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Aspelmeier T, Moore MA. Free-energy barriers in the Sherrington-Kirkpatrick model. Phys Rev E 2022; 105:034138. [PMID: 35428069 DOI: 10.1103/physreve.105.034138] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/12/2021] [Accepted: 02/23/2022] [Indexed: 06/14/2023]
Abstract
The free-energy landscape of the Sherrington-Kirkpatrick (SK) Ising spin glass is simple in the framework of the Thouless-Anderson-Palmer (TAP) equations as each solution (which are minima of the free energy) has associated with it a nearby index-one saddle point. The free-energy barrier to escape the minimum is just the difference between the saddle point free energy and that at its associated minimum. This difference is calculated for the states with free energies f>f_{c}. It is very small for these states, decreasing as 1/N^{2}, where N is the number of spins in the system. These states are not marginally stable. We argue that such small barriers are why numerical studies never find these states when N is large. Instead, the states that are found are those that have marginal stability. For them the barriers are at least of O(1). f_{c} is the free energy per spin below which the states develop broken replica-symmetry-like overlaps with each other. In the regime f<f_{c} we can only offer some possibilities based around scaling arguments. One of these suggest that the barriers might become as large as N^{1/3}. That might be consistent with recent numerical studies on the Viana-Bray model, which were at variance with the expectations of Cugliandolo and Kurchan for the SK model.
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Affiliation(s)
- T Aspelmeier
- Institute for Theoretical Physics, Georg-August-Universität Göttingen, D37077 Göttingen, Germany
| | - M A Moore
- Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
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Aspelmeier T, Moore MA. Realizable solutions of the Thouless-Anderson-Palmer equations. Phys Rev E 2019; 100:032127. [PMID: 31640029 DOI: 10.1103/physreve.100.032127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2019] [Indexed: 06/10/2023]
Abstract
We show that the only solutions of the Thouless-Anderson-Palmer (TAP) equations for the Sherrington-Kirkpatrick model of Ising spin glasses which can be found by iteration are those whose free energy lies on the border between replica-symmetric and broken-replica-symmetric states, when the number of spins N is large. Convergence to this same borderline also happens in quenches from a high-temperature initial state to a locally stable state where each spin is parallel to its local field; both are examples of self-organized criticality. At this borderline the band of eigenvalues of the Hessian associated with a solution extends to zero, so the states reached have marginal stability. We have also investigated the factors which determine the free-energy difference between a stationary solution corresponding to a saddle point and its associated minimum, which is the barrier which has to be surmounted to escape from the vicinity of a TAP minimum or pure state.
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Affiliation(s)
- T Aspelmeier
- Institute for Theoretical Physics, Georg-August-Universität Göttingen, D37077, Göttingen, Germany
| | - M A Moore
- Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
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Lundow PH, Campbell IA. Ising spin glasses in two dimensions: Universality and nonuniversality. Phys Rev E 2017; 95:042107. [PMID: 28505857 DOI: 10.1103/physreve.95.042107] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2017] [Indexed: 11/07/2022]
Abstract
Following numerous earlier studies, extensive simulations and analyses were made on the continuous interaction distribution Gaussian model and the discrete bimodal interaction distribution Ising spin glass (ISG) models in two dimensions [Lundow and Campbell, Phys. Rev. E 93, 022119 (2016)1539-375510.1103/PhysRevE.93.022119]. Here we further analyze the bimodal and Gaussian data together with data on two other continuous interaction distribution two-dimensional ISG models, the uniform and the Laplacian models, and three other discrete interaction distribution models, a diluted bimodal model, an "antidiluted" model, and a more exotic symmetric Poisson model. Comparisons between the three continuous distribution models show that not only do they share the same exponent η≡0 but that to within the present numerical precision they share the same critical exponent ν also, and so lie in a single universality class. On the other hand the critical exponents of the four discrete distribution models are not the same as those of the continuous distributions, and the present data strongly indicate that they differ from one discrete distribution model to another. This is evidence that discrete distribution ISG models in two dimensions have nonzero values of the critical exponent η and do not lie in a single universality class.
