686 research outputs found
Collective Atomic Recoil Laser as a synchronization transition
We consider here a model previously introduced to describe the collective
behavior of an ensemble of cold atoms interacting with a coherent
electromagnetic field. The atomic motion along the self-generated
spatially-periodic force field can be interpreted as the rotation of a phase
oscillator. This suggests a relationship with synchronization transitions
occurring in globally coupled rotators. In fact, we show that whenever the
field dynamics can be adiabatically eliminated, the model reduces to a
self-consistent equation for the probability distribution of the atomic
"phases". In this limit, there exists a formal equivalence with the Kuramoto
model, though with important differences in the self-consistency conditions.
Depending on the field-cavity detuning, we show that the onset of synchronized
behavior may occur through either a first- or second-order phase transition.
Furthermore, we find a secondary threshold, above which a periodic self-pulsing
regime sets in, that is immediately followed by the unlocking of the
forward-field frequency. At yet higher, but still experimentally meaningful,
input intensities, irregular, chaotic oscillations may eventually appear.
Finally, we derive a simpler model, involving only five scalar variables, which
is able to reproduce the entire phenomenology exhibited by the original model
Statistical Mechanics of finite arrays of coupled bistable elements
We discuss the equilibrium of a single collective variable characterizing a
finite set of coupled, noisy, bistable systems as the noise strength, the size
and the coupling parameter are varied. We identify distinct regions in
parameter space. The results obtained in prior works in the asymptotic infinite
size limit are significantly different from the finite size results. A
procedure to construct approximate 1-dimensional Langevin equation is adopted.
This equation provides a useful tool to understand the collective behavior even
in the presence of an external driving force
On low-sampling-rate Kramers-Moyal coefficients
We analyze the impact of the sampling interval on the estimation of
Kramers-Moyal coefficients. We obtain the finite-time expressions of these
coefficients for several standard processes. We also analyze extreme situations
such as the independence and no-fluctuation limits that constitute useful
references. Our results aim at aiding the proper extraction of information in
data-driven analysis.Comment: 9 pages, 4 figure
A field theoretic approach to master equations and a variational method beyond the Poisson ansatz
We develop a variational scheme in a field theoretic approach to a stochastic
process. While various stochastic processes can be expressed using master
equations, in general it is difficult to solve the master equations exactly,
and it is also hard to solve the master equations numerically because of the
curse of dimensionality. The field theoretic approach has been used in order to
study such complicated master equations, and the variational scheme achieves
tremendous reduction in the dimensionality of master equations. For the
variational method, only the Poisson ansatz has been used, in which one
restricts the variational function to a Poisson distribution. Hence, one has
dealt with only restricted fluctuation effects. We develop the variational
method further, which enables us to treat an arbitrary variational function. It
is shown that the variational scheme developed gives a quantitatively good
approximation for master equations which describe a stochastic gene regulatory
network.Comment: 13 pages, 2 figure
Three-state herding model of the financial markets
We propose a Markov jump process with the three-state herding interaction. We
see our approach as an agent-based model for the financial markets. Under
certain assumptions this agent-based model can be related to the stochastic
description exhibiting sophisticated statistical features. Along with power-law
probability density function of the absolute returns we are able to reproduce
the fractured power spectral density, which is observed in the high-frequency
financial market data. Given example of consistent agent-based and stochastic
modeling will provide background for the further developments in the research
of complex social systems.Comment: 11 pages, 3 figure
Additive-multiplicative stochastic models of financial mean-reverting processes
We investigate a generalized stochastic model with the property known as mean
reversion, that is, the tendency to relax towards a historical reference level.
Besides this property, the dynamics is driven by multiplicative and additive
Wiener processes. While the former is modulated by the internal behavior of the
system, the latter is purely exogenous. We focus on the stochastic dynamics of
volatilities, but our model may also be suitable for other financial random
variables exhibiting the mean reversion property. The generalized model
contains, as particular cases, many early approaches in the literature of
volatilities or, more generally, of mean-reverting financial processes. We
analyze the long-time probability density function associated to the model
defined through a It\^o-Langevin equation. We obtain a rich spectrum of shapes
for the probability function according to the model parameters. We show that
additive-multiplicative processes provide realistic models to describe
empirical distributions, for the whole range of data.Comment: 8 pages, 3 figure
Equilibration problem for the generalized Langevin equation
We consider the problem of equilibration of a single oscillator system with
dynamics given by the generalized Langevin equation. It is well-known that this
dynamics can be obtained if one considers a model where the single oscillator
is coupled to an infinite bath of harmonic oscillators which are initially in
equilibrium. Using this equivalence we first determine the conditions necessary
for equilibration for the case when the system potential is harmonic. We then
give an example with a particular bath where we show that, even for parameter
values where the harmonic case always equilibrates, with any finite amount of
nonlinearity the system does not equilibrate for arbitrary initial conditions.
We understand this as a consequence of the formation of nonlinear localized
excitations similar to the discrete breather modes in nonlinear lattices.Comment: 5 pages, 2 figure
Tunable nonlinearity in atomic response to a bichromatic field
Atomic response to a probe beam can be tailored, by creating coherences
between atomic levels with help of another beam. Changing parameters of the
control beam will change the nature of coherences and hence the nature of
atomic response as well. Such change can depend upon intensity of both probe
and control beams, in a nonlinear fashion. We present a situation where this
nonlinearity in dependence can be precisely controlled, as to obtain different
variations as desired. We also present a detailed analysis of how this
nonlinear dependency arises and show that this is an interesting effect of
several Coherent Population Trap(CPT) states that exist and a competition among
them to trap atomic population in them.Comment: 16 pages and 6 figure
Irreversible spherical model and its stationary entropy production rate
The nonequilibrium stationary state of an irreversible spherical model is
investigated on hypercubic lattices. The model is defined by Langevin equations
similar to the reversible case, but with asymmetric transition rates. In spite
of being irreversible, we have succeeded in finding an explicit form for the
stationary probability distribution, which turns out to be of the
Boltzmann-Gibbs type. This enables one to evaluate the exact form of the
entropy production rate at the stationary state, which is non-zero if the
dynamical rules of the transition rates are asymmetric
Multiple time-scale approach for a system of Brownian particles in a non-uniform temperature field
The Smoluchowsky equation for a system of interacting Brownian particles in a
temperature gradient is derived from the Kramers equation by means of a
multiple time-scale method. The interparticle interactions are assumed to be
represented by a mean-field description. We present numerical results that
compare well with the theoretical prediction together with an extensive
discussion on the prescription of the Langevin equation in overdamped systems.Comment: 8 pages, 2 figure
- …