Distance traveled by random walkers before absorption in a random medium

We consider the penetration length $l$ of random walkers diffusing in a medium of perfect or imperfect absorbers of number density $\rho$. We solve this problem on a lattice and in the continuum in all dimensions $D$, by means of a mean-field renormalization group. For a homogeneous system in $D>2$, we find that $l\sim \max(\xi,\rho^{-1/2})$, where $\xi$ 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 $l$ 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 $A + B \to B$ reactions taking place in three dimensional systems, in which an annihilation of diffusive $A$ particles by diffusive traps $B$ may happen only if the encounter of an $A$ with any of the $B$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 $P(t)$ of an $A$ particle diffusing in $d$-dimensional media in the presence of a concentration $\rho$ of traps $B$ that move sub-diffusively, such that the mean square displacement of each trap grows as $t^{\gamma}$ with $0\leq \gamma \leq 1$. 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 $P(t)$ and show that for $\gamma \leq 2/(d+2)$ 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 $A$ particle sees the traps as essentially immobile, and Lifshitz or trapping tails remain unchanged. For $\gamma > 2/(d+2)$ and $d\leq 2$ 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 $d>2$, however, the upper and lower bounds in this $\gamma$ regime no longer coincide and the decay law for the survival probability of the $A$ 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-$T$ 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 $T^* \propto \sqrt N$, where $N$ 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 $\exp [-Ct^{1/3}\log^{2/3}(t)]$. 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 $d$-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