58 research outputs found
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 , 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 , where 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
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
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
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
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