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Affiliation(s)
- P H Lundow
- Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden
| | - I A Campbell
- Laboratoire Charles Coulomb, Université Montpellier II, 34095 Montpellier, France
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Sharma A, Yeo J, Moore MA. Metastable minima of the Heisenberg spin glass in a random magnetic field. Phys Rev E 2016; 94:052143. [PMID: 27967114 DOI: 10.1103/physreve.94.052143] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2016] [Indexed: 11/07/2022]
Abstract
We have studied zero-temperature metastable minima in classical m-vector component spin glasses in the presence of m-component random fields for two models, the Sherrington-Kirkpatrick (SK) model and the Viana-Bray (VB) model. For the SK model we have calculated analytically its complexity (the log of the number of minima) for both the annealed case where one averages the number of minima before taking the log and the quenched case where one averages the complexity itself, both for fields above and below the de Almeida-Thouless (AT) field, which is finite for m>2. We have done numerical quenches starting from a random initial state (infinite temperature state) by putting spins parallel to their local fields until there is no further decrease of the energy and found that in zero field it always produces minima that have zero overlap with each other. For the m=2 and m=3 cases in the SK model the final energy reached in the quench is very close to the energy E_{c} at which the overlap of the states would acquire replica symmetry-breaking features. These minima have marginal stability and will have long-range correlations between them. In the SK limit we have analytically studied the density of states ρ(λ) of the Hessian matrix in the annealed approximation. Despite the fact that in the presence of a random field there are no continuous symmetries, the spectrum extends down to zero with the usual sqrt[λ] form for the density of states for fields below the AT field. However, when the random field is larger than the AT field, there is a gap in the spectrum, which closes up as the AT field is approached. The VB model behaves differently and seems rather similar to studies of the three-dimensional Heisenberg spin glass in a random vector field.
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Affiliation(s)
- Auditya Sharma
- Department of Physics, Indian Institute of Science Education and Research, Bhopal, India
| | - Joonhyun Yeo
- Department of Physics, Konkuk University, Seoul 143-701, Korea
| | - M A Moore
- School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
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Moore MA, Katzgraber HG. Dealing with correlated choices: how a spin-glass model can help political parties select their policies. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042117. [PMID: 25375448 DOI: 10.1103/physreve.90.042117] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/02/2014] [Indexed: 06/04/2023]
Abstract
Starting from preferences on N proposed policies obtained via questionnaires from a sample of the electorate, an Ising spin-glass model in a field can be constructed from which a political party could find the subset of the proposed policies which would maximize its appeal, form a coherent choice in the eyes of the electorate, and have maximum overlap with the party's existing policies. We illustrate the application of the procedure by simulations of a spin glass in a random field on scale-free networks.
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Affiliation(s)
- M A Moore
- School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
| | - Helmut G Katzgraber
- Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA and Materials Science and Engineering, Texas A&M University, College Station, Texas 77843, USA and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501
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Waclaw B, Burda Z. Counting metastable states of Ising spin glasses on arbitrary graphs. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:041114. [PMID: 18517585 DOI: 10.1103/physreve.77.041114] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2008] [Indexed: 05/26/2023]
Abstract
Using a field-theoretical representation of the Tanaka-Edwards integral, we develop a method to systematically compute the number Ns of one-spin stable states (local energy minima) of a glassy Ising system with nearest-neighbor interactions and random Gaussian couplings on an arbitrary graph. In particular, we use this method to determine Ns for K -regular random graphs and d -dimensional regular lattices for d=2,3 . The method works also for other graphs. Excellent accuracy of the results allows us to observe that the number of local energy minima depends mainly on local properties of the graph on which the spin glass is defined.
