13 research outputs found
Universal properties of branching random walks in confined geometries
Characterizing the occupation statistics of a radiation flow through confined
geometries is key to such technological issues as nuclear reactor design and
medical diagnosis. This amounts to assessing the distribution of the travelled
length and the number of collisions performed by the underlying
stochastic transport process, for which remarkably simple Cauchy-like formulas
were established in the case of branching Pearson random walks with
exponentially distributed jumps. In this Letter, we show that such formulas
strikingly carry over to the much broader class of branching processes with
arbitrary jumps, provided that scattering is isotropic and the average jump
size is finite.Comment: 5 pages, 3 figure
The critical catastrophe revisited
The neutron population in a prototype model of nuclear reactor can be
described in terms of a collection of particles confined in a box and
undergoing three key random mechanisms: diffusion, reproduction due to
fissions, and death due to absorption events. When the reactor is operated at
the critical point, and fissions are exactly compensated by absorptions, the
whole neutron population might in principle go to extinction because of the
wild fluctuations induced by births and deaths. This phenomenon, which has been
named critical catastrophe, is nonetheless never observed in practice: feedback
mechanisms acting on the total population, such as human intervention, have a
stabilizing effect. In this work, we revisit the critical catastrophe by
investigating the spatial behaviour of the fluctuations in a confined geometry.
When the system is free to evolve, the neutrons may display a wild patchiness
(clustering). On the contrary, imposing a population control on the total
population acts also against the local fluctuations, and may thus inhibit the
spatial clustering. The effectiveness of population control in quenching
spatial fluctuations will be shown to depend on the competition between the
mixing time of the neutrons (i.e., the average time taken for a particle to
explore the finite viable space) and the extinction time.Comment: 16 pages, 6 figure
The Stochastic complexity of spin models: Are pairwise models really simple?
Models can be simple for different reasons: because they yield a simple and
computationally efficient interpretation of a generic dataset (e.g. in terms of
pairwise dependences) - as in statistical learning - or because they capture
the essential ingredients of a specific phenomenon - as e.g. in physics -
leading to non-trivial falsifiable predictions. In information theory and
Bayesian inference, the simplicity of a model is precisely quantified in the
stochastic complexity, which measures the number of bits needed to encode its
parameters. In order to understand how simple models look like, we study the
stochastic complexity of spin models with interactions of arbitrary order. We
highlight the existence of invariances with respect to bijections within the
space of operators, which allow us to partition the space of all models into
equivalence classes, in which models share the same complexity. We thus found
that the complexity (or simplicity) of a model is not determined by the order
of the interactions, but rather by their mutual arrangements. Models where
statistical dependencies are localized on non-overlapping groups of few
variables (and that afford predictions on independencies that are easy to
falsify) are simple. On the contrary, fully connected pairwise models, which
are often used in statistical learning, appear to be highly complex, because of
their extended set of interactions
Cauchy's formulas for random walks in bounded domains
Cauchy's formula was originally established for random straight paths
crossing a body and basically relates the average
chord length through to the ratio between the volume and the surface of the
body itself. The original statement was later extended in the context of
transport theory so as to cover the stochastic paths of Pearson random walks
with exponentially distributed flight lengths traversing a bounded domain. Some
heuristic arguments suggest that Cauchy's formula may also hold true for
Pearson random walks with arbitrarily distributed flight lengths. For such a
broad class of stochastic processes, we rigorously derive a generalized
Cauchy's formula for the average length travelled by the walkers in the body,
and show that this quantity depends indeed only on the ratio between the volume
and the surface, provided that some constraints are imposed on the entrance
step of the walker in . Similar results are obtained also for the average
number of collisions performed by the walker in , and an extension to
absorbing media is discussed.Comment: 12 pages, 6 figure
Efficient simulations of epidemic models with tensor networks: application to the one-dimensional SIS model
The contact process is an emblematic model of a non-equilibrium system,
containing a phase transition between inactive and active dynamical regimes. In
the epidemiological context, the model is known as the
susceptible-infected-susceptible (SIS) model, and widely used to describe
contagious spreading. In this work, we demonstrate how accurate and efficient
representations of the full probability distribution over all configurations of
the contact process on a one-dimensional chain can be obtained by means of
Matrix Product States (MPS). We modify and adapt MPS methods from many-body
quantum systems to study the classical distributions of the driven contact
process at late times. We give accurate and efficient results for the
distribution of large gaps, and illustrate the advantage of our methods over
Monte Carlo simulations. Furthermore, we study the large deviation statistics
of the dynamical activity, defined as the total number of configuration changes
along a trajectory, and investigate quantum-inspired entropic measures, based
on the second R\'enyi entropy.Comment: 17 pages, 11 figures. Submitted for publication to PR
Asymmetric Lévy flights in the presence of absorbing boundaries
16 pages, 12 figures. Ref. addedInternational audienceWe consider a one dimensional asymmetric random walk whose jumps are identical, independent and drawn from a distribution \phi(\eta) displaying asymmetric power law tails (i.e. \phi(\eta) \sim c/\eta^{\alpha +1} for large positive jumps and \phi(\eta) \sim c/(\gamma |\eta|^{\alpha +1}) for large negative jumps, with 0 < \alpha < 2). In absence of boundaries and after a large number of steps n, the probability density function (PDF) of the walker position, x_n, converges to an asymmetric Lévy stable law of stability index \alpha and skewness parameter \beta=(\gamma-1)/(\gamma+1). In particular the right tail of this PDF decays as c n/x_n^{1+\alpha}. Much less is known when the walker is confined, or partially confined, in a region of the space. In this paper we first study the case of a walker constrained to move on the positive semi-axis and absorbed once it changes sign. In this case, the persistence exponent \theta_+, which characterizes the algebraic large time decay of the survival probability, can be computed exactly and we show that the tail of the PDF of the walker position decays as c \, n/[(1-\theta_+) \, x_n^{1+\alpha}]. This last result can be generalized in higher dimensions such as a planar Lévy walker confined in a wedge with absorbing walls. Our results are corroborated by precise numerical simulations
Fractality in nonequilibrium steady states of quasiperiodic systems
We investigate the nonequilibrium response of quasiperiodic systems to boundary driving. In particular, we focus on the Aubry-André-Harper model at its metal-insulator transition and the diagonal Fibonacci model. We find that opening the system at the boundaries provides a viable experimental technique to probe its underlying fractality, which is reflected in the fractal spatial dependence of simple observables (such as magnetization) in the nonequilibrium steady state. We also find that the dynamics in the nonequilibrium steady state depends on the length of the chain chosen: generic length chains harbour qualitatively slower transport (different scaling exponent) than Fibonacci length chains, which is in turn slower than in the closed system. We conjecture that such fractal nonequilibrium steady states should arise in generic driven critical systems that have fractal properties
Generalisation of opacity formulas for neutron transport
In the context of reactor physics, one is often called to relate the physical properties of a medium to the statistics of the random trajectories of the neutrons flowing through it. For instance, Cauchy’s formula establishes a link between the average length of the neutron paths (which is proportional to the medium opacity) and the volume-to-surface ratio of the traversed medium. In this work, we consider some extensions of such results for neutrons undergoing scattering, absorption and branching. A validation of the proposed formulas via Monte Carlo simulations is discussed