1,553 research outputs found
Fractional derivatives of random walks: Time series with long-time memory
We review statistical properties of models generated by the application of a
(positive and negative order) fractional derivative operator to a standard
random walk and show that the resulting stochastic walks display
slowly-decaying autocorrelation functions. The relation between these
correlated walks and the well-known fractionally integrated autoregressive
(FIGARCH) models, commonly used in econometric studies, is discussed. The
application of correlated random walks to simulate empirical financial times
series is considered and compared with the predictions from FIGARCH and the
simpler FIARCH processes. A comparison with empirical data is performed.Comment: 10 pages, 14 figure
Time evolution of the reaction front in a subdiffusive system
Using the quasistatic approximation, we show that in a subdiffusion--reaction
system the reaction front evolves in time according to the formula
, with being the subdiffusion parameter. The
result is derived for the system where the subdiffusion coefficients of
reactants differ from each other. It includes the case of one static reactant.
As an application of our results, we compare the time evolution of reaction
front extracted from experimental data with the theoretical formula and we find
that the transport process of organic acid particles in the tooth enamel is
subdiffusive.Comment: 18 pages, 3 figure
Squeezed States and Hermite polynomials in a Complex Variable
Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec
[J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of
coherent states, related to the Hermite polynomials in a complex variable which
are orthogonal with respect to a non-rotationally invariant measure. We
investigate relations between these coherent states and obtain the relationship
between them and the squeezed states of quantum optics. We also obtain a second
realization of the canonical coherent states in the Bargmann space of analytic
functions, in terms of a squeezed basis. All this is done in the flavor of the
classical approach of V. Bargmann [Commun. Pur. Appl. Math. 14, 187 (1961)].Comment: 15 page
Using the fractional interaction law to model the impact dynamics in arbitrary form of multiparticle collisions
Using the molecular dynamics method, we examine a discrete deterministic
model for the motion of spherical particles in three-dimensional space. The
model takes into account multiparticle collisions in arbitrary forms. Using
fractional calculus we proposed an expression for the repulsive force, which is
the so called fractional interaction law. We then illustrate and discuss how to
control (correlate) the energy dissipation and the collisional time for an
individual article within multiparticle collisions. In the multiparticle
collisions we included the friction mechanism needed for the transition from
coupled torsion-sliding friction through rolling friction to static friction.
Analysing simple simulations we found that in the strong repulsive state binary
collisions dominate. However, within multiparticle collisions weak repulsion is
observed to be much stronger. The presented numerical results can be used to
realistically model the impact dynamics of an individual particle in a group of
colliding particles.Comment: 17 pages, 8 figures, 1 table; In review process of Physical Review
Non-Markovian Levy diffusion in nonhomogeneous media
We study the diffusion equation with a position-dependent, power-law
diffusion coefficient. The equation possesses the Riesz-Weyl fractional
operator and includes a memory kernel. It is solved in the diffusion limit of
small wave numbers. Two kernels are considered in detail: the exponential
kernel, for which the problem resolves itself to the telegrapher's equation,
and the power-law one. The resulting distributions have the form of the L\'evy
process for any kernel. The renormalized fractional moment is introduced to
compare different cases with respect to the diffusion properties of the system.Comment: 7 pages, 2 figure
Spectral Asymptotics of Eigen-value Problems with Non-linear Dependence on the Spectral Parameter
We study asymptotic distribution of eigen-values of a quadratic
operator polynomial of the following form ,
where is a second order differential positive elliptic operator
with quadratic dependence on the spectral parameter . We derive
asymptotics of the spectral density in this problem and show how to compute
coefficients of its asymptotic expansion from coefficients of the asymptotic
expansion of the trace of the heat kernel of . The leading term in
the spectral asymptotics is the same as for a Laplacian in a cavity. The
results have a number of physical applications. We illustrate them by examples
of field equations in external stationary gravitational and gauge backgrounds.Comment: latex, 20 page
Measuring subdiffusion parameters
We propose a method to extract from experimental data the subdiffusion
parameter and subdiffusion coefficient which are defined by
means of the relation where
denotes a mean square displacement of a random walker starting from
at the initial time . The method exploits a membrane system where a
substance of interest is transported in a solvent from one vessel to another
across a thin membrane which plays here only an auxiliary role. Using such a
system, we experimentally study a diffusion of glucose and sucrose in a gel
solvent. We find a fully analytic solution of the fractional subdiffusion
equation with the initial and boundary conditions representing the system under
study. Confronting the experimental data with the derived formulas, we show a
subdiffusive character of the sugar transport in gel solvent. We precisely
determine the parameter , which is smaller than 1, and the subdiffusion
coefficient .Comment: 17 pages, 9 figures, revised, to appear in Phys. Rev.
Fractional Quantum Mechanics
A path integral approach to quantum physics has been developed. Fractional
path integrals over the paths of the L\'evy flights are defined. It is shown
that if the fractality of the Brownian trajectories leads to standard quantum
and statistical mechanics, then the fractality of the L\'evy paths leads to
fractional quantum mechanics and fractional statistical mechanics. The
fractional quantum and statistical mechanics have been developed via our
fractional path integral approach. A fractional generalization of the
Schr\"odinger equation has been found. A relationship between the energy and
the momentum of the nonrelativistic quantum-mechanical particle has been
established. The equation for the fractional plane wave function has been
obtained. We have derived a free particle quantum-mechanical kernel using Fox's
H function. A fractional generalization of the Heisenberg uncertainty relation
has been established. Fractional statistical mechanics has been developed via
the path integral approach. A fractional generalization of the motion equation
for the density matrix has been found. The density matrix of a free particle
has been expressed in terms of the Fox's H function. We also discuss the
relationships between fractional and the well-known Feynman path integral
approaches to quantum and statistical mechanics.Comment: 27 page
Anomalous Rotational Relaxation: A Fractional Fokker-Planck Equation Approach
In this study we obtained analytically relaxation function in terms of
rotational correlation functions based on Brownian motion for complex
disordered systems in a stochastic framework. We found out that rotational
relaxation function has a fractional form for complex disordered systems, which
indicates relaxation has non-exponential character obeys to
Kohlrausch-William-Watts law, following the Mittag-Leffler decay.Comment: Revtex4, 9 pages. Paper was revised. References adde
Time in treatment: examining mental illness trajectories across inpatient psychiatric treatment
Early discharge or reduced length of stay for inpatient psychiatric patients is related to increased readmission rates and worse clinical outcomes including increased risk for suicide. Trajectories of mental illness outcomes have been identified as an important method for predicting the optimal length of stay but the distinguishing factors that separate trajectories remain unclear. We sought to identify the distinct classes of patients who demonstrated similar trajectories of mental illness over the course of inpatient treatment, and we explore the patient characteristics associated with these mental illness trajectories. We used data (N = 3406) from an inpatient psychiatric hospital with intermediate lengths of stay. Using growth mixture modeling, latent mental illness scores were derived from six mental illness indicators: psychological flexibility, emotion regulation problems, anxiety, depression, suicidal ideation, and disability. The patients were grouped into three distinct trajectory classes: (1) High-Risk, Rapid Improvement (HR-RI); (2) Low-Risk, Gradual Improvement (LR-GI); and (3) High-Risk, Gradual Improvement (HR-GI). The HR-GI was significantly younger than the other two classes. The HR-GI had significantly more female patients than males, while the LR-GI had more male patients than females. Our findings indicated that younger females had more severe mental illness at admission and only gradual improvement during the inpatient treatment period, and they remained in treatment for longer lengths of stay, than older males
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