Dynamics of coupled oscillator systems in presence of a local potential

We consider a long-range model of coupled phase-only oscillators subject to a local potential and evolving in presence of thermal noise. The model is a non-trivial generalization of the celebrated Kuramoto model of collective synchronization. We demonstrate by exact results and numerics a surprisingly rich long-time behavior, in which the system settles into either a stationary state that could be in or out of equilibrium and supports either global synchrony or absence of it, or, in a time-periodic synchronized state. The system shows both continuous and discontinuous phase transitions, as well as an interesting reentrant transition in which the system successively loses and gains synchrony on steady increase of the relevant tuning parameter.Comment: v2: close to the published versio

Microcanonical solution of the mean-field ϕ4\phi^4 model: comparison with time averages at finite size

We solve the mean-field $\phi^4$ model in an external magnetic field in the microcanonical ensemble using two different methods. The first one is based on Rugh's microcanonical formalism and leads to express macroscopic observables, such as temperature, specific heat, magnetization and susceptibility, as time averages of convenient functions of the phase-space. The approach is applicable for any finite number of particles $N$. The second method uses large deviation techniques and allows us to derive explicit expressions for microcanonical entropy and for macroscopic observables in the $N \to\infty$ limit. Assuming ergodicity, we evaluate time averages in molecular dynamics simulations and, using Rugh's approach, we determine the value of macroscopic observables at finite $N$. These averages are affected by a slow time evolution, often observed in systems with long-range interactions. We then show how the finite $N$ time averages of macroscopic observables converge to their corresponding $N\to\infty$ values as $N$ is increased. As expected, finite size effects scale as $N^{-1}$.Comment: 18 pages, 1 figur

Long time behavior of quasi-stationary states of the Hamiltonian Mean-Field model

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, quasi-stationary 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 new numerical tool, based on the fluctuations of the phase of the instantaneous magnetization of the system. Using this tool, we study the quasi-stationary 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 quasi-stationary 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.Comment: Submitted to Phys. Rev.

Long-range interacting classical systems: universality in mixing weakening

Through molecular dynamics, we study the $d=2,3$ classical model of $N$ coupled rotators (inertial XY model) assuming a coupling constant which decays with distance as $r_{ij}^{-\alpha}$ ($\alpha \ge 0$). The total energy $$ is asymptotically $\propto N {\tilde N}$ with ${\tilde N} \equiv [N^{1-\alpha/d}-(\alpha/d)]/[1-\alpha/d]$, hence the model is thermodynamically extensive if $\alpha/d>1$ and nonextensive otherwise. We numerically show that, for energies above some threshold, the (appropriately scaled) maximum Lyapunov exponent is $\propto N^{-\kappa}$ where $\kappa$ is an {\it universal} (one and the same for $d=1,2$ and 3, and all energies) function of $\alpha/d$, which monotonically decreases from 1/3 to zero when $\alpha/d$ increases from 0 to 1, and identically vanishes above 1. These features are consistent with the nonextensive statistical mechanics scenario, where thermodynamic extensivity is associated with {\it exponential} mixing in phase space, whereas {\it weaker} (possibly {\it power-law} in the present case) mixing emerges at the $N \to \infty$ limit whenever nonextensivity is observed.Comment: 4 pages, 3 figures; Submitted to Physical Review Letter

Dynamics and thermodynamics of rotators interacting with both long and short range couplings

The effect of nearest-neighbor coupling on the thermodynamic and dynamical properties of the ferromagnetic Hamiltonian Mean Field model (HMF) is studied. For a range of antiferromagnetic nearest-neighbor coupling, a canonical first order transition is observed, and the canonical and microcanonical ensembles are non-equivalent. In studying the relaxation time of non-equilibrium states it is found that as in the HMF model, a class of non-magnetic states is quasi-stationary, with an algebraic divergence of their lifetime with the number of degrees of freedom $N$. The lifetime of metastable states is found to increase exponentially with $N$ as expected.Comment: 12 pages, 6 figure

Statistical Mechanics of systems with long range interactions

Recent theoretical studies of statistical mechanical properties of systems with long range interactions are briefly reviewed. In these systems the interaction potential decays with a rate slower than 1/r^d at large distances r in d dimensions. As a result, these systems are non-additive and they display unusual thermodynamic and dynamical properties which are not present in systems with short range interactions. In particular, the various statistical mechanical ensembles are not equivalent and the microcanonical specific heat may be negative. Long range interactions may also result in breaking of ergodicity, making the maximal entropy state inaccessible from some regions of phase space. In addition, in many cases long range interactions result in slow relaxation processes, with time scales which diverge in the thermodynamic limit. Various models which have been found to exhibit these features are discussed.Comment: Published in AIP Conference Proceedings 970 "Dynamics and Thermodynamics of Systems with Long-Range Interactions: Theory and Experiments", Assisi, Italy 4-8 July 2007, editors A. Campa, A. Giansanti, G. Morigi and F. Sylos Labini, p. 22 (2008

Metastable states in a class of long-range Hamiltonian systems

We numerically show that metastable states, similar to the Quasi Stationary States found in the so called Hamiltonian Mean Field Model, are also present in a generalized model in which $N$ classical spins (rotators) interact through ferromagnetic couplings decaying as $r^{-\alpha}$, where $r$ is their distance over a regular lattice. Scaling laws with $N$ are briefly discussed.Comment: Latex 2e, 11 pages, 3 eps figures, contributed paper to the conf. "NEXT 2001", 23-30 May 2001, Cagliari (Italy), submitted to Physica