82 research outputs found
Distance traveled by random walkers before absorption in a random medium
We consider the penetration length of random walkers diffusing in a
medium of perfect or imperfect absorbers of number density . We solve
this problem on a lattice and in the continuum in all dimensions , by means
of a mean-field renormalization group. For a homogeneous system in , we
find that , where is the absorber density
correlation length. The cases of D=1 and D=2 are also treated. In the presence
of long-range correlations, we estimate the temporal decay of the density of
random walkers not yet absorbed. These results are illustrated by exactly
solvable toy models, and extensive numerical simulations on directed
percolation, where the absorbers are the active sites. Finally, we discuss the
implications of our results for diffusion limited aggregation (DLA), and we
propose a more effective method to measure in DLA clusters.Comment: Final version: also considers the case of imperfect absorber
Kinetics of diffusion-limited catalytically-activated reactions: An extension of the Wilemski-Fixman approach
We study kinetics of diffusion-limited catalytically-activated
reactions taking place in three dimensional systems, in which an annihilation
of diffusive particles by diffusive traps may happen only if the
encounter of an with any of the s happens within a special catalytic
subvolumen, these subvolumens being immobile and uniformly distributed within
the reaction bath. Suitably extending the classical approach of Wilemski and
Fixman (G. Wilemski and M. Fixman, J. Chem. Phys. \textbf{58}:4009, 1973) to
such three-molecular diffusion-limited reactions, we calculate analytically an
effective reaction constant and show that it comprises several terms associated
with the residence and joint residence times of Brownian paths in finite
domains. The effective reaction constant exhibits a non-trivial dependence on
the reaction radii, the mean density of catalytic subvolumens and particles'
diffusion coefficients. Finally, we discuss the fluctuation-induced kinetic
behavior in such systems.Comment: To appear in J. Chem. Phy
Survival probability of a particle in a sea of mobile traps: A tale of tails
We study the long-time tails of the survival probability of an
particle diffusing in -dimensional media in the presence of a concentration
of traps that move sub-diffusively, such that the mean square
displacement of each trap grows as with .
Starting from a continuous time random walk (CTRW) description of the motion of
the particle and of the traps, we derive lower and upper bounds for and
show that for these bounds coincide asymptotically, thus
determining asymptotically exact results. The asymptotic decay law in this
regime is exactly that obtained for immobile traps. This means that for
sufficiently subdiffusive traps, the moving particle sees the traps as
essentially immobile, and Lifshitz or trapping tails remain unchanged. For
and the upper and lower bounds again coincide,
leading to a decay law equal to that of a stationary particle. Thus, in this
regime the moving traps see the particle as essentially immobile. For ,
however, the upper and lower bounds in this regime no longer coincide
and the decay law for the survival probability of the particle remains
ambiguous
Low-temperature specific heat of real crystals: Possibility of leading contribution of optical and short-wavelength acoustical vibrations
We point out that the repeatedly reported glass-like properties of
crystalline materials are not necessarily associated with localized (or
quasilocalized) excitations. In real crystals, optical and short-wavelength
acoustical vibrations remain damped due to defects down to zero temperature. If
such a damping is frequency-independent, e.g. due to planar defects or charged
defects, these optical and short-wavelength acoustical vibrations yield a
linear-in- contribution to the low-temperature specific heat of the crystal
lattices. At low enough temperatures such a contribution will prevail over that
of the long-wavelength acoustical vibrations (Debye contribution). The
crossover between the linear and the Debye regime takes place at , where is the concentration of the defects responsible for the
damping. Estimates show that this crossover could be observable.Comment: 5 pages. v4: Error in Appendix corrected, which does not change the
main results of the pape
Effective conductivity of 2D isotropic two-phase systems in magnetic field
Using the linear fractional transformation, connecting effective
conductivities sigma_{e} of isotropic two-phase systems with and without
magnetic field, explicit approximate expressions for sigma_{e} in a magnetic
field are obtained. They allow to describe sigma_{e} of various inhomogeneous
media at arbitrary phase concentrations x and magnetic fields. the x-dependence
plots of sigma_e at some values of inhomogeneity and magnetic field are
constructed. Their behaviour is qualitatively compatible with the existing
experimental data. The obtained results are applicable for different two-phase
