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Register a new userHard-core collisional dynamics in the hamiltonian mean-field model
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We consider a modification of the well studied Hamiltonian Mean-Field model by introducing a hard-core point-like repulsive interaction and propose a numerical integration scheme to integrate numerically its dynamics. Our results show that the outcome of the initial violent relaxation is altered, and also that the phase-diagram is modified with a critical temperature at a higher value than in the non-collisional counterpart.
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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 $Nto infty$ 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. A finite system will eventually reach thermal equilibrium, while an infinite system will remain trapped in the quasi-stationary state 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 is invariant under time reversal so that it does not naturally describe a relaxational dynamics. However, as time grows the dynamics affects 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 (HMF). The approach described here generalizes the one proposed in G. Giachetti and L. Casetti, J. Stat. Mech.: Theory Exp. 2019, 043201 (2019), that was limited to cold initial conditions, to generic initial conditions, allowing us to 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.
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 $Ntoinfty$ limit. However, a faster and 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 systems size. After the damping of such oscillations the system is found in a quasi-stationary state that survives 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. We propose an approximation scheme to describe collisionless relaxation, based on the introduction of moments of the distribution function, and apply it to 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; generic initial conditions will be considered in a forthcoming paper.
We study the dynamics of the N-particle system evolving in the XY hamiltonian mean field (HMF) model for a repulsive potential, when no phase transition occurs. Starting from a homogeneous distribution, particles evolve in a mean field created by the interaction with all others. This interaction does not change the homogeneous state of the system, and particle motion is approximately ballistic with small corrections. For initial particle data approaching a waterbag, it is explicitly proved that corrections to the ballistic velocities are in the form of independent brownian noises over a time scale diverging not slower than $N^{2/5}$ as $N to infty$, which proves the propagation of molecular chaos. Molecular dynamics simulations of the XY-HMF model confirm our analytical findings.
We determine the phase diagram for a generalisation of two-and three-dimensional hard spheres: a classical system with three-body interactions realised as a hard cut-off on the mean-square distance for each triplet of particles. Quant
We perform extensive MD simulations of two-dimensional systems of hard disks, focusing on the emph{on}-collision statistical properties. We analyze the distribution functions of velocity, free flight time and free path length for packing fractions ranging from the fluid to the solid phase. The behaviors of the mean free flight time and path length between subsequent collisions are found to drastically change in the coexistence phase. We show that single particle dynamical properties behave analogously in collisional and continuous time representations, exhibiting apparent crossovers between the fluid and the solid phase. We find that, both in collisional and continuous time representation, the mean square displacement, velocity autocorrelation functions, intermediate scattering functions and self part of the van Hove function (propagator), closely reproduce the same behavior exhibited by the corresponding quantities in granular media, colloids and supercooled liquids close to the glass or jamming transition.
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