35 research outputs found
Finite-velocity diffusion on a comb
A Cattaneo equation for a comb structure is considered. We present a rigorous
analysis of the obtained fractional diffusion equation, and corresponding
solutions for the probability distribution function are obtained in the form of
the Fox -function and its infinite series. The mean square displacement
along the backbone is obtained as well in terms of the infinite series of the
Fox -function. The obtained solutions describe the transition from normal
diffusion to subdiffusion, which results from the comb geometry.Comment: 7 page
Anomalous diffusion on a fractal mesh
An exact analytical analysis of anomalous diffusion on a fractal mesh is
presented. The fractal mesh structure is a direct product of two fractal sets
which belong to a main branch of backbones and side branch of fingers. The
fractal sets of both backbones and fingers are constructed on the entire
(infinite) and axises. To this end we suggested a special algorithm of
this special construction. The transport properties of the fractal mesh is
studied, in particular, subdiffusion along the backbones is obtained with the
dispersion relation , where the transport
exponent is determined by the fractal dimensions of both backbone and
fingers. Superdiffusion with has been observed as well when the
environment is controlled by means of a memory kernel
Heterogeneous diffusion in comb and fractal grid structures
We give an exact analytical results for diffusion with a power-law position
dependent diffusion coefficient along the main channel (backbone) on a comb and
grid comb structures. For the mean square displacement along the backbone of
the comb we obtain behavior , where
is the power-law exponent of the position dependent diffusion
coefficient . Depending on the value of we
observe different regimes, from anomalous subdiffusion, superdiffusion, and
hyperdiffusion. For the case of the fractal grid we observe the mean square
displacement, which depends on the fractal dimension of the structure of the
backbones, i.e., , where
is the fractal dimension of the backbones structure. The reduced
probability distribution functions for both cases are obtained by help of the
Fox -functions
First encounters on Bethe Lattices and Cayley Trees
In this work we consider the first encounter problems between a fixed and/or
mobile target A and a moving trap B on Bethe Lattices and Cayley trees. The
survival probability (SP) of the target A on the both kinds of structures are
analyzed analytically and compared. On Bethe Lattices, the results show that
the fixed target will still prolong its survival time, whereas, on Cayley
trees, there are some initial positions where the target should move to prolong
its survival time. The mean first encounter time (MFET) for mobile target A is
evaluated numerically and compared with the mean first passage time (MFPT) for
the fixed target A. Different initial settings are addressed and clear
boundaries are obtained. These findings are helpful for optimizing the strategy
to prolong the survival time of the target or to speed up the search process on
Cayley trees, in relation to the target's movement and the initial position
configuration of the two walkers. We also present a new method, which uses a
small amount of memory, for simulating random walks on Cayley trees.Comment: 25 pages, 11 figure