400 research outputs found
On second order elliptic equations with a small parameter
The Neumann problem with a small parameter
is
considered in this paper. The operators and are self-adjoint second
order operators. We assume that has a non-negative characteristic form
and is strictly elliptic. The reflection is with respect to inward
co-normal unit vector . The behavior of
is effectively described via
the solution of an ordinary differential equation on a tree. We calculate the
differential operators inside the edges of this tree and the gluing condition
at the root. Our approach is based on an analysis of the corresponding
diffusion processes.Comment: 28 pages, 1 figure, revised versio
Numerical simulations versus theoretical predictions for a non-Gaussian noise induced escape problem in application to full counting statistics
A theoretical approach for characterizing the influence of asymmetry of noise distribution on the escape rate
of a multistable system is presented. This was carried out via the estimation of an action, which is defined as
an exponential factor in the escape rate, and discussed in the context of full counting statistics paradigm. The
approach takes into account all cumulants of the noise distribution and demonstrates an excellent agreement with
the results of numerical simulations. An approximation of the third-order cumulant was shown to have limitations
on the range of dynamic stochastic system parameters. The applicability of the theoretical approaches developed
so far is discussed for an adequate characterization of the escape rate measured in experiments
Universality of residence-time distributions in non-adiabatic stochastic resonance
We present mathematically rigorous expressions for the residence-time and
first-passage-time distributions of a periodically forced Brownian particle in
a bistable potential. For a broad range of forcing frequencies and amplitudes,
the distributions are close to periodically modulated exponential ones.
Remarkably, the periodic modulations are governed by universal functions,
depending on a single parameter related to the forcing period. The behaviour of
the distributions and their moments is analysed, in particular in the low- and
high-frequency limits.Comment: 8 pages, 1 figure New version includes distinction between
first-passage-time and residence-time distribution
Noise-induced escape in an excitable system
We consider the stochastic dynamics of escape in an excitable system, the FitzHugh-Nagumo (FHN) neuronal model, for different classes of excitability. We discuss, first, the threshold structure of the FHN model as an example of a system without a saddle state. We then develop a nonlinear (nonlocal) stability approach based on the theory of large fluctuations, including a finite-noise correction, to describe noise-induced escape in the excitable regime. We show that the threshold structure is revealed via patterns of most probable (optimal) fluctuational paths. The approach allows us to estimate the escape rate and the exit location distribution. We compare the responses of a monostable resonator and monostable integrator to stochastic input signals and to a mixture of periodic and stochastic stimuli. Unlike the commonly used local analysis of the stable state, our nonlocal approach based on optimal paths yields results that are in good agreement with direct numerical simulations of the Langevin equation
Variable-range Projection Model for Turbulence-driven Collisions
We discuss the probability distribution of relative speed of
inertial particles suspended in a highly turbulent gas when the Stokes numbers,
a dimensionless measure of their inertia, is large. We identify a mechanism
giving rise to the distribution
(for some constant ). Our conclusions are supported by numerical simulations
and the analytical solution of a model equation of motion. The results
determine the rate of collisions between suspended particles. They are relevant
to the hypothesised mechanism for formation of planets by aggregation of dust
particles in circumstellar nebula.Comment: 4 pages, 2 figure
Dispersion and reaction in random flows: Single realization vs ensemble average
We examine the dispersion of a passive scalar released in an incompressible
fluid flow in an unbounded domain. The flow is assumed to be spatially
periodic, with zero spatial average, and random in time, in the manner of the
random-phase alternating sine flow which we use as an exemplar. In the
long-time limit, the scalar concentration takes the same, predictable form for
almost all realisations of the flow, with a Gaussian core characterised by an
effective diffusivity, and large-deviation tails characterised by a rate
function (which can be evaluated by computing the largest Lyapunov exponent of
a family of random-in-time partial differential equations). We contrast this
single-realisation description with that which applies to the average of the
concentration over an ensemble of flow realisations. We show that the
single-realisation and ensemble-average effective diffusivities are identical
but that the corresponding rate functions are not, and that the
ensemble-averaged description overestimates the concentration in the tails
compared with that obtained for single-flow realisations. This difference has a
marked impact for scalars reacting according to the
Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) model. Such scalars form an
expanding front whose shape is approximately independent of the flow
realisation and can be deduced from the single-realisation large-deviation rate
function. We test our predictions against numerical simulations of the
alternating sine flow
Scattering solutions in a network of thin fibers: small diameter asymptotics
Small diameter asymptotics is obtained for scattering solutions in a network
of thin fibers. The asymptotics is expressed in terms of solutions of related
problems on the limiting quantum graph. We calculate the Lagrangian gluing
conditions at vertices for the problems on the limiting graph. If the frequency
of the incident wave is above the bottom of the absolutely continuous spectrum,
the gluing conditions are formulated in terms of the scattering data for each
individual junction of the network
Scaling and crossovers in activated escape near a bifurcation point
Near a bifurcation point a system experiences critical slowing down. This
leads to scaling behavior of fluctuations. We find that a periodically driven
system may display three scaling regimes and scaling crossovers near a
saddle-node bifurcation where a metastable state disappears. The rate of
activated escape scales with the driving field amplitude as , where is the bifurcational value of . With
increasing field frequency the critical exponent changes from
for stationary systems to a dynamical value and then again to
. The analytical results are in agreement with the results of
asymptotic calculations in the scaling region. Numerical calculations and
simulations for a model system support the theory.Comment: 18 page
Lagrangian phase transitions in nonequilibrium thermodynamic systems
In previous papers we have introduced a natural nonequilibrium free energy by
considering the functional describing the large fluctuations of stationary
nonequilibrium states. While in equilibrium this functional is always convex,
in nonequilibrium this is not necessarily the case. We show that in
nonequilibrium a new type of singularities can appear that are interpreted as
phase transitions. In particular, this phenomenon occurs for the
one-dimensional boundary driven weakly asymmetric exclusion process when the
drift due to the external field is opposite to the one due to the external
reservoirs, and strong enough.Comment: 10 pages, 2 figure
Convergence of invariant densities in the small-noise limit
This paper presents a systematic numerical study of the effects of noise on
the invariant probability densities of dynamical systems with varying degrees
of hyperbolicity. It is found that the rate of convergence of invariant
densities in the small-noise limit is frequently governed by power laws. In
addition, a simple heuristic is proposed and found to correctly predict the
power law exponent in exponentially mixing systems. In systems which are not
exponentially mixing, the heuristic provides only an upper bound on the power
law exponent. As this numerical study requires the computation of invariant
densities across more than 2 decades of noise amplitudes, it also provides an
opportunity to discuss and compare standard numerical methods for computing
invariant probability densities.Comment: 27 pages, 19 figures, revised with minor correction
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