Quantitative verification of constrained-potential (Franz-Parisi) predictions for dynamical correlations in finite-dimensional supercooled liquids: rounding below $T_c$, the fate of the spinodal exponent $\gamma = 1/2$, and the curvature-to-length mapping

Franz, Donati, Parisi and Glotzer (1999) showed that the growth of the four-point dynamical susceptibility $\chi_4$ and of its correlation length near the mode-coupling temperature $T_c$ of a supercooled liquid can be understood within the constrained-potential (Franz-Parisi) framework: the overlap susceptibility equals the inverse curvature of the constrained potential $V(q)$, and this curvature vanishes as a square root at $T_c$, as it does at a spinodal, predicting $\chi \sim |T-T_c|^{-1/2}$ and a correlation length $\xi \sim |T-T_c|^{-1/4}$. Below $T_c$, finite metastable lifetimes round the divergence into a finite maximum reached at a finite observation time. The paper left open the verification of these predictions in numerical simulations and the quantitative relation between the constrained-potential curvature and a measurable real-space correlation length. Later work observed the predicted non-monotonic temperature evolution of a point-to-set-type dynamical length below $T_c$ (Kob et al. 2012), measured four-point susceptibilities and lengths above and below $T_c$ (Flenner and Szamel 2013), recovered the predicted exponents within idealized inhomogeneous mode-coupling theory, and measured overlap fluctuations and point-to-set lengths in model liquids. What remains open is the integrated quantitative verification in a realistic finite-dimensional liquid: determining the effective scaling that replaces the mean-field square-root divergence, quantitatively characterizing the below-$T_c$ rounding, and testing the curvature-to-length mapping within a single system.

Background

When a liquid is cooled below its melting point without crystallizing, it becomes supercooled: its structural relaxation time grows by many orders of magnitude until, near the laboratory glass transition, it falls out of equilibrium on experimental timescales. No thermodynamic singularity accompanies this slowdown, so its origin is sought in growing correlations between the motion of different particles, a phenomenon called dynamical heterogeneity. A standard diagnostic is the four-point dynamical susceptibility $\chi_4(t)$: the variance (multiplied by system size) of the spatially averaged self-intermediate scattering function, i.e. of the fraction of particles that have not yet moved a distance of order the interaction range within a time interval $t$. Its maximum measures the number of particles in a cooperatively rearranging region, and an associated four-point correlation length $\xi_4(t)$ measures the size of dynamically correlated domains. Mode-coupling theory (MCT), a mean-field-type theory expressing the dynamics in terms of static structure factors, predicts that the relaxation time diverges at a critical temperature $T_c$; in finite dimensions this singularity is avoided (rounded by activated, hopping and nucleation-like processes), but $T_c$ still marks a sharp crossover in many simulated liquids. Franz, Parisi and coworkers developed a static, landscape-based description of the transition using an overlap order parameter: given two configurations $X$ and $Y$ of the same system, their overlap is $q(X,Y) = (1/N)\sum_i w(|\mathbf{r}_i - \mathbf{r}'_{\sigma(i)}|)$, where $w$ is a smooth mask function equal to 1 when a particle of $Y$ sits near the position of a particle of $X$ (matched by the permutation $\sigma$) and decaying to 0 beyond a distance of order the particle size. The constrained potential or Franz-Parisi potential is $V(q) = -k_B T \ln P(q)$, where $P(q)$ is the probability that an equilibrium configuration at temperature $T$ has overlap $q$ with a fixed reference equilibrium configuration. In mean-field glass models of the p-spin type (exactly solvable models of disordered, randomly coupled spins with a discontinuous dynamic transition, which serve as prototypes for structured glasses), $V(q)$ develops a secondary minimum at large $q$ below a dynamic transition temperature identified with $T_c$: the secondary minimum describes metastable glass states, while the low-$q$ (primary) minimum describes the liquid. Fluctuations of $q$ within the large-$q$ minimum ('instate' fluctuations, as opposed to transitions between the liquid and glass minima, 'interstate' fluctuations) define an overlap susceptibility $\chi = N(\langle q^2 \rangle - \langle q \rangle^2)$, related to the curvature of the constrained potential by $\chi = 1/V''(q_{\min})$ in appropriate units. Near $T_c$ the curvature of the glass minimum vanishes as a square root, $V''(q_{\min}) \propto (T_c - T)^{1/2}$, exactly as an order-parameter curvature vanishes at a spinodal (the endpoint of a first-order transition line where a metastable minimum disappears); the constrained potential at the dynamic transition is