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Yamaguchi YY. Strange scaling and relaxation of finite-size fluctuation in thermal equilibrium. Phys Rev E 2016; 94:012133. [PMID: 27575102 DOI: 10.1103/physreve.94.012133] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2016] [Indexed: 11/07/2022]
Abstract
We numerically exhibit two strange phenomena of finite-size fluctuation in thermal equilibrium of a paradigmatic long-range interacting system having a second-order phase transition. One is a nonclassical finite-size scaling at the critical point, which differs from the prediction by statistical mechanics. With the aid of this strange scaling, the scaling theory for infinite-range models conjectures the nonclassical values of critical exponents for the correlation length. The other is relaxation of the fluctuation strength from one level to another in spite of being in thermal equilibrium. A scenario is proposed to explain these phenomena from the viewpoint of the Casimir invariants and their nonexactness in finite-size systems, where the Casimir invariants are conserved in the Vlasov dynamics describing the long-range interacting systems in the limit of large population. This scenario suggests appearance of the reported phenomena in a wide class of isolated long-range interacting systems.
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Affiliation(s)
- Yoshiyuki Y Yamaguchi
- Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, 606-8501 Kyoto, Japan
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Colonna-Romano L, Gould H, Klein W. Anomalous mean-field behavior of the fully connected Ising model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042111. [PMID: 25375442 DOI: 10.1103/physreve.90.042111] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2014] [Indexed: 06/04/2023]
Abstract
Although the fully connected Ising model does not have a length scale, we show that the critical exponents for thermodynamic quantities such as the mean magnetization and the susceptibility can be obtained using finite size scaling with the scaling variable equal to N, the number of spins. Surprisingly, the mean value and the most probable value of the magnetization are found to scale differently with N at the critical temperature of the infinite system, and the magnetization probability distribution is not a Gaussian, even for large N. Similar results inconsistent with the usual understanding of mean-field theory are found at the spinodal. We relate these results to the breakdown of hyperscaling and show that hyperscaling can be restored by increasing N while holding the Ginzburg parameter rather than the temperature fixed, or by doing finite size scaling at the pseudocritical temperature where the susceptibility is a maximum for a given value of N. We conclude that finite size scaling for the fully connected Ising model yields different results depending on how the mean-field limit is approached.
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Affiliation(s)
| | - Harvey Gould
- Department of Physics, Clark University, Worcester, Massachusetts 01610, USA and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - W Klein
- Department of Physics and Center for Computational Science, Boston University, Boston, Massachusetts 02215, USA
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Ginelli F, Takeuchi KA, Chaté H, Politi A, Torcini A. Chaos in the Hamiltonian mean-field model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:066211. [PMID: 22304182 DOI: 10.1103/physreve.84.066211] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/28/2011] [Revised: 12/07/2011] [Indexed: 05/31/2023]
Abstract
We study the dynamical properties of the canonical ordered phase of the Hamiltonian mean-field (HMF) model, in which N particles, globally coupled via pairwise attractive interactions, form a rotating cluster. Using a combination of numerical and analytical arguments, we first show that the largest Lyapunov exponent remains strictly positive in the infinite-size limit, converging to its asymptotic value with 1/lnN corrections. We then elucidate the scaling laws ruling the behavior of this asymptotic value in the critical region separating the ordered, clustered phase and the disordered phase present at high-energy densities. We also show that the full spectrum of Lyapunov exponents consists of a bulk component converging to the (zero) value taken by a test oscillator forced by the mean field, plus subextensive bands of O(ln N) exponents taking finite values. We finally investigate the robustness of these results by studying a "2D" extension of the HMF model where each particle is endowed with 4 degrees of freedom, thus allowing the emergence of chaos at the level of a single particle. Altogether, these results illustrate the subtle effects of global (or long-range) coupling and the importance of the order in which the infinite-time and infinite-size limits are taken: For an infinite-size HMF system represented by the Vlasov equation, no chaos is present, while chaos exists and subsists for any finite system size.
