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Andriushchenko P, Kapitan D, Kapitan V. A New Look at the Spin Glass Problem from a Deep Learning Perspective. ENTROPY 2022; 24:e24050697. [PMID: 35626580 PMCID: PMC9141424 DOI: 10.3390/e24050697] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/03/2022] [Revised: 05/10/2022] [Accepted: 05/10/2022] [Indexed: 02/04/2023]
Abstract
Spin glass is the simplest disordered system that preserves the full range of complex collective behavior of interacting frustrating elements. In the paper, we propose a novel approach for calculating the values of thermodynamic averages of the frustrated spin glass model using custom deep neural networks. The spin glass system was considered as a specific weighted graph whose spatial distribution of the edges values determines the fundamental characteristics of the system. Special neural network architectures that mimic the structure of spin lattices have been proposed, which has increased the speed of learning and the accuracy of the predictions compared to the basic solution of fully connected neural networks. At the same time, the use of trained neural networks can reduce simulation time by orders of magnitude compared to other classical methods. The validity of the results is confirmed by comparison with numerical simulation with the replica-exchange Monte Carlo method.
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Affiliation(s)
- Petr Andriushchenko
- National Center for Cognitive Research, ITMO University, bldg. A, Kronverksky Pr. 49, 197101 Saint Petersburg, Russia;
- Correspondence: ; Tel.: +7-964-442-2112
| | - Dmitrii Kapitan
- National Center for Cognitive Research, ITMO University, bldg. A, Kronverksky Pr. 49, 197101 Saint Petersburg, Russia;
- Department of Theoretical Physics and Smart Technologies, Far Eastern Federal University, Russky Island, 10 Ajax Bay, 690922 Vladivostok, Russia;
- Institute of Applied Mathematics, Far Eastern Branch, Russian Academy of Science, 7 Radio, 690041 Vladivostok, Russia
| | - Vitalii Kapitan
- Department of Theoretical Physics and Smart Technologies, Far Eastern Federal University, Russky Island, 10 Ajax Bay, 690922 Vladivostok, Russia;
- Institute of Applied Mathematics, Far Eastern Branch, Russian Academy of Science, 7 Radio, 690041 Vladivostok, Russia
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Peyrard M, Garden JL. Memory effects in glasses: Insights into the thermodynamics of out-of-equilibrium systems revealed by a simple model of the Kovacs effect. Phys Rev E 2020; 102:052122. [PMID: 33327132 DOI: 10.1103/physreve.102.052122] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2020] [Accepted: 11/03/2020] [Indexed: 11/07/2022]
Abstract
Glasses are interesting materials because they allow us to explore the puzzling properties of out-of-equilibrium systems. One of them is the Kovacs effect in which a glass, brought to an out-of-equilibrium state in which all its thermodynamic variables are identical to those of an equilibrium state, nevertheless evolves, showing a hump in some global variable before the thermodynamic variables come back to their starting point. We show that a simple three-state system is sufficient to study this phenomenon using numerical integrations and exact analytical calculations. It also brings some light on the concept of fictive temperature, often used to extend standard thermodynamics to the out-of-equilibrium properties of glasses. We confirm that the concept of a unique fictive temperature is not valid, an show it can be extended to make a connection with the various relaxation processes in the system. The model also brings further insights on the thermodynamics of out-of-equilibrium systems. Moreover, we show that the three-state model is able to describe various effects observed in glasses such as the asymmetric relaxation to equilibrium discussed by Kovacs, or the reverse crossover measured on B_{2}O_{3}.
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Affiliation(s)
- Michel Peyrard
- Université de Lyon, Ecole Normale Supérieure de Lyon, Laboratoire de Physique CNRS UMR 5672, 46 allée d'Italie, F-69364 Lyon Cedex 7, France
| | - Jean-Luc Garden
- Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut NÉEL, 38000 Grenoble, France
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Gujrati PD. Hierarchy of Relaxation Times and Residual Entropy: A Nonequilibrium Approach. ENTROPY 2018; 20:e20030149. [PMID: 33265240 PMCID: PMC7512666 DOI: 10.3390/e20030149] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/29/2018] [Revised: 02/21/2018] [Accepted: 02/22/2018] [Indexed: 11/17/2022]
Abstract
We consider nonequilibrium (NEQ) states such as supercooled liquids and glasses that are described with the use of internal variables. We classify the latter by the state-dependent hierarchy of relaxation times to assess their relevance for irreversible contributions. Given an observation time τobs, we determine the window of relaxation times that divide the internal variables into active and inactive groups, the former playing a central role in the NEQ thermodynamics. Using this thermodynamics, we determine (i) a bound on the NEQ entropy and on the residual entropy and (ii) the nature of the isothermal relaxation of the entropy and the enthalpy in accordance with the second law. A theory that violates the second law such as the entropy loss view is shown to be internally inconsistent if we require it to be consistent with experiments. The inactive internal variables still play an indirect role in determining the temperature T(t) and the pressure P(t) of the system, which deviate from their external values.
