Quasi-equilibrium (Boltzmann pseudodynamics) theory of glassy aging: closures, overlap-sampling validity, gauge fixing, and quantitative prediction for glassy liquids

The Boltzmann pseudodynamics (BPD) construction of Franz, Parisi, Ricci-Tersenghi and Urbani (2015) recasts the long-time limit of glassy dynamics as a discrete-time Markov chain in which each new configuration is drawn with its Boltzmann weight at a fixed overlap with the previous configuration; it reproduces the exact mean-field aging equations of the spherical p-spin model and, for structural glasses, yields dynamical Ornstein-Zernike equations for two-time correlation and response functions that must be closed (hypernetted-chain or Szamel closures, both producing mode-coupling-like equations). The open problem is to turn this construction into an internally consistent and quantitatively predictive theory of aging in glassy liquids: an input-free (or quantitatively reliable) closure; a system-independent prescription for the overlap constraint together with validation of the equal-reachability hypothesis in finite-dimensional glass formers; a practical gauge-fixing prescription for the time-reparametrization invariance of the equations; and tested predictions of aging observables for a realistic supercooled liquid. No published closure is simultaneously input-free and valid for aging: the existing closures still require the static structure factor as input and address equilibrium rather than aging, and the memory-loss premise of the quasi-equilibrium picture is now contested even within mean-field models.

Background

When a liquid is cooled fast enough to avoid crystallization, equilibration times grow so quickly that below a crossover temperature the system never equilibrates on experimental timescales: observables keep drifting with the age of the sample (aging), equilibrium relations such as the fluctuation-dissipation theorem (FDT) — which links the response to a small perturbation with the correlation of spontaneous fluctuations — are violated, and slow degrees of freedom can be assigned an effective temperature. The only fully solved aging dynamics are mean-field: for the spherical p-spin model (a spin system with random p-body interactions, the prototype of a structural glass in random-first-order-transition theory), Cugliandolo and Kurchan derived in 1993 exact closed equations for the two-time correlation $C(t,t_w)$ and response $R(t,t_w)$, exhibiting aging, effective temperatures, and covariance under monotone reparametrizations of time. The mean-field solution rests on weak ergodicity breaking: the system keeps aging without ever reaching equilibrium, yet loses all memory of its initial condition. For realistic finite-dimensional liquids no such equations exist. Mode-coupling theory (MCT) gives an approximate equation for the equilibrium density autocorrelation function, but it needs the equilibrium static structure factor $S(q)$ (the wavevector-resolved spectrum of density fluctuations) as external input, is formulated near the dynamical transition temperature $T_d$ where it predicts an idealized ergodicity-breaking transition, and does not describe aging; in two and three dimensions this sharp transition is moreover avoided by activated, barrier-crossing processes. The quasi-equilibrium route of Franz, Parisi and collaborators aims to connect these two schools. Its central objects are the overlap $q(S,S')=\sum_i S_i S'_i/N$, a measure of similarity between configurations, and the Franz-Parisi potential: the free energy of one replica constrained at fixed overlap with a pinned reference configuration, which develops a secondary minimum at the Edwards-Anderson overlap $q_{EA}$, the typical overlap between two configurations trapped in the same metastable glass state. The Boltzmann pseudodynamics (BPD) is a discrete-time Markov chain in which, given the configuration $S(t)$, the next configuration is drawn with probability proportional to $e^{-\beta H[S(t+1)]}$ among configurations at a prescribed overlap $\widetilde{C}$ with $S(t)$: fast relaxation inside a metastable state is coarse-grained into a single step, and the chain is hypothesized to describe the slow aging dynamics. In the spherical p-spin model the long-chain limit reproduces the exact Cugliandolo-Kurchan equations, and marginality of the dynamics — the vanishing of the free-energy barriers that would block quasi-equilibrium exploration — is a necessary condition for having non-trivial time-dependent solutions and, in last analysis, for the equivalence of the slow real dynamics and the Boltzmann pseudodynamics. For particle systems, promoting replica indices to pseudo-time indices turns the replicated Ornstein-Zernike (OZ) relation of liquid theory — which links the total correlation function $h$ to the direct correlation function $c$ — into dynamical OZ equations for two-time correlations and responses. These equations are not closed. Two schemes borrowed from equilibrium liquid theory have been studied: the hypernetted-chain (HNC) closure, a variational approximation obtained by a partial diagrammatic resummation of the free energy, and Szamel's closure, a truncation of the replicated BBGKY hierarchy (the hierarchy of equations for reduced distribution functions). Both produce mode-coupling-like dynamical equations; Szamel's reproduces the standard MCT equation in the time-translation-invariant (TTI) regime where FDT holds, while HNC additionally allows an asymptotic description of aging below $T_d$. The source work states the limitations of this framework explicitly: HNC is known to be quantitatively unreliable for the glass transition of realistic liquids; Szamel's closure requires the static structure factor (or the static direct correlation $c_0(q)$) as external input; the chain construction depends on the choice of $\widetilde{C}$ (set by hand near $q_{EA}$) and on the assumption that all configurations at fixed overlap are equally reachable by the fast relaxation processes; the resulting equations are covariant under monotone reparametrizations of time, a gauge symmetry that must be fixed to extract physical observables (done so far only in the TTI/FDT regime); and the scheme is mean-field in spirit, leaving activated processes, finite-dimensional corrections, and replica-symmetry-breaking (RSB — the hierarchical organization of the glass phase in which the overlap order parameter is not self-averaging) structures outside the replica-symmetric BPD parametrization. These interlocking gaps define the open problem.

