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arXiv:1705.05623

Stationary ordered non-equilibrium states of long-range interacting systems

Joyce, Michael, Jules Morand, Pascal Viot

Published May 16, 2017• 0 citations
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Abstract

Long-range interacting Hamiltonian systems are believed to relax generically towards non-equilibrium states called "quasi-stationary" because they evolve towards thermodynamic equilibrium very slowly, on a time-scale diverging with particle number. We show here that, by applying a suitable perturbation operator for a finite time interval, we obtain, in a family of long-range systems, non-equilibrium states which appear to be strictly stationary. They exist even in the case of a harmonic potential, and are characterised by an ordered microscopic phase space structure. We give some simple heuristic arguments which predict reasonably well some properties of these states.

DOI
10.48550/arxiv.1705.05623

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Long-range interacting Hamiltonian systems are believed to relax generically towards non-equilibrium states called "quasi-stationary" because they evolve towards thermodynamic equilibrium very slowly, on a time-scale diverging with particle number. We show here that, by applying a suitable perturbation operator for a finite time interval, we obtain, in a family of long-range systems, non-equilibrium states which appear to be strictly stationary. They exist even in the case of a harmonic potential, and are characterised by an ordered microscopic phase space structure. We give some simple heuristic arguments which predict reasonably well some properties of these states.
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.
In this Thesis we describe the theoretical-computational study performed on the behavior of isolated systems, far from thermodynamic equilibrium. Analyzing models well-known in literature we follow a path bringing to the classification of different behaviors in function of the interaction range of the systems' particles. In the case of systems with long-range interaction we studied the "Quasi-Stationary states" (QSSs) which emerge at short times when the system evolves with Hamiltonian dynamics. Their interest is in the fact that in many physical systems, such as self-gravitating systems, plasmas and systems characterized by wave-particle interaction, QSSs are the only experimentally accessible regime. QSS are defined as stable solutions of the Vlasov equation and, as their duration diverges with the system size, for large systems' size they can be seen as the true equilibria. They do not follow the Boltzmann statistics, and it does not exists a general theory which describes them. Anyway it is possible to give an approximate description using Lynden-Bell theory. One part of the thesis is devoted to shed light on the characteristics of the phase diagram of the "Hamiltonian mean field" model (HMF), during the QSS, calculated with the Lynden-Bell theory. The results of our work allowed to confirm numerically the presence of a phase re-entrance. In the Thesis is present also a detailed description on the system's caloric curves and on the metastability. Still in this context we show an analysis of the equivalence of the statistical ensembles, confirmed in almost the totality of the phase diagram (except for a small region), although the presence of negative specific heat in the microcanonical ensemble, which in Boltzmannian systems implies the non-equivalence of statistical ensembles. This result allowed us to arrive to a surprising conclusion: the presence of negative specific heat in the canonical ensemble. Still in the context of long-range interacting systems we analyze the linear stability of the non-homogeneous QSSs with respect to the Vlasov equation. Since the study of QSS find an application in the Free-electron laser (FEL) and other light sources, which are characterized by wave-particle interaction, we analyze, in the last chapter, the experimental perspectives of our work in this context. The other class of systems we studied are short-range interacting systems. Here the behavior of the components of the system is strongly influenced by the neighbors, and if one takes a system in a disordered state (a zero magnetization state for magnetic systems), which relaxes towards an ordered equilibrium state, one sees that the ordering process first develops locally and then extends to the whole system forming domains of opposed magnetization which grow in size. This process is called "coarsening". Our work in this field consisted in investigating numerically the laws of scale, and in the Thesis we characterize the temporal dependence of the domain sizes for different interaction ranges and we show a comparison between Hamiltonian and Langevin dynamics. This work inserts in the open debate on the equivalence of different dynamics where we found that, at least for times not too large, the two dynamics give different scaling laws.
We introduce a Hamiltonian dynamics for the description of long-range interacting systems in contact with a thermal bath (i.e., in the canonical ensemble). The dynamics confirms statistical mechanics equilibrium predictions for the Hamiltonian Mean Field model and the equilibrium ensemble equivalence. We find that long-lasting quasi-stationary states persist in presence of the interaction with the environment. Our results indicate that quasi-stationary states are indeed reproducible in real physical experiments.
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.
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.