92 research outputs found

### Generation-by-generation dissection of the response function in long memory epidemic processes

In a number of natural and social systems, the response to an exogenous shock relaxes back to the average level according to a long-memory kernel ~1/t1+Î¸ with 0 â‰¤ Î¸ 1 and we find in this case that the total renormalized response is a constant for t < 1/(1-n) followed by a cross-over to ~1/t1+Î¸ for t â‰« 1/(1-n

### From Diffusion to Anomalous Diffusion: A Century after Einstein's Brownian Motion

Einstein's explanation of Brownian motion provided one of the cornerstones
which underlie the modern approaches to stochastic processes. His approach is
based on a random walk picture and is valid for Markovian processes lacking
long-term memory. The coarse-grained behavior of such processes is described by
the diffusion equation. However, many natural processes do not possess the
Markovian property and exhibit to anomalous diffusion. We consider here the
case of subdiffusive processes, which are semi-Markovian and correspond to
continuous-time random walks in which the waiting time for a step is given by a
probability distribution with a diverging mean value. Such a process can be
considered as a process subordinated to normal diffusion under operational time
which depends on this pathological waiting-time distribution. We derive two
different but equivalent forms of kinetic equations, which reduce to know
fractional diffusion or Fokker-Planck equations for waiting-time distributions
following a power-law. For waiting time distributions which are not pure power
laws one or the other form of the kinetic equation is advantageous, depending
on whether the process slows down or accelerates in the course of time

### Models of Passive and Reactive Tracer Motion: an Application of Ito Calculus

By means of Ito calculus it is possible to find, in a straight-forward way,
the analytical solution to some equations related to the passive tracer
transport problem in a velocity field that obeys the multidimensional Burgers
equation and to a simple model of reactive tracer motion.Comment: revised version 7 pages, Latex, to appear as a letter to J. of
Physics

### Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes

We present a stability study of the class of multivariate self-excited Hawkes
point processes, that can model natural and social systems, including
earthquakes, epileptic seizures and the dynamics of neuron assemblies, bursts
of exchanges in social communities, interactions between Internet bloggers,
bank network fragility and cascading of failures, national sovereign default
contagion, and so on. We present the general theory of multivariate generating
functions to derive the number of events over all generations of various types
that are triggered by a mother event of a given type. We obtain the stability
domains of various systems, as a function of the topological structure of the
mutual excitations across different event types. We find that mutual triggering
tends to provide a significant extension of the stability (or subcritical)
domain compared with the case where event types are decoupled, that is, when an
event of a given type can only trigger events of the same type.Comment: 27 pages, 8 figure

### Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

We present a numerical method for the Monte Carlo simulation of uncoupled
continuous-time random walks with a Levy alpha-stable distribution of jumps in
space and a Mittag-Leffler distribution of waiting times, and apply it to the
stochastic solution of the Cauchy problem for a partial differential equation
with fractional derivatives both in space and in time. The one-parameter
Mittag-Leffler function is the natural survival probability leading to
time-fractional diffusion equations. Transformation methods for Mittag-Leffler
random variables were found later than the well-known transformation method by
Chambers, Mallows, and Stuck for Levy alpha-stable random variables and so far
have not received as much attention; nor have they been used together with the
latter in spite of their mathematical relationship due to the geometric
stability of the Mittag-Leffler distribution. Combining the two methods, we
obtain an accurate approximation of space- and time-fractional diffusion
processes almost as easy and fast to compute as for standard diffusion
processes.Comment: 7 pages, 5 figures, 1 table. Presented at the Conference on Computing
in Economics and Finance in Montreal, 14-16 June 2007; at the conference
"Modelling anomalous diffusion and relaxation" in Jerusalem, 23-28 March
2008; et

### An Alternative Method for Solving a Certain Class of Fractional Kinetic Equations

An alternative method for solving the fractional kinetic equations solved
earlier by Haubold and Mathai (2000) and Saxena et al. (2002, 2004a, 2004b) is
recently given by Saxena and Kalla (2007). This method can also be applied in
solving more general fractional kinetic equations than the ones solved by the
aforesaid authors. In view of the usefulness and importance of the kinetic
equation in certain physical problems governing reaction-diffusion in complex
systems and anomalous diffusion, the authors present an alternative simple
method for deriving the solution of the generalized forms of the fractional
kinetic equations solved by the aforesaid authors and Nonnenmacher and Metzler
(1995). The method depends on the use of the Riemann-Liouville fractional
calculus operators. It has been shown by the application of Riemann-Liouville
fractional integral operator and its interesting properties, that the solution
of the given fractional kinetic equation can be obtained in a straight-forward
manner. This method does not make use of the Laplace transform.Comment: 7 pages, LaTe

### Turbulence and passive scalar transport in a free-slip surface

We consider the two-dimensional (2D) flow in a flat free-slip surface that
bounds a three-dimensional (3D) volume in which the flow is turbulent. The
equations of motion for the two-dimensional flow in the surface are neither
compressible nor incompressible but strongly influenced by the 3D flow
underneath the surface. The velocity correlation functions in the 2D surface
and in the 3D volume scale with the same exponents. In the viscous subrange the
amplitudes are the same, but in the inertial subrange the 2D one is reduced to
2/3 of the 3D amplitude. The surface flow is more strongly intermittent than
the 3D volume flow. Geometric scaling theory is used to derive a relation
between the scaling of the velocity field and the density fluctuations of a
passive scalar advected on the surface.Comment: 11 pages, 10 Postscript 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

### Fractional transport equations for Levy stable processes

The influence functional method of Feynman and Vernon is used to obtain a
quantum master equation for a Brownian system subjected to a Levy stable random
force. The corresponding classical transport equations for the Wigner function
are then derived, both in the limit of weak and strong friction. These are
fractional extensions of the Klein-Kramers and the Smoluchowski equations. It
is shown that the fractional character acquired by the position in the
Smoluchowski equation follows from the fractional character of the momentum in
the Klein-Kramers equation. Connections among fractional transport equations
recently proposed are clarified.Comment: 4 page

### Residence Time Statistics for Normal and Fractional Diffusion in a Force Field

We investigate statistics of occupation times for an over-damped Brownian
particle in an external force field. A backward Fokker-Planck equation
introduced by
Majumdar and Comtet describing the distribution of occupation times is
solved. The solution gives a general relation between occupation time
statistics and probability currents which are found from solutions of the
corresponding problem of first passage time. This general relationship between
occupation times and first passage times, is valid for normal Markovian
diffusion and for non-Markovian sub-diffusion, the latter modeled using the
fractional Fokker-Planck equation. For binding potential fields we find in the
long time limit ergodic behavior for normal diffusion, while for the fractional
framework weak ergodicity breaking is found, in agreement with previous results
of Bel and Barkai on the continuous time random walk on a lattice. For
non-binding potential rich physical behaviors are obtained, and classification
of occupation time statistics is made possible according to whether or not the
underlying random walk is recurrent and the averaged first return time to the
origin is finite. Our work establishes a link between fractional calculus and
ergodicity breaking.Comment: 12 page

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