3,387 research outputs found
Initial conditions, Discreteness and non-linear structure formation in cosmology
In this lecture we address three different but related aspects of the initial
continuous fluctuation field in standard cosmological models. Firstly we
discuss the properties of the so-called Harrison-Zeldovich like spectra. This
power spectrum is a fundamental feature of all current standard cosmological
models. In a simple classification of all stationary stochastic processes into
three categories, we highlight with the name ``super-homogeneous'' the
properties of the class to which models like this, with , belong. In
statistical physics language they are well described as glass-like. Secondly,
the initial continuous density field with such small amplitude correlated
Gaussian fluctuations must be discretised in order to set up the initial
particle distribution used in gravitational N-body simulations. We discuss the
main issues related to the effects of discretisation, particularly concerning
the effect of particle induced fluctuations on the statistical properties of
the initial conditions and on the dynamical evolution of gravitational
clustering.Comment: 28 pages, 1 figure, to appear in Proceedings of 9th Course on
Astrofundamental Physics, International School D. Chalonge, Kluwer, eds N.G.
Sanchez and Y.M. Pariiski, uses crckapb.st pages, 3 figure, ro appear in
Proceedings of 9th Course on Astrofundamental Physics, International School
D. Chalonge, Kluwer, Eds. N.G. Sanchez and Y.M. Pariiski, uses crckapb.st
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
Diffusion, super-diffusion and coalescence from single step
From the exact single step evolution equation of the two-point correlation
function of a particle distribution subjected to a stochastic displacement
field \bu(\bx), we derive different dynamical regimes when \bu(\bx) is
iterated to build a velocity field. First we show that spatially uncorrelated
fields \bu(\bx) lead to both standard and anomalous diffusion equation. When
the field \bu(\bx) is spatially correlated each particle performs a simple
free Brownian motion, but the trajectories of different particles result to be
mutually correlated. The two-point statistical properties of the field
\bu(\bx) induce two-point spatial correlations in the particle distribution
satisfying a simple but non-trivial diffusion-like equation. These
displacement-displacement correlations lead the system to three possible
regimes: coalescence, simple clustering and a combination of the two. The
existence of these different regimes, in the one-dimensional system, is shown
through computer simulations and a simple theoretical argument.Comment: RevTeX (iopstyle) 19 pages, 5 eps-figure
Gravitational Dynamics of an Infinite Shuffled Lattice: Particle Coarse-grainings, Non-linear Clustering and the Continuum Limit
We study the evolution under their self-gravity of infinite ``shuffled
lattice'' particle distributions, focussing specifically on the comparison of
this evolution with that of ``daughter'' particle distributions, defined by a
simple coarse-graining procedure. We consider both the case that such
coarse-grainings are performed (i) on the initial conditions, and (ii) at a
finite time with a specific additional prescription. In numerical simulations
we observe that, to a first approximation, these coarse-grainings represent
well the evolution of the two-point correlation properties over a significant
range of scales. We note, in particular, that the form of the two-point
correlation function in the original system, when it is evolving in the
asymptotic ``self-similar'' regime, may be reproduced well in a daughter
coarse-grained system in which the dynamics are still dominated by two-body
(nearest neighbor) interactions. Using analytical results on the early time
evolution of these systems, however, we show that small observed differences
between the evolved system and its coarse-grainings at the initial time will in
fact diverge as the ratio of the coarse-graining scale to the original
inter-particle distance increases. The second coarse-graining studied,
performed at a finite time in a specified manner, circumvents this problem. It
also makes more physically transparent why gravitational dynamics from these
initial conditions tends toward a ``self-similar'' evolution. We finally
discuss the precise definition of a limit in which a continuum (specifically
Vlasov-like) description of the observed linear and non-linear evolution should
be applicable.Comment: 21 pages, 8 eps figures, 2 jpeg figures (available in high resolution
at http://pil.phys.uniroma1.it/~sylos/PRD_dec_2006/
Stochastic interacting particle systems out of equilibrium
This paper provides an introduction to some stochastic models of lattice
gases out of equilibrium and a discussion of results of various kinds obtained
in recent years. Although these models are different in their microscopic
features, a unified picture is emerging at the macroscopic level, applicable,
in our view, to real phenomena where diffusion is the dominating physical
mechanism. We rely mainly on an approach developed by the authors based on the
study of dynamical large fluctuations in stationary states of open systems. The
outcome of this approach is a theory connecting the non equilibrium
thermodynamics to the transport coefficients via a variational principle. This
leads ultimately to a functional derivative equation of Hamilton-Jacobi type
