2,514 research outputs found
A stochastic approximation scheme and convergence theorem for particle interactions with perfectly reflecting boundaries
We prove the existence of a solution to an equation governing the number
density within a compact domain of a discrete particle system for a prescribed
class of particle interactions taking into account the effects of the diffusion
and drift of the set of particles. Each particle carries a number of internal
coordinates which may evolve continuously in time, determined by what we will
refer to as the internal drift, or discretely via the interaction kernels.
Perfectly reflecting boundary conditions are imposed on the system and all the
processes may be spatially and temporally inhomogeneous. We use a relative
compactness argument to construct a sequence of measures that converge weakly
to a solution of the governing equation. Since the proof of existence is a
constructive one, it provides a stochastic approximation scheme that can be
used for the numerical study of molecular dynamics.Comment: 43 page
Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries
Well posedness is established for a family of equations modelling particle
populations undergoing delocalised coagulation, advection, inflow and outflow
in a externally specified velocity field. Very general particle types are
allowed while the spatial domain is a bounded region of -dimensional space
for which every point lies on exactly one streamline associated with the
velocity field. The problem is formulated as a semi-linear ODE in the Banach
space of bounded measures on particle position and type space. A local
Lipschitz property is established in total variation norm for the propagators
(generalised semi-groups) associated with the problem and used to construct a
Picard iteration that establishes local existence and global uniqueness for any
initial condition. The unique weak solution is shown further to be a
differentiable or at least bounded variation strong solution under smoothness
assumptions on the parameters of the coagulation interaction. In the case of
one spatial dimension strong differentiability is established even for
coagulation parameters with a particular bounded variation structure in space.
This one dimensional extension establishes the convergence of the simulation
processes studied in [Patterson, Stoch. Anal. Appl. 31, 2013] to a unique and
differentiable limit
DNA-Protein Binding Rates: Bending Fluctuation and Hydrodynamic Coupling Effects
We investigate diffusion-limited reactions between a diffusing particle and a
target site on a semiflexible polymer, a key factor determining the kinetics of
DNA-protein binding and polymerization of cytoskeletal filaments. Our theory
focuses on two competing effects: polymer shape fluctuations, which speed up
association, and the hydrodynamic coupling between the diffusing particle and
the chain, which slows down association. Polymer bending fluctuations are
described using a mean field dynamical theory, while the hydrodynamic coupling
between polymer and particle is incorporated through a simple heuristic
approximation. Both of these we validate through comparison with Brownian
dynamics simulations. Neither of the effects has been fully considered before
in the biophysical context, and we show they are necessary to form accurate
estimates of reaction processes. The association rate depends on the stiffness
of the polymer and the particle size, exhibiting a maximum for intermediate
persistence length and a minimum for intermediate particle radius. In the
parameter range relevant to DNA-protein binding, the rate increase is up to
100% compared to the Smoluchowski result for simple center-of-mass motion. The
quantitative predictions made by the theory can be tested experimentally.Comment: 21 pages, 11 figures, 1 tabl
On the Inversion of High Energy Proton
Inversion of the K-fold stochastic autoconvolution integral equation is an
elementary nonlinear problem, yet there are no de facto methods to solve it
with finite statistics. To fix this problem, we introduce a novel inverse
algorithm based on a combination of minimization of relative entropy, the Fast
Fourier Transform and a recursive version of Efron's bootstrap. This gives us
power to obtain new perspectives on non-perturbative high energy QCD, such as
probing the ab initio principles underlying the approximately negative binomial
distributions of observed charged particle final state multiplicities, related
to multiparton interactions, the fluctuating structure and profile of proton
and diffraction. As a proof-of-concept, we apply the algorithm to ALICE
proton-proton charged particle multiplicity measurements done at different
center-of-mass energies and fiducial pseudorapidity intervals at the LHC,
available on HEPData. A strong double peak structure emerges from the
inversion, barely visible without it.Comment: 29 pages, 10 figures, v2: extended analysis (re-projection ratios,
2D
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