1,189 research outputs found
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
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
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
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
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.
Non-equilibrium Phase Transitions with Long-Range Interactions
This review article gives an overview of recent progress in the field of
non-equilibrium phase transitions into absorbing states with long-range
interactions. It focuses on two possible types of long-range interactions. The
first one is to replace nearest-neighbor couplings by unrestricted Levy flights
with a power-law distribution P(r) ~ r^(-d-sigma) controlled by an exponent
sigma. Similarly, the temporal evolution can be modified by introducing waiting
times Dt between subsequent moves which are distributed algebraically as P(Dt)~
(Dt)^(-1-kappa). It turns out that such systems with Levy-distributed
long-range interactions still exhibit a continuous phase transition with
critical exponents varying continuously with sigma and/or kappa in certain
ranges of the parameter space. In a field-theoretical framework such
algebraically distributed long-range interactions can be accounted for by
replacing the differential operators nabla^2 and d/dt with fractional
derivatives nabla^sigma and (d/dt)^kappa. As another possibility, one may
introduce algebraically decaying long-range interactions which cannot exceed
the actual distance to the nearest particle. Such interactions are motivated by
studies of non-equilibrium growth processes and may be interpreted as Levy
flights cut off at the actual distance to the nearest particle. In the
continuum limit such truncated Levy flights can be described to leading order
by terms involving fractional powers of the density field while the
differential operators remain short-ranged.Comment: LaTeX, 39 pages, 13 figures, minor revision
The W51 Giant Molecular Cloud
We present 45"-47" angular resolution maps at 50" sampling of the 12CO and
13CO J=1-0 emission toward a 1.39 deg x 1.33 deg region in the W51 HII region
complex. These data permit the spatial and kinematic separation of several
spectral features observed along the line of sight to W51, and establish the
presence of a massive (1.2 x 10^6 Mo), large (83 pc x 114 pc) giant molecular
cloud (GMC), defined as the W51 GMC, centered at (l,b,V) = (49.5 deg, -0.2 deg,
61 km/s). A second massive (1.9 x 10^5 Mo), elongated (136 pc x 22 pc)
molecular cloud is found at velocities of about 68 km/s along the southern edge
of the W51 GMC. Of the five radio continuum sources that classically define the
W51 region, the brightest source at lambda 6cm (G49.5-0.4) is spatially and
kinematically coincident with the W51 GMC and three (G48.9-0.3, G49.1-0.4, and
G49.2-0.4) are associated with the 68 km/s cloud. Published absorption line
spectra indicate that the fifth prominent continuum source (G49.4-0.3) is
located behind the W51 molecular cloud. The W51 GMC is among the upper 1% of
clouds in the Galactic disk by size and the upper 5-10% by mass. While the W51
GMC is larger and more massive than any nearby molecular cloud, the average H2
column density is not unusual given its size and the mean H2 volume density is
comparable to that in nearby clouds. The W51 GMC is also similar to other
clouds in that most of the molecular mass is contained in a diffuse envelope
that is not currently forming massive stars. We speculate that much of the
massive star formation activity in this region has resulted from a collision
between the 68 km/s cloud and the W51 GMC.Comment: Accepted for publication by the Astronomical Journal. 21 pages, plus
7 figures and 1 tabl
Infrared spectroscopy of diatomic molecules - a fractional calculus approach
The eigenvalue spectrum of the fractional quantum harmonic oscillator is
calculated numerically solving the fractional Schr\"odinger equation based on
the Riemann and Caputo definition of a fractional derivative. The fractional
approach allows a smooth transition between vibrational and rotational type
spectra, which is shown to be an appropriate tool to analyze IR spectra of
diatomic molecules.Comment: revised + extended version, 9 pages, 6 figure
Factorial Moments of Continuous Order
The normalized factorial moments are continued to noninteger values of
the order , satisfying the condition that the statistical fluctuations
remain filtered out. That is, for Poisson distribution for all .
The continuation procedure is designed with phenomenology and data analysis in
mind. Examples are given to show how can be obtained for positive and
negative values of . With being continuous, multifractal analysis is
made possible for multiplicity distributions that arise from self-similar
dynamics. A step-by-step procedure of the method is summarized in the
conclusion.Comment: 15 pages + 9 figures (figures available upon request), Late
Variational Problems with Fractional Derivatives: Euler-Lagrange Equations
We generalize the fractional variational problem by allowing the possibility
that the lower bound in the fractional derivative does not coincide with the
lower bound of the integral that is minimized. Also, for the standard case when
these two bounds coincide, we derive a new form of Euler-Lagrange equations. We
use approximations for fractional derivatives in the Lagrangian and obtain the
Euler-Lagrange equations which approximate the initial Euler-Lagrange equations
in a weak sense
- …