166 research outputs found
Dynamics of continuous-time quantum walks in restricted geometries
We study quantum transport on finite discrete structures and we model the
process by means of continuous-time quantum walks. A direct and effective
comparison between quantum and classical walks can be attained based on the
average displacement of the walker as a function of time. Indeed, a fast growth
of the average displacement can be advantageously exploited to build up
efficient search algorithms. By means of analytical and numerical
investigations, we show that the finiteness and the inhomogeneity of the
substrate jointly weaken the quantum walk performance. We further highlight the
interplay between the quantum-walk dynamics and the underlying topology by
studying the temporal evolution of the transfer probability distribution and
the lower bound of long time averages.Comment: 25 pages, 13 figure
On the influence of reflective boundary conditions on the statistics of Poisson-Kac diffusion processes
We analyze the influence of reflective boundary conditions on the statistics
of Poisson-Kac diffusion processes, and specifically how they modify the
Poissonian switching-time statistics. After addressing simple cases such as
diffusion in a channel, and the switching statistics in the presence of a
polarization potential, we thoroughly study Poisson-Kac diffusion in fractal
domains. Diffusion in fractal spaces highlights neatly how the modification in
the switching-time statistics associated with reflections against a complex and
fractal boundary induces new emergent features of Poisson-Kac diffusion leading
to a transition from a regular behavior at shorter timescales to emerging
anomalous diffusion properties controlled by walk dimensionality of the fractal
set
Geometry-controlled kinetics
It has long been appreciated that transport properties can control reaction
kinetics. This effect can be characterized by the time it takes a diffusing
molecule to reach a target -- the first-passage time (FPT). Although essential
to quantify the kinetics of reactions on all time scales, determining the FPT
distribution was deemed so far intractable. Here, we calculate analytically
this FPT distribution and show that transport processes as various as regular
diffusion, anomalous diffusion, diffusion in disordered media and in fractals
fall into the same universality classes. Beyond this theoretical aspect, this
result changes the views on standard reaction kinetics. More precisely, we
argue that geometry can become a key parameter so far ignored in this context,
and introduce the concept of "geometry-controlled kinetics". These findings
could help understand the crucial role of spatial organization of genes in
transcription kinetics, and more generally the impact of geometry on
diffusion-limited reactions.Comment: Submitted versio
Random walks on graphs: ideas, techniques and results
Random walks on graphs are widely used in all sciences to describe a great
variety of phenomena where dynamical random processes are affected by topology.
In recent years, relevant mathematical results have been obtained in this
field, and new ideas have been introduced, which can be fruitfully extended to
different areas and disciplines. Here we aim at giving a brief but
comprehensive perspective of these progresses, with a particular emphasis on
physical aspects.Comment: LateX file, 34 pages, 13 jpeg figures, Topical Revie
Exact calculations of first-passage quantities on recursive networks
We present general methods to exactly calculate mean-first passage quantities
on self-similar networks defined recursively. In particular, we calculate the
mean first-passage time and the splitting probabilities associated to a source
and one or several targets; averaged quantities over a given set of sources
(e.g., same-connectivity nodes) are also derived. The exact estimate of such
quantities highlights the dependency of first-passage processes with respect to
the source-target distance, which has recently revealed to be a key parameter
to characterize transport in complex media. We explicitly perform calculations
for different classes of recursive networks (finitely ramified fractals,
scale-free (trans)fractals, non-fractals, mixtures between fractals and
non-fractals, non-decimable hierarchical graphs) of arbitrary size. Our
approach unifies and significantly extends the available results in the field.Comment: 16 pages, 10 figure
Single-file diffusion on self-similar substrates
We study the single file diffusion problem on a one-dimensional lattice with
a self-similar distribution of hopping rates. We find that the time dependence
of the mean-square displacement of both a tagged particle and the center of
mass of the system present anomalous power laws modulated by logarithmic
periodic oscillations. The anomalous exponent of a tagged particle is one half
of the exponent of the center of mass, and always smaller than 1/4. Using
heuristic arguments, the exponents and the periods of oscillation are
analytically obtained and confirmed by Monte Carlo simulations.Comment: 12 pages, 6 figure
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