therefore a spinodal-like object. This yields the predictions at the center of this problem: approaching $T_c$ from above, the four-point dynamical susceptibility $\chi_4$ diverges as $\chi_4 \sim |T-T_c|^{-\gamma}$ with $\gamma = 1/2$ (the spinodal value, instead of the mean-field Ising value $\gamma = 1$) and the associated correlation length diverges as $\xi \sim |T-T_c|^{-1/4}$. Below $T_c$ the mean-field glass states have finite lifetimes in finite dimensions, so the divergence is rounded into a finite maximum, the four-point susceptibility reaches its maximum at a finite observation time set by the escape time from the metastable state, and dynamical correlations decrease on further cooling while static (instate, overlap-based) correlations continue to grow. This framework is the static backbone of the random first-order transition (RFOT) theory of the glass transition, in which the low-temperature phase is a mosaic of glass domains whose typical size is defined by point-to-set correlations (the sensitivity of the equilibrium statistics in a spatial region to the constraint imposed by fixing the surrounding configuration). An inhomogeneous extension of MCT (inhomogeneous MCT) recovers the same exponents at Gaussian level, with an upper critical dimension of 8, so finite-dimensional fluctuations are expected to matter qualitatively. The 1999 paper of Franz, Donati, Parisi and Glotzer combined this theory with numerical results on dynamical correlations in a binary Lennard-Jones liquid and left open: (a) numerical verification of the below-$T_c$ predictions, (b) how the spinodal exponent $\gamma = 1/2$ is modified by finite-dimensional fluctuations, finite-size effects and activated processes, and (c) the quantitative mapping from the constrained-potential curvature to a real-space correlation length and its temperature dependence.

Problem Statement

Determine, quantitatively and within a single consistent framework, whether the constrained-potential (Franz-Parisi) description correctly predicts the behavior of dynamical correlations in finite-dimensional model supercooled liquids. Concretely, in one or more realistic finite-dimensional model glass formers, equilibrated at a sequence of temperatures above and below the mode-coupling temperature $T_c$, establish the following three relations. (i) Rounding of the mean-field divergence: measure the four-point dynamical susceptibility $\chi_4(t)$ and its correlation length through the avoided transition, and give a quantitative account of the rounding below $T_c$ — in particular whether $\chi_4$ reaches its maximum at a finite observation time set by the finite lifetime of metastable states, and whether the dynamical correlations decrease below $T_c$ while a static, overlap-based (instate) correlation measure continues to grow, with fitted maxima, characteristic times and growth rates compared to the constrained-potential rounding predictions in their explicit mean-field form: a maximum of $\chi_4(t)$ at a finite observation time below $T_c$ set by the finite escape time from metastable states, the smoothing of the $\chi_4 \sim a/(T-T_c)^{1/2} + b$ divergence across $T_c$, and a peak time that grows on approaching $T_c$. (ii) Fate of the spinodal exponent: determine whether the growth of the four-point dynamical susceptibility $\chi_4$ and correlation length approaching $T_c$ from above follows, over any identifiable scaling window, the predicted $|T-T_c|^{-1/2}$ and $|T-T_c|^{-1/4}$ behavior, or is replaced by a different effective scaling law, with a quantitative assessment of finite-size and pre-asymptotic corrections sufficient to distinguish these alternatives. (iii) Curvature-to-length mapping: measure the constrained potential $V(q)$ (or its curvature at its minimum, or equivalently the overlap susceptibility $\chi = N(\langle q^2 \rangle - \langle q \rangle^2)$) and the spatial correlation lengths — dynamical (from four-point correlations) and static (overlap-based or point-to-set) — in the same system, and test quantitatively the relation between them and its temperature dependence as predicted by the constrained-potential framework. A quantitative account of these three relations in a well-characterized finite-dimensional liquid requires error bars, stated scaling windows and controlled systematics, and it establishes whether the constrained-potential predictions are quantitatively confirmed or quantitatively refuted, characterizing in the latter case what replaces them, for example an alternative effective scaling or a demonstrated breakdown of the curvature-to-length mapping. This requires measurements on both sides of $T_c$ and comparison with measured quantities in a finite-dimensional system; purely qualitative observations (for example merely noting that $\chi_4$ is non-monotonic below $T_c$), measurements confined to the high-temperature side, or purely theoretical arguments without such comparison are not a quantitative verification of the predicted relations.

