On weak ergodicity breaking in mean-field spin glasses. (English) Zbl 07905044
Summary: The weak ergodicity breaking hypothesis postulates that out-of-equilibrium glassy systems lose memory of their initial state despite being unable to reach an equilibrium stationary state. It is a milestone of glass physics, and has provided a lot of insight on the physical properties of glass aging. Despite its undoubted usefulness as a guiding principle, its general validity remains a subject of debate. Here, we present evidence that this hypothesis does not hold for a class of mean-field spin glass models. While most of the qualitative physical picture of aging remains unaffected, our results suggest that some important technical aspects should be revisited.
MSC:
| 82Dxx | Applications of statistical mechanics to specific types of physical systems |
| 82Cxx | Time-dependent statistical mechanics (dynamic and nonequilibrium) |
| 60Kxx | Special processes |
References:
| [1] | A. P. Young, Spin glasses and random fields, World Scientific, Singapore, ISBN 9789810231835 (1997), doi:10.1142/3517. · doi:10.1142/3517 |
| [2] | K. Binder and A. P. Young, Spin glasses: Experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys. 58, 801 (1986), doi:10.1103/RevModPhys.58.801. · doi:10.1103/RevModPhys.58.801 |
| [3] | M. Mezard, G. Parisi and M. Virasoro, Spin glass theory and beyond, World Scientific, Singapore, ISBN 9789971501167 (1986), doi:10.1142/0271. · doi:10.1142/0271 |
| [4] | A. Cavagna, Supercooled liquids for pedestrians, Phys. Rep. 476, 51 (2009), doi:10.1016/j.physrep.2009.03.003. · doi:10.1016/j.physrep.2009.03.003 |
| [5] | S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi, Optimization by simulated annealing, Science 220, 671 (1983), doi:10.1126/science.220.4598.671. · Zbl 1225.90162 · doi:10.1126/science.220.4598.671 |
| [6] | G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto and L. Zdeborová, Machine learning and the physical sciences, Rev. Mod. Phys. 91, 045002 (2019), doi:10.1103/RevModPhys.91.045002. · doi:10.1103/RevModPhys.91.045002 |
| [7] | G. Folena, A. Manacorda and F. Zamponi, Introduction to the dynamics of disordered systems: Equilibrium and gradient descent, Phys. A: Stat. Mech. Appl. 128152 (2022), doi:10.1016/j.physa.2022.128152. · Zbl 07791850 · doi:10.1016/j.physa.2022.128152 |
| [8] | L. F. Cugliandolo and J. Kurchan, Analytical solution of the off-equilibrium dy-namics of a long-range spin-glass model, Phys. Rev. Lett. 71, 173 (1993), doi:10.1103/PhysRevLett.71.173. · doi:10.1103/PhysRevLett.71.173 |
| [9] | L. F. Cugliandolo and J. Kurchan, Weak ergodicity breaking in mean-field spin-glass models, Philos. Mag. B 71, 501 (1995), doi:10.1080/01418639508238541. · doi:10.1080/01418639508238541 |
| [10] | J. Kurchan, G. Parisi and M. A. Virasoro, Barriers and metastable states as saddle points in the replica approach, J. Phys. I France 3, 1819 (1993), doi:10.1051/jp1:1993217. · doi:10.1051/jp1:1993217 |
