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arXiv:cond-mat/9911030 · Physical Review Letters

Rotators with Long-Range Interactions: Connection with the Mean-Field Approximation

Francisco A. Tamarit, Celia Anteneodo

Published Jan 10, 2000• 149 citations
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Abstract

We analyze the equilibrium properties of a chain of ferromagnetically coupled rotators which interact through a force that decays as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ where $r$ is the interparticle distance and $\ensuremath{\alpha}\ensuremath{\ge}0$. By integrating the equations of motion we obtain the microcanonical time averages of both the magnetization and the kinetic energy. We detect three different regimes depending on whether $\ensuremath{\alpha}$ belongs to the intervals $[0,1)$, $(1,2)$, or $(2,\ensuremath{\infty})$. For $0<\ensuremath{\alpha}<1$, the microcanonical averages agree, after a scaling, with those obtained in the canonical ensemble for the mean-field case $(\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0)$. This correspondence offers a mathematically tractable way of dealing with systems governed by slowly decaying long-range interactions.

DOI
10.1103/physrevlett.84.208

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We analyze the equilibrium properties of a chain of ferromagnetically coupled rotators which interact through a force that decays as r\ensuremath−\ensuremathα{r}^{\ensuremath{-}\ensuremath{\alpha}}r\ensuremath−\ensuremathα where rrr is the interparticle distance and \ensuremathα\ensuremath≥0\ensuremath{\alpha}\ensuremath{\ge}0\ensuremathα\ensuremath≥0. By integrating the equations of motion we obtain the microcanonical time averages of both the magnetization and the kinetic energy. We detect three different regimes depending on whether \ensuremathα\ensuremath{\alpha}\ensuremathα belongs to the intervals [0,1)[0,1)[0,1), (1,2)(1,2)(1,2), or (2,\ensuremath∞)(2,\ensuremath{\infty})(2,\ensuremath∞). For 0<\ensuremathα<10<\ensuremath{\alpha}<10<\ensuremathα<1, the microcanonical averages agree, after a scaling, with those obtained in the canonical ensemble for the mean-field case (\ensuremathα=0)(\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0)(\ensuremathα=0). This correspondence offers a mathematically tractable way of dealing with systems governed by slowly decaying long-range interactions.
An N-sized inertial classical Heisenberg ferromagnet, which consists of a modification of the well-known standard model, where the spins are replaced by classical rotators, is studied in the limit of infinite-range interactions. The usual canonical-ensemble mean-field solution of the inertial classical n-vector ferromagnet (for which n=3 recovers the particular Heisenberg model considered herein) is briefly reviewed, showing the well-known second-order phase transition. This Heisenberg model is studied numerically within the microcanonical ensemble through molecular dynamics. In what concerns the caloric curve, it is shown that, far from criticality, the kinetic temperature obtained at the long-time-limit microcanonical-ensemble simulation recovers well the equilibrium canonical-ensemble estimate, whereas, close to criticality, a discrepancy (presumably due to finite-size effects) is found. The time evolution of the kinetic temperature indicates that a basin of attraction exists for the initial conditions for which the system evolves into a metastable state, whose duration diverges as N--> infinity, before attaining the terminal thermal equilibrium. Such a metastable state is observed for a whole range of energies, which starts right below criticality and extends up to very high energies (in fact, the gap between the kinetic temperatures associated with the metastable and the terminal-equilibrium states is expected to disappear only as one approaches infinite energy). To the best our knowledge, this has never before been observed on similar Hamiltonian models, in a noticeable way, for such a large range of energies. For example, for the XY (n=2) version of the present model, such a behavior was observed only near criticality. It is shown also that the (metastable state) maximum Lyapunov exponent decreases with N like lambda(max) approximately N-kappa, where for the initial conditions employed herein (maximal magnetization), kappa=0.225+/-0.030, both above and below the critical point.
The Ising model with ferromagnetic interactions that decay as $\frac{1}{{r}^{\ensuremath{\alpha}}}$ is analyzed in the nonextensive regime $0<~\ensuremath{\alpha}<~d$, where the thermodynamic limit is not defined. In order to study the asymptotic properties of the model in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit (NNN being the number of spins) we propose a generalization of the Curie-Weiss model, for which the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit is well defined for all $\ensuremath{\alpha}>~0$. We conjecture that mean-field theory is exact in the last model for all $0<~\ensuremath{\alpha}<~d$. This conjecture is supported by Monte Carlo heat-bath simulations in the d=1d=1d=1 case. Moreover, we confirm a recently conjectured scaling by Tsallis that allows for a unification of extensive ($\ensuremath{\alpha}>d$) and nonextensive ($0<~\ensuremath{\alpha}<~d$) regimes.
The thermodynamic and dynamical properties of an Ising model with both short-range and long-range, mean-field-like, interactions are studied within the microcanonical ensemble. It is found that the relaxation time of thermodynamically unstable states diverges logarithmically with system size. This is in contrast with the case of short-range interactions where this time is finite. Moreover, at sufficiently low energies, gaps in the magnetization interval may develop to which no microscopic configuration corresponds. As a result, in local microcanonical dynamics the system cannot move across the gap, leading to breaking of ergodicity even in finite systems. These are general features of systems with long-range interactions and are expected to be valid even when the interaction is slowly decaying with distance.
In this thesis we study the dynamics of the long-range interacting Ising chain using a cluster mean-field approach, a mean-field theory that accounts for short range correlations. The principal result is that, if the system is isolated from an external environment, the model exhibits two different behaviors depending on the range of the interactions. Whenever the system is truly long-range, it exhibits a sharp mean-field dynamical phase transition from a dynamical ferromagnetic to a dynamical paramagnetic phase. Reducing the range of the interactions a critical region, showing hypersensitivity to initial conditions, appears. Interestingly, this chaotic region shares the same physics of a classical tossed coin that is allowed to bounce on the floor. In a second part of the work we derive the cluster mean-field equations of motion describing the dynamics of a fully connected Ising chain connected to an external bath. In particular we focused on the dynamics in presence of a dissipation generated by string of Glauber operators acting on one or more sites of the chain. This model, in presence of global dissipative processes, exhibits persistent oscillations in time revealing the existence of a boundary time-crystal and we studied the stability of the boundary time-crystal showing that global dissipative processes are a key ingredient for their existence.
Abstract We study, by means of first-principle molecular-dynamics, the temperature T QSS of the quasi-stationary state (QSS) of a d -dimensional classical inertial α -Heisenberg ferromagnetic model, consisting of a set of rotators with interactions decaying with distance as r − α ( α ⩾ 0 ) . We find, through detailed finite-size calculations, the scaling relation T QSS − T ∞ ∝ N − β , where T ∞ ≡ U − 1 / 2 , with energy per particle U < U c ( U c = 5 / 6 ), and total number of particles N , which holds for the very long-range interactions regime 0 ⩽ α / d ⩽ 1 , in which the QSS exists. We show that the scaling exponent β does not depend on U but only in the quotient α / d , being β ( α / d ) ≃ 1 for 0 ⩽ α / d ⩽ 0.5 , and decaying linearly for α / d > 0.5 , so <