arXiv subjects

R. Metzler

Publications and source records attributed to R. Metzler.

47 records · Page 3Linked to original sources

arXiv2013

Trajectory-to-trajectory fluctuations in first-passage phenomena in bounded domains

We study the statistics of the first passage of a random walker to absorbing subsets of the boundary of compact domains in different spatial dimensions. We describe a novel diagnostic method to quantify the trajectory-to-trajectory fluctuations of the first passage, based on the distribution of the so-called uniformity index $ω$, measuring the similarity of the first passage times of two independent walkers starting at the same location. We show that the characteristic shape of $P(ω)$ exhibits a transition from unimodal to bimodal, depending on the starting point of the trajectories. From the study of different geometries in one, two and three dimensions, we conclude that this transition is a generic property of first passage phenomena in bounded domains. Our results show that, in general, the Mean First Passage Time (MFPT) is a meaningful characteristic measure of the first passage behaviour only when the Brownian walkers start sufficiently far from the absorbing boundary. Strikingly, in the opposite case, the first passage statistics exhibit large trajectory-to-trajectory fluctuations and the MFPT is not representative of the actual behaviour.

cond-mat.stat-mech↗

arXiv2011

First passage behaviour of fractional Brownian motion in two-dimensional wedge domains

We study the survival probability and the corresponding first passage time density of fractional Brownian motion confined to a two-dimensional open wedge domain with absorbing boundaries. By analytical arguments and numerical simulation we show that in the long time limit the first passage time density scales as t**{-1+pi*(2H-2)/(2*Theta)} in terms of the Hurst exponent H and the wedge angle Theta. We discuss this scaling behaviour in connection with the reaction kinetics of FBM particles in a one-dimensional domain.

cond-mat.stat-mech↗

arXiv2010

Generalized Khinchin Theorem for a Class of Aging Processes

The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by non-stationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and provide a generalization of Khinchin's theorem. Our work quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics.

cond-mat.stat-mech↗

arXiv2008

Random Time-Scale Invariant Diffusion and Transport Coefficients

Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement $\overline{δ^2}$ of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that $\overline{δ^2}$ differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable $\overline{δ^2}$. Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed.

cond-mat.stat-mech↗

arXiv2005

Directed motion emerging from two coupled random processes: Translocation of a chain through a membrane nanopore driven by binding proteins

We investigate the translocation of a stiff polymer consisting of M monomers through a nanopore in a membrane, in the presence of binding particles (chaperones) that bind onto the polymer, and partially prevent backsliding of the polymer through the pore. The process is characterized by the rates: k for the polymer to make a diffusive jump through the pore, q for unbinding of a chaperone, and the rate q kappa for binding (with a binding strength kappa); except for the case of no binding kappa=0 the presence of the chaperones give rise to an effective force that drives the translocation process. Based on a (2+1) variate master equation, we study in detail the coupled dynamics of diffusive translocation and (partial) rectification by the binding proteins. In particular, we calculate the mean translocation time as a function of the various physical parameters.

q-bio.BM↗

arXiv2004

Lévy flights as subordination process: first passage times

We obtain the first passage time density for a Lévy flight random process from a subordination scheme. By this method, we infer the asymptotic behavior directly from the Brownian solution and the Sparre Andersen theorem, avoiding explicit reference to the fractional diffusion equation. Our results corroborate recent findings for Markovian Lévy flights and generalize to broad waiting times.

cond-mat.stat-mech↗

arXiv2002

The dynamical foundation of fractal stream chemistry: The origin of extremely long retention times

We present a physical model to explain the behavior of long-term, time series measurements of chloride, a natural passive tracer, in rainfall and runoff in catchments [Kirchner et al., Nature 403(524), 2000]. A spectral analysis of the data shows the chloride concentrations in rainfall to have a white noise spectrum, while in streamflow, the spectrum exhibits a fractal $1/f$ scaling. The empirically derived distribution of tracer travel times $h(t)$ follows a power-law, indicating low-level contaminant delivery to streams for a very long time. Our transport model is based on a continuous time random walk (CTRW) with an event time distribution governed by $ψ(t)\sim A_βt^{-1-β}$. The CTRW using this power-law $ψ(t)$ (with $0<β<1$) is interchangeable with the time-fractional advection-dispersion equation (FADE) and has accounted for the universal phenomenon of anomalous transport in a broad range of disordered and complex systems. In the current application, the events can be realized as transit times on portions of the catchment network. The travel time distribution is the first passage time distribution $F(t; l)$ at a distance $l$ from a pulse input (at $t=0$) at the origin. We show that the empirical $h(t)$ is the catchment areal composite of $F(t; l)$ and that the fractal $1/f$ spectral response found in many catchments is an example of the larger class of transport phenomena cited above. The physical basis of $ψ(t)$, which determines $F(t; l)$, is the origin of the extremely long chemical retention times in catchments.

cond-mat.stat-mech↗

arXiv2000

On time's arrow in Ehrenfest models with reversible deterministic dynamics

We introduce a deterministic, time-reversible version of the Ehrenfest urn model. The distribution of first-passage times from equilibrium to non-equilibrium states and vice versa is calculated. We find that average times for transition to non-equilibrium always scale exponentially with the system size, whereas the time scale for relaxation to equilibrium depends on microscopic dynamics. To illustrate this, we also look at deterministic and stochastic versions of the Ehrenfest model with a distribution of microscopic relaxation times.

cond-mat.stat-mech↗

arXiv2000

A New Stochastic Strategy for the Minority Game

We present a variant of the Minority Game in which players who where successful in the previous timestep stay with their decision, while the losers change their decision with a probability $p$. Analytical results for different regimes of $p$ and the number of players $N$ are given and connections to existing models are discussed. It is shown that for $p \propto 1/N$ the average loss $σ^2$ is of the order of 1 and does not increase with $N$ as for other known strategies.

cond-mat.stat-mech↗

arXiv2000

Interacting Neural Networks

Several scenarios of interacting neural networks which are trained either in an identical or in a competitive way are solved analytically. In the case of identical training each perceptron receives the output of its neighbour. The symmetry of the stationary state as well as the sensitivity to the used training algorithm are investigated. Two competitive perceptrons trained on mutually exclusive learning aims and a perceptron which is trained on the opposite of its own output are examined analytically. An ensemble of competitive perceptrons is used as decision-making algorithms in a model of a closed market (El Farol Bar problem or Minority Game); each network is trained on the history of minority decisions. This ensemble of perceptrons relaxes to a stationary state whose performance can be better than random.

cond-mat.dis-nn↗

arXiv2000

Dynamics of Interacting Neural Networks

The dynamics of interacting perceptrons is solved analytically. For a directed flow of information the system runs into a state which has a higher symmetry than the topology of the model. A symmetry breaking phase transition is found with increasing learning rate. In addition it is shown that a system of interacting perceptrons which is trained on the history of its minority decisions develops a good strategy for the problem of adaptive competition known as the Bar Problem or Minority Game.

cond-mat.dis-nn↗