Stochastic beta-relaxation (SBR) is a beyond-mean-field effective theory of glassy dynamics in which the deviation of the density correlator from its plateau obeys a stochastic equation with quenched Gaussian fluctuations of the distance to the mode-coupling transition; these fluctuations turn the ideal mean-field dynamical arrest into the crossover observed in real supercooled liquids. Its parameters were computed analytically and validated without adjustable parameters on the Bethe-lattice Ising p-spin glass and on kinetically constrained models, and Rizzo (arXiv:2209.09689) posed the open problem of establishing the theory's relevance to actual supercooled liquids by computing its coefficients from realistic liquid models. Recent work has transformed the landscape: Laudicina, Janssen and Szamel derived SBR microscopically from a fluctuation-extended mode-coupling theory, expressing all coupling constants in terms of the static structure factor and evaluating them for Percus-Yevick hard spheres; and Laudicina, Pihlajamaa, Janssen, Voigtmann and Rizzo performed the first application of SBR to molecular-dynamics data of a supercooled liquid, reproducing a newly discovered non-monotonic beta-regime dynamical correlation length, but with most parameters fitted to the data. The surviving open core is the fully parameter-free validation: computing the complete SBR parameter set from the equilibrium static structure of a realistic glass former and testing the resulting no-adjustable-parameter predictions for beta-regime observables against molecular-dynamics data across the avoided mode-coupling crossover.

Background

When a liquid is cooled fast enough to avoid crystallization it enters the supercooled regime, where the structural (alpha) relaxation time grows by many orders of magnitude with only subtle changes in static structure. Mode-coupling theory (MCT), a microscopic mean-field theory formulated in terms of the intermediate scattering function F(k,t)F(k,t) (the equilibrium density-density correlator at wavevector kk), predicts that this slowdown ends at a critical temperature TcT_c in a dynamical transition to an arrested, ideal-glass state. Real experiments and simulations show no sharp transition: the power-law increase of the relaxation time crosses over to weaker, roughly Arrhenius growth. The transition is said to be avoided, and identifying the mechanism that restores ergodicity in finite dimensions is a central question of glass physics.

Mean-field spin glasses provide the cleanest laboratory for this question. Discontinuous spin-glass models, that is models with one-step replica-symmetry breaking (1RSB, the mean-field spin-glass scenario in which the order parameter jumps discontinuously at the transition), such as Ising spins interacting in triplets (p=3p=3) on a Bethe lattice (an infinite random graph of fixed connectivity, exactly solvable by the cavity method, a message-passing technique that becomes exact on such locally tree-like graphs), display the same dynamical transition at a temperature TdT_d, smoothed in finite systems the same way as in liquids. Rizzo showed (EPL 106, 56003 (2014); Phys. Rev. B 94, 014202 (2016)) that resumming the leading divergent terms of the loop expansion around the MCT/1RSB critical point yields a dynamical stochastic equation, stochastic beta relaxation (SBR), for the deviation g(x,t)g(x,t) of the correlator from its plateau value qdq_d in the beta regime (the intermediate time window, between the microscopic transient and the final structural alpha relaxation, in which the correlator lingers on its plateau):

σ+s(x)=−Γ∇2g(x,t)−λ g(x,t)2+ddt∫0tdτ g(x,t−τ) g(x,τ)\sigma+s(x)=-\Gamma\nabla^2 g(x,t)-\lambda\, g(x,t)^2+\frac{d}{dt}\int_0^t d\tau\, g(x,t-\tau)\, g(x,\tau)