systems (regular and nonregular as well as random), satisfying the symmetry and
self-duality conditions, and admit a direct experimental checking.Comment: 9 pages, 2 figures, Latex2e, small corrections and new figure
Optimal Fluctuations and Tail States of non-Hermitian Operators
A statistical field theory is developed to explore the density of states and
spatial profile of `tail states' at the edge of the spectral support of a
general class of disordered non-Hermitian operators. These states, which are
identified with symmetry broken, instanton field configurations of the theory,
are closely related to localized sub-gap states recently identified in
disordered superconductors. By focusing separately on the problems of a quantum
particle propagating in a random imaginary scalar potential, and a random
imaginary vector potential, we discuss the methodology of our approach and the
universality of the results. Finally, we address potential physical
applications of our findings.Comment: 27 pages AMSLaTeX (with LaTeX2e), 12 eps figures (J. Phys. A, to
appear
Field theory of self-avoiding walks in random media
Based on the analogy with the quantum mechanics of a particle propagating in
a {\em complex} potential, we develop a field-theoretical description of the
statistical properties of a self-avoiding polymer chain in a random
environment. We show that the account of the non-Hermiticity of the quantum
Hamiltonian results in a qualitatively different structure of the effective
action, compared to previous studies. Applying the renormalisation group
analysis, we find a transition between the weak-disorder regime, where the
quenched randomness is irrelevant, and the strong-disorder regime, where the
polymer chain collapses. However, the fact that the renormalised interaction
constants and the chiral symmetry breaking regularisation parameter flow
towards strong coupling raises questions about the applicability of the
perturbative analysis.Comment: RevTeX, 9 pages; accepted for publication in J. Phys.
Analytical results for random walks in the presence of disorder and traps
In this paper, we study the dynamics of a random walker diffusing on a
disordered one-dimensional lattice with random trappings. The distribution of
escape probabilities is computed exactly for any strength of the disorder.
These probabilities do not display any multifractal properties contrary to
previous numerical claims. The explanation for this apparent multifractal
behavior is given, and our conclusion are supported by numerical calculations.
These exact results are exploited to compute the large time asymptotics of the
survival probability (or the density) which is found to decay as . An exact lower bound for the density is found to
decay in a similar way.Comment: 21 pages including 3 PS figures. Submitted to Phys. Rev.
Kinetics of stochastically-gated diffusion-limited reactions and geometry of random walk trajectories
In this paper we study the kinetics of diffusion-limited, pseudo-first-order
A + B -> B reactions in situations in which the particles' intrinsic
reactivities vary randomly in time. That is, we suppose that the particles are
bearing "gates" which interchange randomly and independently of each other
between two states - an active state, when the reaction may take place, and a
blocked state, when the reaction is completly inhibited. We consider four
different models, such that the A particle can be either mobile or immobile,
gated or ungated, as well as ungated or gated B particles can be fixed at
random positions or move randomly. All models are formulated on a
-dimensional regular lattice and we suppose that the mobile species perform
independent, homogeneous, discrete-time lattice random walks. The model
involving a single, immobile, ungated target A and a concentration of mobile,
gated B particles is solved exactly. For the remaining three models we
determine exactly, in form of rigorous lower and upper bounds, the large-N
asymptotical behavior of the A particle survival probability. We also realize
that for all four models studied here such a probalibity can be interpreted as
the moment generating function of some functionals of random walk trajectories,
such as, e.g., the number of self-intersections, the number of sites visited
exactly a given number of times, "residence time" on a random array of lattice
sites and etc. Our results thus apply to the asymptotical behavior of the
corresponding generating functions which has not been known as yet.Comment: Latex, 45 pages, 5 ps-figures, submitted to PR
Optimal Fluctuations and Tail States of non-Hermitian Operators
We develop a general variational approach to study the statistical properties
of the tail states of a wide class of non-Hermitian operators. The utility of
the method, which is a refinement of the instanton approach introduced by
Zittartz and Langer, is illustrated in detail by reference to the problem of a
quantum particle propagating in an imaginary scalar potential.Comment: 4 pages, 2 figures, to appear in PR
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