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

University of Florence, Florens, Tuscany, Italy
Physical Review Letters (Impact Factor: 7.51). 05/2007; 98(15):150602. DOI: 10.1103/PHYSREVLETT.98.150602
Source: PubMed


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.

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    • "Prior studies of the HMF model have revealed non-trivial collective oscillations [10] and thermodynamical properties that pertain to a large class of long-range interacting systems [9] [11] [12] [13] [14] [15]. "
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    ABSTRACT: We propose an efficient method to compute Lyapunov exponents and Lyapunov eigenvectors of long-range interacting many-particle systems, whose dynamics is described by the Vlasov equation. We show that an expansion of a distribution function using Hermite modes (in momentum variable) and Fourier modes (in configuration variable) converges fast if an appropriate scaling parameter is introduced and identified with the inverse of the system temperature. As a consequence, dynamics and linear stability properties of many-particle states, both in the close-to and in the far-from equilibrium regimes can be predicted using a small number of expansion coefficients. As an example of a long-range interacting system we investigate stability properties of stationary states of the Hamiltonian mean-field model.
    Physical Review E 09/2009; 80(3 Pt 2):036402. DOI:10.1103/PhysRevE.80.036402 · 2.29 Impact Factor
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    ABSTRACT: We present a molecular dynamics test of the Central Limit Theorem (CLT) in a paradigmatic long-range-interacting many-body classical Hamiltonian system, the HMF model. We calculate sums of velocities at equidistant times along deterministic trajectories for different sizes and energy densities. We show that, when the system is in a chaotic regime (specifically, at thermal equilibrium), ergodicity is essentially verified, and the Pdfs of the sums appear to be Gaussians, consistently with the standard CLT. When the system is, instead, only weakly chaotic (specifically, along longstanding metastable Quasi-Stationary States), nonergodicity (i.e., discrepant ensemble and time averages) is observed, and robust $q$-Gaussian attractors emerge, consistently with recently proved generalizations of the CLT.
    EPL (Europhysics Letters) 06/2007; 802(2). DOI:10.1209/0295-5075/80/26002 · 2.10 Impact Factor
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