directly relate to the finite-temperature relaxation dynamics. If ASEER accurately captures these excitations, then the long-time relaxation dynamics should be predictable from the ASEER-identified excitations.

Density of States

We applied ASEER to a polydisperse three-dimensional system of $N=2,000$ soft repulsive particles (44) that can be equilibrated up to experimentally comparable temperatures through the “swap” algorithm (45–48). In recent work (40), we investigated this model under the assumption that the structural relaxation time varies as $\tau ={\tau }_0{e}^{E_a(T)/T}$. Here, E_a is the activation energy, while ${\tau }_0={\tilde{\tau }}_0{e}^{S_a}$ is the combination of a microscopic time scale ${\tilde{\tau }}_0$ and S_a is the entropy of the activation barrier. We found with good approximation that τ₀ is temperature-independent, an inference based on measuring the relaxation time right after sudden changes of temperatures. It implies that in the considered liquid, the fragility is controlled by the change of activation energy, not its entropy. It also leads to an evaluation of the activation energy regulating structural relaxation $E_a(T)=Tlog (\tau (T)/{\tau }_0)$. For the two lowest temperatures (open circle), we estimate τ through the time-temperature superposition principle ( 40, 49; SI Appendix). In the Materials and Methods, we provide numerical details on the numerical model and the measure of the relaxation time.

Fig. 2A illustrates the energy E dependence of the density of excitations $N(E)$ uncovered by SEER, and its cumulative $F(E)$ (Inset). At each temperature T, we average over ten samples. Notably, $N(E)$ and $F(E)$ approximately shifts toward higher energy values as T decreases: the energy of local barriers grows under cooling. Indeed, before its maximum $N(E)$ is well described by

$$N(E)\propto {(E-E_g(T))}^a$$ [1]

Fig. 2.

(A) Density of excitations $N(E)$, normalized by the system size $\mathcal{N}$, and its cumulative distribution $F(E)$ (Inset). The solid dots mark the values of the activation energy (40), $E_a(T)=Tlog \left(\tau /{\tau }_0\right)$, and the cross-shaped line has been obtained with the SEER algorithm at $T=0.5$. (B) The $N(E)$ curves collapse when the energy is shifted by $E_g(T)$. The thick green curves is $N(E)\approx g_1\times {(E-E_g)}^{2.7\pm 0.1}$, where $g_1=(4.5\pm 0.5)\times {10}^{-4}$. (C) The shift of E_g on cooling matches the shift in activation energy evaluated via SEER (40). (D) The increase in E_g matches the increase in activation energy measured from the relaxation dynamics, with $T=0.5$ an arbitrary reference temperature. Open circles correspond to the low-temperature values at which we use the time-temperature superposition to estimate the relaxation time and E_a.

with $a\approx 2.7$, as demonstrated by the data collapse in Fig. 2B.

The energy gap E_g characterizing the system’s stability increases on cooling, as shown in Fig. 2C. Fig. 2D shows that the variation of E_g on cooling is consistent with that of the activation energy, indicating that local barriers control the dynamics in this liquid. The increase of a gap E_g is reminiscent of the mean-field prediction according to which, for $T<T_c$, the density of vibrational modes is gapped up to a frequency ${\omega }_{min}$ indeed increasing on cooling (50, 51).

In Fig. 2 A and C, we also present data previously obtained using the SEER algorithm (40) (see SI Appendix, Fig. S1 for a more in depth comparison). The algorithms yield consistent results for the variation of E_g (and other quantities shown in SI Appendix, Fig. S2), demonstrating that ASEER effectively captures excitations relevant to finite-temperature relaxation dynamics. Panel (A) highlights ASEER’s capability to detect excitations on the scale of E_a, which can drive relaxation on long time scales-excitations that thermal cycling algorithms like SEER are unable to expose.

Our studies below on propensity and on the architecture of excitations at the activation energy scale thus require ASEER and we use this algorithm alone to study propensity. However, an algorithm such as SEER, which is biased toward low-energy excitations and extracts ∼40 less excitations in a given sample, is about 40 times faster to extract those. Thus in our study on the architecture of excitations, we combine a dataset of 10 configurations analyzed with ASEER (necessary to obtain statistics on high-energy excitations) and 1,000 configurations analyzed with SEER (necessary to obtain statistics on the less numerous low-energy excitations). Further evidence on the consistency of the two approaches is shown in SI Appendix, Fig. S2.

