104 research outputs found
Classical Heisenberg spins with long-range interactions: Relaxation to equilibrium for finite systems
Systems with long-range interactions often relax towards statistical
equilibrium over timescales that diverge with , the number of particles. A
recent work [S. Gupta and D. Mukamel, J. Stat. Mech.: Theory Exp. P03015
(2011)] analyzed a model system comprising globally coupled classical
Heisenberg spins and evolving under classical spin dynamics. It was numerically
shown to relax to equilibrium over a time that scales superlinearly with .
Here, we present a detailed study of the Lenard-Balescu operator that accounts
at leading order for the finite- effects driving this relaxation. We
demonstrate that corrections at this order are identically zero, so that
relaxation occurs over a time longer than of order , in agreement with the
reported numerical results.Comment: 20 pages, 3 figures; v2: minor changes, published versio
Bifurcations and singularities for coupled oscillators with inertia and frustration
We prove that any non zero inertia, however small, is able to change the
nature of the synchronization transition in Kuramoto-like models, either from
continuous to discontinuous, or from discontinuous to continuous. This result
is obtained through an unstable manifold expansion in the spirit of J.D.
Crawford, which features singularities in the vicinity of the bifurcation. Far
from being unwanted artifacts, these singularities actually control the
qualitative behavior of the system. Our numerical tests fully support this
picture.Comment: 10 pages, 2 figure
On rigidity, orientability and cores of random graphs with sliders
Suppose that you add rigid bars between points in the plane, and suppose that
a constant fraction of the points moves freely in the whole plane; the
remaining fraction is constrained to move on fixed lines called sliders. When
does a giant rigid cluster emerge? Under a genericity condition, the answer
only depends on the graph formed by the points (vertices) and the bars (edges).
We find for the random graph the threshold value of
for the appearance of a linear-sized rigid component as a function of ,
generalizing results of Kasiviswanathan et al. We show that this appearance of
a giant component undergoes a continuous transition for and a
discontinuous transition for . In our proofs, we introduce a
generalized notion of orientability interpolating between 1- and
2-orientability, of cores interpolating between 2-core and 3-core, and of
extended cores interpolating between 2+1-core and 3+2-core; we find the precise
expressions for the respective thresholds and the sizes of the different cores
above the threshold. In particular, this proves a conjecture of Kasiviswanathan
et al. about the size of the 3+2-core. We also derive some structural
properties of rigidity with sliders (matroid and decomposition into components)
which can be of independent interest.Comment: 32 pages, 1 figur
Non equilibrium phase transition with gravitational-like interaction in a cloud of cold atoms
We propose to use a cloud of laser cooled atoms in a quasi two dimensional
trap to investigate a non equilibrium collapse phase transition in presence of
gravitational-like interaction. Using theoretical arguments and numerical
simulations, we show that, like in two dimensional gravity, a transition to a
collapsed state occurs below a critical temperature. In addition and as a
signature of the non equilibrium nature of the system, persistent particles
currents, dramatically increasing close to the phase transition, are observed.Comment: 5 pages, 4 figure
Microcanonical solution of lattice models with long range interactions
We present a general method to obtain the microcanonical solution of lattice
models with long range interactions. As an example, we apply it to the long
range Ising chain, focusing on the role of boundary conditions.Comment: 6 pages, proceedings of the NEXT 2001 conferenc
Stability of trajectories for N -particles dynamics with singular potential
We study the stability in finite times of the trajectories of interacting
particles. Our aim is to show that in average and uniformly in the number of
particles, two trajectories whose initial positions in phase space are close,
remain close enough at later times. For potential less singular than the
classical electrostatic kernel, we are able to prove such a result, for initial
positions/velocities distributed according to the Gibbs equilibrium of the
system
From Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck Systems to Incompressible Euler Equations: the case with finite charge
We study the asymptotic regime of strong electric fields that leads from the
Vlasov-Poisson system to the Incompressible Euler equations. We also deal with
the Vlasov-Poisson-Fokker-Planck system which induces dissipative effects. The
originality consists in considering a situation with a finite total charge
confined by a strong external field. In turn, the limiting equation is set in a
bounded domain, the shape of which is determined by the external confining
potential. The analysis extends to the situation where the limiting density is
non-homogeneous and where the Euler equation is replaced by the Lake Equation,
also called Anelastic Equation.Comment: 39 pages, 3 figure
Dynamical pattern formations in two dimensional fluid and Landau pole bifurcation
A phenomenological theory is proposed to analyze the asymptotic dynamics of
perturbed inviscid Kolmogorov shear flows in two dimensions. The phase diagram
provided by the theory is in qualitative agreement with numerical observations,
which include three phases depending on the aspect ratio of the domain and the
size of the perturbation: a steady shear flow, a stationary dipole, and four
traveling vortices. The theory is based on a precise study of the inviscid
damping of the linearized equation and on an analysis of nonlinear effects. In
particular, we show that the dominant Landau pole controlling the inviscid
damping undergoes a bifurcation, which has important consequences on the
asymptotic fate of the perturbation.Comment: 9 pages, 7 figure
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