Systems with long-range interactions, while relaxing towards equilibrium, sometimes get trapped in long-lived non-Boltzmann quasistationary states (QSS) which have lifetimes that grow algebraically with the system size. Such states have been observed in models of globally coupled particles that move under Hamiltonian dynamics either on a unit circle or on a unit spherical surface. Here, we address the ubiquity of QSS in long-range systems by considering a different dynamical setting. Thus, we consider an anisotropic Heisenberg model consisting of classical Heisenberg spins with mean-field interactions and evolving under classical spin dynamics. Our analysis of the corresponding Vlasov equation for time evolution of the phase space distribution shows that in a certain energy interval, relaxation of a class of initial states occurs over a timescale which grows algebraically with the system size. We support these findings by extensive numerical simulations. This work further supports the generality of occurrence of QSS in long-range systems evolving under Hamiltonian dynamics.
Systems with long-range interactions, while relaxing towards equilibrium, sometimes get trapped in long-lived non-Boltzmann quasistationary states (QSS) which have lifetimes that grow algebraically with the system size. Such states have been observed in models of globally coupled particles that move under Hamiltonian dynamics either on a unit circle or on a unit spherical surface. Here, we address the ubiquity of QSS in long-range systems by considering a different dynamical setting. Thus, we consider an anisotropic Heisenberg model consisting of classical Heisenberg spins with mean-field interactions and evolving under classical spin dynamics. Our analysis of the corresponding Vlasov equation for time evolution of the phase space distribution shows that in a certain energy interval, relaxation of a class of initial states occurs over a timescale which grows algebraically with the system size. We support these findings by extensive numerical simulations. This work further supports the generality of occurrence of QSS in long-range systems evolving under Hamiltonian dynamics.
We consider a long-range interacting system of N particles moving on a spherical surface under an attractive Heisenberg-like interaction of infinite range and evolving under deterministic Hamilton dynamics. The system may also be viewed as one of globally coupled Heisenberg spins. In equilibrium, the system has a continuous phase transition from a low-energy magnetized phase, in which the particles are clustered on the spherical surface, to a high-energy homogeneous phase. The dynamical behavior of the model is studied analytically by analyzing the Vlasov equation for the evolution of the single-particle distribution and numerically by direct simulations. The model is found to exhibit long-lived nonmagnetized quasistationary states (QSSs) which in the thermodynamic limit are dynamically stable within an energy range where the equilibrium state is magnetized. For finite N, these states relax to equilibrium over a time that increases algebraically with N. In the dynamically unstable regime, nonmagnetized states relax fast to equilibrium over a time that scales as lnN. These features are retained in presence of a global anisotropy in the magnetization.
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.
The duration of the quasistationary states (QSSs) emerging in the d-dimensional classical inertial $\ensuremath{\alpha}$-Heisenberg model, i.e., N three-dimensional rotators whose interactions decay with distance rij as $1/{r}_{ij}^{\ensuremath{\alpha}}$ ($\ensuremath{\alpha}\ensuremath{\ge}0$), is studied through first-principle molecular dynamics. These QSSs appear for the very-long-range interaction regime ($0\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}1$), for an average energy per rotator U<Uc (Uc=5/6), and they do not exist for U>Uc. They are characterized by a kinetic temperature TQSS, before a crossover to a second plateau occurring at the Boltzmann-Gibbs temperature TBG>TQSS. We investigate here the behavior of their duration tQSS when U approaches Uc from below, for large values of N. The QSS gradually disappears as $U\ensuremath{\rightarrow}{U}_{c}$, while its duration undergoes a critical phenomenon, namely ${t}_{\text{QSS}}\ensuremath{\propto}{({U}_{c}\ensuremath{-}U)}^{\ensuremath{-}\ensuremath{\xi}}$. Universality is found for the critical exponent $\ensuremath{\xi}=1\text{.}67\ifmmode\pm\else\textpm\fi{}0\text{.}02$ throughout the very-long-range interaction regime.
The dynamics of quasistationary states of long-range interacting systems with N particles can be described by kinetic equations such as the Balescu-Lenard and Landau equations. In the case of one-dimensional homogeneous systems, two-body contributions vanish as two-body collisions in one dimension only exchange momentum and thus cannot change the one-particle distribution. Using a Kac factor in the interparticle potential implies a scaling of the dynamics proportional to N(δ) with δ=1 except for one-dimensional homogeneous systems. For the latter different values for δ were reported for a few models. Recently it was shown by Rocha Filho and collaborators [Phys. Rev. E 90, 032133 (2014)] for the Hamiltonian mean-field model that δ=2 provided that N is sufficiently large, while small N effects lead to δ≈1.7. More recently, Gupta and Mukamel [J. Stat. Mech. (2011) P03015] introduced a classical spin model with an anisotropic interaction with a scaling in the dynamics proportional to N(1.7) for a homogeneous state. We show here that this model reduces to a one-dimensional Hamiltonian system and that the scaling of the dynamics approaches N(2) with increasing N. We also explain from theoretical consideration why usual kinetic theory fails for small N values, which ultimately is the origin of noninteger exponents in the scaling.
Isolated long-range interacting particle systems appear generically to relax to nonequilibrium states ("quasistationary states" or QSSs) which are stationary in the thermodynamic limit. A fundamental open question concerns the "robustness" of these states when the system is not isolated. In this paper we explore, using both analytical and numerical approaches to a paradigmatic one-dimensional model, the effect of a simple class of perturbations. We call them "internal local perturbations" in that the particle energies are perturbed at collisions in a way which depends only on the local properties. Our central finding is that the effect of the perturbations is to drive all the very different QSSs we consider towards a unique QSS. The latter is thus independent of the initial conditions of the system, but determined instead by both the long-range forces and the details of the perturbations applied. Thus in the presence of such a perturbation the long-range system evolves to a unique nonequilibrium stationary state, completely different from its state in absence of the perturbation, and it remains in this state when the perturbation is removed. We argue that this result may be generic for long-range interacting systems subject to perturbations which are dependent on the local properties (e.g., spatial density or velocity distribution) of the system itself.
The duration of the quasistationary states (QSSs) emerging in the d-dimensional classical inertial $\ensuremath{\alpha}\text{\ensuremath{-}}XY$ model, i.e., N planar rotators whose interactions decay with the distance rij as $1/{r}_{ij}^{\ensuremath{\alpha}}$ ($\ensuremath{\alpha}\ensuremath{\ge}0$), is studied through first-principles molecular dynamics. These QSSs appear along the whole long-range interaction regime ($0\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}1$), for an average energy per rotator U<Uc (Uc=3/4), and they do not exist for U>Uc. They are characterized by a kinetic temperature TQSS, before a crossover to a second plateau occurring at the Boltzmann-Gibbs temperature TBG>TQSS. We investigate here the behavior of their duration tQSS when U approaches Uc from below, for large values of N. Contrary to the usual belief that the QSS merely disappears as $U\ensuremath{\rightarrow}{U}_{c}$, we show that its duration goes through a critical phenomenon, namely ${t}_{\text{QSS}}\ensuremath{\propto}{({U}_{c}\ensuremath{-}U)}^{\ensuremath{-}\ensuremath{\xi}}$. Universality is found for the critical exponent $\ensuremath{\xi}\ensuremath{\simeq}5/3$ throughout the whole long-range interaction regime.