Here σ∝Tc−T\sigma\propto T_c-T (or the distance in packing fraction) is the separation parameter, s(x)s(x) is a quenched Gaussian random field of variance Δσ2\Delta\sigma^2 representing self-generated, effectively frozen fluctuations of the local distance from the transition, Γ\Gamma is a stiffness, and λ\lambda is the exponent parameter of MCT, which fixes the power-law exponents aa and bb of the approach to and departure from the plateau through λ=ΓE(1−a)2/ΓE(1−2a)=ΓE(1+b)2/ΓE(1+2b)\lambda=\Gamma_{\rm E}(1-a)^2/\Gamma_{\rm E}(1-2a)=\Gamma_{\rm E}(1+b)^2/\Gamma_{\rm E}(1+2b), where ΓE(⋅)\Gamma_{\rm E}(\cdot) denotes the Euler gamma function (distinct from the stiffness Γ\Gamma). Setting Δσ2=0\Delta\sigma^2=0 recovers the ideal mean-field transition; quenched fluctuations nucleate rare liquid-like regions that cut the transition into a crossover and produce dynamical heterogeneity, the transient coexistence of fast- and slow-moving regions customarily quantified by four-point correlation functions (G4(r,t)G_4(r,t), which measures the correlation between the local relaxation at two points separated by rr, and its spatial integral χ4(t)\chi_4(t)) and a dynamical correlation length ξd(t)\xi_d(t).

SBR is a universal effective theory: its form is fixed by symmetry, while a handful of model-dependent constants (λ\lambda, Γ\Gamma, Δσ2\Delta\sigma^2, σ(T)\sigma(T), and a microscopic time unit t0t_0) carry all microscopic information. The paradigm for using it is: compute the constants analytically, solve the stochastic equations, and compare the parameter-free predictions with simulation data. This program was executed for the Ising 3-spin Bethe-lattice spin glass (constants from the static cavity method, agreement with simulations up to N=1.8×106N=1.8\times 10^6 spins; arXiv:2209.09689) and for kinetically constrained models, lattice models whose moves are blocked by local occupation rules engineered to reproduce glassy dynamics (Fredrickson-Andersen and Kob-Andersen models on Bethe lattices; Phys. Rev. Lett. 124, 195501 (2020), SciPost Phys. 18, 020 (2025), Phys. Rev. E 112, 045402 (2025)). Whether the same theory, with constants computed the same way, describes particle-based supercooled liquids was posed in arXiv:2209.09689 as an open problem. Recently, Laudicina, Janssen and Szamel (arXiv:2512.13082; arXiv:2512.13092) derived SBR microscopically from a fluctuation-extended diagrammatic MCT, expressing all coupling constants through the static structure factor and computing them for Percus-Yevick (an analytic closure approximation for the hard-sphere structure factor) hard spheres; the stiffness had previously been computed in the same framework by Biroli, Bouchaud, Miyazaki and Reichman (Phys. Rev. Lett. 97, 195701 (2006)). Laudicina, Pihlajamaa, Janssen, Voigtmann and Rizzo (arXiv:2608.21219) then compared SBR with molecular-dynamics data of a polydisperse (particle sizes drawn from a broad continuous distribution) soft-sphere glass former, finding that SBR reproduces a newly discovered non-monotonic beta-regime dynamical correlation length, but with most parameters fitted to the data rather than computed. The final step, a genuinely parameter-free computation from the liquid's own static structure tested against its dynamics, is the subject of this problem.

Problem Statement

Stochastic beta-relaxation theory has now been derived microscopically for particle-based liquids, and all of its coupling constants can be expressed through the equilibrium static structure factor. What remains open, and what this problem asks, is the quantitative, parameter-free establishment of SBR's relevance to actual supercooled liquids.

For a realistic (finite-dimensional, particle-based) glass-forming liquid, compute the complete set of SBR parameters, namely the exponent parameter λ\lambda, the stiffness Γ\Gamma, the disorder strength Δσ2\Delta\sigma^2, the temperature-dependent separation parameter σ(T)\sigma(T), and the microscopic time and amplitude scales (the amplitude being the critical-decay amplitude h(k)h(k) that relates the coarse-grained SBR field to the measured density correlator), from the liquid's own equilibrium static structure (the simulated static structure factor and direct correlation function, the latter related to the structure factor by the Ornstein-Zernike relation, inserted into the published microscopic expressions), without fitting to any dynamical data (fixing the microscopic time unit by matching short-time microscopic behavior, rather than adjusting it to the measured relaxation data, is not such fitting) and without idealized analytic input such as the Percus-Yevick approximation. Then test the resulting no-adjustable-parameter SBR predictions against independent molecular-dynamics data for the beta-regime observables, namely the departure of the density correlator from its plateau, the four-point susceptibility χ4(t)\chi_4(t), the four-point spatial correlation function G4(r,t)G_4(r,t), and the dynamical correlation length ξd(t)\xi_d(t), over a range of temperatures spanning the avoided mode-coupling crossover TcT_c.

