30 research outputs found
First-passage time of a Brownian motion: two unexpected journeys
The distribution of the first-passage time for a Brownian particle with
drift subject to hitting an absorber at a level is well-known and
given by its density , which is normalized only if . In this article,
we show that there are two other families of diffusion processes (the first
with one parameter and the second with two parameters) having the same first
passage-time distribution when . In both cases we establish the
propagators and study in detail these new processes. An immediate consequence
is that the distribution of the first-passage time does not allow us to know if
the process comes from a drifted Brownian motion or from one of these new
processes.Comment: 16 pages, 2 figures; improved mathematical notatio
Counting statistics: a Feynman-Kac perspective
By building upon a Feynman-Kac formalism, we assess the distribution of the
number of hits in a given region for a broad class of discrete-time random
walks with scattering and absorption. We derive the evolution equation for the
generating function of the number of hits, and complete our analysis by
examining the moments of the distribution, and their relation to the walker
equilibrium density. Some significant applications are discussed in detail: in
particular, we revisit the gambler's ruin problem and generalize to random
walks with absorption the arcsine law for the number of hits on the half-line.Comment: 10 pages, 6 figure
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
Probability density function for random photon steps in a binary (isotropic-Poisson) statistical mixture
Monte Carlo (MC) simulations allowing to describe photons propagation in
statistical mixtures represent an interest that goes way beyond the domain of
optics, and can cover, e.g., nuclear reactor physics, image analysis or life
science just to name a few. MC simulations are considered a ``gold standard''
because they give exact solutions (in the statistical sense), however, in the
case of statistical mixtures they are enormously time consuming and their
implementation is often extremely complex. For this reason, the aim of the
present contribution is to propose a new approach that should allow us in the
future to simplify the MC approach. This is done through an explanatory
example, i.e.; by deriving the `exact' analytical expression for the
probability density function of photons' random steps (single step function,
SSF) propagating in a medium represented as a binary (isotropic-Poisson)
statistical mixture. The use of the SSF reduces the problem to an `equivalent'
homogeneous medium behaving exactly as the original binary statistical mixture.
This will reduce hundreds time-consuming MC simulations to only one equivalent
simple MC simulation. To the best of our knowledge the analytically `exact' SSF
for a binary (isotropic-Poisson) statistical mixture has never been derived
before
Properties of branching exponential flights in bounded domains
Branching random flights are key to describing the evolution of many physical
and biological systems, ranging from neutron multiplication to gene mutations.
When their paths evolve in bounded regions, we establish a relation between the
properties of trajectories starting on the boundary and those starting inside
the domain. Within this context, we show that the total length travelled by the
walker and the number of performed collisions in bounded volumes can be
assessed by resorting to the Feynman-Kac formalism. Other physical observables
related to the branching trajectories, such as the survival and escape
probability, are derived as well.Comment: 5 pages, 2 figure
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