5,351 research outputs found
A perturbative approach to the Bak-Sneppen Model
We study the Bak-Sneppen model in the probabilistic framework of the Run Time
Statistics (RTS). This model has attracted a large interest for its simplicity
being a prototype for the whole class of models showing Self-Organized
Criticality. The dynamics is characterized by a self-organization of almost all
the species fitnesses above a non-trivial threshold value, and by a lack of
spatial and temporal characteristic scales. This results in {\em avalanches} of
activity power law distributed. In this letter we use the RTS approach to
compute the value of , the value of the avalanche exponent and the
asymptotic distribution of minimal fitnesses.Comment: 4 pages, 3 figures, to be published on Physical Review Letter
1-d gravity in infinite point distributions
The dynamics of infinite, asymptotically uniform, distributions of
self-gravitating particles in one spatial dimension provides a simple toy model
for the analogous three dimensional problem. We focus here on a limitation of
such models as treated so far in the literature: the force, as it has been
specified, is well defined in infinite point distributions only if there is a
centre of symmetry (i.e. the definition requires explicitly the breaking of
statistical translational invariance). The problem arises because naive
background subtraction (due to expansion, or by "Jeans' swindle" for the static
case), applied as in three dimensions, leaves an unregulated contribution to
the force due to surface mass fluctuations. Following a discussion by
Kiessling, we show that the problem may be resolved by defining the force in
infinite point distributions as the limit of an exponentially screened pair
interaction. We show that this prescription gives a well defined (finite) force
acting on particles in a class of perturbed infinite lattices, which are the
point processes relevant to cosmological N-body simulations. For identical
particles the dynamics of the simplest toy model is equivalent to that of an
infinite set of points with inverted harmonic oscillator potentials which
bounce elastically when they collide. We discuss previous results in the
literature, and present new results for the specific case of this simplest
(static) model starting from "shuffled lattice" initial conditions. These show
qualitative properties (notably its "self-similarity") of the evolution very
similar to those in the analogous simulations in three dimensions, which in
turn resemble those in the expanding universe.Comment: 20 pages, 8 figures, small changes (section II shortened, added
discussion in section IV), matches final version to appear in PR
Generalized model of blockage in particulate flow limited by channel carrying capacity
We investigate stochastic models of particles entering a channel with a
random time distribution. When the number of particles present in the channel
exceeds a critical value , a blockage occurs and the particle flux is
definitively interrupted. By introducing an integral representation of the
particle survival probabilities, we obtain exact expressions for the survival
probability, the distribution of the number of particles that pass before
failure, the instantaneous flux of exiting particle and their time correlation.
We generalize previous results for to an arbitrary distribution of entry
times and obtain new, exact solutions for for a Poisson distribution and
partial results for .Comment: 13 pages, 9 figure
Two-point correlation properties of stochastic "cloud processes''
We study how the two-point density correlation properties of a point particle
distribution are modified when each particle is divided, by a stochastic
process, into an equal number of identical "daughter" particles. We consider
generically that there may be non-trivial correlations in the displacement
fields describing the positions of the different daughters of the same "mother"
particle, and then treat separately the cases in which there are, or are not,
correlations also between the displacements of daughters belonging to different
mothers. For both cases exact formulae are derived relating the structure
factor (power spectrum) of the daughter distribution to that of the mother.
These results can be considered as a generalization of the analogous equations
obtained in ref. [1] (cond-mat/0409594) for the case of stochastic displacement
fields applied to particle distributions. An application of the present results
is that they give explicit algorithms for generating, starting from regular
lattice arrays, stochastic particle distributions with an arbitrarily high
degree of large-scale uniformity.Comment: 14 pages, 3 figure
The Role of Bedrock Groundwater in Headwater Catchments: Processes, Patterns, Storage and Transit Time
Understanding the role of deep aquifer contributions to headwater rainfall-runoff processes, storages and transit times remains a major challenge in hydrology. Bedrock groundwater contributions to the stream channel can significantly augment streamflow, mediate water quality and control the age of water discharging from catchments. Yet, the hydroclimatic and bedrock characteristics that control these dynamics are not fully understood. Direct observation of bedrock groundwater dynamics, storages and surface water connections remain limited, challenging our ability to fully constrain new catchment scale models that are needed to aid future resource management decisions. I undertook a large field campaign at a well-studied research site in New Zealand. Bedrock groundwater dynamics were monitored for one year and combined with bedrock characterization, tritium-based age dating and hydrochemical analysis to constrain a new conceptual model of the headwater aquifer. Findings were used to develop a new index to identify the controls of bedrock permeability and landscape structure on the time scales of catchment storage-release processes. The three major findings of this research were firstly, that unfractured low-permeability bedrock underlying the research catchment limited to deep flowpaths. Minimal bedrock groundwater flux combined with large bedrock storage resulted in significantly older bedrock groundwater that contributed minimally to catchment discharge. Second, unfractured low-permeability bedrock was a primary control on bedrock groundwater recharge seasonality. Groundwater movement occurred as matrix flow, requiring long durations of high catchment-wetness for considerable recharge to occur, a condition that was only attained during cold-season months when evapotranspiration rates were low and catchment wetness was high. Third, permeability contrasts at the soil-bedrock interface and landscape structure were highly correlated with mean transit time for eight catchments in geologically diverse regions, suggesting that subsurface anisotropy is a major control on setting streamwater age. Overall, through the coupled analysis of the processes, patterns, storages and transit times, this research has advanced our understanding of the role of bedrock groundwater in headwaters. The findings presented here offer new insights into the function of deeper hydrologic layers and have implications for future models of headwater catchment function â models that need to better incorporate the influence of deep flowpaths and storages in groundwater-surface water and rainfall-runoff predictions
Statistical properties of fractures in damaged materials
We introduce a model for the dynamics of mud cracking in the limit of of
extremely thin layers. In this model the growth of fracture proceeds by
selecting the part of the material with the smallest (quenched) breaking
threshold. In addition, weakening affects the area of the sample neighbour to
the crack. Due to the simplicity of the model, it is possible to derive some
analytical results. In particular, we find that the total time to break down
the sample grows with the dimension L of the lattice as L^2 even though the
percolating cluster has a non trivial fractal dimension. Furthermore, we obtain
a formula for the mean weakening with time of the whole sample.Comment: 5 pages, 4 figures, to be published in Europhysics Letter
Invasion Percolation with Temperature and the Nature of SOC in Real Systems
We show that the introduction of thermal noise in Invasion Percolation (IP)
brings the system outside the critical point. This result suggests a possible
definition of SOC systems as ordinary critical systems where the critical point
correspond to set to 0 one of the parameters. We recover both IP and EDEN
model, for , and respectively. For small we find a
dynamical second order transition with correlation length diverging when .Comment: 4 pages, 2 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
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
- âŠ