5,055 research outputs found
Spectral properties of fractional Fokker-Plank operator for the L\'evy flight in a harmonic potential
We present a detailed analysis of the eigenfunctions of the Fokker-Planck
operator for the L\'evy-Ornstein-Uhlenbeck process, their asymptotic behavior
and recurrence relations, explicit expressions in coordinate space for the
special cases of the Ornstein-Uhlenbeck process with Gaussian and with Cauchy
white noise and for the transformation kernel, which maps the fractional
Fokker-Planck operator of the Cauchy-Ornstein-Uhlenbeck process to the
non-fractional Fokker-Planck operator of the usual Gaussian Ornstein-Uhlenbeck
process. We also describe how non-spectral relaxation can be observed in
bounded random variables of the L\'evy-Ornstein-Uhlenbeck process and their
correlation functions.Comment: 10 pages, 5 figures, submitted to Euro. Phys. J.
Brownian yet non-Gaussian diffusion in heterogeneous media: from superstatistics to homogenization
We discuss the situations under which Brownian yet non-Gaussian (BnG)
diffusion can be observed in the model of a particle's motion in a random
landscape of diffusion coefficients slowly varying in space. Our conclusion is
that such behavior is extremely unlikely in the situations when the particles,
introduced into the system at random at , are observed from the
preparation of the system on. However, it indeed may arise in the case when the
diffusion (as described in Ito interpretation) is observed under equilibrated
conditions. This paradigmatic situation can be translated into the model of the
diffusion coefficient fluctuating in time along a trajectory, i.e. into a kind
of the "diffusing diffusivity" model.Comment: 12 pages; 10 figure
On reaction-subdiffusion equations
To analyze possible generalizations of reaction-diffusion schemes for the
case of subdiffusion we discuss a simple monomolecular conversion A --> B. We
derive the corresponding kinetic equations for local A and B concentrations.
Their form is rather unusual: The parameters of reaction influence the
diffusion term in the equation for a component A, a consequence of the
nonmarkovian nature of subdiffusion. The equation for a product contains a term
which depends on the concentration of A at all previous times. Our discussion
shows that reaction-subdiffusion equations may not resemble the corresponding
reaction-diffusion ones and are not obtained by a trivial change of the
diffusion operator for a subdiffusion one
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