Rare Trajectories in a Prototypical Mean-field Disordered Model: Insights into Landscape and Instantons

TL;DR

This work tackles how activated relaxation pathways emerge in mean-field disordered models within the RFOT class by introducing a dynamical potential $V_{t_f}(q)$ that constrains the Langevin evolution to reach a fixed overlap $q$ with a reference metastable state after time $t_f$. Using a replica-path-integral framework, it derives a set of self-consistent integro-differential equations for two-time order parameters under a RS ansatz, and demonstrates, both analytically and via simulations of the spherical $p$-spin model, that the phase space around metastable states is convex for high overlaps but becomes fibered as overlap decreases, with an irreversibility threshold $q_{irr}$ delimiting the onset of true instantonic relaxation. The results reveal that, within the RS (convex) regime, the most probable escape paths are time-reversed relaxations toward equilibrium and that the dynamical potential converges to the Franz–Parisi free-energy landscape as $t_f o o o o o o o o o o ofty$. In the fibered regime, a few dominant fibers control the escape, while for $q_f<q_{irr}$ the dynamics become irreversible and may involve hub states connected by low-index saddles, suggesting a nuanced, hub-and-fiber mechanism for RFOT relaxation. The study advances a dynamical, landscape-agnostic route to RFOT instantons and proposes the dynamical potential as a practical observable for simulations of structural glasses and related disordered systems, with implications for understanding aging, heterogeneity, and the connection between static landscapes and dynamic pathways.

Abstract

For disordered systems within the random first-order transition (RFOT) universality class, such as structural glasses and certain spin glasses, the role played by activated relaxation processes is rich to the point of perplexity. Over the last decades, various efforts have attempted to formalize and systematize such processes in terms of instantons similar to the nucleation droplets of first-order phase transitions. In particular, Kirkpatrick, Thirumalai, and Wolynes proposed in the late '80s an influential nucleation theory of relaxation in structural glasses. Already within this picture, however, the resulting structures are far from the compact objects expected from the classical droplet description. In addition, an altogether different type of single-particle hopping-like instantons has recently been isolated in molecular simulations. Landscape studies of mean-field spin glass models have further revealed that simple saddle crossing does not capture relaxation in these systems. We present here a landscape-agnostic study of rare dynamical events, which delineates the richness of instantons in these systems. Our work not only captures the structure of metastable states, but also identifies the point of irreversibility, beyond which activated relaxation processes become a fait accompli. An interpretation of the associated landscape features is articulated, thus charting a path toward a complete understanding of RFOT instantons.

Figures (13)