Excitations Predict the Spatiotemporal Relaxation Dynamics

To demonstrate the relevance of excitations to the relaxation dynamics further, we examine their ability to predict structural relaxation. As a measure of structural relaxation, we focus on the propensity of motion (41). While particles with a high propensity are interpreted as more prone to structural relaxation, a particle’s propensity also depends on its vibrational motion. Indeed, by comparing the standard and the inherent structure mean square displacements, both averaged over configurations and isoconfigurational trajectories, Fig. 3A demonstrates that the vibrational contribution dominates the mean square displacements for a long transient. To ensure the propensity informs on structural relaxation, here we filter out the vibrational contribution by defining it from the inherent structure mean square displacement instead of the previously used finite temperature displacement. Henceforth, the propensity of particle i at time t is $p_i(t)={⟨{\mathrm{\Delta }\mathbf{r}}_{i,\mathrm{I}S}^2(t)⟩}_{\mathrm{isoconf}}$, with ${\mathrm{\Delta }\mathbf{r}}_{i,\mathrm{I}S}(t)={\mathbf{r}}_{i,\mathrm{I}S}(t)-{\mathbf{r}}_{i,\mathrm{I}S}(0)$, and ${\mathbf{r}}_{i,\mathrm{I}S}(t)$ the position of the particle in the IS associated with the configuration visited by the system at time t.

Fig. 3.

(A) Symbols identify the finite-temperature and IS mean square displacements averaged over 10 configurations and isoconfigurational runs. Lines are examples of IS mean square displacement of individual trajectories. (B) Time dependence of the Spearman correlation coefficient between the particle propensity and the excitation-based predictor, for varying number of considered excitations. All excitations were extracted by ASEER. $n\approx 70$ is approximately the number of excitations SEER extracts for each configuration, and leads to suboptimal performance. ASEER extracts many more, and $n\approx 1,000$ are those with activation energy smaller than $Tlog (\tau /t_0)$. (C) Time dependence of the Spearman correlation coefficient between the particle propensity and various predictors: the Debye–Waller factor (T-DWF) and its Harmonic approximation (H-DWF), the packing capability (PC), and the excitation-based predictor. All data refer to $T=0.45$, below the estimated mode-coupling critical temperature for this model, $T_c\in [0.51:0.54]$.

We define a predictor ${\mathrm{\Lambda }}_i^2(t,n)$ for the propensity of motion by analyzing the n excitations with the smallest activation energy associated with the initial configuration. To this end, we treat an excitation as a double potential well and consider it inactive when the system is in the minima of the $t=0$ configuration; it is active otherwise. Our proposed excitation-based predictor is

$${\mathrm{\Lambda }}_i^2(t,T_p,n)={\displaystyle \sum _{k=1}^n}{\mathbf{u}}_{i,k}^2{P}_k(T_p,t),$$ [2]

where ${\mathbf{u}}_{i,k}$ is the displacement of particle i in excitation k. Here, $P_k(T_p,t)=P_{\mathrm{eq}}\left[1-e^{-({\mathrm{\Gamma }}_f+{\mathrm{\Gamma }}_b)t}\right]$ is the probability that the excitation is active at time t within the approximation that excitations do not interact, ${\mathrm{\Gamma }}_{f,b}=t_0^{-1}{e}^{-E_{f,b}/T_p}$ are the forward and backward transitions rates, with $E_{f,b}$ the corresponding activation energies, and $P_{k,\mathrm{e}q}=\frac{{\mathrm{\Gamma }}_f}{{\mathrm{\Gamma }}_f+{\mathrm{\Gamma }}_b}$ is the equilibrium activation probability. We average the results over five independent initial configurations and consider 20 isoconfigurational trajectories for each configuration.

Fig. 3 A and B illustrates the time dependence of the Spearman’s rank correlation coefficient between ${\mathrm{\Lambda }}_i^2(t,n)$ and the IS-propensity $p_i(t)$ at $T=0.45$. We consider $n\approx 70$, the number of excitations per configuration typically extracted by the SEER algorithm (40). Other considered n values are bounded by the number of excitations with activation energy smaller than the activation energy E_a regulating structural relaxation, which is $n\approx 1,000$ according to Fig. 2B. We find the correlation coefficient is n independent up to its maximum value of ≃0.7, which occurs at $t\simeq {10}^{-3}\tau$, when the mean square displacement crossovers toward the diffusive regime. This result demonstrates that only low-energy excitations contribute to the short-time dynamics. The correlation coefficient decreases at long times, and better predictions are obtained on increasing n. Considering that the excitations are ordered according to their activation energy, this finding shows that at later times, high-energy excitations contribute to the relaxation process.

