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Dynamical stability criterion for inhomogeneous quasi-stationary states in long-range systems  [PDF]
Alessandro Campa,Pierre-Henri Chavanis
Statistics , 2010, DOI: 10.1088/1742-5468/2010/06/P06001
Abstract: We derive a necessary and sufficient condition of linear dynamical stability for inhomogeneous Vlasov stationary states of the Hamiltonian Mean Field (HMF) model. The condition is expressed by an explicit disequality that has to be satisfied by the stationary state, and it generalizes the known disequality for homogeneous stationary states. In addition, we derive analogous disequalities that express necessary and sufficient conditions of formal stability for the stationary states. Their usefulness, from the point of view of linear dynamical stability, is that they are simpler, although they provide only sufficient criteria of linear stability. We show that for homogeneous stationary states the relations become equal, and therefore linear dynamical stability and formal stability become equivalent.
Quasi-stationary states and the range of pair interactions  [PDF]
Andrea Gabrielli,Michael Joyce,Bruno Marcos
Physics , 2010, DOI: 10.1103/PhysRevLett.105.210602
Abstract: "Quasi-stationary" states are approximately time-independent out of equilibrium states which have been observed in a variety of systems of particles interacting by long-range interactions. We investigate here the conditions of their occurrence for a generic pair interaction V(r \rightarrow \infty) \sim 1/r^a with a > 0, in d>1 dimensions. We generalize analytic calculations known for gravity in d=3 to determine the scaling parametric dependences of their relaxation rates due to two body collisions, and report extensive numerical simulations testing their validity. Our results lead to the conclusion that, for a < d-1, the existence of quasi-stationary states is ensured by the large distance behavior of the interaction alone, while for a > d-1 it is conditioned on the short distance properties of the interaction, requiring the presence of a sufficiently large soft-core in the interaction potential.
Out-of-equilibrium dynamics in systems with long-range interactions: characterizing quasi-stationary states  [PDF]
Pierre de Buyl
Physics , 2012, DOI: 10.1007/978-3-319-00395-5_18
Abstract: Systems with long-range interactions (LRI) display unusual thermodynamical and dynamical properties that stem from the non-additive character of the interaction potential. We focus in this work on the lack of relaxation to thermal equilibrium when a LRI system is started out-of-equilibrium. Several attempts have been made at predicting the so-called quasi-stationary state (QSS) reached by the dynamics and at characterizing the resulting transition between magnetized and non-magnetized states. We review in this work recent theories and interpretations about the QSS. Several theories exist but none of them has provided yet a full account of the dynamics found in numerical simulations.
Scaling quasi-stationary states in long range systems with dissipation  [PDF]
Michael Joyce,Jules Morand,Fran?ois Sicard,Pascal Viot
Physics , 2013, DOI: 10.1103/PhysRevLett.112.070602
Abstract: Hamiltonian systems with long-range interactions give rise to long lived out of equilibrium macroscopic states, so-called quasi-stationary states. We show here that, in a suitably generalized form, this result remains valid for many such systems in the presence of dissipation. Using an appropriate mean-field kinetic description, we show that models with dissipation due to a viscous damping or due to inelastic collisions admit "scaling quasi-stationary states", i.e., states which are quasi-stationary in rescaled variables. A numerical study of one dimensional self-gravitating systems confirms both the relevance of these solutions, and gives indications of their regime of validity in line with theoretical predictions. We underline that the velocity distributions never show any tendency to evolve towards a Maxwell-Boltzmann form.
A maximum entropy principle explains quasi-stationary states in systems with long-range interactions: the example of the Hamiltonian Mean Field model  [PDF]
Andrea Antoniazzi,Duccio Fanelli,Julien Barré,Pierre-Henri Chavanis,Thierry Dauxois,Stefano Ruffo
Physics , 2006, DOI: 10.1103/PhysRevE.75.011112
Abstract: A generic feature of systems with long-range interactions is the presence of {\it quasi-stationary} states with non-Gaussian single particle velocity distributions. For the case of the Hamiltonian Mean Field (HMF) model, we demonstrate that a maximum entropy principle applied to the associated Vlasov equation explains known features of such states for a wide range of initial conditions. We are able to reproduce velocity distribution functions with an analytical expression which is derived from the theory with no adjustable parameters. A normal diffusion of angles is detected and a new dynamical effect, two oscillating clusters surrounded by a halo, is also found and theoretically justified.
Condensation in the zero range process: stationary and dynamical properties  [PDF]
Stefan Grosskinsky,Gunter M. Schuetz,Herbert Spohn
Statistics , 2003,
Abstract: The zero range process is of particular importance as a generic model for domain wall dynamics of one-dimensional systems far from equilibrium. We study this process in one dimension with rates which induce an effective attraction between particles. We rigorously prove that for the stationary probability measure there is a background phase at some critical density and for large system size essentially all excess particles accumulate at a single, randomly located site. Using random walk arguments supported by Monte Carlo simulations, we also study the dynamics of the clustering process with particular attention to the difference between symmetric and asymmetric jump rates. For the late stage of the clustering we derive an effective master equation, governing the occupation number at clustering sites.
Existence of Quasi-stationary states at the Long Range threshold  [PDF]
Alessio Turchi,Duccio Fanelli,Xavier Leoncini
Physics , 2010, DOI: 10.1016/j.cnsns.2011.03.013
Abstract: In this paper the lifetime of quasi-stationary states (QSS) in the $\alpha-$HMF model are investigated at the long range threshold ($\alpha=1$). It is found that QSS exist and have a diverging lifetime $\tau(N)$ with system size which scales as $\mbox{\ensuremath{\tau}(N)\ensuremath{\sim}}\log N$, which contrast to the exhibited power law for $\alpha<1$ and the observed finite lifetime for $\alpha>1$. Another feature of the long range nature of the system beyond the threshold ($\alpha>1$) namely a phase transition is displayed for $\alpha=1.5$. The definition of a long range system is as well discussed.
Dynamical anomalies and the role of initial conditions in the HMF model  [PDF]
Alessandro Pluchino,Vito Latora,Andrea Rapisarda
Physics , 2004, DOI: 10.1016/j.physa.2004.02.025
Abstract: We discuss the role of the initial conditions for the dynamical anomalies observed in the quasi-stationary states of the Hamiltonian Mean Field (HMF) model.
Hamiltonian dynamics reveals the existence of quasi-stationary states for long-range systems in contact with a reservoir  [PDF]
Fulvio Baldovin,Enzo Orlandini
Physics , 2006, DOI: 10.1103/PhysRevLett.96.240602
Abstract: 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.
Quasi-stationary states and incomplete violent relaxation in systems with long-range interactions  [PDF]
P. H. Chavanis
Physics , 2005, DOI: 10.1016/j.physa.2006.01.006
Abstract: We discuss the nature of quasi-stationary states (QSS) with non-Boltzmannian distribution in systems with long-range interactions in relation with a process of incomplete violent relaxation based on the Vlasov equation. We discuss several attempts to characterize these QSS and explain why their prediction is difficult in general.
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