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Affiliation(s)
- B Waclaw
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, 04009 Leipzig, Germany
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Nakagawa N. Conformational temperature characterizing the folding of a protein. PHYSICAL REVIEW LETTERS 2007; 98:128104. [PMID: 17501160 DOI: 10.1103/physrevlett.98.128104] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2006] [Indexed: 05/15/2023]
Abstract
The time sequences of the molecular dynamics simulation for the folding process of a protein is analyzed with the inherent structure landscape which focuses on the configurational dynamics of the system. Time-dependent energy and entropy for inherent structures are introduced, and from these quantities a conformational temperature is defined. The conformational temperature follows the time evolution of a slow relaxation process and reaches the bath temperature when the system is equilibrated. We show that the nonequilibrium system is described by two temperatures, one for fast vibration and the other for slow configurational relaxation, while the equilibrium system is described by one temperature. The proposed formalism is applicable widely for systems with many metastable states.
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Affiliation(s)
- Naoko Nakagawa
- College of Science, Ibaraki University, Mito, Ibaraki 310-8512, Japan
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Nakagawa N, Peyrard M. Modeling protein thermodynamics and fluctuations at the mesoscale. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:041916. [PMID: 17155105 DOI: 10.1103/physreve.74.041916] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/2005] [Indexed: 05/12/2023]
Abstract
We use an extended Go model, in unfrustrated and frustrated variants, to study the energy landscape and the fluctuations of a model protein. The model exhibits two transitions, folding and dynamical transitions, when changing the temperature. The inherent structures corresponding to the minima of the landscape are analyzed and we show how their energy density can be obtained from simulations around the folding temperature. The scaling of this energy density is found to reflect the folding transition. Moreover, this approach allows us to build a reduced thermodynamics in the inherent structure landscape. Equilibrium studies, from full molecular dynamics (MD) simulations and from the reduced thermodynamics, detect the features of a dynamical transition at low temperature and we analyze the location and time scale of the fluctuations of the protein, showing the need of some frustration in the model to get realistic results. The frustrated model also shows the presence of a kinetic trap which strongly affects the dynamics of folding.
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Affiliation(s)
- Naoko Nakagawa
- Department of Mathematical Sciences, Ibaraki University, Mito, Ibaraki 310-8512, Japan
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Abstract
Using the Gō model of a real protein, we explore the landscape of its metastable structures. First, we show how the inherent structure energy density can be obtained from the probability density determined by sampling molecular dynamics trajectories and quenching. The analysis of the inherent structure landscape can characterize the folding transition. Then we show how thermodynamics of the inherent states can be established to study the equilibrium properties of proteins. Our work brings some elements into the current discussion about the protein dynamical transition. The study uses a simplified model to illustrate the ideas, but, as the inherent structure landscape is much simpler than the free energy surface of the protein, it appears to be accessible for an all-atom model of a small protein, at the expense of much longer calculations.
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Affiliation(s)
- Naoko Nakagawa
- Department of Mathematical Sciences, Ibaraki University, Mito, Ibaraki 310-8512, Japan.
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Heylen R, Skantzos NS, Blanco JB, Bollé D. Metastable configurations of small-world networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:016138. [PMID: 16486247 DOI: 10.1103/physreve.73.016138] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/24/2005] [Indexed: 05/06/2023]
Abstract
We calculate the number of metastable configurations of Ising small-world networks that are constructed upon superimposing sparse Poisson random graphs onto a one-dimensional chain. Our solution is based on replicated transfer-matrix techniques. We examine the denegeracy of the ground state and find a jump in the entropy of metastable configurations exactly at the crossover between the small-world and the Poisson random graph structures. We also examine the difference in entropy between metastable and all possible configurations, for both ferromagnetic and bond-disordered long-range couplings.
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Affiliation(s)
- R Heylen
- Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium.