for the non equilibrium free energy in which local thermodynamic variables are
the independent arguments. In the first part of the paper we give a detailed
introduction to the microscopic dynamics considered, while the second part,
devoted to the macroscopic properties, illustrates many consequences of the
Hamilton-Jacobi equation. In both parts several novelties are included.Comment: 36 page
Gravitational dynamics of an infinite shuffled lattice: early time evolution and universality of non-linear correlations
In two recent articles a detailed study has been presented of the out of
equilibrium dynamics of an infinite system of self-gravitating points initially
located on a randomly perturbed lattice. In this article we extend the
treatment of the early time phase during which strong non-linear correlations
first develop, prior to the onset of ``self-similar'' scaling in the two point
correlation function. We establish more directly, using appropriate
modifications of the numerical integration, that the development of these
correlations can be well described by an approximation of the evolution in two
phases: a first perturbative phase in which particles' displacements are small
compared to the lattice spacing, and a subsequent phase in which particles
interact only with their nearest neighbor. For the range of initial amplitudes
considered we show that the first phase can be well approximated as a
transformation of the perturbed lattice configuration into a Poisson
distribution at the relevant scales. This appears to explain the
``universality'' of the spatial dependence of the asymptotic non-linear
clustering observed from both shuffled lattice and Poisson initial conditions.Comment: 11 pages, 11 figures, shortened introductory sections and other minor
modifications, version to appear in Phys. Rev.
Galaxy distribution and extreme value statistics
We consider the conditional galaxy density around each galaxy, and study its
fluctuations in the newest samples of the Sloan Digital Sky Survey Data Release
7. Over a large range of scales, both the average conditional density and its
variance show a nontrivial scaling behavior, which resembles to criticality.
The density depends, for 10 < r < 80 Mpc/h, only weakly (logarithmically) on
the system size. Correspondingly, we find that the density fluctuations follow
the Gumbel distribution of extreme value statistics. This distribution is
clearly distinguishable from a Gaussian distribution, which would arise for a
homogeneous spatial galaxy configuration. We also point out similarities
between the galaxy distribution and critical systems of statistical physics
Gravitational force in an infinite one-dimensional Poisson distribution
We consider the statistical properties of the gravitational field F in an
infinite one-dimensional homogeneous Poisson distribution of particles, using
an exponential cut-off of the pair interaction to control and study the
divergences which arise. Deriving an exact analytic expression for the
probability density function (PDF) P(F), we show that it is badly defined in
the limit in which the well known Holtzmark distribution is obtained in the
analogous three-dimensional case. A well defined P(F) may, however, be obtained
in the infinite range limit by an appropriate renormalization of the coupling
strength, giving a Gaussian form. Calculating the spatial correlation
properties we show that this latter procedure has a trivial physical meaning.
Finally we calculate the PDF and correlation properties of differences of
forces (at separate spatial points), which are well defined without any
renormalization. We explain that the convergence of these quantities is in fact
sufficient to allow a physically meaningful infinite system limit to be defined
for the clustering dynamics from Poissonian initial conditions.Comment: 9 pages, minor changes, final version published in Phys. Rev.
Very large scale correlations in the galaxy distribution
We characterize galaxy correlations in the Sloan Digital Sky Survey by
measuring several moments of galaxy counts in spheres. We firstly find that the
average counts grows as a power-law function of the distance with an exponent
D= 2.1+- 0.05 for r in [0.5,20] Mpc/h and D = 2.8+-0.05 for r in [30,150]
Mpc/h. In order to estimate the systematic errors in these measurements we
consider the counts variance finding that it shows systematic finite size
effects which depend on the samples sizes. We clarify, by making specific
tests, that these are due to galaxy long-range correlations extending up to the
largest scales of the sample. The analysis of mock galaxy catalogs, generated
from cosmological N-body simulations of the standard LCDM model, shows that for
r<20 Mpc/h the counts exponent is D~2.0, weakly dependent on galaxy luminosity,
while D=3 at larger scales. In addition, contrary to the case of the observed
galaxy samples, no systematic finite size effects in the counts variance are
found at large scales, a result that agrees with the absence of large scale,
r~100 Mpc/h, correlations in the mock catalogs. We thus conclude that the
observed galaxy distribution is characterized by correlations, fluctuations and
hence structures, which are larger, both in amplitude and in spatial extension,
than those predicted by the standard model LCDM of galaxy formation.Comment: 6 pages, 7 figures to be published in Europhysics Letter
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