694 research outputs found
First-passage distributions for the one-dimensional Fokker-Planck equation
We present an analytical framework to study the first-passage (FP) and
first-return (FR) distributions for the broad family of models described by the
one-dimensional Fokker-Planck equation in finite domains, identifying general
properties of these distributions for different classes of models. When in the
Fokker-Planck equation the diffusion coefficient is positive (nonzero) and the
drift term is bounded, as in the case of a Brownian walker, both distributions
may exhibit a power-law decay with exponent -3/2 for intermediate times. We
discuss how the influence of an absorbing state changes this exponent. The
absorbing state is characterized by a vanishing diffusion coefficient and/or a
diverging drift term. Remarkably, the exponent of the Brownian walker class of
models is still found, as long as the departure and arrival regions are far
enough from the absorbing state, but the range of times where the power law is
observed narrows. Close enough to the absorbing point, though, a new exponent
may appear. The particular value of the exponent depends on the behavior of the
diffusion and the drift terms of the Fokker-Planck equation. We focus on the
case of a diffusion term vanishing linearly at the absorbing point. In this
case, the FP and FR distributions are similar to those of the voter model,
characterized by a power law with exponent -2. As an illustration of the
general theory, we compare it with exact analytical solutions and extensive
numerical simulations of a two-parameter voter-like family models. We study the
behavior of the FP and FR distributions by tuning the importance of the
absorbing points throughout changes of the parameters. Finally, the possibility
of inferring relevant information about the steady-sate probability
distribution of a model from the FP and FR distributions is addressed.Comment: 17 pages, 8 figure
Mean first-passage times for an ac-driven magnetic moment of a nanoparticle
The two-dimensional backward Fokker-Planck equation is used to calculate the
mean first-passage times (MFPTs) of the magnetic moment of a nanoparticle
driven by a rotating magnetic field. It is shown that a magnetic field that is
rapidly rotating in the plane {\it perpendicular} to the easy axis of the
nanoparticle governs the MFPTs just in the same way as a static magnetic field
that is applied {\it along} the easy axis. Within this framework, the features
of the magnetic relaxation and net magnetization of systems composed of
ferromagnetic nanoparticles arising from the action of the rotating field are
revealed.Comment: 7 pages, 1 figur
Constructing solutions for a kinetic model of angiogenesis in annular domains
We prove existence and stability of solutions for a model of angiogenesis set
in an annular region. Branching, anastomosis and extension of blood vessel tips
are described by an integrodifferential kinetic equation of Fokker-Planck type
supplemented with nonlocal boundary conditions and coupled to a diffusion
problem with Neumann boundary conditions through the force field created by the
tumor induced angiogenic factor and the flux of vessel tips. Our technique
exploits balance equations, estimates of velocity decay and compactness results
for kinetic operators, combined with gradient estimates of heat kernels for
Neumann problems in non convex domains.Comment: to appear in Applied Mathematical Modellin
Rate description of Fokker-Planck processes with time-periodic parameters
The large time dynamics of a periodically driven Fokker-Planck process
possessing several metastable states is investigated. At weak noise transitions
between the metastable states are rare. Their dynamics then represent a
discrete Markovian process characterized by time dependent rates. Apart from
the occupation probabilities, so-called specific probability densities and
localizing functions can be associated to each metastable state. Together,
these three sets of functions uniquely characterize the large time dynamics of
the conditional probability density of the original process. Exact equations of
motion are formulated for these three sets of functions and strategies are
discussed how to solve them. These methods are illustrated and their usefulness
is demonstrated by means of the example of a bistable Brownian oscillator
within a large range of driving frequencies from the slow semiadiabatic to the
fast driving regime
Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization
We describe methods for proving upper and lower bounds on infinite-time
averages in deterministic dynamical systems and on stationary expectations in
stochastic systems. The dynamics and the quantities to be bounded are assumed
to be polynomial functions of the state variables. The methods are
computer-assisted, using sum-of-squares polynomials to formulate sufficient
conditions that can be checked by semidefinite programming. In the
deterministic case, we seek tight bounds that apply to particular local
attractors. An obstacle to proving such bounds is that they do not hold
globally; they are generally violated by trajectories starting outside the
local basin of attraction. We describe two closely related ways past this
obstacle: one that requires knowing a subset of the basin of attraction, and
another that considers the zero-noise limit of the corresponding stochastic
system. The bounding methods are illustrated using the van der Pol oscillator.
We bound deterministic averages on the attracting limit cycle above and below
to within 1%, which requires a lower bound that does not hold for the unstable
fixed point at the origin. We obtain similarly tight upper and lower bounds on
stochastic expectations for a range of noise amplitudes. Limitations of our
methods for certain types of deterministic systems are discussed, along with
prospects for improvement.Comment: 25 pages; Added new Section 7.2; Added references; Corrected typos;
Submitted to SIAD
- …