Problem Statement

Establish the quasi-equilibrium (Boltzmann pseudodynamics) construction as an internally consistent, quantitatively predictive theory of the long-time aging dynamics of glassy liquids. Establishing this requires, with explicit mutual consistency: (i) a closure scheme for the dynamical Ornstein-Zernike equations of the BPD construction that is internally consistent — generating the static input self-consistently rather than requiring the equilibrium static structure factor $S(q)$ or direct correlation $c_0(q)$ from outside — and/or a quantitatively accurate improvement of the hypernetted-chain closure for aging dynamics, whose known static inaccuracies for the glass transition of realistic liquids currently block quantitative prediction; (ii) a system-independent prescription for the overlap constraint $\widetilde{C}$ (currently set by hand near $q_{EA}$) together with a validation of the quasi-equilibrium sampling hypothesis — that all configurations at fixed overlap are equally reachable by the fast relaxation processes — in finite-dimensional, realistic glass formers where activated processes operate, including a precise determination of the regimes in which it holds, in the light of recent mean-field evidence that memory of the initial condition can persist (contesting weak ergodicity breaking); (iii) a practical prescription to fix the time-reparametrization gauge of the BPD equations in general aging calculations, beyond imposing time-translation invariance and the fluctuation-dissipation theorem, together with the mapping from the reparametrization-fixed solution to physical (experimental or simulational) time scales; (iv) solutions of the closed equations for a realistic supercooled liquid and quantitative predictions for aging observables — two-time correlation functions, FDT violation and effective temperatures, aging exponents — tested against simulation or experiment; and, where required for consistency in the low-temperature or finite-dimensional regimes, (v) extensions beyond the replica-symmetric BPD parametrization (systematic inclusion of replica-symmetry-breaking structures) and incorporation of finite-dimensional and activated corrections. The unifying completion boundary is (iv): a quantitatively validated prediction of aging observables for a realistic glass former obtained from the quasi-equilibrium construction; items (i)-(iii) are the jointly required prerequisites that currently prevent it, and item (v) enters only where required for consistency in the low-temperature or finite-dimensional regimes.