References

1. S. Franz, C. Donati, G. Parisi, S. C. Glotzer, 'On dynamical correlations in supercooled liquids', Philosophical Magazine B 79, 1827-1831 (1999). DOI: 10.1080/13642819908223066, URL: https://doi.org/10.1080/13642819908223066
2. S. Franz, G. Parisi, 'Phase diagram of coupled glassy systems: a mean-field study', Physical Review Letters 79, 2486-2489 (1997). DOI: 10.1103/PhysRevLett.79.2486, URL: https://doi.org/10.1103/PhysRevLett.79.2486
3. S. Franz, G. Parisi, 'Effective potential in glassy systems: theory and simulations', Physica A: Statistical Mechanics and its Applications 261, 317-339 (1998). DOI: 10.1016/S0378-4371(98)00315-X, URL: https://doi.org/10.1016/S0378-4371(98)00315-X
4. S. Franz, G. Parisi, 'On non-linear susceptibility in supercooled liquids', Journal of Physics: Condensed Matter 12, 6335-6342 (2000). DOI: 10.1088/0953-8984/12/29/305, URL: https://doi.org/10.1088/0953-8984/12/29/305
5. C. Donati, S. Franz, S. C. Glotzer, G. Parisi, 'Theory of non-linear susceptibility and correlation length in glasses and liquids', Journal of Non-Crystalline Solids 307-310, 215-224 (2002). DOI: 10.1016/S0022-3093(02)01461-8, arXiv:cond-mat/9905433, URLs: https://doi.org/10.1016/S0022-3093(02)01461-8, https://arxiv.org/abs/cond-mat/9905433
6. L. Berthier, G. Biroli, 'Theoretical perspective on the glass transition and amorphous materials', Reviews of Modern Physics 83, 587-645 (2011). DOI: 10.1103/RevModPhys.83.587, arXiv:1011.2578, URLs: https://doi.org/10.1103/RevModPhys.83.587, https://arxiv.org/abs/1011.2578
7. G. Biroli, J.-P. Bouchaud, 'Diverging length scale and upper critical dimension in the Mode-Coupling Theory of the glass transition', Europhysics Letters 67, 21-27 (2004). DOI: 10.1209/epl/i2004-10044-6, URL: https://doi.org/10.1209/epl/i2004-10044-6
8. G. Biroli, J.-P. Bouchaud, K. Miyazaki, 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, URL: https://doi.org/10.1103/PhysRevLett.97.195701
9. L. Berthier, G. Biroli, J.-P. Bouchaud, W. Kob, K. Miyazaki, D. R. Reichman, 'Spontaneous and induced dynamic fluctuations in glass formers. I. General results and dependence on ensemble and dynamics', The Journal of Chemical Physics 126, 184503 (2007). DOI: 10.1063/1.2721554, URL: https://doi.org/10.1063/1.2721554
10. L. Berthier, G. Biroli, J.-P. Bouchaud, W. Kob, K. Miyazaki, D. R. Reichman, 'Spontaneous and induced dynamic correlations in glass formers. II. Model calculations and comparison to numerical simulations', The Journal of Chemical Physics 126, 184504 (2007). DOI: 10.1063/1.2721555, URL: https://doi.org/10.1063/1.2721555
11. J.-P. Bouchaud, G. Biroli, 'On the Adam-Gibbs-Kirkpatrick-Thirumalai-Wolynes scenario for the viscosity increase in glasses', The Journal of Chemical Physics 121, 7347-7354 (2004). DOI: 10.1063/1.1796231, URL: https://doi.org/10.1063/1.1796231
