Statistics of Long-Range Force Fields in Random Environments: Beyond Holtsmark

Since the times of Holtsmark (1911), statistics of fields in random environments have been widely studied, for example in astrophysics, active matter, and line-shape broadening. The power-law decay of the two-body interaction, of the form $1/|r|^\delta$, and assuming spatial uniformity of the medium particles exerting the forces, imply that the fields are fat-tailed distributed, and in general are described by stable L\'evy distributions. With this widely used framework, the variance of the field diverges, which is non-physical, due to finite size cutoffs. We find a complementary statistical law to the L\'evy-Holtsmark distribution describing the large fields in the problem, which is related to the finite size of the tracer particle. We discover bi-scaling, with a sharp statistical transition of the force moments taking place when the order of the moment is $d/\delta$, where $d$ is the dimension. The high-order moments, including the variance, are described by the framework presented in this paper, which is expected to hold for many systems. The new scaling solution found here is non-normalized similar to infinite invariant densities found in dynamical systems.Comment: 9 pages 2 figure

A Random Walk to a Non-Ergodic Equilibrium Concept

Random walk models, such as the trap model, continuous time random walks, and comb models exhibit weak ergodicity breaking, when the average waiting time is infinite. The open question is: what statistical mechanical theory replaces the canonical Boltzmann-Gibbs theory for such systems? In this manuscript a non-ergodic equilibrium concept is investigated, for a continuous time random walk model in a potential field. In particular we show that in the non-ergodic phase the distribution of the occupation time of the particle on a given lattice point, approaches U or W shaped distributions related to the arcsin law. We show that when conditions of detailed balance are applied, these distributions depend on the partition function of the problem, thus establishing a relation between the non-ergodic dynamics and canonical statistical mechanics. In the ergodic phase the distribution function of the occupation times approaches a delta function centered on the value predicted based on standard Boltzmann-Gibbs statistics. Relation of our work with single molecule experiments is briefly discussed.Comment: 14 pages, 6 figure

Anti-Semitism and apostasy in Nineteenth-Century France: A response to Jonathan Helfand

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43005/1/10835_2005_Article_BF01668931.pd

Some exact results for the trapping of subdiffusive particles in one dimension

We study a generalization of the standard trapping problem of random walk theory in which particles move subdiffusively on a one-dimensional lattice. We consider the cases in which the lattice is filled with a one-sided and a two-sided random distribution of static absorbing traps with concentration c. The survival probability Phi(t) that the random walker is not trapped by time t is obtained exactly in both versions of the problem through a fractional diffusion approach. Comparison with simulation results is madeComment: 15 pages, 2 figure

Subdiffusion-limited reactions

We consider the coagulation dynamics A+A -> A and A+A A and the annihilation dynamics A+A -> 0 for particles moving subdiffusively in one dimension. This scenario combines the "anomalous kinetics" and "anomalous diffusion" problems, each of which leads to interesting dynamics separately and to even more interesting dynamics in combination. Our analysis is based on the fractional diffusion equation

Quantitative analysis of single particle trajectories: mean maximal excursion method

An increasing number of experimental studies employ single particle tracking to probe the physical environment in complex systems. We here propose and discuss new methods to analyze the time series of the particle traces, in particular, for subdiffusion phenomena. We discuss the statistical properties of mean maximal excursions, i.e., the maximal distance covered by a test particle up to time t. Compared to traditional methods focusing on the mean squared displacement we show that the mean maximal excursion analysis performs better in the determination of the anomalous diffusion exponent. We also demonstrate that combination of regular moments with moments of the mean maximal excursion method provides additional criteria to determine the exact physical nature of the underlying stochastic subdiffusion processes. We put the methods to test using experimental data as well as simulated time series from different models for normal and anomalous dynamics, such as diffusion on fractals, continuous time random walks, and fractional Brownian motion.Comment: 10 pages, 7 figures, 2 tables. NB: Supplementary material may be found in the downloadable source file

Anomalous versus slowed-down Brownian diffusion in the ligand-binding equilibrium

Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) which converges back to a Brownian motion with reduced diffusion coefficient at long times, after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed-down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the reversible ligand-binding reaction in 2d. We compare the (long-time) equilibrium properties obtained with transient anomalous diffusion due to obstacle hindrance or power-law distributed residence times (continuous-time random walks) to those obtained with space-dependent slowed-down Brownian motion. Using theoretical arguments and Monte-Carlo simulations, we show that those three scenarios have distinctive effects on the apparent affinity of the reaction. While continuous-time random walks decrease the apparent affinity of the reaction, locally slowed-down Brownian motion and local hinderance by obstacles both improve it. However, only in the case of slowed-down Brownian motion, the affinity is maximal when the slowdown is restricted to a subregion of the available space. Hence, even at long times (equilibrium), these processes are different and exhibit irreconcilable behaviors when the area fraction of reduced mobility changes.Comment: Biophysical Journal (2013

Reaction Front in an A+B -> C Reaction-Subdiffusion Process

We study the reaction front for the process A+B -> C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular its unusual behavior at the center of the reaction zone

Inelastically scattering particles and wealth distribution in an open economy

Using the analogy with inelastic granular gasses we introduce a model for wealth exchange in society. The dynamics is governed by a kinetic equation, which allows for self-similar solutions. The scaling function has a power-law tail, the exponent being given by a transcendental equation. In the limit of continuous trading, closed form of the wealth distribution is calculated analytically.Comment: 8 pages 5 figure

Upscaling Flow and Transport Processes

Peer reviewe