References

1. S. Franz, G. Parisi, F. Ricci-Tersenghi, P. Urbani, 'Quasi equilibrium construction for the long time limit of glassy dynamics', J. Stat. Mech. (2015) P10010, DOI: 10.1088/1742-5468/2015/10/P10010, arXiv:1509.06872, https://arxiv.org/abs/1509.06872
2. S. Franz, G. Parisi, 'Quasi-equilibrium in glassy dynamics: an algebraic view', J. Stat. Mech. (2013) P02003, DOI: 10.1088/1742-5468/2013/02/P02003, arXiv:1206.4067, https://arxiv.org/abs/1206.4067
3. S. Franz, G. Parisi, P. Urbani, 'Quasi-equilibrium in glassy dynamics: a liquid theory approach', J. Phys. A: Math. Theor. 48, 19FT01 (2015), DOI: 10.1088/1751-8113/48/19/19FT01, arXiv:1212.4291, https://arxiv.org/abs/1212.4291
4. G. Szamel, 'Dynamic glass transition: Bridging the gap between mode-coupling theory and the replica approach', EPL 91, 56004 (2010), DOI: 10.1209/0295-5075/91/56004, https://doi.org/10.1209/0295-5075/91/56004
5. S. Franz, H. Jacquin, G. Parisi, P. Urbani, F. Zamponi, 'Quantitative field theory of the glass transition', Proc. Natl. Acad. Sci. USA 109, 18725 (2012), DOI: 10.1073/pnas.1216578109, arXiv:1206.2482, https://arxiv.org/abs/1206.2482
6. L. F. Cugliandolo, J. Kurchan, 'Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model', Phys. Rev. Lett. 71, 173 (1993), DOI: 10.1103/PhysRevLett.71.173, https://doi.org/10.1103/PhysRevLett.71.173
7. A. Altieri, G. Biroli, C. Cammarota, 'Dynamical mean-field theory and aging dynamics', J. Phys. A: Math. Theor. 53, 375006 (2020), DOI: 10.1088/1751-8121/aba3dd, arXiv:2005.05118, https://arxiv.org/abs/2005.05118
8. G. Folena, F. Zamponi, 'On weak ergodicity breaking in mean-field spin glasses', SciPost Phys. 15, 109 (2023), DOI: 10.21468/SciPostPhys.15.3.109, arXiv:2303.00026, https://arxiv.org/abs/2303.00026
9. V. Citro, F. Ricci-Tersenghi, 'Strong ergodicity breaking in dynamical mean-field equations for mixed p-spin glasses', Phys. Rev. Lett. 135, 247102 (2025), DOI: 10.1103/lvmp-pydk, arXiv:2504.12367, https://arxiv.org/abs/2504.12367
10. B. Li, D. Pan, T. Qu, Y. Jin, 'Universal activated aging and weak ergodicity breaking in spin and structural glasses', Science Advances 12, eaec4416 (2026), DOI: 10.1126/sciadv.aec4416, arXiv:2501.00338, https://arxiv.org/abs/2501.00338
11. F. Ghimenti, L. Berthier, J. Kurchan, F. van Wijland, 'Clever algorithms for glasses work by time reparameterization', Proc. Natl. Acad. Sci. USA 123, e2520818123 (2026), DOI: 10.1073/pnas.2520818123, arXiv:2409.17121, https://arxiv.org/abs/2409.17121
12. G. Szamel, 'An alternative, dynamic density functional-like theory for time-dependent density fluctuations in glass-forming fluids', J. Chem. Phys. 156, 191102 (2022), DOI: 10.1063/5.0091385, arXiv:2203.09543, https://arxiv.org/abs/2203.09543
13. T. Maimbourg, J. Kurchan, F. Zamponi, 'Solution of the dynamics of liquids in the large-dimensional limit', Phys. Rev. Lett. 116, 015902 (2016), DOI: 10.1103/PhysRevLett.116.015902, arXiv:1507.03421, https://arxiv.org/abs/1507.03421
14. J. Kurchan, T. Maimbourg, F. Zamponi, 'Statics and dynamics of infinite-dimensional liquids and glasses: a parallel, compact derivation', J. Stat. Mech. (2016) 033210, DOI: 10.1088/1742-5468/2016/03/033210, arXiv:1512.02186, https://arxiv.org/abs/1512.02186
15. E. Agoritsas, T. Maimbourg, F. Zamponi, 'Out-of-equilibrium dynamical equations of infinite-dimensional particle systems. I. The isotropic case', J. Phys. A: Math. Theor. 52, 144002 (2019), DOI: 10.1088/1751-8121/ab099d, arXiv:1808.00236, https://arxiv.org/abs/1808.00236
16. C. Rainone, P. Urbani, 'Following the evolution of glassy states under external perturbations: the full replica symmetry breaking solution', J. Stat. Mech. (2016) 053302, DOI: 10.1088/1742-5468/2016/05/053302, arXiv:1512.00341, https://arxiv.org/abs/1512.00341
17. H. Frusawa, 'Replica Field Theory for a Generalized Franz-Parisi Potential of Inhomogeneous Glassy Systems: New Closure and the Associated Self-Consistent Equation', Entropy 26, 241 (2024), DOI: 10.3390/e26030241, arXiv:2403.11720, https://arxiv.org/abs/2403.11720
18. P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, F. Zamponi, 'Glass and Jamming Transitions: From Exact Results to Finite-Dimensional Descriptions', Annu. Rev. Condens. Matter Phys. 8, 265 (2017), DOI: 10.1146/annurev-conmatphys-031016-025334, arXiv:1605.03008, https://arxiv.org/abs/1605.03008
19. L. F. Cugliandolo, 'Recent Applications of Dynamical Mean-Field Methods', Annu. Rev. Condens. Matter Phys. 15, 177 (2024), DOI: 10.1146/annurev-conmatphys-040721-022848, arXiv:2305.01229, https://arxiv.org/abs/2305.01229