Relaxation dynamics of stochastic long-range interacting systems

Gupta, Shamik; Mukamel, David

doi:10.1088/1742-5468/2010/08/p08026

Cited by 14 publications

(15 citation statements)

References 44 publications

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“…For practical purposes, the Vlasov equation is used as a good approximation for sufficiently large N , its validity being limited for short times. Although the lifetime of a QSS have been extensively studied in the literature [19][20][21][22][23][24][25] many aspects of its physical interpretation and phenomenology remain unclear. In the present paper we discuss how to properly define a timescale for its long-time evolution which is governed by collisional corrections to the Vlasov equation leading to the Landau or Balescu-Lenard kinetic equations (see Ref.…”

Section: Introductionmentioning

confidence: 99%

Dynamics and physical interpretation of quasistationary states in systems with long-range interactions

et al. 2014

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Although the Vlasov equation is used as a good approximation for a sufficiently large N , Braun and Hepp have showed that the time evolution of the one particle distribution function of a N particle classical Hamiltonian system with long range interactions satisfies the Vlasov equation in the limit of infinite N . Here we rederive this result using a different approach allowing a discussion of the role of inter-particle correlations on the system dynamics. Otherwise for finite N collisional corrections must be introduced. This has allowed the a quite comprehensive study of the Quasi Stationary States (QSS) but many aspects of the physical interpretations of these states remain unclear. In this paper a proper definition of timescale for long time evolution is discussed and several numerical results are presented, for different values of N . Previous reports indicates that the lifetimes of the QSS scale as N 1.7 or even the system properties scales with exp(N ). However, preliminary results presented here shows indicates that time scale goes as N 2 for a different type of initial condition. We also discuss how the form of the inter-particle potential determines the convergence of the N -particle dynamics to the Vlasov equation. The results are obtained in the context of following models: the Hamiltonian Mean Field, the Self Gravitating Ring Model, and a 2-D Systems of Gravitating Particles. We have also provided information of the validity of the Vlasov equation for finite N , i. e. how the dynamics converges to the mean-field (Vlasov) description as N increases and how inter-particle correlations arise.

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Section: Introductionmentioning

confidence: 99%

Dynamics and physical interpretation of quasistationary states in systems with long-range interactions

et al. 2014

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“…The robustness of quasistationarity to stochastic dynamical processes has also been analyzed, where it is found that QSS exist only as a crossover phenomenon. Under such dynamics, these states have a finite relaxation time which is determined by the rate of the stochastic process [28][29][30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Quasistationarity in a model of classical spins with long-range interactions

Gupta¹,

Mukamel²

2011

J. Stat. Mech.

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Abstract. 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 longrange 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.

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“…It is interesting to compare our results with related previous work in the literature. In [11,12] the stochastic perturbation applied to the long-range system (the HMF model) permutes the momenta of triplets of particles chosen randomly in the system, and drives the system to relax to thermal equilibrium efficiently. Applied to a one dimensional self-gravitating system, we have checked that we observe the same behaviour.…”

Section: Discussionmentioning

confidence: 99%

“…However, much more generally, it has been suggested on the basis of study of the one dimensional HMF model (see e.g. [11,12]) that QSS will disappear in the presence of any generic stochastic perturbation to the dynamics. A study of the effect of external stochastic fields with spatial correlation applied to the same model (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Attractor nonequilibrium stationary states in perturbed long-range interacting systems

2016

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Isolated long-range interacting particle systems appear generically to relax to non-equilibrium states ("quasi-stationary states" or QSS) 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 QSS 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 non-equilibrium stationary state, completely different to 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.

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Relaxation dynamics of stochastic long-range interacting systems

Cited by 14 publications

References 44 publications

Dynamics and physical interpretation of quasistationary states in systems with long-range interactions

Dynamics and physical interpretation of quasistationary states in systems with long-range interactions

Quasistationarity in a model of classical spins with long-range interactions

Attractor nonequilibrium stationary states in perturbed long-range interacting systems

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