Previous works have performed related studies by considering predictors defined by analyzing the structural and elastic properties of the initial configuration; see ref. 52 for a review. In the following, we compare our excitation-based predictor to physically motivated [as opposed to machine-learning-derived (53)] ones: i) The Debye–Waller factor T-DWF = $⟨\mathrm{\Delta }{r}_i^2(t_{\text{DWF}})⟩$, with $t_{\text{DWF}}$ the estimated DWF time (54); ii) The Debye–Waller factor evaluated in the harmonic approximation H-DWF = $\sum _k{\mathbf{e}}_{i,k}^2{\omega }_k^{-2}$, where the sum is over the modes of the dynamical matrix associated with the $t=0$ inherent structure, ${\mathbf{e}}_{i,k}$ is the displacement of particle i in mode k, and ω_k the mode eigenfrequency (55); iii) The local PC, which measures how well the particles surrounding the considered one are well packed (56). iv) The particle radius. We highlight our approach is the only one whose predictions have a time dependence.

Fig. 3C demonstrates that the excitation-based predictor outperforms all other predictors, except in the very short time ballistic regime. The finite temperature Debye–Waller factor demonstrates a similar predictive ability close to the relaxation time, while the Harmonic DWF has a poorer predictive ability (27, 57). The PC and particle radius have the worst predictive ability, with the radius’s predictive power trivially increasing over time since smaller particles diffuse more than larger ones. This suggests that polydispersity disrupts the correlation between local geometrical properties and structural relaxation observed in mono- and bidisperse systems. Our predictor behaves similarly to recently proposed ones focusing on the local plastic rather than elastic response (28). However, while these works find predictors attaining their maximum value roughly at the relaxation time, our predictor peaks at earlier times. This possibly occurs as our predictor is based on a two-state model of independent excitations. In reality, excitations are coupled, as the occurrence of an excitation may affect others’ energy barrier.

Architecture of Excitations

We study the dependence of the geometrical properties of excitations on the temperature T and activation energy E by analyzing their displacement field $\mathbf{dr}$. We focus on: i) the characteristic number of involved particles, estimated by the participation ratio $V\equiv N{P}_r={(\sum {\mathbf{dr}}_i^2)}^2/(\sum {\mathbf{dr}}_i^4)$, where the sums run over all particles i; ii) the squared norm of the displacement field, ${|dr|}^2=\sum _i{\mathbf{dr}}_i^2$; iii) the maximum particle displacement, $\delta = \underset{i}{max}||{\mathbf{dr}}_i||$. Furthermore, we resolve the excitations’ spatial properties by investigating the decay of ${|dr|}^2(r)$ as a function of the distance r from the excitation center, defined as $\mathbf{r}=\sum _i\delta (\mathbf{r}-{\mathbf{r}}_i)$, ${\mathbf{r}}_i={\mathbf{dr}}_i-\sum _j{m}_j{\mathbf{dr}}_j$ where $m_i={\mathbf{dr}}_i^2/{|dr|}^2$ (33). Note that we exclude string-like excitations (SLEs) from the geometrical analysis (they are defined as excitations involving the exchange of particles; see below). In general, including them only affects results for the largest energies considered, as shown in SI Appendix, Fig. S5.

Architecture of Lowest-Energy Excitations.

We first focus on the excitations with the minimal activation energy $E_{\min }$. The Random First-Order Transition theory of the glass transition (22) predicts a dynamical transition (avoided in finite dimension) where the dynamics stops in large spatial dimension, closely related to the mode coupling theory of liquids. When heating a system toward this threshold temperature, an elastic instability occurs: vibrational modes become unstable and the elastic moduli soften. This transition is predicted to come with a diverging dynamical length scale (30–32). The authors of ref. 33 argued that these latter results could be used to characterize lowest-energy excitations, and predicted that for those:

$$V_{min}(T)\sim \frac{1}{{\delta }_{min}(T)}\sim \frac{1}{|d{r}_{min}{(T)|}^2}\sim \frac{1}{{E_{min}(T)}^{\frac{1}{3}}}.$$ [3]

