2,652 research outputs found

### Weyl Quantization of Fractional Derivatives

The quantum analogs of the derivatives with respect to coordinates q_k and
momenta p_k are commutators with operators P_k and $Q_k. We consider quantum
analogs of fractional Riemann-Liouville and Liouville derivatives. To obtain
the quantum analogs of fractional Riemann-Liouville derivatives, which are
defined on a finite interval of the real axis, we use a representation of these
derivatives for analytic functions. To define a quantum analog of the
fractional Liouville derivative, which is defined on the real axis, we can use
the representation of the Weyl quantization by the Fourier transformation.Comment: 9 pages, LaTe

### Fractional Chemotaxis Diffusion Equations

We introduce mesoscopic and macroscopic model equations of chemotaxis with
anomalous subdiffusion for modelling chemically directed transport of
biological organisms in changing chemical environments with diffusion hindered
by traps or macro-molecular crowding. The mesoscopic models are formulated
using Continuous Time Random Walk master equations and the macroscopic models
are formulated with fractional order differential equations. Different models
are proposed depending on the timing of the chemotactic forcing.
Generalizations of the models to include linear reaction dynamics are also
derived. Finally a Monte Carlo method for simulating anomalous subdiffusion
with chemotaxis is introduced and simulation results are compared with
numerical solutions of the model equations. The model equations developed here
could be used to replace Keller-Segel type equations in biological systems with
transport hindered by traps, macro-molecular crowding or other obstacles.Comment: 25page

### 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 $x_{f}$ evolves in time according to the formula
$x_{f} \sim t^{\alpha/2}$, with $\alpha$ 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

### On the Consistency of the Solutions of the Space Fractional Schr\"odinger Equation

Recently it was pointed out that the solutions found in literature for the
space fractional Schr\"odinger equation in a piecewise manner are wrong, except
the case with the delta potential. We reanalyze this problem and show that an
exact and a proper treatment of the relevant integral proves otherwise. We also
discuss effective potential approach and present a free particle solution for
the space and time fractional Schr\"odinger equation in general coordinates in
terms of Fox's H-functions

### Dynamics of Fractal Solids

We describe the fractal solid by a special continuous medium model. We
propose to describe the fractal solid by a fractional continuous model, where
all characteristics and fields are defined everywhere in the volume but they
follow some generalized equations which are derived by using integrals of
fractional order. The order of fractional integral can be equal to the fractal
mass dimension of the solid. Fractional integrals are considered as an
approximation of integrals on fractals. We suggest the approach to compute the
moments of inertia for fractal solids. The dynamics of fractal solids are
described by the usual Euler's equations. The possible experimental test of the
continuous medium model for fractal solids is considered.Comment: 12 pages, LaTe

### Measuring subdiffusion parameters

We propose a method to extract from experimental data the subdiffusion
parameter $\alpha$ and subdiffusion coefficient $D_\alpha$ which are defined by
means of the relation $=2D_\alpha/\Gamma(1+\alpha) t^\alpha$ where
$$ denotes a mean square displacement of a random walker starting from
$x=0$ at the initial time $t=0$. 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 $\alpha$, which is smaller than 1, and the subdiffusion
coefficient $D_\alpha$.Comment: 17 pages, 9 figures, revised, to appear in Phys. Rev.

### Hyperbolic subdiffusive impedance

We use the hyperbolic subdiffusion equation with fractional time derivatives
(the generalized Cattaneo equation) to study the transport process of
electrolytes in media where subdiffusion occurs. In this model the flux is
delayed in a non-zero time with respect to the concentration gradient. In
particular, we obtain the formula of electrochemical subdiffusive impedance of
a spatially limited sample in the limit of large and of small pulsation of the
electric field. The boundary condition at the external wall of the sample are
taken in the general form as a linear combination of subdiffusive flux and
concentration of the transported particles. We also discuss the influence of
the equation parameters (the subdiffusion parameter and the delay time) on the
Nyquist impedance plots.Comment: 10 pages, 5 figure

### 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

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