| [11] | J. Kurchan, Time-reparametrization invariances, multithermalization and the Parisi scheme, SciPost Phys. Core 6, 001 (2023), doi:10.21468/SciPostPhysCore.6.1.001. · doi:10.21468/SciPostPhysCore.6.1.001 |
| [12] | L. F. Cugliandolo, J. Kurchan and L. Peliti, Energy flow, partial equilibration, and ef-fective temperatures in systems with slow dynamics, Phys. Rev. E 55, 3898 (1997), doi:10.1103/PhysRevE.55.3898. · doi:10.1103/PhysRevE.55.3898 |
| [13] | J. P. Bouchaud, Weak ergodicity breaking and aging in disordered systems, J. Phys. I France 2, 1705 (1992), doi:10.1051/jp1:1992238. · doi:10.1051/jp1:1992238 |
| [14] | L. Berthier and G. Biroli, Theoretical perspective on the glass transition and amorphous materials, Rev. Mod. Phys. 83, 587 (2011), doi:10.1103/RevModPhys.83.587. · doi:10.1103/RevModPhys.83.587 |
| [15] | M. Baity-Jesi, L. Sagun, M. Geiger, S. Spigler, G. B. Arous, C. Cammarota, Y. LeCun, M. Wyart and G. Biroli, Comparing dynamics: Deep neural networks versus glassy systems, J. Stat. Mech. 124013 (2019), doi:10.1088/1742-5468/ab3281. · Zbl 1459.82317 · doi:10.1088/1742-5468/ab3281 |
| [16] | F. Mignacco, F. Krzakala, P. Urbani and L. Zdeborová, Dynamical mean-field theory for stochastic gradient descent in Gaussian mixture classification, Adv. Neural Inf. Process. Syst. 33, 9540 (2020), doi:10.1088/1742-5468/ac3a80. · Zbl 1539.62298 · doi:10.1088/1742-5468/ac3a80 |
| [17] | F. Mignacco, P. Urbani and L. Zdeborová, Stochasticity helps to navigate rough landscapes: Comparing gradient-descent-based algorithms in the phase retrieval problem, Mach. Learn.: Sci. Technol. 2, 035029 (2021), doi:10.1088/2632-2153/ac0615. · doi:10.1088/2632-2153/ac0615 |
| [18] | A. Sclocchi and P. Urbani, High-dimensional optimization under nonconvex excluded vol-ume constraints, Phys. Rev. E 105, 024134 (2022), doi:10.1103/PhysRevE.105.024134. · doi:10.1103/PhysRevE.105.024134 |
| [19] | G. Folena and P. Urbani, Marginal stability of soft anharmonic mean field spin glasses, J. Stat. Mech. 053301 (2022), doi:10.1088/1742-5468/ac6253. · Zbl 1539.82376 · doi:10.1088/1742-5468/ac6253 |
| [20] | F. Mignacco and P. Urbani, The effective noise of stochastic gradient descent, J. Stat. Mech. 083405 (2022), doi:10.1088/1742-5468/ac841d. · Zbl 1539.68312 · doi:10.1088/1742-5468/ac841d |
| [21] | A. Manacorda and F. Zamponi, Gradient descent dynamics and the jamming transition in infinite dimensions, J. Phys. A: Math. Theor. 55, 334001 (2022), doi:10.1088/1751-8121/ac7f06. · Zbl 1511.82036 · doi:10.1088/1751-8121/ac7f06 |
| [22] | T. Rizzo, Replica-symmetry-breaking transitions and off-equilibrium dynamics, Phys. Rev. E 88, 032135 (2013), doi:10.1103/PhysRevE.88.032135. · doi:10.1103/PhysRevE.88.032135 |
| [23] | M. Bernaschi, A. Billoire, A. Maiorano, G. Parisi and F. Ricci-Tersenghi, Strong ergodicity breaking in aging of mean-field spin glasses, Proc. Natl. Acad. Sci. U.S.A. 117, 17522 (2020), doi:10.1073/pnas.1910936117. · Zbl 1485.82012 · doi:10.1073/pnas.1910936117 |
| [24] | L. F. Cugliandolo and J. Kurchan, On the out-of-equilibrium relaxation of the Sherrington-Kirkpatrick model, J. Phys. A: Math. Gen. 27, 5749 (1994), doi:10.1088/0305-4470/27/17/011. · Zbl 0850.82056 · doi:10.1088/0305-4470/27/17/011 |
