94 research outputs found
An encounter-based approach for restricted diffusion with a gradient drift
We develop an encounter-based approach for describing restricted diffusion
with a gradient drift towards a partially reactive boundary. For this purpose,
we introduce an extension of the Dirichlet-to-Neumann operator and use its
eigenbasis to derive a spectral decomposition for the full propagator, i.e.,
the joint probability density function for the particle position and its
boundary local time. This is the central quantity that determines various
characteristics of diffusion-influenced reactions such as conventional
propagators, survival probability, first-passage time distribution, boundary
local time distribution, and reaction rate. As an illustration, we investigate
the impact of a constant drift onto the boundary local time for restricted
diffusion on an interval. More generally, this approach accesses how external
forces may influence the statistics of encounters of a diffusing particle with
the reactive boundary
Encounter-based approach to target search problems: a review
In this review, we present the encounter-based approach to target search
problems, in which the diffusive dynamics is described by the joint probability
of the position of the particle and the number of its encounters with a given
target set. The knowledge of the statistics of encounters allows one to
implement various mechanisms of reactions on the target set, beyond
conventional reaction schemes. We formulate this approach for three relevant
settings: discrete random walks, Brownian motion with bulk reactions, and
reflected Brownian motion with surface reactions. In all cases, we discuss the
advantages of this approach, its recent applications and possible extensions
An encounter-based approach to the escape problem
We revise the encounter-based approach to imperfect diffusion-controlled
reactions, which employs the statistics of encounters between a diffusing
particle and the reactive region to implement surface reactions. We extend this
approach to deal with a more general setting, in which the reactive region is
surrounded by a reflecting boundary with an escape region. We derive a spectral
expansion for the full propagator and investigate the behavior and
probabilistic interpretations of the associated probability flux density. In
particular, we obtain the joint probability density of the escape time and the
number of encounters with the reactive region before escape, and the
probability density of the first-crossing time of a prescribed number of
encounters. We briefly discuss generalizations of the conventional
Poissonian-type surface reaction mechanism described by Robin boundary
condition and potential applications of this formalism in chemistry and
biophysics
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