Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model

We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field (HMF) model, a prototype for long-range interactions in $N$-particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated $N ~$ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite $N$, dynamics. We then propose and verify numerically a scenario for the relaxation process, relying on the Vlasov equation. When starting from a non stationary or a Vlasov unstable stationary initial state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov equation via non stationary states: we characterize numerically this dynamical instability in the finite $N$ system by introducing appropriate indicators. This first step of the evolution towards Boltzmann-Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov equation. If the finite $N$ system is initialized in a Vlasov stable homogenous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law $N^{1.7}$. Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the $q$-exponential distributions derived from Tsallis statistics.Comment: To appear in Physica

Ensemble inequivalence: A formal approach

Ensemble inequivalence has been observed in several systems. In particular it has been recently shown that negative specific heat can arise in the microcanonical ensemble in the thermodynamic limit for systems with long-range interactions. We display a connection between such behaviour and a mean-field like structure of the partition function. Since short-range models cannot display this kind of behaviour, this strongly suggests that such systems are necessarily non-mean field in the sense indicated here. We further show that a broad class of systems with non-integrable interactions are indeed of mean-field type in the sense specified, so that they are expected to display ensemble inequivalence as well as the peculiar behaviour described above in the microcanonical ensemble.Comment: 4 pages, no figures, given at the NEXT2001 conference on non-extensive thermodynamic

Inhomogeneous Quasi-stationary States in a Mean-field Model with Repulsive Cosine Interactions

The system of N particles moving on a circle and interacting via a global repulsive cosine interaction is well known to display spatially inhomogeneous structures of extraordinary stability starting from certain low energy initial conditions. The object of this paper is to show in a detailed manner how these structures arise and to explain their stability. By a convenient canonical transformation we rewrite the Hamiltonian in such a way that fast and slow variables are singled out and the canonical coordinates of a collective mode are naturally introduced. If, initially, enough energy is put in this mode, its decay can be extremely slow. However, both analytical arguments and numerical simulations suggest that these structures eventually decay to the spatially uniform equilibrium state, although this can happen on impressively long time scales. Finally, we heuristically introduce a one-particle time dependent Hamiltonian that well reproduces most of the observed phenomenology.Comment: to be published in J. Phys.

Ensemble inequivalence in systems with long-range interactions

Ensemble inequivalence has been observed in several systems. In particular it has been recently shown that negative specific heat can arise in the microcanonical ensemble in the thermodynamic limit for systems with long-range interactions. We display a connection between such behaviour and a mean-field like structure of the partition function. Since short-range models cannot display this kind of behaviour, this strongly suggests that such systems are necessarily non-mean field in the sense indicated here. We illustrate our results showing an application to the Blume-Emery-Griffiths model. We further show that a broad class of systems with non-integrable interactions are indeed of mean-field type in the sense specified, so that they are expected to display ensemble inequivalence as well as the peculiar behaviour described above in the microcanonical ensemble.Comment: 12 pages, no figure

The genus Maera (Crustacea: Amphipoda: Melitidae) from Bermuda

Traditionally,the Bermudian amphipod fauna has included three species of Maera. After examining collections that span more than 10 years, we retain one species, M. tinkerensis; report a second species, M. quadrimana, as a verified record for the Atlantic; and describe four new species: M. ceres, M. miranda, M. ariel, and M. caliban. Discussion of M. quadrimana sensu lato, M. pacifica and M. rathbunae clarifies their taxonomic status and their relationship to the Bermudian fauna. Maera tinkerensis resides within the grossimana complex of species, and the other five species reside within the quadrimana complex. The zoogeographical implications of these morphological complex alignments are briefly considered. We provide data on habitat preferences and a key to the six species of Maera now recognized from Bermuda

Obese patients with a binge eating disorder have an unfavorable metabolic and inflammatory profile

