Article

Exploring the Thermodynamic Limit of Hamiltonian Models: Convergence to the Vlasov Equation

University of Florence, Florens, Tuscany, Italy
Physical Review Letters (Impact Factor: 7.73). 05/2007; 98(15):150602. DOI: 10.1103/PHYSREVLETT.98.150602
Source: PubMed

ABSTRACT We here discuss the emergence of quasistationary states (QSS), a universal feature of systems with long-range interactions. With reference to the Hamiltonian mean-field 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 particle correlations.

Full-text

Available from: Duccio Fanelli, Jun 03, 2015
1 Follower
 · 
102 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We study the core-halo structure of low-energy quasi-stationary states in the Hamiltonian mean field model. The core-halo structure results in the superposition of two independent Lynden-Bell distributions. We examine the completeness of the Lynden-Bell relaxation and the relaxation between these two Lynden-Bell distributions.
    Physical Review E 11/2014; 91(3-1). DOI:10.1103/PhysRevE.91.032144 · 2.33 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Systems with long-range interactions display a short-time relaxation towards Quasi-Stationary States (QSSs), whose lifetime increases with system size. With reference to the Hamiltonian Mean Field (HMF) model, we here review Lynden-Bell's theory of ``violent relaxation''. The latter results in a maximum entropy scheme for a water-bag initial profile which predicts the presence of out-of-equilibrium phase transitions} separating homogeneous (zero magnetization) from inhomogeneous (non-zero magnetization) QSSs. Two different parametric representations of the initial condition are analyzed and the features of the phase diagram are discussed. In both representations we find a second order and a first order line of phase transitions that merge at a tricritical point. Particular attention is payed to the condition of existence and stability of the homogenous phase.
    Chaos, Complexity and Transport - Theory and Applications - The CCT '07; 01/2008
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The nonlinear propagation of coherent optical fields has been extensively explored in the framework of nonlinear optics, while the linear propagation of incoherent fields has been widely studied in the framework of statistical optics. However, these two fundamental fields of optics have been mostly developed independently of each other, so that a satisfactory understanding of statistical nonlinear optics is still lacking. This article is aimed at reviewing a unified theoretical formulation of statistical nonlinear optics on the basis of the wave turbulence theory, which provides a nonequilibrium thermodynamic description of the system of incoherent nonlinear waves. We consider the nonlinear Schrödinger equation as a representative model accounting either for a nonlocal or a noninstantaneous nonlinearity, as well as higher-order dispersion effects. Depending on the amount of nonlocal (noninstantaneous) nonlinear interaction and the amount of inhomogeneous (nonstationary) statistics of the incoherent wave, different types of kinetic equations are derived and discussed. In the spatial domain, when the incoherent wave exhibits inhomogeneous statistical fluctuations, different forms of the (Hamiltonian) Vlasov equation are obtained depending on the amount of nonlocality. This Vlasov approach describes the processes of incoherent modulational instability and localized incoherent soliton structures. In the temporal domain, the causality property inherent to the response function leads to a kinetic formulation analogous to the weak Langmuir turbulence equation, which describes nonlocalized spectral incoherent solitons. In the presence of a highly noninstantaneous response, this formulation reduces to a family of singular integro-differential kinetic equations (e.g., Benjamin–Ono equation), which describe incoherent dispersive shock waves. Conversely, a non-stationary statistics leads to a (non-Hamiltonian) long-range Vlasov formulation, whose self-consistent potential is constrained by the causality condition of the response function. The spatio-temporal domain will be considered in the limit of an intertial nonlinearity. We review different theories developed to describe bright and dark spatial incoherent solitons experimentally observed in slowly responding nonlinear media: the coherent density method, the mutual coherence function approach, the modal theory and the Wigner–Moyal formulation. When the incoherent wave exhibits homogeneous fluctuations, the relevant kinetic equation is the wave turbulence (Hasselmann) equation. It describes wave condensation and the underlying irreversible process of thermalization to thermodynamic equilibrium, as well as genuine nonequilibrium turbulent regimes. In this way different remarkable phenomena associated to wave thermalization are reviewed, e.g., wave condensation or supercontinuum generation in photonic crystal fibers, as well as different mechanisms of breakdown of thermalization. Finally, recent developments aimed at providing a wave turbulence formulation of Raman fiber lasers and passive optical cavities are reviewed in relation with condensation-like phenomena.
    Physics Reports 09/2014; 542(1). DOI:10.1016/j.physrep.2014.03.002 · 22.91 Impact Factor