Moreover, the spatial decay $G(r)$ of their squared magnitude is predicted to follow in three dimensions the scaling relation:

$$G(r)\xi \sim h(r/\xi ),$$ [4]

where h is a scaling function and the length scale $\xi (T)\sim \sqrt{V_{min}(T)}\sim {E_{min}(T)}^{-\frac{1}{6}}$. $\xi (T)$ is empirically known to characterize the linear response to an imposed dipole (58, 59). Ref. 33 verified the relationships between $V_{min}$, ${\delta }_{min}(T)$ and $|d{r}_{min}{(T)|}^2$, but the activation energy $E_{min}$ was not measured; instead a proxy corresponding to the energy difference between the two IS was used. Eq. 4 was not tested. We validate the scaling of the excitation architecture with $E_{min}$ in Fig. 4A–C (green stars) by investigating, at each temperature, the features of the lowest-energy excitation of 1,000 systems. Fig. 5A illustrates the spatial dependence of the squared norm of the displacement field ${|dr|}^2(r)$, normalized by its value at the excitation center, revealing the presence of far-field contributions which are larger at higher T. Fig. 5 A and B demonstrates that the different curves collapse as expected from Eq. 4.

Fig. 4.

Dependence of (A) volume, (B) largest particle displacement, and (C) squared norm of the displacement field on the excitation’s activation energy, for various temperatures. The black line $c_0{E}^{1/3}$ corresponds to the mean field prediction for the excitations with the smallest activation energy, illustrated in green. In panel (D), the square norm of the induced displacement field, $|d{r}_{\mathrm{i}}{|}^2={|dr|}^2-{|d{r}_p|}^2$ are collapsed by a model postulating that the excitations have a primary field inducing a T-dependent far-field displacement. Here, $|d{r}_p{|}^2=c_0{E}^{1/3}$, where c₀ is estimated as ${|dr|}^2(E=E_{min})/E_{\min }^{1/3}$. In all panels, we plot median values (see SI Appendix, Fig. S3 for mean values).

Fig. 5.

Spatial decay of the displacement of (A) the primary displacement, corresponding to the displacement of excitations with minimal energy, at each temperature, and of the overall (B) and induced (C) displacements multiplied by $r^4$ as a function of temperature at fixed E. Panels (D) and (E) demonstrate the corresponding data collapse obtained by rescaling distances by $\xi (T)=E_{\min }^{-1/6}$. Panel (F) demonstrates that the maxima seen in panel (E) occur at the same value of $r/\xi$, marked by a vertical line. In all panels, the displacements are rescaled by their value at the center of the excitation, ${\widehat{|dr|}}^2(r)\equiv {|dr|}^2(r)/{|dr|}^2(r=0)$.

Architecture of High-Energy Excitations.

Fig. 4 illustrates how the excitations’ geometric properties depend on E and T, for $E>E_{min}$. Panel (B) shows a remarkable result: the largest displacement in an excitation depends on the excitation energy but not on its temperature so that $\delta \propto E^{1/3}$. By contrast, panels (A) and (C) demonstrate that, at fixed energy, the volume of an excitation and its squared norm increase with temperature. Example displacement fields are presented in SI Appendix, Fig. S6.

We rationalize these observations by assuming that excitations consist of i) a primary field that depends only on energy E and ii) an induced elastic response. In this view, the norm square of an excitation consists of two parts ${|dr|}^2(T,E)=|d{r}_{\mathrm{p}}{|}^2+{|d{r}_{\mathrm{i}}|}^2$, which we now estimate.

Primary field: Since the primary field has energy E but no temperature dependence, its properties can be estimated at any temperature $T^{\prime }$. We thus estimate them at temperature $T^{\prime }$ at which $E_{\min }(T^{\prime })=E$, finding $|d{r}_{\mathrm{p}}{|}^2(E,T)={|d{r}_{\mathrm{p}}|}^2(E=E_{\min },T^{\prime })$. It follows that the primary fields comply with the scaling description of Eq. 3. The maximal particle displacement, and therefore of the whole excitation, behave as $\delta \sim E^{1/3}$, the number of particles in a primary field follows $V_{\mathrm{p}}(E)\sim E^{-1/3}$, and its the square norm follows $|d{r}_{\mathrm{p}}{|}^2(E)\propto {\delta }^2(E)V_{\mathrm{p}}(E)=c_0{E}^{1/3}$.