To evaluate whether obese patients with a binge eating disorder (BED) have an altered metabolic and inflammatory profile related to their eating behaviors compared with non-BED obese.A total of 115 White obese patients consecutively recruited underwent biochemical, anthropometrical evaluation, and a 75-g oral glucose tolerance test. Patients answered the Binge Eating Scale and were interviewed by a psychiatrist. The patients were subsequently divided into 2 groups according to diagnosis: non-BED obese (n = 85) and BED obese (n = 30). Structural equation modeling analysis was performed to elucidate the relation between eating behaviors and metabolic and inflammatory profile.BED obese exhibited significantly higher percentages of altered eating behaviors, body mass index (P < 0.001), waist circumference (P < 0.01), fat mass (P < 0.001), and a lower lean mass (P < 0.001) when compared with non-BED obese. Binge eating disorder obese also had a worse metabolic and inflammatory profile, exhibiting significantly lower high-density lipoprotein cholesterol levels (P < 0.05), and higher levels of glycated hemoglobin (P < 0.01), uric acid (P < 0.05), erythrocyte sedimentation rate (P < 0.001), high-sensitive C-reactive protein (P < 0.01), and white blood cell counts (P < 0.01). Higher fasting insulin (P < 0.01) and higher insulin resistance (P < 0.01), assessed by homeostasis model assessment index and visceral adiposity index (P < 0.001), were observed among BED obese. All differences remained significant after adjusting for body mass index. No significant differences in fasting plasma glucose or 2-hour postchallenge plasma glucose were found. Structural equation modeling analysis confirmed the relation between the altered eating behaviors of BED and the metabolic and inflammatory profile.Binge eating disorder obese exhibited an unfavorable metabolic and inflammatory profile, which is related to their characteristic eating habits

Kinetic theory for non-equilibrium stationary states in long-range interacting systems

We study long-range interacting systems perturbed by external stochastic forces. Unlike the case of short-range systems, where stochastic forces usually act locally on each particle, here we consider perturbations by external stochastic fields. The system reaches stationary states where external forces balance dissipation on average. These states do not respect detailed balance and support non-vanishing fluxes of conserved quantities. We generalize the kinetic theory of isolated long-range systems to describe the dynamics of this non-equilibrium problem. The kinetic equation that we obtain applies to plasmas, self-gravitating systems, and to a broad class of other systems. Our theoretical results hold for homogeneous states, but may also be generalized to apply to inhomogeneous states. We obtain an excellent agreement between our theoretical predictions and numerical simulations. We discuss possible applications to describe non-equilibrium phase transitions.Comment: 11 pages, 2 figures; v2: small changes, close to the published versio

Exploring the thermodynamic limit of Hamiltonian models: convergence to the Vlasov equation

We here discuss the emergence of Quasi Stationary States (QSS), a universal feature of systems with long-range interactions. With reference to the Hamiltonian Mean Field (HMF) model, numerical simulations are performed based on both the original $N$-body setting and the continuum Vlasov model which is supposed to hold in the thermodynamic limit. A detailed comparison unambiguously demonstrates that the Vlasov-wave system provides the correct framework to address the study of QSS. Further, analytical calculations based on Lynden-Bell's theory of violent relaxation are shown to result in accurate predictions. Finally, in specific regions of parameters space, Vlasov numerical solutions are shown to be affected by small scale fluctuations, a finding that points to the need for novel schemes able to account for particles correlations.Comment: 5 pages, 3 figure

Stochastic quantum Zeno by large deviation theory

Quantum measurements are crucial for observing the properties of a quantum system, which, however, unavoidably perturb its state and dynamics in an irreversible way. Here we study the dynamics of a quantum system being subjected to a sequence of projective measurements applied at random times. In the case of independent and identically distributed intervals of time between consecutive measurements, we analytically demonstrate that the survival probability of the system to remain in the projected state assumes a large deviation (exponentially decaying) form in the limit of an infinite number of measurements. This allows us to estimate the typical value of the survival probability, which can therefore be tuned by controlling the probability distribution of the random time intervals. Our analytical results are numerically tested for Zeno-protected entangled states, which also demonstrate that the presence of disorder in the measurement sequence further enhances the survival probability when the Zeno limit is not reached (as it happens in experiments). Our studies provide a new tool for protecting and controlling the amount of quantum coherence in open complex quantum systems by means of tunable stochastic measurements. \ua9 2016 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

Algebraic Correlation Function and Anomalous Diffusion in the HMF model

In the quasi-stationary states of the Hamiltonian Mean-Field model, we numerically compute correlation functions of momenta and diffusion of angles with homogeneous initial conditions. This is an example, in a N-body Hamiltonian system, of anomalous transport properties characterized by non exponential relaxations and long-range temporal correlations. Kinetic theory predicts a striking transition between weak anomalous diffusion and strong anomalous diffusion. The numerical results are in excellent agreement with the quantitative predictions of the anomalous transport exponents. Noteworthy, also at statistical equilibrium, the system exhibits long-range temporal correlations: the correlation function is inversely proportional to time with a logarithmic correction instead of the usually expected exponential decay, leading to weak anomalous transport properties