Anomalous diffusion and stretched exponentials in heterogeneous glass-forming liquids: Low-temperature behavior

We propose a model of a heterogeneous glass forming liquid and compute the low-temperature behavior of a tagged molecule moving within it. This model exhibits stretched-exponential decay of the wavenumber-dependent, self intermediate scattering function in the limit of long times. At temperatures close to the glass transition, where the heterogeneities are much larger in extent than the molecular spacing, the time dependence of the scattering function crosses over from stretched-exponential decay with an index $b=1/2$ at large wave numbers to normal, diffusive behavior with $b = 1$ at small wavenumbers. There is a clear separation between early-stage, cage-breaking $\beta$ relaxation and late-stage $\alpha$ relaxation. The spatial representation of the scattering function exhibits an anomalously broad exponential (non-Gaussian) tail for sufficiently large values of the molecular displacement at all finite times.Comment: 9 pages, 6 figure

Theory of Single File Diffusion in a Force Field

The dynamics of hard-core interacting Brownian particles in an external potential field is studied in one dimension. Using the Jepsen line we find a very general and simple formula relating the motion of the tagged center particle, with the classical, time dependent single particle reflection ${\cal R}$ and transmission ${\cal T}$ coefficients. Our formula describes rich physical behaviors both in equilibrium and the approach to equilibrium of this many body problem.Comment: 4 Phys. Rev. page

Migration and proliferation dichotomy in tumor cell invasion

We propose a two-component reaction-transport model for the migration-proliferation dichotomy in the spreading of tumor cells. By using a continuous time random walk (CTRW) we formulate a system of the balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration. The transport process is formulated in terms of the CTRW with an arbitrary waiting time distribution law. Proliferation is modeled by a standard logistic growth. We apply hyperbolic scaling and Hamilton-Jacobi formalism to determine the overall rate of tumor cell invasion. In particular, we take into account both normal diffusion and anomalous transport (subdiffusion) in order to show that the standard diffusion approximation for migration leads to overestimation of the overall cancer spreading rate.Comment: 9 page

Dimers on two-dimensional lattices

We consider close-packed dimers, or perfect matchings, on two-dimensional regular lattices. We review known results and derive new expressions for the free energy, entropy, and the molecular freedom of dimers for a number of lattices including the simple-quartic (4^4), honeycomb (6^3), triangular (3^6), kagome (3.6.3.6), 3-12 (3.12^2) and its dual [3.12^2], and 4-8 (4.8^2) and its dual Union Jack [4.8^2] Archimedean tilings. The occurrence and nature of phase transitions are also analyzed and discussed.Comment: Typos corrections in Eqs. (28), (32) and (43

Multifractals of Normalized First Passage Time in Sierpinski Gasket

The multifractal behavior of the normalized first passage time is investigated on the two dimensional Sierpinski gasket with both absorbing and reflecting barriers. The normalized first passage time for Sinai model and the logistic model to arrive at the absorbing barrier after starting from an arbitrary site, especially obtained by the calculation via the Monte Carlo simulation, is discussed numerically. The generalized dimension and the spectrum are also estimated from the distribution of the normalized first passage time, and compared with the results on the finitely square lattice.Comment: 10 pages, Latex, with 3 figures and 1 table. to be published in J. Phys. Soc. Jpn. Vol.67(1998

Mean Exit Time and Survival Probability within the CTRW Formalism

An intense research on financial market microstructure is presently in progress. Continuous time random walks (CTRWs) are general models capable to capture the small-scale properties that high frequency data series show. The use of CTRW models in the analysis of financial problems is quite recent and their potentials have not been fully developed. Here we present two (closely related) applications of great interest in risk control. In the first place, we will review the problem of modelling the behaviour of the mean exit time (MET) of a process out of a given region of fixed size. The surveyed stochastic processes are the cumulative returns of asset prices. The link between the value of the MET and the timescale of the market fluctuations of a certain degree is crystal clear. In this sense, MET value may help, for instance, in deciding the optimal time horizon for the investment. The MET is, however, one among the statistics of a distribution of bigger interest: the survival probability (SP), the likelihood that after some lapse of time a process remains inside the given region without having crossed its boundaries. The final part of the article is devoted to the study of this quantity. Note that the use of SPs may outperform the standard "Value at Risk" (VaR) method for two reasons: we can consider other market dynamics than the limited Wiener process and, even in this case, a risk level derived from the SP will ensure (within the desired quintile) that the quoted value of the portfolio will not leave the safety zone. We present some preliminary theoretical and applied results concerning this topic.Comment: 10 pages, 2 figures, revtex4; corrected typos, to appear in the APFA5 proceeding