Induced response: Considering that the square displacement of the primary field scales with the energy as that of minimal energy excitations, the square displacement of the induced field $|d{r}_{\mathrm{i}}{|}^2$ must by definition correspond to the additional displacement over the line $c_0{E}^{1/3}$ seen in Fig. 4C. To estimate the magnitude of this induced displacement, two effects must be considered. First, we expect that the primary field effectively acts as a local dipole (60), of square magnitude $|d{r}_{\mathrm{p}}{|}^2(E)\propto E^{1/3}$. Second, the response to a dipolar perturbation depends on the stability of the material: it is more extended at large rather than small temperatures (58, 59), consistently with our observation in SI Appendix, Fig. S6. Quantitatively, we expect the induced field to have a volume proportional to $V_{min}(T)$ (33, 58), which is of order $\sim E_{min}{(T)}^{-1/3}$ according to Eq. 3. Putting these two effects together, we obtain

$$|d{r}_{\mathrm{i}}{|}^2\propto f\left(\frac{E}{E_{min}}\right){|d{r}_{\mathrm{p}}|}^2(E)V_{min}(T)\propto f\left(\frac{E}{E_{min}}\right){\left(\frac{E}{E_{min}}\right)}^{1/3},$$ [5]

where we introduced the interpolating function $f(x=E/E_{\min })$ capturing the fact that as $E\to E_{\min }$, the induced field must by definition vanish. Thus one must have $f(1)=0$ and $\underset{x\to \infty }{lim}f(x)=C>0$. Fig. 4D validates this theoretical prediction for a natural choice of interpolating function $f(E/E_{\min })\propto (x-1)/(x+1)$. We detail the determination of $f(x)$ in SI Appendix, Fig. S7. In SI Appendix, Fig. S4, we show that Eq. 5 also holds if $E_{min}$ is replaced by E₂ (the median of the second-lowest energy excitation) or E_g.

To test our description of excitations further, we consider their spatial properties. Fig. 5C illustrates ${|dr|}^2(E,T)(r)=|d{r}_{\mathrm{p}}{|}^2(E)(r)+{|d{r}_{\mathrm{i}}|}^2(E,T)(r)$ at fixed $E=E_{\min }(T=0.3)$ for various temperatures. Since $|d{r}_{\mathrm{p}}{|}^2$ is temperature independent, the increase in ${|dr|}^2$ with T demonstrates the increase of the induced response. Fig. 5D shows that these data also collapse when distances are rescaled by $\xi (T)$, indicating that this length scale controls the induced response, as predicted.

We finally consider the spatial properties of the induced response alone. Fig. 5E illustrates that $r^4\times {|d{r}_{\mathrm{i}}|}^2(E,T)(r)$ reaches a plateau at large r, as expected for fields induced by a dipolar response. At shorter distances, it exhibits a maximum, whose position is also controlled by the predicted length scale ξ, as demonstrated in Fig. 5F.

SLEs.

Numerical simulations (28, 39, 61, 62) and recent experiments (63) show string-like particle motion in low-temperature structural relaxation. String-like rearrangements involve one or more particles swapping positions in the core of the primary field, with displacements comparable to interparticle separation (33) (see examples in Dataset S5). We thus expect that i) High-energy excitations are more likely string-like due to their larger displacements. ii) The probability of a SLE is T-independent as those of the primary fields.

We identify SLEs by assuming that particle i ends up in the position originally occupied by particle $j\ne i$ if $|{\mathbf{r}}_j^{\ast }-{\mathbf{r}}_i^0|<\mathrm{\Delta }$, ${\mathbf{r}}_i^0$ and ${\mathbf{r}}_j^{\ast }$ being the positions in the initial and final configurations. The choice of Δ is not critical as long as its value corresponds to a small fraction of the interparticle distance. Here, we fix $\mathrm{\Delta }=0.1$. The number of particles involved in a string defines a string length $n_p=\sum _i\sum _{j,j\ne i}\theta \left(0.1-|{\mathbf{r}}_j^{\ast }-{\mathbf{r}}_i^0|\right)$. An excitation is string-like provided $n_p\ge 1$. An example SLE is in Movie S1. Fig. 6A shows that an excitation is a string with a T-independent probability $F_s=⟨\theta (n_p-1)⟩$ that increases with E, consistently with our expectations. The string length similarly increases with E and is approximately T-independent, as we illustrate in Fig. 6B.

Fig. 6.

Energy dependence of (A) the fraction of SLEs and (B) the average string length, n_p, conditioned to having a string.

