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

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.

Authors:
;  [1];  [2];  [1];  [3]
  1. Dipartimento di Energetica and CSDC, Universita di Firenze, and INFN, via S. Marta, 3, 50139 Florence (Italy)
  2. Dipartimento di Fisica 'E.Fermi' and CNISM, Universita di Pisa, Largo Pontecorvo, 3 56127 Pisa (Italy)
  3. (United Kingdom)
Publication Date:
OSTI Identifier:
20951225
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review Letters; Journal Volume: 98; Journal Issue: 15; Other Information: DOI: 10.1103/PhysRevLett.98.150602; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOLTZMANN-VLASOV EQUATION; COMPARATIVE EVALUATIONS; CONVERGENCE; CORRELATIONS; FLUCTUATIONS; HAMILTONIANS; INTERACTION RANGE; MATHEMATICAL SPACE; MEAN-FIELD THEORY; NUMERICAL SOLUTION; RELAXATION; SIMULATION

Citation Formats

Antoniazzi, Andrea, Ruffo, Stefano, Califano, Francesco, Fanelli, Duccio, and Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL. Exploring the Thermodynamic Limit of Hamiltonian Models: Convergence to the Vlasov Equation. United States: N. p., 2007. Web. doi:10.1103/PHYSREVLETT.98.150602.
Antoniazzi, Andrea, Ruffo, Stefano, Califano, Francesco, Fanelli, Duccio, & Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL. Exploring the Thermodynamic Limit of Hamiltonian Models: Convergence to the Vlasov Equation. United States. doi:10.1103/PHYSREVLETT.98.150602.
Antoniazzi, Andrea, Ruffo, Stefano, Califano, Francesco, Fanelli, Duccio, and Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL. Fri . "Exploring the Thermodynamic Limit of Hamiltonian Models: Convergence to the Vlasov Equation". United States. doi:10.1103/PHYSREVLETT.98.150602.
@article{osti_20951225,
title = {Exploring the Thermodynamic Limit of Hamiltonian Models: Convergence to the Vlasov Equation},
author = {Antoniazzi, Andrea and Ruffo, Stefano and Califano, Francesco and Fanelli, Duccio and Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL},
abstractNote = {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.},
doi = {10.1103/PHYSREVLETT.98.150602},
journal = {Physical Review Letters},
number = 15,
volume = 98,
place = {United States},
year = {Fri Apr 13 00:00:00 EDT 2007},
month = {Fri Apr 13 00:00:00 EDT 2007}
}
  • When the Vlasov equation is investigated numerically using the method of test particles, the particle-particle interactions that inevitably arise in the simulation (but are not present in the Vlasov equation itself) result in an accumulation of errors which eventually drive the collection of test particles toward a state of classical thermal equilibrium. We estimate the rate at which these errors accumulate. {copyright} {ital 1996 The American Physical Society.}
  • Morrison and Pfirsch derived expressions for plasma wave energy using the formalism of Van Kampen modes, apparently unaware of earlier work of Best on the same subject by the same method. It is stressed in this Comment that energy density, rather than total energy, is the relevant physical quantity.
  • The energy density expression Best presents is not (when integrated) equal to the energy of a linear perturbation in Vlasov theory. The exact expression is given by either Eq.(42) or Eq.(48) of Ref.2. The authors comment on this and other discrepancies. (AIP)
  • Expressions for the energy content of one-dimensional electrostatic perturbations about homogeneous equilibria are revisited. The well-known dielectric energy, E{sub {ital D}}, is compared with the exact plasma free energy expression, {delta}{sup 2}{ital F}, that is conserved by the Vlasov--Poisson system (Phys. Rev. A {bold 40}, 3898 (1989) and Phys. Fluids B {bold 2}, 1105 (1990)). The former is an expression in terms of the perturbed electric field amplitude, while the latter is determined by a generating function, which describes perturbations of the distribution function that respect the important constraint of {ital dynamical} {ital accessibility} of the system. Thus the comparisonmore » requires solving the Vlasov equation for such a perturbation of the distribution function in terms of the electric field. This is done for neutral modes of oscillation that occur for equilibria with stationary inflection points, and it is seen that for these special modes {delta}{sup 2}{ital F}=E{sub {ital D}}. In the case of unstable and corresponding damped modes it is seen that {delta}{sup 2}{ital F}{ne}E{sub {ital D}}; in fact {delta}{sup 2}{ital F}{equivalent to}0. This failure of the dielectric energy expression persists even for arbitrarily small growth and damping rates since E{sub {ital D}} is nonzero in this limit, whereas {delta}{sup 2}{ital F} remains zero.« less
  • From the Hamiltonian structure of the Vlasov equation, we build a Hamiltonian model for the first three moments of the Vlasov distribution function, namely, the density, the momentum density and the specific internal energy. We derive the Poisson bracket of this model from the Poisson bracket of the Vlasov equation, and we discuss the associated Casimir invariants.