| [25] | G. Folena, S. Franz and F. Ricci-Tersenghi, Rethinking mean-field glassy dynamics and its relation with the energy landscape: The surprising case of the spherical mixed p-spin model, Phys. Rev. X 10, 031045 (2020), doi:10.1103/PhysRevX.10.031045. · doi:10.1103/PhysRevX.10.031045 |
| [26] | S. Franz, E. Marinari and G. Parisi, Series expansion of the off-equilibrium mode coupling equations, J. Phys. A: Math. Gen. 28, 5437 (1995), doi:10.1088/0305-4470/28/19/001. · Zbl 0868.60105 · doi:10.1088/0305-4470/28/19/001 |
| [27] | G. Parisi, Off-equilibrium fluctuation-dissipation relation in fragile glasses, Phys. Rev. Lett. 79, 3660 (1997), doi:10.1103/PhysRevLett.79.3660. · doi:10.1103/PhysRevLett.79.3660 |
| [28] | P. Charbonneau and P. K. Morse, Memory formation in jammed hard spheres, Phys. Rev. Lett. 126, 088001 (2021), doi:10.1103/PhysRevLett.126.088001. · doi:10.1103/PhysRevLett.126.088001 |
| [29] | Y. Nishikawa, M. Ozawa, A. Ikeda, P. Chaudhuri and L. Berthier, Relaxation dynam-ics in the energy landscape of glass-forming liquids, Phys. Rev. X 12, 021001 (2022), doi:10.1103/PhysRevX.12.021001. · doi:10.1103/PhysRevX.12.021001 |
| [30] | T. R. Kirkpatrick and P. G. Wolynes, Connections between some kinetic and equilibrium theo-ries of the glass transition, Phys. Rev. A 35, 3072 (1987), doi:10.1103/PhysRevA.35.3072. · doi:10.1103/PhysRevA.35.3072 |
| [31] | T. Maimbourg, J. Kurchan and F. Zamponi, Solution of the dynamics of liq-uids in the large-dimensional limit, Phys. Rev. Lett. 116, 015902 (2016), doi:10.1103/PhysRevLett.116.015902. · doi:10.1103/PhysRevLett.116.015902 |
| [32] | G. Szamel, Simple theory for the dynamics of mean-field-like models of glass-forming fluids, Phys. Rev. Lett. 119, 155502 (2017), doi:10.1103/PhysRevLett.119.155502. · doi:10.1103/PhysRevLett.119.155502 |
| [33] | E. Agoritsas, T. Maimbourg and 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. · Zbl 1509.82106 · doi:10.1088/1751-8121/ab099d |
| [34] | S. Franz and J. Kurchan, The relation between Parisi scheme and multi-thermalized dynam-ics in finite dimensions, (arXiv preprint) doi:10.48550/arXiv.2207.05720. · doi:10.48550/arXiv.2207.05720 |
| [35] | G. Folena, S. Franz and F. Ricci-Tersenghi, Gradient descent dynamics in the mixed p-spin spherical model: Finite-size simulations and comparison with mean-field integration, J. Stat. Mech. 033302 (2021), doi:10.1088/1742-5468/abe29f. · Zbl 1504.82051 · doi:10.1088/1742-5468/abe29f |
| [36] | T. M. Nieuwenhuizen, Exactly solvable model of a quantum spin glass, Phys. Rev. Lett. 74, 4289 (1995), doi:10.1103/PhysRevLett.74.4289. · doi:10.1103/PhysRevLett.74.4289 |
| [37] | A. Auffinger and Y. Zhou, The spherical p+s spin glass at zero temperature, (arXiv preprint) doi:10.48550/arXiv.2209.03866. · doi:10.48550/arXiv.2209.03866 |
| [38] | A. Crisanti and L. Leuzzi, Spherical 2+p spin-glass model: An exactly solvable model for glass to spin-glass transition, Phys. Rev. Lett. 93, 217203 (2004), doi:10.1103/PhysRevLett.93.217203. · doi:10.1103/PhysRevLett.93.217203 |