• Figure 1: Schematics of the basin of attraction in the free energy landscape around an equilibrium configuration, as obtained from our work. At the center lies the reference equilibrium configuration, ${\cal C}$ (red dot). A typical other configuration within the cluster, ${\cal C}'$, has an overlap $q_\mathrm{eq}$ with ${\cal C}$ (green circle). The free energy landscape remains convex up to the marginal overlap $q_\mathrm{mg}$ (dashed blue circle). Beyond this point, the landscape becomes fibered into numerous channels, but only a few of these channels contribute significantly to the measure. The free energy keeps increasing monotonically until $q_\mathrm{irr}$, at which point the dominant channels encounter their respective saddle points (dotted purple circle). Any trajectory starting within this basin that preserves $q> q_\mathrm{irr}$ typically returns to $q_\mathrm{eq}$ near ${\cal C}$. However, if a trajectory successfully crosses the irreversible$q_\mathrm{irr}$, with high probability it never returns.
• Figure 2: Long-time limit of the dynamical potential, $V(m)$ (colored lines), along with the Landau free energy $\beta f(m)$ (dashed lines), for the fully-connected Ising model with $h=0$ at various sub-critical $T=4/5$ (blue), 2/3 (orange), and 4/7 (green). (The critical temperature is $T_c=1$.) The finite-time $V_{t_\mathrm{f}}(m)$ for $t_\mathrm{f}=1, 2, 4, 6, 10, 15,$ and $20$ (pale blue lines, from top to bottom) for $T=4/5$ is obtained by numerically integrating Eq. \ref{['eq:dynpot']}. The harmonic approximation to the right minimum is given for $T=4/5$ (dotted line).
• Figure 3: Optimal trajectories of the magnetization (in a harmonic well) around $m_\mathrm{min}\approx0.71$ for $T=4/5$. The dynamics is initiated at $m_\mathrm{i}=1$ and set to reach $m_\mathrm{f}=1/2$ at times $t_\mathrm{f}=1,2,..,10$ (green lines). For long enough $t_\mathrm{f}$, the optimal relaxation trajectories from $m_i$ to $m_\mathrm{min}$ (dashed blue line) and the mirrored ones for $t_\mathrm{f}-t$ at from $m_\mathrm{f}$ to $m_\mathrm{min}$ (red dashed lines) recapitulate approach to and depart from $m_\mathrm{min}$, respectively. (Inset) Representation of the forward relaxation (blue arrow), the jump to the final magnetization $m_\mathrm{f}$ (green arrow), and its mirroring relaxation (red arrow) within the harmonic potential.
• Figure 4: (a) Dynamical potential $V(q_\mathrm{f})\equiv V_\mathrm{t_\mathrm{f}}(q)$ and (b) final energy $E_\mathrm{f}\equiv E(\mathrm{t_\mathrm{f}})$ at $T=1/1.695$ for $t_\mathrm{f}=10, 20, 40,$ and $60$ (solid lines). As reference, the (shifted) static FP potential and the corresponding static energy (thick black lines) are included, along with the RS (thin dash-dotted line) and 1RSB (dashed thin line) extensions as well as the highest free-energy fibers (thick dash-dotted line). Between this last line and the 1RSB line lays the fibered phase of the basin, in which an exponential number (in $N$) of fibers is expected (see Appendix \ref{['sec::CompFib']}). The equilibrium $E_\mathrm{eq}$ and $q_\mathrm{eq}$ (blue dotted lines), the threshold energy $E_{\mathrm{th}}$ (dashed red line) as well as the transition from the convex (RS) to the fibered (1RSB) regime at $q_\mathrm{mg}$ (cross) and the irreversibility onset at $q_\mathrm{irr}$ (dot) are also indicated. The second minimum of the three replica potential ($M_2$, star) cavagna1997, corresponds to the second replica being at $q_\mathrm{irr}$, thus defining the hub state. As expected, as $t_\mathrm{f}$ increases the dynamical potential converges to the FP potential in the convex (RS) regime. Already for $t_\mathrm{f}=60$, an unphysical discrepancy is noted for the energy in the fibered (1RSB) regime. (c): Static FP energy profile at $T = 1/2.5<T_\mathrm{d}=1/\sqrt{2e}$, within the clustered phase, for the $p$-spin model in the limit $p \to \infty$ (black line). Even in this case, which corresponds to the (spherical) REM, the 1RSB solution (dashed line) remains below the threshold energy (red dotted line).
• Figure 5: (a): Time parametric plot of $E_\mathrm{Q}(t)$ and $Q(t)$ with the reference equilibrium configuration $\tau$, for initial configurations sampled from the static FP potential (thick black line) Barrat1998 at different overlap with $\tau$, $q_\mathrm{f}=q_\mathrm{irr}, 0.35, 0.4, 0.45, 0.5,$ and $0.55$ (dashed colored lines). For all $q_\mathrm{f}>q_\mathrm{irr}$ the trajectory remains within the basin of attraction of $\tau$, with $q_\mathrm{eq}$ and $E_\mathrm{eq}$ (dotted blue lines). (b): Dynamical evolution of $E(t)$ under the same conditions. Note that the relaxation time diverges as $q_\mathrm{f}$ approaches $q_\mathrm{irr}$, and that the dynamics then remains stuck in a metastable state (star) of energy $E_\mathrm{Q}(\infty)=E_\mathrm{hub} > E_\mathrm{eq}$.