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Affiliation(s)
- Purushottam D. Gujrati
- Department of Physics, The University of Akron, Akron, OH 44325, USA;
- Department of Polymer Science, The University of Akron, Akron, OH 44325, USA
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Toledo-Marín JQ, Naumis GG. Short time dynamics determine glass forming ability in a glass transition two-level model: A stochastic approach using Kramers’ escape formula. J Chem Phys 2017. [DOI: 10.1063/1.4977517] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022] Open
Affiliation(s)
- J. Quetzalcóatl Toledo-Marín
- Departamento de Sistemas Complejos, Instituto de Física, Universidad Nacional Autónoma de México (UNAM), Apartado Postal 20-364, 01000 México, Distrito Federal, México
| | - Gerardo G. Naumis
- Departamento de Sistemas Complejos, Instituto de Física, Universidad Nacional Autónoma de México (UNAM), Apartado Postal 20-364, 01000 México, Distrito Federal, México
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Naumis GG. Simple solvable energy-landscape model that shows a thermodynamic phase transition and a glass transition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:061505. [PMID: 23005102 DOI: 10.1103/physreve.85.061505] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2012] [Indexed: 06/01/2023]
Abstract
When a liquid melt is cooled, a glass or phase transition can be obtained depending on the cooling rate. Yet, this behavior has not been clearly captured in energy-landscape models. Here, a model is provided in which two key ingredients are considered in the landscape, metastable states and their multiplicity. Metastable states are considered as in two level system models. However, their multiplicity and topology allows a phase transition in the thermodynamic limit for slow cooling, while a transition to the glass is obtained for fast cooling. By solving the corresponding master equation, the minimal speed of cooling required to produce the glass is obtained as a function of the distribution of metastable states.
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Affiliation(s)
- Gerardo G Naumis
- Instituto de Física, Universidad Nacional Autónoma de México (UNAM), Apartado Postal 20-364, 01000 México, Distrito Federal, Mexico
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Johari GP. Mechanical relaxation and the notion of time-dependent extent of ergodicity during the glass transition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:021501. [PMID: 21928991 DOI: 10.1103/physreve.84.021501] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2011] [Indexed: 05/31/2023]
Abstract
A postulate that ergodicity and entropy continuously decrease to zero on cooling a liquid to a glassy state was used to support the view that glass has no residual entropy, and the features of mechanical relaxation spectra were cited as proof for the decrease. We investigate whether such spectra and the relaxation isochrones can serve as the proof. We find that an increase in the real component of elastic moduli with an increase in spectral frequency does not indicate continuous loss of ergodicity and entropy, and the spectra do not confirm isothermal glass transition or loss of entropy. Variation in ergodicity and entropy with the spectral frequency has untenable consequences for both thermodynamics and molecular dynamics and implies that, despite a broad distribution of its relaxation times, an equilibrium liquid can be considered as always ergodic. Perturbation from equilibrium used to obtain a spectrum does not have the effect of dynamic freezing and unfreezing, and Maxwell-Voigt models for the mechanical response function have neither the characteristic irreversibility of liquid-glass transition nor are commutable to ergodicity or entropy.
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Affiliation(s)
- G P Johari
- Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada
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Goldstein M. On the reality of the residual entropies of glasses and disordered crystals: counting microstates, calculating fluctuations, and comparing averages. J Chem Phys 2011; 134:124502. [PMID: 21456671 DOI: 10.1063/1.3570615] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023] Open
Abstract
In the course of an on-going debate on whether glasses or disordered crystals should have zero entropy at 0 K, i.e., whether the "residual entropy" assigned to them by calorimetric measurements is real, the view has been expressed by some who hold the zero entropy view that to measure entropy, all or an appreciable number of the microstates that contribute to the entropy must be visited. We show here that the entropy calculated on the basis of the number of microstates visited during any conceivable time of measurement would be underestimated by at least 20 orders of magnitude. We also examine and refute the claim that an ensemble average for glassy systems, which predicts a finite residual entropy, also predicts physically impossible properties. We conclude that calorimetrically measured residual entropies are real.
