43 research outputs found
Dynamical stability criterion for inhomogeneous quasi-stationary states in long-range systems
We derive a necessary and sufficient condition of linear dynamical stability
for inhomogeneous Vlasov stationary states of the Hamiltonian Mean Field (HMF)
model. The condition is expressed by an explicit disequality that has to be
satisfied by the stationary state, and it generalizes the known disequality for
homogeneous stationary states. In addition, we derive analogous disequalities
that express necessary and sufficient conditions of formal stability for the
stationary states. Their usefulness, from the point of view of linear dynamical
stability, is that they are simpler, although they provide only sufficient
criteria of linear stability. We show that for homogeneous stationary states
the relations become equal, and therefore linear dynamical stability and formal
stability become equivalent.Comment: Submitted to Journal of Statistical Mechanics: Theory and Experimen
Relaxation to thermal equilibrium in the self-gravitating sheet model
We revisit the issue of relaxation to thermal equilibrium in the so-called
"sheet model", i.e., particles in one dimension interacting by attractive
forces independent of their separation. We show that this relaxation may be
very clearly detected and characterized by following the evolution of order
parameters defined by appropriately normalized moments of the phase space
distribution which probe its entanglement in space and velocity coordinates.
For a class of quasi-stationary states which result from the violent relaxation
of rectangular waterbag initial conditions, characterized by their virial ratio
R_0, we show that relaxation occurs on a time scale which (i) scales
approximately linearly in the particle number N, and (ii) shows also a strong
dependence on R_0, with quasi-stationary states from colder initial conditions
relaxing much more rapidly. The temporal evolution of the order parameter may
be well described by a stretched exponential function. We study finally the
correlation of the relaxation times with the amplitude of fluctuations in the
relaxing quasi-stationary states, as well as the relation between temporal and
ensemble averages.Comment: 37 pages, 24 figures; some additional discussion of previous
literature and other minor modifications, final published versio
Self-gravitating Brownian particles in two dimensions: the case of N=2 particles
We study the motion of N=2 overdamped Brownian particles in gravitational
interaction in a space of dimension d=2. This is equivalent to the simplified
motion of two biological entities interacting via chemotaxis when time delay
and degradation of the chemical are ignored. This problem also bears some
similarities with the stochastic motion of two point vortices in viscous
hydrodynamics [Agullo & Verga, Phys. Rev. E, 63, 056304 (2001)]. We
analytically obtain the density probability of finding the particles at a
distance r from each other at time t. We also determine the probability that
the particles have coalesced and formed a Dirac peak at time t (i.e. the
probability that the reduced particle has reached r=0 at time t). Finally, we
investigate the variance of the distribution and discuss the proper form
of the virial theorem for this system. The reduced particle has a normal
diffusion behaviour for small times with a gravity-modified diffusion
coefficient =r_0^2+(4k_B/\xi\mu)(T-T_*)t, where k_BT_{*}=Gm_1m_2/2 is a
critical temperature, and an anomalous diffusion for large times
~t^(1-T_*/T). As a by-product, our solution also describes the growth of
the Dirac peak (condensate) that forms in the post-collapse regime of the
Smoluchowski-Poisson system (or Keller-Segel model) for T<T_c=GMm/(4k_B). We
find that the saturation of the mass of the condensate to the total mass is
algebraic in an infinite domain and exponential in a bounded domain.Comment: Revised version (20/5/2010) accepted for publication in EPJ
Phase transitions of quasistationary states in the Hamiltonian Mean Field model
The out-of-equilibrium dynamics of the Hamiltonian Mean Field (HMF) model is
studied in presence of an externally imposed magnetic field h. Lynden-Bell's
theory of violent relaxation is revisited and shown to adequately capture the
system dynamics, as revealed by direct Vlasov based numerical simulations in
the limit of vanishing field. This includes the existence of an
out-of-equilibrium phase transition separating magnetized and non magnetized
phases. We also monitor the fluctuations in time of the magnetization, which
allows us to elaborate on the choice of the correct order parameter when
challenging the performance of Lynden-Bell's theory. The presence of the field
h removes the phase transition, as it happens at equilibrium. Moreover, regions
with negative susceptibility are numerically found to occur, in agreement with
the predictions of the theory.Comment: 6 pages, 7 figure
A dynamical classification of the range of pair interactions
We formalize a classification of pair interactions based on the convergence
properties of the {\it forces} acting on particles as a function of system
size. We do so by considering the behavior of the probability distribution
function (PDF) P(F) of the force field F in a particle distribution in the
limit that the size of the system is taken to infinity at constant particle
density, i.e., in the "usual" thermodynamic limit. For a pair interaction
potential V(r) with V(r) \rightarrow \infty) \sim 1/r^a defining a {\it
bounded} pair force, we show that P(F) converges continuously to a well-defined
and rapidly decreasing PDF if and only if the {\it pair force} is absolutely
integrable, i.e., for a > d-1, where d is the spatial dimension. We refer to
this case as {\it dynamically short-range}, because the dominant contribution
to the force on a typical particle in this limit arises from particles in a
finite neighborhood around it. For the {\it dynamically long-range} case, i.e.,
a \leq d-1, on the other hand, the dominant contribution to the force comes
from the mean field due to the bulk, which becomes undefined in this limit. We
discuss also how, for a \leq d-1 (and notably, for the case of gravity, a=d-2)
P(F) may, in some cases, be defined in a weaker sense. This involves a
regularization of the force summation which is generalization of the procedure
employed to define gravitational forces in an infinite static homogeneous
universe. We explain that the relevant classification in this context is,
however, that which divides pair forces with a > d-2 (or a < d-2), for which
the PDF of the {\it difference in forces} is defined (or not defined) in the
infinite system limit, without any regularization. In the former case dynamics
can, as for the (marginal) case of gravity, be defined consistently in an
infinite uniform system.Comment: 12 pages, 1 figure; significantly shortened and focussed, additional
references, version to appear in J. Stat. Phy
Invariant measures of the 2D Euler and Vlasov equations
We discuss invariant measures of partial differential equations such as the
2D Euler or Vlasov equations. For the 2D Euler equations, starting from the
Liouville theorem, valid for N-dimensional approximations of the dynamics, we
define the microcanonical measure as a limit measure where N goes to infinity.