12. N. Lacevic, S. C. Glotzer, 'Approach to the glass transition studied by higher order correlation functions', Journal of Physics: Condensed Matter 15, S2437-S2446 (2003). DOI: 10.1088/0953-8984/15/31/318, URL: https://doi.org/10.1088/0953-8984/15/31/318
13. L. Berthier, 'Finite-size scaling analysis of the glass transition', Physical Review Letters 91, 055701 (2003). DOI: 10.1103/PhysRevLett.91.055701, URL: https://doi.org/10.1103/PhysRevLett.91.055701
14. G. Szamel, E. Flenner, 'Four-point susceptibility of a glass-forming binary mixture: Brownian dynamics', Physical Review E 74, 021507 (2006). DOI: 10.1103/PhysRevE.74.021507, arXiv:cond-mat/0605244, URLs: https://doi.org/10.1103/PhysRevE.74.021507, https://arxiv.org/abs/cond-mat/0605244
15. D. Chandler, J. P. Garrahan, R. L. Jack, L. Maibaum, A. C. Pan, 'Lengthscale dependence of dynamic four-point susceptibilities in glass formers', Physical Review E 74, 051501 (2006). DOI: 10.1103/PhysRevE.74.051501, arXiv:cond-mat/0605084, URLs: https://doi.org/10.1103/PhysRevE.74.051501, https://arxiv.org/abs/cond-mat/0605084
16. C. Dalle-Ferrier, C. Thibierge, C. Alba-Simionesco, L. Berthier, G. Biroli, J.-P. Bouchaud, F. Ladieu, D. L'Hôte, G. Tarjus, 'Spatial correlations in the dynamics of glassforming liquids: experimental determination of their temperature dependence', Physical Review E 76, 041510 (2007). DOI: 10.1103/PhysRevE.76.041510, arXiv:0706.1906, URLs: https://doi.org/10.1103/PhysRevE.76.041510, https://arxiv.org/abs/0706.1906
17. T. Sarlat, A. Billoire, G. Biroli, J.-P. Bouchaud, 'Predictive power of MCT: numerical testing and finite size scaling for a mean field spin glass', Journal of Statistical Mechanics: Theory and Experiment 2009, P08014 (2009). DOI: 10.1088/1742-5468/2009/08/P08014, arXiv:0905.3333 (preprint title: 'Predictive power of MCT: Numerics and Finite size scaling for a mean field spin glass'), URLs: https://doi.org/10.1088/1742-5468/2009/08/P08014, https://arxiv.org/abs/0905.3333
18. G. Szamel, E. Flenner, 'Diverging length scale of the inhomogeneous mode-coupling theory: a numerical investigation', Physical Review E 81, 031507 (2010). DOI: 10.1103/PhysRevE.81.031507, arXiv:0912.1318, URLs: https://doi.org/10.1103/PhysRevE.81.031507, https://arxiv.org/abs/0912.1318
19. E. Flenner, G. Szamel, 'Dynamic heterogeneity in a glass forming fluid: susceptibility, structure factor, and correlation length', Physical Review Letters 105, 217801 (2010). DOI: 10.1103/PhysRevLett.105.217801, arXiv:1005.3794, URLs: https://doi.org/10.1103/PhysRevLett.105.217801, https://arxiv.org/abs/1005.3794
20. W. Kob, S. Roldán-Vargas, L. Berthier, 'Non-monotonic temperature evolution of dynamic correlations in glass-forming liquids', Nature Physics 8, 164-167 (2012). DOI: 10.1038/nphys2133, arXiv:1107.3928, URLs: https://doi.org/10.1038/nphys2133, https://arxiv.org/abs/1107.3928
