65 research outputs found
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
Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot
We introduce a new set of generalized Fokker-Planck equations that conserve
energy and mass and increase a generalized entropy until a maximum entropy
state is reached. The concept of generalized entropies is rigorously justified
for continuous Hamiltonian systems undergoing violent relaxation. Tsallis
entropies are just a special case of this generalized thermodynamics.
Application of these results to stellar dynamics, vortex dynamics and Jupiter's
great red spot are proposed. Our prime result is a novel relaxation equation
that should offer an easily implementable parametrization of geophysical
turbulence. This relaxation equation depends on a single key parameter related
to the skewness of the fine-grained vorticity distribution. Usual
parametrizations (including a single turbulent viscosity) correspond to the
infinite temperature limit of our model. They forget a fundamental systematic
drift that acts against diffusion as in Brownian theory. Our generalized
Fokker-Planck equations may have applications in other fields of physics such
as chemotaxis for bacterial populations. We propose the idea of a
classification of generalized entropies in classes of equivalence and provide
an aesthetic connexion between topics (vortices, stars, bacteries,...) which
were previously disconnected.Comment: Submitted to Phys. Rev.
Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution
Using a Maximum Entropy Production Principle (MEPP), we derive a new type of
relaxation equations for two-dimensional turbulent flows in the case where a
prior vorticity distribution is prescribed instead of the Casimir constraints
[Ellis, Haven, Turkington, Nonlin., 15, 239 (2002)]. The particular case of a
Gaussian prior is specifically treated in connection to minimum enstrophy
states and Fofonoff flows. These relaxation equations are compared with other
relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776
(1992)] and Chavanis [Physica D, 237, 1998 (2008)]. They can provide a
small-scale parametrization of 2D turbulence or serve as numerical algorithms
to compute maximum entropy states with appropriate constraints. We perform
numerical simulations of these relaxation equations in order to illustrate
geometry induced phase transitions in geophysical flows.Comment: 21 pages, 9 figure
Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions
We address the thermodynamics (equilibrium density profiles, phase diagram,
instability analysis...) and the collapse of a self-gravitating gas of Brownian
particles in D dimensions, in both canonical and microcanonical ensembles. In
the canonical ensemble, we derive the analytic form of the density scaling
profile which decays as f(x)=x^{-\alpha}, with alpha=2. In the microcanonical
ensemble, we show that f decays as f(x)=x^{-\alpha_{max}}, where \alpha_{max}
is a non-trivial exponent. We derive exact expansions for alpha_{max} and f in
the limit of large D. Finally, we solve the problem in D=2, which displays
rather rich and peculiar features
Kinetic theory of point vortices: diffusion coefficient and systematic drift
We develop a kinetic theory for point vortices in two-dimensional
hydrodynamics. Using standard projection operator technics, we derive a
Fokker-Planck equation describing the relaxation of a ``test'' vortex in a bath
of ``field'' vortices at statistical equilibrium. The relaxation is due to the
combined effect of a diffusion and a drift. The drift is shown to be
responsible for the organization of point vortices at negative temperatures. A
description that goes beyond the thermal bath approximation is attempted. A new
kinetic equation is obtained which respects all conservation laws of the point
vortex system and satisfies a H-theorem. Close to equilibrium this equation
reduces to the ordinary Fokker-Planck equation.Comment: 50 pages. To appear in Phys. Rev.
On the effective velocity created by a point vortex in two-dimensional hydrodynamics
We complete previous investigations on the statistics of velocity
fluctuations arising from a random distribution of point vortices in
two-dimensional hydrodynamics. We show that, on a statistical sense, the
velocity created by a point vortex is shielded by cooperative effects on a
distance , the inter-vortex separation. For , the ``effective'' velocity decays as instead of the ordinary
law recovered for . These results are similar to those
obtained by Agekyan [Sov. Astron. 5 (1962) 809] in his investigations on the
fluctuations of the gravitational field. They give further support to our
previous observation that the statistics of velocity fluctuations are
(marginally) dominated by the contribution of the nearest neighbor.Comment: Submitted to Phys. Rev.
Statistical mechanics of Fofonoff flows in an oceanic basin
We study the minimization of potential enstrophy at fixed circulation and
energy in an oceanic basin with arbitrary topography. For illustration, we
consider a rectangular basin and a linear topography h=by which represents
either a real bottom topography or the beta-effect appropriate to oceanic
situations. Our minimum enstrophy principle is motivated by different arguments
of statistical mechanics reviewed in the article. It leads to steady states of
the quasigeostrophic (QG) equations characterized by a linear relationship
between potential vorticity q and stream function psi. For low values of the
energy, we recover Fofonoff flows [J. Mar. Res. 13, 254 (1954)] that display a
strong westward jet. For large values of the energy, we obtain geometry induced
phase transitions between monopoles and dipoles similar to those found by
Chavanis and Sommeria [J. Fluid Mech. 314, 267 (1996)] in the absence of
topography. In the presence of topography, we recover and confirm the results
obtained by Venaille and Bouchet [Phys. Rev. Lett. 102, 104501 (2009)] using a
different formalism. In addition, we introduce relaxation equations towards
minimum potential enstrophy states and perform numerical simulations to
illustrate the phase transitions in a rectangular oceanic basin with linear
topography (or beta-effect).Comment: 26 pages, 28 figure
Collapses and explosions in self-gravitating systems
Collapse and reverse to collapse explosion transition in self-gravitating
systems are studied by molecular dynamics simulations. A microcanonical
ensemble of point particles confined to a spherical box is considered; the
particles interact via an attractive soft Coulomb potential. It is observed
that the collapse in the particle system indeed takes place when the energy of
the uniform state is put near or below the metastability-instability threshold
(collapse energy), predicted by the mean-field theory. Similarly, the explosion
in the particle system occurs when the energy of the core-halo state is
increased above the explosion energy, where according to the mean field
predictions the core-halo state becomes unstable. For a system consisting of
125 -- 500 particles, the collapse takes about single particle crossing
times to complete, while a typical explosion is by an order of magnitude
faster. A finite lifetime of metastable states is observed. It is also found
that the mean-field description of the uniform and the core-halo states is
exact within the statistical uncertainty of the molecular dynamics data.Comment: 9 pages, 14 figure
Statistics of the gravitational force in various dimensions of space: from Gaussian to Levy laws
We discuss the distribution of the gravitational force created by a
Poissonian distribution of field sources (stars, galaxies,...) in different
dimensions of space d. In d=3, it is given by a Levy law called the Holtsmark
distribution. It presents an algebraic tail for large fluctuations due to the
contribution of the nearest neighbor. In d=2, it is given by a marginal
Gaussian distribution intermediate between Gaussian and Levy laws. In d=1, it
is exactly given by the Bernouilli distribution (for any particle number N)
which becomes Gaussian for N>>1. Therefore, the dimension d=2 is critical
regarding the statistics of the gravitational force. We generalize these
results for inhomogeneous systems with arbitrary power-law density profile and
arbitrary power-law force in a d-dimensional universe
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
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