Conclusion

Through the ASEER algorithm, we have measured the density of excitations in a model glass-former up to the activation energy. We confirmed that a shift in this density under cooling predicts with the change of activation energy of the liquid (40) and demonstrated that the excitations successfully correlate with the spatiotemporal relaxation dynamics. In particular, excitations correlate with particle propensity with great accuracy. Most importantly, we have clarified how the geometry of excitations depends on both their energy and temperature. Excitations display a primary field whose properties scale with their energy and an induced component whose length scale is governed by temperature. These insights align with known observations on the geometry of relaxation in liquids, such as the increased predominance of strings at lower temperatures.

These results support the idea that excitations are influenced by a dynamical transition. On the one hand, the shift in the density of excitations is reminiscent of the shift in the Hessian of the energy landscape predicted in high dimensions, for which a gap opens for temperatures below some dynamical transition. Note that such a shift of the density of excitations differs from the change that would be obtained with a simple rescaling of the energy scale. Such a rescaling would be expected from a naive interpretation of elastic models, where energies are scaled by a temperature-dependent elastic modulus. On the other hand, the excitation primary fields follow scaling laws expected in the vicinity of a dynamical transition. Overall, these points suggest that hopping processes govern the dynamics at all low temperatures and prevent the divergence of the relaxation time at T_c, a divergence that only occurs in infinite dimensions. However, these hopping processes are significantly affected by the presence of the elastic instability associated with T_c. In this view, the latter thus plays a key role in controlling activation even at lower temperatures.

Materials and Methods

We consider a three-dimensional system of soft repulsive particles (44) with size σ distributed as $p(\sigma )\propto {\sigma }^{-3}$ in the range [${\sigma }_{\min }$:$2.2{\sigma }_{\min }$]. This modern numerical model can be equilibrated up to experimentally comparable temperatures through the swap algorithm (45–48). The pair interaction is given by

$$U(r_{\mathit{ij}})=ϵ\left[{\left(\frac{{\sigma }_{\mathit{ij}}}{r_{\mathit{ij}}}\right)}^{10}+{\displaystyle \sum _{l=0}^3}{c}_{2l}{\left(\frac{r_{\mathit{ij}}}{{\sigma }_{\mathit{ij}}}\right)}^{2l}\right]$$ [6]

for $r_{\mathit{ij}}<x_c=1.4$. We use a nonadditive particle size ${\sigma }_{\mathit{ij}}=\frac{1}{2}({\sigma }_i+{\sigma }_j)(1-0.1|{\sigma }_i-{\sigma }_j|)$ to prevent crystallization and set $c_{2l}$ to enforce continuity at x_c up to three derivatives. We studied systems of $\mathcal{N}=2,000$ particles of mass m at number density $\rho =0.58$ in cubic simulation boxes with periodic boundary conditions. We express mass in units of m, temperature in units of ϵ, lengths in units ${\rho }^{-1/3}$, and time in units of $\sqrt{m{\sigma }_{\min }^2/ϵ}$. We minimize the energy of configurations equilibrated at temperature T to produce ISs that we investigate with ASEER.

For this model system, in a previous work (40), we have i) investigated the relaxation dynamics and shown that the self-scattering correlation function evaluated at the first peak of the static structure factor, a self-overlap function, and a total-overlap function, give consistent measures for the temperature dependence of the relaxation time τ. All correlation functions satisfy the time-temperature superposition principle, which we exploit to measure the relaxation time at very low temperatures. ii) Estimated the microscopic time τ₀ influencing the relaxation time, $\tau ={\tau }_0{e}^{E_a(T)/T}$.

The evaluation of τ and of τ₀ allows us to measure the activation energy regulating structural relaxation, $E_a(T)=Tlog (\tau /{\tau }_0)$. $E_a(T)$ fixes the scale of the activation energy of the excitations that have a nonnegligible probability of being activated during the relaxation dynamics. The ASEER algorithm crucially allows us to extract excitations with activation energy up and beyond E_a.

Supplementary Material

Appendix 01 (PDF)

Dataset S01 (TXT)

Dataset S02 (TXT)

Dataset S03 (TXT)

Dataset S04 (TXT)

Code S01 (TXT)

Movie S1.

Evolution of the system along a string-like excitation. The video illustrates the sequence of configurations the system visits along the minimum energy path. Particles are scaled in size for clarity.

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information.