When only the energy and enstrophy invariants are taken into account, we give
an explicit computation to prove the following result: the microcanonical
measure is actually a Young measure corresponding to the maximization of a
mean-field entropy. We explain why this result remains true for more general
microcanonical measures, when all the dynamical invariants are taken into
account. We give an explicit proof that these microcanonical measures are
invariant measures for the dynamics of the 2D Euler equations. We describe a
more general set of invariant measures, and discuss briefly their stability and
their consequence for the ergodicity of the 2D Euler equations. The extension
of these results to the Vlasov equations is also discussed, together with a
proof of the uniqueness of statistical equilibria, for Vlasov equations with
repulsive convex potentials. Even if we consider, in this paper, invariant
measures only for Hamiltonian equations, with no fluxes of conserved
quantities, we think this work is an important step towards the description of
non-equilibrium invariant measures with fluxes.Comment: 40 page
Collective dynamics of colloids at fluid interfaces
The evolution of an initially prepared distribution of micron sized colloidal
particles, trapped at a fluid interface and under the action of their mutual
capillary attraction, is analyzed by using Brownian dynamics simulations. At a
separation \lambda\ given by the capillary length of typically 1 mm, the
distance dependence of this attraction exhibits a crossover from a logarithmic
decay, formally analogous to two-dimensional gravity, to an exponential decay.
We discuss in detail the adaption of a particle-mesh algorithm, as used in
cosmological simulations to study structure formation due to gravitational
collapse, to the present colloidal problem. These simulations confirm the
predictions, as far as available, of a mean-field theory developed previously
for this problem. The evolution is monitored by quantitative characteristics
which are particularly sensitive to the formation of highly inhomogeneous
structures. Upon increasing \lambda\ the dynamics show a smooth transition from
the spinodal decomposition expected for a simple fluid with short-ranged
attraction to the self-gravitational collapse scenario.Comment: 13 pages, 12 figures, revised, matches version accepted for
publication in the European Physical Journal
Algebraic damping in the one-dimensional Vlasov equation
We investigate the asymptotic behavior of a perturbation around a spatially
non homogeneous stable stationary state of a one-dimensional Vlasov equation.
Under general hypotheses, after transient exponential Landau damping, a
perturbation evolving according to the linearized Vlasov equation decays
algebraically with the exponent -2 and a well defined frequency. The
theoretical results are successfully tested against numerical -body
simulations, corresponding to the full Vlasov dynamics in the large limit,
in the case of the Hamiltonian mean-field model. For this purpose, we use a
weighted particles code, which allows us to reduce finite size fluctuations and
to observe the asymptotic decay in the -body simulations.Comment: 26 pages, 8 figures; text slightly modified, references added, typos
correcte
Dynamical stability of infinite homogeneous self-gravitating systems: application of the Nyquist method
We complete classical investigations concerning the dynamical stability of an
infinite homogeneous gaseous medium described by the Euler-Poisson system or an
infinite homogeneous stellar system described by the Vlasov-Poisson system
(Jeans problem). To determine the stability of an infinite homogeneous stellar
system with respect to a perturbation of wavenumber k, we apply the Nyquist
method. We first consider the case of single-humped distributions and show
that, for infinite homogeneous systems, the onset of instability is the same in
a stellar system and in the corresponding barotropic gas, contrary to the case
of inhomogeneous systems. We show that this result is true for any symmetric
single-humped velocity distribution, not only for the Maxwellian. If we
specialize on isothermal and polytropic distributions, analytical expressions
for the growth rate, damping rate and pulsation period of the perturbation can
be given. Then, we consider the Vlasov stability of symmetric and asymmetric
double-humped distributions (two-stream stellar systems) and determine the
stability diagrams depending on the degree of asymmetry. We compare these
results with the Euler stability of two self-gravitating gaseous streams.
Finally, we determine the corresponding stability diagrams in the case of
plasmas and compare the results with self-gravitating systems