| [39] | A. Crisanti and L. Leuzzi, Spherical 2+p spin-glass model: An analytically solv-able model with a glass-to-glass transition, Phys. Rev. B 73, 014412 (2006), doi:10.1103/PhysRevB.73.014412. · doi:10.1103/PhysRevB.73.014412 |
| [40] | L. F. Cugliandolo and D. S. Dean, Full dynamical solution for a spherical spin-glass model, J. Phys. A: Math. Gen. 28, 4213 (1995), doi:10.1088/0305-4470/28/15/003. · Zbl 0925.82197 · doi:10.1088/0305-4470/28/15/003 |
| [41] | A. Crisanti and H.-J. Sommers, The sphericalp-spin interaction spin glass model: The stat-ics, Z. Phys. B Condens. Matter 87, 341 (1992), doi:10.1007/BF01309287. · doi:10.1007/BF01309287 |
| [42] | A. Crisanti and H.-J. Sommers, Thouless-Anderson-Palmer approach to the spherical p-spin spin glass model, J. Phys. I France 5, 805 (1995), doi:10.1051/jp1:1995164. · doi:10.1051/jp1:1995164 |
| [43] | A. Cavagna, I. Giardina and G. Parisi, Stationary points of the Thouless-Anderson-Palmer free energy, Phys. Rev. B 57, 11251 (1998), doi:10.1103/PhysRevB.57.11251. · doi:10.1103/PhysRevB.57.11251 |
| [44] | J. Kent-Dobias and J. Kurchan, How to count in hierarchical landscapes: A ’full’ solution to mean-field complexity, Phyr. Rev. E 107, 064111 (2023), doi:10.1103/PhysRevE.107.064111. · doi:10.1103/PhysRevE.107.064111 |
| [45] | E. Subag, Following the ground states of full-RSB spherical spin glasses, Comm. Pure Appl. Math. 74, 1021 (2020), doi:10.1002/cpa.21922. · Zbl 1467.82097 · doi:10.1002/cpa.21922 |
| [46] | A. Montanari, Optimization of the Sherrington-Kirkpatrick Hamiltonian, in IEEE 60th annual symposium on foundations of computer science, Baltimore, US (2019), doi:10.1109/FOCS.2019.00087. · Zbl 1528.82043 · doi:10.1109/FOCS.2019.00087 |
| [47] | A. El Alaoui, A. Montanari and M. Sellke, Optimization of mean-field spin glasses, Ann. Probab. 49, 2922 (2021), doi:10.1214/21-AOP1519. · Zbl 07467487 · doi:10.1214/21-AOP1519 |
| [48] | A. E. Alaoui and A. Montanari, Algorithmic thresholds in mean field spin glasses, (arXiv preprint) doi:10.48550/arXiv.2009.11481. · doi:10.48550/arXiv.2009.11481 |
| [49] | G. Folena, The mixed p-spin model: Selecting, following and losing states, PhD thesis, Uni-versité Paris-Saclay, Paris, France; Università degli studi La Sapienza, Rome, Italy (2020). |
| [50] | G. B. Arous, E. Subag and O. Zeitouni, Geometry and temperature chaos in mixed spherical spin glasses at low temperature: The perturbative regime, Comm. Pure Appl. Math. 73, 1732 (2019), doi:10.1002/cpa.21875. · Zbl 1453.82089 · doi:10.1002/cpa.21875 |
| [51] | A. Montanari and F. Ricci-Tersenghi, On the nature of the low-temperature phase in dis-continuous mean-field spin glasses, Eur. Phys. J. B -Condens. Matter 33, 339 (2003), doi:10.1140/epjb/e2003-00174-7. · doi:10.1140/epjb/e2003-00174-7 |
| [52] | A. Montanari and F. Ricci-Tersenghi, Cooling-schedule dependence of the dynamics of mean-field glasses, Phys. Rev. B 70, 134406 (2004), doi:10.1103/PhysRevB.70.134406. · doi:10.1103/PhysRevB.70.134406 |
