14,786 research outputs found
Rapidly driven nanoparticles: Mean first-passage times and relaxation of the magnetic moment
We present an analytical method of calculating the mean first-passage times
(MFPTs) for the magnetic moment of a uniaxial nanoparticle which is driven by a
rapidly rotating, circularly polarized magnetic field and interacts with a heat
bath. The method is based on the solution of the equation for the MFPT derived
from the two-dimensional backward Fokker-Planck equation in the rotating frame.
We solve these equations in the high-frequency limit and perform precise,
numerical simulations which verify the analytical findings. The results are
used for the description of the rates of escape from the metastable domains
which in turn determine the magnetic relaxation dynamics. A main finding is
that the presence of a rotating field can cause a drastic decrease of the
relaxation time and a strong magnetization of the nanoparticle system. The
resulting stationary magnetization along the direction of the easy axis is
compared with the mean magnetization following from the stationary solution of
the Fokker-Planck equation.Comment: 24 pages, 4 figure
A short-time drift propagator approach to the Fokker-Planck equation
The Fokker-Planck equation is a partial differential equation that describes
the evolution of a probability distribution over time. It is used to model a
wide range of physical and biological phenomena, such as diffusion, chemical
reactions, and population dynamics. Solving the Fokker-Planck equation is a
difficult task, as it involves solving a system of coupled nonlinear partial
differential equations. In general, analytical solutions are not available and
numerical methods must be used. In this research, we propose a novel approach
to the solution of the Fokker-Planck equation in a short time interval. The
numerical solution to the equation can be obtained iteratively using a new
technique based on the short-time drift propagator. This new approach is
different from the traditional methods, as the state-dependent drift function
has been removed from the multivariate Gaussian integral component and is
instead presented as a state-shifted element. We evaluated our technique
employing a fundamental Wiener process with constant drift components in both
one- and two-dimensional space. The results of the numerical calculation were
found to be consistent with the exact solution. The proposed approach offers a
promising new direction for research in this area.Comment: 6 pages, 2 figures, accepted for publication in J. Korean Phys. So
Structure preserving schemes for Fokker–Planck equations with nonconstant diffusion matrices
In this work we consider an extension of a recently proposed structure preserving numerical scheme for nonlinear Fokker–Planck-type equations to the case of nonconstant full diffusion matrices. While in existing works the schemes are formulated in a one-dimensional setting, here we consider exclusively the two-dimensional case. We prove that the proposed schemes preserve fundamental structural properties like nonnegativity of the solution without restriction on the size of the mesh and entropy dissipation. Moreover, all the methods presented here are at least second order accurate in the transient regimes and arbitrarily high order for large times in the hypothesis in which the flux vanishes at the stationary state. Suitable numerical tests will confirm the theoretical results
Numerical self-consistent stellar models of thin disks
We find a numerical self-consistent stellar model by finding the distribution
function of a thin disk that satisfies simultaneously the Fokker-Planck and
Poisson equations. The solution of the Fokker-Planck equation is found by a
direct numerical solver using finite differences and a variation of Stone's
method. The collision term in the Fokker-Planck equation is found using the
local approximation and the Rosenbluth potentials. The resulting diffusion
coefficients are explicitly evaluated using a Maxwellian distribution for the
field stars. As a paradigmatic example, we apply the numerical formalism to
find the distribution function of a Kuzmin-Toomre thin disk. This example is
studied in some detail showing that the method applies to a large family of
actual galaxies.Comment: 12 pages, 9 figures, version accepted in Astronomy & Astrophysic
Solving procedure for a twenty-five diagonal coefficient matrix: direct numerical solutions of the three dimensional linear Fokker-Planck equation
We describe an implicit procedure for solving linear equation systems
resulting from the discretization of the three dimensional (seven variables)
linear Fokker-Planck equation. The discretization of the Fokker-Planck equation
is performed using a twenty-five point molecule that leads to a coefficient
matrix with equal number of diagonals. The method is an extension of Stone's
implicit procedure, includes a vast class of collision terms and can be applied
to stationary or non stationary problems with different discretizations in
time. Test calculations and comparisons with other methods are presented in two
stationary examples, including an astrophysical application for the
Miyamoto-Nagai disk potential for a typical galaxy.Comment: 20 pages, RevTex, no proofreading, accepted in Journal of
Computational Physic
Stochastic analysis of a full system of two competing populations in a chemostat
This paper formulates two 3D stochastic differential equations (SDEs) of two
microbial populations in a chemostat competing over a single substrate. The two
models have two distinct noise sources. One is general noise whereas the other
is dilution rate induced noise. Nonlinear Monod growth rates are assumed and
the paper is mainly focused on the parameter values where coexistence is
present deterministically. Nondimensionalising the equations around the point
of intersection of the two growth rates leads to a large parameter which is the
nondimensional substrate feed. This in turn is used to perform an asymptotic
analysis leading to a reduced 2D system of equations describing the dynamics of
the populations on and close to a line of steady states retrieved from the
deterministic stability analysis. That reduced system allows the formulation of
a spatially 2D Fokker-Planck equation which when solved numerically admits
results similar to those from simulation of the SDEs. Contrary to previous
suggestions, one particular population becomes dominant at large times.
Finally, we brie y explore the case where death rates are added
Finite element methods for deterministic simulation of polymeric fluids
In this work we consider a finite element method for solving the coupled Navier-Stokes (NS) and Fokker-Planck (FP) multiscale model that describes the dynamics of dilute polymeric fluids. Deterministic approaches such as ours have not received much attention in the literature because they present a formidable computational challenge, due to the fact that the analytical solution to the Fokker-Planck equation may be a function of a large number of independent variables. For instance, to simulate a non-homogeneous flow one must solve the coupled NS-FP system in which (for a 3-dimensional flow, using the dumbbell model for polymers) the Fokker-Planck equation is posed in a 6-dimensional domain. In this work we seek to demonstrate the feasibility of our deterministic approach. We begin by discussing the physical and mathematical foundations of the NS-FP model. We then present a literature review of relevant developments in computational rheology and develop our deterministic finite element based method in detail. Numerical results demonstrating the efficiency of our approach are then given, including some novel results for the simulation of a fully 3-dimensional flow. We utilise parallel computation to perform the large-scale numerical simulations
A Selection Criterion for Patterns in Reaction-Diffusion Systems
Alan Turing's work in Morphogenesis has received wide attention during the
past 60 years. The central idea behind his theory is that two chemically
interacting diffusible substances are able to generate stable spatial patterns,
provided certain conditions are met. Turing's proposal has already been
confirmed as a pattern formation mechanism in several chemical and biological
systems and, due to their wide applicability, there is a great deal of interest
in deciphering how to generate specific patterns under controlled conditions.
However, techniques allowing one to predict what kind of spatial structure will
emerge from Turing systems, as well as generalized reaction-diffusion systems,
remain unknown. Here, we consider a generalized reaction diffusion system on a
planar domain and provide an analytic criterion to determine whether spots or
stripes will be formed. It is motivated by the existence of an associated
energy function that allows bringing in the intuition provided by phase
transitions phenomena. This criterion is proved rigorously in some situations,
generalizing well known results for the scalar equation where the pattern
selection process can be understood in terms of a potential. In more complex
settings it is investigated numerically. Our criterion can be applied to
efficiently design Biotechnology and Developmental Biology experiments, or
simplify the analysis of hypothesized morphogenetic models.Comment: 19 pages, 10 figure
- …