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Affiliation(s)
- Francesco Ginelli
- Istituto dei Sistemi Complessi, CNR, via dei Taurini 19, I-00185 Roma, Italy
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Milotti E. Exactly solved dynamics for an infinite-range spin system. II. Antiferromagnetic interactions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:027102. [PMID: 11863693 DOI: 10.1103/physreve.65.027102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2001] [Revised: 08/13/2001] [Indexed: 05/23/2023]
Abstract
In a previous paper [E. Milotti, Phys. Rev. E 63, 026116 (2001)]. I have shown how to derive both thermodynamical and dynamical properties of an infinite-range Ising spin system with binary ferromagnetic interactions from the master equation for magnetization obtained from a simple spin dynamics. The same method can be adapted to different spin interactions: here I discuss the case of antiferromagnetic interactions. This model permits a study of the static properties of the antiferromagnetic lattice, and it displays very clearly the differences between the antiferromagnetic and the ferromagnetic case with long-range interactions. The dynamical behavior of the antiferromagnetic system is simpler, and the magnetization always relaxes exponentially.
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Affiliation(s)
- Edoardo Milotti
- Dipartimento di Fisica dell'Università di Udine and INFN, Sezione di Trieste, Via delle Scienze 208, I-33100 Udine, Italy
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McKerrell A, Bowers RG. Critical phenomena with many body interactions: an exactly soluble model system. ACTA ACUST UNITED AC 2001. [DOI: 10.1088/0022-3719/5/1/003] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Nagle JF, Bonner JC. Numerical studies of the Ising chain with long-range ferromagnetic interactions. ACTA ACUST UNITED AC 2001. [DOI: 10.1088/0022-3719/3/2/017] [Citation(s) in RCA: 65] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Klein DJ. Exact ground states for a class of antiferromagnetic Heisenberg models with short-range interactions. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/15/2/032] [Citation(s) in RCA: 83] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Brankov JG, Zagrebnov VA. On the description of the phase transition in the Husimi-Temperley model. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/16/10/019] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Oh SK, Yoon CN, Chung JS. Dynamic finite-size effect in the classical spin van der Waals model. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:11557-11561. [PMID: 9982776 DOI: 10.1103/physrevb.53.11557] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Matsen F. Freeon Dynamics: A Novel Theory of Atoms and Molecules. ADVANCES IN QUANTUM CHEMISTRY 1996. [DOI: 10.1016/s0065-3276(08)60250-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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Samson JH. Classical effective Hamiltonians, Wigner functions, and the sign problem. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:223-233. [PMID: 9977081 DOI: 10.1103/physrevb.51.223] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Heisenberg models and a particular isotropic model. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:7125-7133. [PMID: 10006881 DOI: 10.1103/physrevb.48.7125] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Liu JM, Müller G. Dynamics of an integrable two-sublattice spin model with long-range interaction. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 44:12020-12022. [PMID: 9999341 DOI: 10.1103/physrevb.44.12020] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Dekeyser R, Lee MH. Nonequilibrium statistical mechanics of the spin-1/2 van der Waals model. I. Time evolution of a single spin. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 43:8123-8130. [PMID: 9996437 DOI: 10.1103/physrevb.43.8123] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Liu JM, Müller G. Infinite-temperature dynamics of the equivalent-neighbor XYZ model. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 42:5854-5864. [PMID: 9903864 DOI: 10.1103/physreva.42.5854] [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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Kaplan TA, Horsch P. Spontaneous symmetry breaking in the Lieb-Mattis model of antiferromagnetism. PHYSICAL REVIEW. B, CONDENSED MATTER 1990; 42:4663-4669. [PMID: 9995998 DOI: 10.1103/physrevb.42.4663] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Boccara N, Mejdani R, De Seze L. Solvable model exhibiting a first-order phase transition. ACTA ACUST UNITED AC 1977. [DOI: 10.1051/jphys:01977003802014900] [Citation(s) in RCA: 79] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
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Tindemans P, Capel H. An exact calculation of the free energy in systems with separable interactions. II. ACTA ACUST UNITED AC 1974. [DOI: 10.1016/0031-8914(74)90340-1] [Citation(s) in RCA: 39] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Tindemans P, Capel H. An exact calculation of the free energy in systems with separable interactions. ACTA ACUST UNITED AC 1974. [DOI: 10.1016/0031-8914(74)90209-2] [Citation(s) in RCA: 49] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Ruijgrok T, Tjon J. Critical slowing down and nonlinear response in an exactly solvable stochastic model. ACTA ACUST UNITED AC 1973. [DOI: 10.1016/0031-8914(73)90065-7] [Citation(s) in RCA: 35] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Capel H. On the possibility of first-order transitions in Ising systems of triplet ions with zero-field splitting III. ACTA ACUST UNITED AC 1967. [DOI: 10.1016/0031-8914(67)90198-x] [Citation(s) in RCA: 127] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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