This paper studies the duration of quasi-stationary states, typical of long-range-interacting Hamiltonians where negative specific heat emerges. For the classical Heisenberg d-dimensional model, the authors show a universal behavior on the time evolution of the average kinetic energy per particle as a function of dimension, system size, and range of the interactions.
This paper studies the duration of quasi-stationary states, typical of long-range-interacting Hamiltonians where negative specific heat emerges. For the classical Heisenberg d-dimensional model, the authors show a universal behavior on the time evolution of the average kinetic energy per particle as a function of dimension, system size, and range of the interactions.
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.
We numerically study, from first principles, the temperature TQSS and duration tQSS of the longstanding initial quasi-stationary state of the isolated d-dimensional classical inertial $\ensuremath{\alpha}\text{\ensuremath{-}}XY$ ferromagnet with two-body interactions decaying as $1/{r}_{ij}^{\ensuremath{\alpha}}$ $(\ensuremath{\alpha}\ensuremath{\ge}0)$. It is shown that this temperature TQSS (defined proportional to the kinetic energy per particle) depends, for the long-range regime $0\ensuremath{\le}\ensuremath{\alpha}/d\ensuremath{\le}1$, on $(\ensuremath{\alpha},d,U,N)$ with numerically negligible changes for dimensions d=1,2,3, with U the energy per particle and N the number of particles. We verify the finite-size scaling ${T}_{\mathrm{QSS}}\ensuremath{-}{T}_{\ensuremath{\infty}}\ensuremath{\propto}1/{N}^{\ensuremath{\beta}}$ where ${T}_{\ensuremath{\infty}}\ensuremath{\equiv}2U\ensuremath{-}1$ for $U\ensuremath{\lesssim}{U}_{c}$, and $\ensuremath{\beta}$ appears to depend sensibly only on $\ensuremath{\alpha}/d$. Our numerical results indicate that neither the scaling with N of TQSS nor that of tQSS depend on U.
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 <
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.
Long-lived quasi-stationary states (QSSs) are a signature characteristic of long-range interacting systems both in the classical and in the quantum realms. Often, they emerge after a sudden quench of the Hamiltonian internal parameters and present a macroscopic lifetime, which increases with the system size. Despite their ubiquity, the fundamental mechanism at their root remains unknown. Here, we show that the spectrum of systems with power-law decaying couplings remains discrete up to the thermodynamic limit. As a consequence, several traditional results on the chaotic nature of the spectrum in many-body quantum systems are not satisfied in the presence of long-range interactions. In particular, the existence of QSSs may be traced back to the finiteness of Poincaré recurrence times. This picture justifies and extends known results on the anomalous magnetization dynamics in the quantum Ising model with power-law decaying couplings. The comparison between the discrete spectrum of long-range systems and more conventional examples of pure point spectra in the disordered case is also discussed.
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.