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arXiv:1304.2982 · Physical Review E

Non-mean-field critical exponent in a mean-field model: Dynamics versus statistical mechanics

Shun Ogawa, Aurelio Patelli, Yoshiyuki Y. Yamaguchi

Published Mar 25, 2014• 15 citations
arXivHTMLPDF

Abstract

Mean-field theory tells us that the classical critical exponent of susceptibility is twice that of magnetization. However, linear response theory based on the Vlasov equation, which is naturally introduced by the mean-field nature, makes the former exponent half of the latter for families of quasistationary states having second order phase transitions in the Hamiltonian mean-field model and its variances, in the low-energy phase. We clarify that this strange exponent is due to the existence of Casimir invariants which trap the system in a quasistationary state for a time scale diverging with the system size. The theoretical prediction is numerically confirmed by N-body simulations for the equilibrium states and a family of quasistationary states.

DOI
10.1103/physreve.89.032131

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Mean-field theory tells us that the classical critical exponent of susceptibility is twice that of magnetization. However, linear response theory based on the Vlasov equation, which is naturally introduced by the mean-field nature, makes the former exponent half of the latter for families of quasistationary states having second order phase transitions in the Hamiltonian mean-field model and its variances, in the low-energy phase. We clarify that this strange exponent is due to the existence of Casimir invariants which trap the system in a quasistationary state for a time scale diverging with the system size. The theoretical prediction is numerically confirmed by N-body simulations for the equilibrium states and a family of quasistationary states.
An external force dynamically drives an isolated mean-field Hamiltonian system to a long-lasting quasistationary state, whose lifetime increases with population of the system. For second order phase transitions in quasistationary states, two nonclassical critical exponents have been reported individually by using a linear and a nonlinear response theories in a toy model. We provide a simple way to compute the critical exponents all at once, which is an analog of the Landau theory. The present theory extends the universality class of the nonclassical exponents to spatially periodic one-dimensional systems and shows that the exponents satisfy a classical scaling relation inevitably by using a key scaling of momentum.
An external force dynamically drives an isolated mean-field Hamiltonian system to a long-lasting quasistationary state, whose lifetime increases with population of the system. For second order phase transitions in quasistationary states, two non-classical critical exponents have been reported individually by using a linear and a nonlinear response theories in a toy model. We provide a simple way to compute the critical exponents all at once, which is an analog of the Landau theory. The present theory extends universality class of the non-classical exponents to spatially periodic one-dimensional systems, and shows that the exponents satisfy a classical scaling relation inevitably by using a key scaling of momentum.
A nonlinear response theory is provided by use of the transient linearization method in the spatially one-dimensional Vlasov systems. The theory inclusively gives responses to external fields and to perturbations for initial stationary states, and is applicable even to the critical point of a second-order phase transition. We apply the theory to the Hamiltonian mean-field model, a toy model of a ferromagnetic body, and investigate the critical exponent associated with the response to the external field at the critical point in particular. The obtained critical exponent is the nonclassical value 3/2, while the classical value is 3. However, interestingly, one scaling relation holds with another nonclassical critical exponent of susceptibility in the isolated Vlasov systems. Validity of the theory is numerically confirmed by directly simulating temporal evolutions of the Vlasov equation.
The Hamiltonian mean-field model has been investigated, since its introduction about a decade ago, to study the equilibrium and dynamical properties of long-range interacting systems. Here we study the long-time behavior of long-lived, out-of-equilibrium, quasistationary dynamical states, whose lifetime diverges in the thermodynamic limit. The nature of these states has been the object of a lively debate in the recent past. We introduce a numerical tool, based on the fluctuations of the phase of the instantaneous magnetization of the system. Using this tool, we study the quasistationary states that arise when the system is started from different classes of initial conditions, showing that the new observable can be exploited to compute the lifetime of these states. We also show that quasistationary states are present not only below, but also above the critical temperature of the second-order magnetic phase transition of the model. We find that at supercritical temperatures the lifetime is much larger than at subcritical temperatures.
Introduction Our discussion of critical phenomena surrounding a second-order phase transition has thus far focused only on qualitative features. We have now examined three systems, fluids, magnets and random percolation, each of which displays an abruptly sharp transition from a less ordered to a more ordered phase as the respective transition point is encountered. Each shows a similar pattern of developing structure just in advance of the critical point. Fluctuations in the respective order parameter display a self-similar structure that is limited only by a single relevant length scale, the correlation length ξ, which diverges on approach to the transition point. For fluids and magnets, this structure arises from inherent fluctuations that are amplified by either a diverging compressibility or susceptibility, respectively. In this chapter, we explore more quantitative, theoretical approaches taken to understand the features of second-order phase transitions. The simplest of these are the mean field theories , in which the pairwise interaction (needed to produce a phase transition) is introduced in the form of an average field. In this approach, the effects of the growing fluctuations of the order parameter near the critical point are ignored. Although the mean field approach does meet many of the requirements and does predict divergences of certain quantities near the transition, the critical exponents predicted by the theory do not match those seen experimentally. Obtaining correct exponents requires a more advanced approach involving renormalization techniques that exploit the self-similar structure of the fluctuations near the critical point and allow all of the various critical exponents to be inter-related, such that knowledge of any two yields all the others.
We have analyzed the dynamic phase transition of the kinetic Ising model in mean-field approximation by means of an analytical approach. Specifically, we study the evolution of the system under the simultaneous influence of time-dependent and time-independent magnetic fields. We demonstrate that within the approximate analytical treatment of our approach, the dynamic phase transition exhibits power-law dependencies for the order parameter that have the same critical exponents as the mean-field equilibrium case. Moreover we have obtained an equation of state, with which we can prove that the time-independent field component is effectively the conjugate field of the order parameter. Our analysis is limited to the parameter range, in which only second-order phase transitions occur, i.e., for small applied field amplitudes and temperatures close to the Curie point. In order to ensure the reliability of our analytical results we have corroborated them by comparison to numerical evaluations of the same model.