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Mauro JC, Loucks RJ, Gupta PK. Metabasin Approach for Computing the Master Equation Dynamics of Systems with Broken Ergodicity. J Phys Chem A 2007; 111:7957-65. [PMID: 17649986 DOI: 10.1021/jp0731194] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
We propose a technique for computing the master equation dynamics of systems with broken ergodicity. The technique involves a partitioning of the system into components, or metabasins, where the relaxation times within a metabasin are short compared to an observation time scale. In this manner, equilibrium statistical mechanics is assumed within each metabasin, and the intermetabasin dynamics are computed using a reduced set of master equations. The number of metabasins depends upon both the temperature of the system and its derivative with respect to time. With this technique, the integration time step of the master equations is governed by the observation time scale rather than the fastest transition time between basins. We illustrate the technique using a simple model landscape with seven basins and show validation against direct Euler integration. Finally, we demonstrate the use of the technique for a realistic glass-forming system (viz., selenium) where direct Euler integration is not computationally feasible.
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Affiliation(s)
- John C Mauro
- Science & Technology Division, Corning Incorporated, Corning, New York 14831, USA.
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Abstract
The phenomenology of the laboratory glass transition is examined in the enthalpy landscape framework. It is shown that a generic description of the glassy state based on partitioning of the phase space caused by the finiteness of the time of observation explains all universal features of glass transition. Using this description of glass, which is referred to as the extrinsically constrained liquid, expressions are derived for properties of glass and for property changes at the laboratory glass transition. A model enthalpy landscape is used to illustrate the basic concepts of this description. Additional new consequences of this description, such as the role of complexity in glass transition and the zero residual entropy of a glass, are discussed.
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Affiliation(s)
- Prabhat K Gupta
- Department of Materials Science and Engineering, Ohio State University, Columbus, Ohio 43210, USA
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Abstract
A system that is initially ergodic can become nonergodic, i.e., display "broken ergodicity," if the relaxation time scale of the system becomes longer than the observation time over which properties are measured. The phenomenon of broken ergodicity is of vital importance to the study of many condensed matter systems. While previous modeling efforts have focused on systems with a sudden, discontinuous loss of ergodicity, they cannot be applied to study a gradual transition between ergodic and nonergodic behavior. This transition range, where the observation time scale is comparable to that of the structural relaxation process, is especially pertinent for the study of glass transition range behavior, as ergodicity breaking is an inherently continuous process for normal laboratory glass formation. In this paper, we present a general statistical mechanical framework for modeling systems with continuously broken ergodicity. Our approach enables the direct computation of entropy loss upon ergodicity breaking, accounting for actual transition rates between microstates and observation over a specified time interval. In contrast to previous modeling efforts for discontinuously broken ergodicity, we make no assumptions about phase space partitioning or confinement. We present a hierarchical master equation technique for implementing our approach and apply it to two simple one-dimensional landscapes. Finally, we demonstrate the compliance of our approach with the second and third laws of thermodynamics.
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Affiliation(s)
- John C Mauro
- Science and Technology Division, Corning Incorporated, SP-TD-01-1, Corning, New York 14831, USA
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Brey JJ, Prados A. Normal solutions for master equations with time-dependent transition rates: Application to heating processes. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:1541-1545. [PMID: 9960174 DOI: 10.1103/physreve.47.1541] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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12
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Parshin DA, Sahling S. Heat release in glasses at low temperatures. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 47:5677-5688. [PMID: 10004515 DOI: 10.1103/physrevb.47.5677] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Parshin DA, Würger A. Freezing of two-level systems in glasses in an electric field. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 46:762-768. [PMID: 10003259 DOI: 10.1103/physrevb.46.762] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Sethna JP, Shore JD, Huang M. Scaling theory for the glass transition. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 44:4943-4959. [PMID: 9998301 DOI: 10.1103/physrevb.44.4943] [Citation(s) in RCA: 68] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Brey JJ, Prados A. Residual properties of a two-level system. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 43:8350-8361. [PMID: 9996465 DOI: 10.1103/physrevb.43.8350] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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16
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Huang M, Sethna JP. History dependence of a two-level system. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 43:3245-3254. [PMID: 9997633 DOI: 10.1103/physrevb.43.3245] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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17
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Brey JJ, Ruiz-Montero MJ. Continuous cooling of a one-dimensional bonded fluid: A Monte Carlo simulation study. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 43:585-590. [PMID: 9996248 DOI: 10.1103/physrevb.43.585] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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18
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Brey JJ, Prados A. Calculation of the entropy from master equations with time-dependent transition probabilities. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:765-768. [PMID: 9904086 DOI: 10.1103/physreva.42.765] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Grannan ER, Randeria M, Sethna JP. Low-temperature properties of a model glass. I. Elastic dipole model. PHYSICAL REVIEW. B, CONDENSED MATTER 1990; 41:7784-7798. [PMID: 9993077 DOI: 10.1103/physrevb.41.7784] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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