The problem is resolved when such a computation demonstrates quantitative agreement between the parameter-free SBR predictions and the dynamical data for at least one realistic glass former, with discrepancies within stated uncertainties across the observables and temperatures examined, or when it identifies, with quantified deviations, which predictions fail for reasons attributable to the theory itself rather than to the static input, to discretization of the stochastic equations, or to regimes outside the asymptotic beta window.

References

  1. T. Rizzo, 'Stochastic equations and dynamics beyond mean-field theory', preprint (2022), arXiv:2209.09689, https://arxiv.org/abs/2209.09689
  2. T. Rizzo, 'Long-wavelength fluctuations lead to a model of the glass crossover', EPL (Europhysics Letters) 106, 56003 (2014), doi:10.1209/0295-5075/106/56003, arXiv:1405.6557, https://arxiv.org/abs/1405.6557
  3. T. Rizzo, 'Dynamical Landau Theory of the Glass Crossover', Physical Review B 94, 014202 (2016), doi:10.1103/PhysRevB.94.014202, arXiv:1603.02925, https://arxiv.org/abs/1603.02925
  4. T. Rizzo, 'The Glass Crossover from Mean-Field Spin-Glasses to Supercooled Liquids', Philosophical Magazine 96, 636-647 (2016), doi:10.1080/14786435.2016.1158878, arXiv:1507.06351, https://arxiv.org/abs/1507.06351
  5. T. Rizzo and T. Voigtmann, 'Qualitative Features at the Glass Crossover', EPL (Europhysics Letters) 111, 56008 (2015), doi:10.1209/0295-5075/111/56008, arXiv:1403.2764, https://arxiv.org/abs/1403.2764
  6. T. Rizzo and T. Voigtmann, 'On The Nature of the Glass Crossover', preprint (2015), arXiv:1504.06263, https://arxiv.org/abs/1504.06263
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  8. G. Perrupato and T. Rizzo, 'Theory of Kinetically-Constrained-Models Dynamics', SciPost Physics 18, 020 (2025), doi:10.21468/SciPostPhys.18.1.020, arXiv:2212.05132, https://arxiv.org/abs/2212.05132
  9. G. Perrupato and T. Rizzo, 'Solution of the Critical Dynamics of the Mean-Field Kob-Andersen Model', Physical Review E 112, 045402 (2025), doi:10.1103/yms3-km6k, arXiv:2505.16451, https://arxiv.org/abs/2505.16451
  10. C. C. L. Laudicina, L. M. C. Janssen and G. Szamel, 'Microscopic Theory of a Fluctuation-Induced Dynamical Crossover in Supercooled Liquids', preprint (2025, revised 2026), arXiv:2512.13082, https://arxiv.org/abs/2512.13082
  11. C. C. L. Laudicina, L. M. C. Janssen and G. Szamel, 'Theory of the beta-Relaxation Beyond Mode-Coupling Theory: A Microscopic Treatment', preprint (2025, revised 2026), arXiv:2512.13092, https://arxiv.org/abs/2512.13092
  12. C. C. L. Laudicina, I. Pihlajamaa, L. M. C. Janssen, T. Voigtmann and T. Rizzo, 'Non-Monotonic Dynamical Correlations Across The Glass Crossover', preprint (2026), arXiv:2608.21219, https://arxiv.org/abs/2608.21219
  13. G. Biroli, J.-P. Bouchaud, K. Miyazaki and D. R. Reichman, 'Inhomogeneous Mode-Coupling Theory and Growing Dynamic Length in Supercooled Liquids', Physical Review Letters 97, 195701 (2006), doi:10.1103/PhysRevLett.97.195701, https://doi.org/10.1103/PhysRevLett.97.195701