804 research outputs found
Diffusion of particles with short-range interactions
A system of interacting Brownian particles subject to short-range repulsive
potentials is considered. A continuum description in the form of a nonlinear
diffusion equation is derived systematically in the dilute limit using the
method of matched asymptotic expansions. Numerical simulations are performed to
compare the results of the model with those of the commonly used mean-field and
Kirkwood-superposition approximations, as well as with Monte Carlo simulation
of the stochastic particle system, for various interaction potentials. Our
approach works best for very repulsive short-range potentials, while the
mean-field approximation is suitable for long-range interactions. The Kirkwood
superposition approximation provides an accurate description for both short-
and long-range potentials, but is considerably more computationally intensive
From Brownian Dynamics to Markov Chain: an Ion Channel Example
A discrete rate theory for general multi-ion channels is presented, in which
the continuous dynamics of ion diffusion is reduced to transitions between
Markovian discrete states. In an open channel, the ion permeation process
involves three types of events: an ion entering the channel, an ion escaping
from the channel, or an ion hopping between different energy minima in the
channel. The continuous dynamics leads to a hierarchy of Fokker-Planck
equations, indexed by channel occupancy. From these the mean escape times and
splitting probabilities (denoting from which side an ion has escaped) can be
calculated. By equating these with the corresponding expressions from the
Markov model the Markovian transition rates can be determined. The theory is
illustrated with a two-ion one-well channel. The stationary probability of
states is compared with that from both Brownian dynamics simulation and the
hierarchical Fokker-Planck equations. The conductivity of the channel is also
studied, and the optimal geometry maximizing ion flux is computed.Comment: submitted to SIAM Journal on Applied Mathematic
Reactive Boundary Conditions as Limits of Interaction Potentials for Brownian and Langevin Dynamics
A popular approach to modeling bimolecular reactions between diffusing
molecules is through the use of reactive boundary conditions. One common model
is the Smoluchowski partial absorption condition, which uses a Robin boundary
condition in the separation coordinate between two possible reactants. This
boundary condition can be interpreted as an idealization of a reactive
interaction potential model, in which a potential barrier must be surmounted
before reactions can occur. In this work we show how the reactive boundary
condition arises as the limit of an interaction potential encoding a steep
barrier within a shrinking region in the particle separation, where molecules
react instantly upon reaching the peak of the barrier. The limiting boundary
condition is derived by the method of matched asymptotic expansions, and shown
to depend critically on the relative rate of increase of the barrier height as
the width of the potential is decreased. Limiting boundary conditions for the
same interaction potential in both the overdamped Fokker-Planck equation
(Brownian Dynamics), and the Kramers equation (Langevin Dynamics) are
investigated. It is shown that different scalings are required in the two
models to recover reactive boundary conditions that are consistent in the high
friction limit where the Kramers equation solution converges to the solution of
the Fokker-Planck equation.Comment: 23 pages, 2 figure
Multiscale reaction-diffusion algorithms: PDE-assisted Brownian dynamics
Two algorithms that combine Brownian dynamics (BD) simulations with
mean-field partial differential equations (PDEs) are presented. This
PDE-assisted Brownian dynamics (PBD) methodology provides exact particle
tracking data in parts of the domain, whilst making use of a mean-field
reaction-diffusion PDE description elsewhere. The first PBD algorithm couples
BD simulations with PDEs by randomly creating new particles close to the
interface which partitions the domain and by reincorporating particles into the
continuum PDE-description when they cross the interface. The second PBD
algorithm introduces an overlap region, where both descriptions exist in
parallel. It is shown that to accurately compute variances using the PBD
simulation requires the overlap region. Advantages of both PBD approaches are
discussed and illustrative numerical examples are presented.Comment: submitted to SIAM Journal on Applied Mathematic
- …