Coarse-grained collisionless dynamics with long-range interactions

Giachetti, Guido; Santini, Alessandro; Casetti, Lapo

doi:10.1103/physrevresearch.2.023379

Cited by 7 publications

(31 citation statements)

References 38 publications

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“…In this section we review the symplectic coarse graining procedure introduced by Giachetti et al (2020). The main idea is to describe VR using an evolution equation which includes an effective dissipative term.…”

Section: Symplectic Coarse Grainingmentioning

confidence: 99%

“…The functions µ k in the above equation can be calculated explicitly for one-dimensional systems. The equation derived from a symplectic coarse-graining for one-dimensional systems in Giachetti et al (2020) has the general structure (8) with a coefficient µ2(H) given by…”

Section: Symplectic Coarse Grainingmentioning

confidence: 99%

“…More recently Giachetti et al (2020) introduced a novel coarse graining procedure that preserves the symplectic struc-ture of the phase-space 3 . Using this formalism, they were able to formulate an explicit effective evolution equation for the coarse grained f of one-dimensional systems with general long-range interactions, without making assumptions on the specific form of the initial state f0.…”

Section: Introductionmentioning

confidence: 99%

“…In this work we extend the effective evolution equation derived by Giachetti et al (2020) to higher-dimensional systems whose mean-field Hamiltonian is instantaneously integrable. In this framework we investigate the VR process of a spherical gravitational dissipationless collapse.…”

Section: Introductionmentioning

confidence: 99%

“…The paper is structured as follows: in Section 2 we briefly review symplectic coarse graining as discussed in Giachetti et al (2020). In Section 3 we generalize it to more dimensions and we derive the effective coarse-grained evolution equation for the phase-space distribution and we evaluate the frequencies and the damping times of the Fourier modes of f k .…”

Section: Introductionmentioning

confidence: 99%

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Symplectic coarse graining approach to the dynamics of spherical self-gravitating systems

Barbieri,

Di Cintio,

Giachetti

et al. 2021

Preprint

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We investigate the evolution of the phase-space distribution function around slightly perturbed stationary states and the process of violent relaxation in the context of the dissipationless collapse of an isolated spherical self-gravitating system. By means of the recently introduced symplectic coarse graining technique, we obtain an effective evolution equation that allows us to compute the scaling of the frequencies around a stationary state, as well as the damping times of Fourier modes of the distribution function, with the magnitude of the Fourier k−vectors themselves. We compare our analytical results with N -body simulations.

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Section: Symplectic Coarse Grainingmentioning

confidence: 99%

Section: Symplectic Coarse Grainingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Symplectic coarse graining approach to the dynamics of spherical self-gravitating systems

Barbieri,

Di Cintio,

Giachetti

et al. 2021

Preprint

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Violent relaxation in the Hamiltonian mean field model: I. Cold collapse and effective dissipation

Giachetti

Casetti

2019

J. Stat. Mech.

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In N -body systems with long-range interactions mean-field effects dominate over binary interactions (collisions), so that relaxation to thermal equilibrium occurs on time scales that grow with N , diverging in the N → ∞ limit. However, a much faster and completely non-collisional relaxation process, referred to as violent relaxation, sets in when starting from generic initial conditions: collective oscillations (referred to as virial oscillations) develop and damp out on timescales not depending on the system's size. After the damping of such oscillations the system is found in a quasi-stationary state that may be very far from a thermal one, and that survives until the slow relaxation driven by two-body interactions becomes effective, that is, virtually forever when the system is very large. During violent relaxation the distribution function obeys the collisionless Boltzmann (or Vlasov) equation, that, being invariant under time reversal, does not "naturally" describe a relaxation process. Indeed, the dynamics is moved to smaller and smaller scales in phase space as time goes on, so that observables that do not depend on small-scale details appear as relaxed after a short time.Here we propose an approximation scheme to describe the collisionless relaxation process, based on the introduction of suitable moments of the distribution function, and apply it to a simple toy model, the Hamiltonian Mean Field (HMF) model. To the leading order, virial oscillations are equivalent to the motion of a particle in a one-dimensional potential. Inserting higher-order contributions in an effective way, inspired by the Caldeira-Leggett model of quantum dissipation, we derive a dissipative equation describing the damping of the oscillations, including a renormalization of the effective potential and yielding predictions for collective properties of the system after the damping in very good agreement with numerical simulations. Here we restrict ourselves to "cold" initial conditions, i.e., where the velocities of all the particles are set to zero: generic initial conditions will be considered in a forthcoming paper. 1 For self-gravitating systems a proper thermal equilibrium state in the usual sense does not exist, at least in three dimensions, because gravity is non-confining so that a Maxwellian velocity distribution would lead to the evaporation of the system, unless the latter has infinite mass. The relaxation time here is the time scale over which binary encounters induce a loss of memory of the initial conditions and the Boltzmann entropy grows. Gravity in spaces with dimension less than three (where a thermal equilibrium state is well defined) also has relaxation times diverging with N . arXiv:1902.02436v2 [cond-mat.stat-mech]

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Violent relaxation in the Hamiltonian mean field model: II. Non-equilibrium phase diagrams

Santini¹,

Giachetti²,

Casetti³

2022

J. Stat. Mech.

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A classical long-range-interacting N-particle system relaxes to thermal equilibrium on time scales growing with N; in the limit N → ∞ such a relaxation time diverges. However, a completely non-collisional relaxation process, known as violent relaxation, takes place on a much shorter time scale independent of N and brings the system towards a non-thermal quasi-stationary state (QSS). A finite system will eventually reach thermal equilibrium, while an infinite system will remain trapped in the QSS forever. For times smaller than the relaxation time, the distribution function of the system obeys the collisionless Boltzmann equation, also known as the Vlasov equation. The Vlasov dynamics are invariant under time reversal so that they do not ‘naturally’ describe the relaxational dynamics. However, as time grows the dynamics affect smaller and smaller scales in phase space, so that observables not depending upon small-scale details appear as relaxed after a short time. Herewith we present an approximation scheme able to describe violent relaxation in a one-dimensional toy-model, the Hamiltonian mean field. The approach described here generalizes the one proposed in Giachetti and Casetti (2019 J. Stat. Mech. 043201), which was limited to ‘cold’ initial conditions, to generic initial conditions, allowing us to predict non-equilibrium phase diagrams that turn out to be in good agreement with those obtained from the numerical integration of the Vlasov equation.

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Coarse-grained collisionless dynamics with long-range interactions

Cited by 7 publications

References 38 publications

Symplectic coarse graining approach to the dynamics of spherical self-gravitating systems

Symplectic coarse graining approach to the dynamics of spherical self-gravitating systems

Violent relaxation in the Hamiltonian mean field model: I. Cold collapse and effective dissipation

Violent relaxation in the Hamiltonian mean field model: II. Non-equilibrium phase diagrams

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