Aging and Rejuvenation with Fractional Derivatives

We discuss a dynamic procedure that makes the fractional derivatives emerge in the time asymptotic limit of non-Poisson processes. We find that two-state fluctuations, with an inverse power-law distribution of waiting times, finite first moment and divergent second moment, namely with the power index mu in the interval 2<mu <3, yields a generalized master equation equivalent to the sum of an ordinary Markov contribution and of a fractional derivative term. We show that the order of the fractional derivative depends on the age of the process under study. If the system is infinitely old, the order of the fractional derivative, ord, is given by ord=3-mu . A brand new system is characterized by the degree ord=mu -2. If the system is prepared at time -ta<0$ and the observation begins at time t=0, we derive the following scenario. For times 0<t<<ta the system is satisfactorily described by the fractional derivative with ord=3-mu . Upon time increase the system undergoes a rejuvenation process that in the time limit t>>ta yields ord=mu -2. The intermediate time regime is probably incompatible with a picture based on fractional derivatives, or, at least, with a mono-order fractional derivative.Comment: 11 pages, 4 figure

Diffusion of Tagged Particle in an Exclusion Process

We study the diffusion of tagged hard core interacting particles under the influence of an external force field. Using the Jepsen line we map this many particle problem onto a single particle one. We obtain general equations for the distribution and the mean square displacement of the tagged center particle valid for rather general external force fields and initial conditions. A wide range of physical behaviors emerge which are very different than the classical single file sub-diffusion $ \sim t^{1/2}$ found for uniformly distributed particles in an infinite space and in the absence of force fields. For symmetric initial conditions and potential fields we find $ = {{\cal R} (1 - {\cal R})\over 2 N {\it r} ^2} $ where $2 N$ is the (large) number of particles in the system, ${\cal R}$ is a single particle reflection coefficient obtained from the single particle Green function and initial conditions, and $r$ its derivative. We show that this equation is related to the mathematical theory of order statistics and it can be used to find even when the motion between collision events is not Brownian (e.g. it might be ballistic, or anomalous diffusion). As an example we derive the Percus relation for non Gaussian diffusion

Anomalous Drude Model

A generalization of the Drude model is studied. On the one hand, the free motion of the particles is allowed to be sub- or superdiffusive; on the other hand, the distribution of the time delay between collisions is allowed to have a long tail and even a non-vanishing first moment. The collision averaged motion is either regular diffusive or L\'evy-flight like. The anomalous diffusion coefficients show complex scaling laws. The conductivity can be calculated in the diffusive regime. The model is of interest for the phenomenological study of electronic transport in quasicrystals.Comment: 4 pages, latex, 2 figures, to be published in Physical Review Letter

Diffusion-limited reactions and mortal random walkers in confined geometries

Motivated by the diffusion-reaction kinetics on interstellar dust grains, we study a first-passage problem of mortal random walkers in a confined two-dimensional geometry. We provide an exact expression for the encounter probability of two walkers, which is evaluated in limiting cases and checked against extensive kinetic Monte Carlo simulations. We analyze the continuum limit which is approached very slowly, with corrections that vanish logarithmically with the lattice size. We then examine the influence of the shape of the lattice on the first-passage probability, where we focus on the aspect ratio dependence: Distorting the lattice always reduces the encounter probability of two walkers and can exhibit a crossover to the behavior of a genuinely one-dimensional random walk. The nature of this transition is also explained qualitatively.Comment: 18 pages, 16 figure