| [53] | L. F. Cugliandolo, Dynamics of glassy systems, (arXiv preprint) doi:10.48550/arXiv.cond-mat/0210312. · doi:10.48550/arXiv.cond-mat/0210312 |
| [54] | T. Castellani and A. Cavagna, Spin-glass theory for pedestrians, J. Stat. Mech. P05012 (2005), doi:10.1088/1742-5468/2005/05/P05012. · Zbl 1456.82490 · doi:10.1088/1742-5468/2005/05/P05012 |
| [55] | G. B. Arous, A. Dembo and A. Guionnet, Cugliandolo-Kurchan equations for dynamics of spin-glasses, Probab. Theory Relat. Fields 136, 619 (2006), doi:10.1007/s00440-005-0491-y. · Zbl 1109.82021 · doi:10.1007/s00440-005-0491-y |
| [56] | F. Caltagirone, G. Parisi and T. Rizzo, Critical off-equilibrium dynamics in glassy systems, Phys. Rev. E 87, 032134 (2013), doi:10.1103/PhysRevE.87.032134. · doi:10.1103/PhysRevE.87.032134 |
| [57] | E. Agoritsas, G. Biroli, P. Urbani and F. Zamponi, Out-of-equilibrium dynamical mean-field equations for the perceptron model, J. Phys. A: Math. Theor. 51, 085002 (2018), doi:10.1088/1751-8121/aaa68d. · Zbl 1391.82047 · doi:10.1088/1751-8121/aaa68d |
| [58] | F. Krzakala and J. Kurchan, Landscape analysis of constraint satisfaction problems, Phys. Rev. E 76, 021122 (2007), doi:10.1103/PhysRevE.76.021122. · doi:10.1103/PhysRevE.76.021122 |
| [59] | S. Franz and G. Parisi, Quasi-equilibrium in glassy dynamics: An algebraic view, J. Stat. Mech. P02003 (2013), doi:10.1088/1742-5468/2013/02/P02003. · doi:10.1088/1742-5468/2013/02/P02003 |
| [60] | S. Franz, G. Parisi and 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. · Zbl 1323.82044 · doi:10.1088/1751-8113/48/19/19FT01 |
| [61] | S. Franz, G. Parisi, F. Ricci-Tersenghi and P. Urbani, Quasi equilibrium construction for the long time limit of glassy dynamics, J. Stat. Mech. P10010 (2015), doi:10.1088/1742-5468/2015/10/P10010. · Zbl 1456.82911 · doi:10.1088/1742-5468/2015/10/P10010 |
| [62] | B. Kim and A. Latz, The dynamics of the spherical p-spin model: From microscopic to asymp-totic, Europhys. Lett. 53, 660 (2001), doi:10.1209/epl/i2001-00202-4. · doi:10.1209/epl/i2001-00202-4 |
| [63] | A. Altieri, G. Biroli and C. Cammarota, Dynamical mean-field theory and aging dynamics, J. Phys. A: Math. Theor. 53, 375006 (2020), doi:10.1088/1751-8121/aba3dd. · Zbl 1519.82134 · doi:10.1088/1751-8121/aba3dd |
| [64] | S. Franz and M. Mézard, On mean field glassy dynamics out of equilibrium, Phys. A: Stat. Mech. Appl. 210, 48 (1994), doi:10.1016/0378-4371(94)00057-3. · doi:10.1016/0378-4371(94)00057-3 |
| [65] | G. A. Baker Jr, Padé approximant, Scholarpedia 7, 9756 (2012), doi:10.4249/scholarpedia.9756. · doi:10.4249/scholarpedia.9756 |
| [66] | K. Bhattacharyya, Asymptotic response of observables from divergent power-series expan-sions, Phys. Rev. A 39, 6124 (1989), doi:10.1103/PhysRevA.39.6124. · doi:10.1103/PhysRevA.39.6124 |
| [67] | R. Monasson, Structural glass transition and the entropy of the metastable states, Phys. Rev. Lett. 75, 2847 (1995), doi:10.1103/PhysRevLett.75.2847. · doi:10.1103/PhysRevLett.75.2847 |
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