21. E. Flenner, G. Szamel, 'Characterizing dynamic length scales in glass-forming liquids' (comment on Kob, Roldán-Vargas and Berthier), Nature Physics 8, 696-697 (2012). DOI: 10.1038/nphys2437, arXiv:1208.6531, URLs: https://doi.org/10.1038/nphys2437, https://arxiv.org/abs/1208.6531
22. W. Kob, S. Roldán-Vargas, L. Berthier, 'Reply to "Characterizing dynamic length scales in glass-forming liquids"', Nature Physics 8, 697 (2012). DOI: 10.1038/nphys2435, arXiv:1208.6414, URLs: https://doi.org/10.1038/nphys2435, https://arxiv.org/abs/1208.6414
23. E. Flenner, G. Szamel, 'Dynamic heterogeneities above and below the mode-coupling temperature: evidence of a dynamic crossover', The Journal of Chemical Physics 138, 12A523 (2013). DOI: 10.1063/1.4773321, arXiv:1212.5174, URLs: https://doi.org/10.1063/1.4773321, https://arxiv.org/abs/1212.5174
24. L. Berthier, 'Overlap fluctuations in glass-forming liquids', Physical Review E 88, 022313 (2013). DOI: 10.1103/PhysRevE.88.022313, arXiv:1306.0425, URLs: https://doi.org/10.1103/PhysRevE.88.022313, https://arxiv.org/abs/1306.0425
25. L. Berthier, P. Charbonneau, S. Yaida, 'Efficient measurement of point-to-set correlations and overlap fluctuations in glass-forming liquids', The Journal of Chemical Physics 144, 024501 (2016). DOI: 10.1063/1.4939640, arXiv:1510.06320, URLs: https://doi.org/10.1063/1.4939640, https://arxiv.org/abs/1510.06320
26. B. Guiselin, L. Berthier, G. Tarjus, 'Statistical mechanics of coupled supercooled liquids in finite dimensions', SciPost Physics 12, 091 (2022). DOI: 10.21468/SciPostPhys.12.3.091, arXiv:2105.08946, URLs: https://doi.org/10.21468/SciPostPhys.12.3.091, https://arxiv.org/abs/2105.08946
27. B. Guiselin, G. Tarjus, L. Berthier, 'Static self-induced heterogeneity in glass-forming liquids: overlap as a microscope', The Journal of Chemical Physics 156, 194503 (2022). DOI: 10.1063/5.0086517, URL: https://doi.org/10.1063/5.0086517
28. G. Biroli, J.-P. Bouchaud, F. Ladieu, 'Amorphous order and nonlinear susceptibilities in glassy materials', The Journal of Physical Chemistry B 125, 7578-7586 (2021). DOI: 10.1021/acs.jpcb.1c00457, URL: https://doi.org/10.1021/acs.jpcb.1c00457
29. K. Shiraishi, L. Berthier, 'Characterizing the slow dynamics of the swap Monte Carlo algorithm', The Journal of Physical Chemistry B 128, 12279-12291 (2024). DOI: 10.1021/acs.jpcb.4c06702, arXiv:2409.13369, URLs: https://doi.org/10.1021/acs.jpcb.4c06702, https://arxiv.org/abs/2409.13369
30. N. Oyama, Y. Hara, T. Kawasaki, K. Kim, 'Potential energy landscape picture of zero-temperature avalanche criticality governing dynamics in supercooled liquids', arXiv:2604.03580 (2026). URL: https://arxiv.org/abs/2604.03580