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Majumdar SN, Dean DS. Exact occupation time distribution in a non-Markovian sequence and its relation to spin glass models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:041102. [PMID: 12443172 DOI: 10.1103/physreve.66.041102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2002] [Indexed: 05/24/2023]
Abstract
We compute exactly the distribution of the occupation time in a discrete non-Markovian toy sequence that appears in various physical contexts such as the diffusion processes and Ising spin glass chains. The non-Markovian property makes the results nontrivial even for this toy sequence. The distribution is shown to have non-Gaussian tails characterized by a nontrivial large deviation function that is computed explicitly. An exact mapping of this sequence to an Ising spin glass chain via a gauge transformation raises an interesting question for a generic finite sized spin glass model; at a given temperature, what is the distribution (over disorder) of the thermally averaged number of spins that are aligned to their local fields? We show that this distribution remains nontrivial even at infinite temperature and can be computed explicitly in few cases such as in the Sherrington-Kirkpatrick model with Gaussian disorder.
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Affiliation(s)
- Satya N Majumdar
- Laboratoire de Physique Quantique (UMR C5626 du CNRS), Université Paul Sabatier, 31062 Toulouse Cedex, France
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Bray AJ, Moore MA. Metastable states, internal field distributions and magnetic excitations in spin glasses. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0022-3719/14/19/013] [Citation(s) in RCA: 72] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Ferreira FF, Fontanari JF, Stadler PF. Landscape statistics of the low-autocorrelation binary string problem. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0305-4470/33/48/304] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Oliveira VMD, Fontanari JF, Stadler PF. Metastable states in short-rangedp-spin glasses. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/32/50/302] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Newman CM, Stein DL. Metastable states in spin glasses and disordered ferromagnets. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:5244-60. [PMID: 11970394 DOI: 10.1103/physreve.60.5244] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/1999] [Indexed: 04/18/2023]
Abstract
We study analytically M-spin-flip stable states in disordered short-ranged Ising models (spin glasses and ferromagnets) in all dimensions and for all M. Our approach is primarily dynamical, and is based on the convergence of sigma(t), a zero-temperature dynamical process with flips of lattice animals up to size M and starting from a deep quench, to a metastable limit sigma(infinity). The results (rigorous and nonrigorous, in infinite and finite volumes) concern many aspects of metastable states: their numbers, basins of attraction, energy densities, overlaps, remanent magnetizations, and relations to thermodynamic states. For example, we show that their overlap distribution is a delta function at zero. We also define a dynamics for M=infinity, which provides a potential tool for investigating ground state structure.
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Affiliation(s)
- C M Newman
- Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, USA
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Jackle J, Kinzel W. Residual magnetic entropy and metastable states of the Edwards-Anderson Ising spin glass. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/16/5/007] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Singh MP, Chengxiang Z, Dasgupta C. Fixed points in a Hopfield model with random asymmetric interactions. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:5261-5272. [PMID: 9964025 DOI: 10.1103/physreve.52.5261] [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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Waugh FR, Marcus CM, Westervelt RM. Reducing neuron gain to eliminate fixed-point attractors in an analog associative memory. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 43:3131-3142. [PMID: 9905382 DOI: 10.1103/physreva.43.3131] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Masui S, Southern BW, Jacobs AE. Metastable states of Ising spin glasses and random ferromagnets. PHYSICAL REVIEW. B, CONDENSED MATTER 1989; 39:6925-6933. [PMID: 9947339 DOI: 10.1103/physrevb.39.6925] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Crisanti A, Sompolinsky H. Dynamics of spin systems with randomly asymmetric bonds: Ising spins and Glauber dynamics. PHYSICAL REVIEW. A, GENERAL PHYSICS 1988; 37:4865-4874. [PMID: 9899634 DOI: 10.1103/physreva.37.4865] [Citation(s) in RCA: 56] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Moore MA. Zero-temperature scaling and combinatorial optimization. PHYSICAL REVIEW LETTERS 1987; 58:1703-1706. [PMID: 10034513 DOI: 10.1103/physrevlett.58.1703] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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