We investigate majority rule dynamics in a population with two classes of people, each with two opinion states $pm 1$, and with tunable interactions between people in different classes. In an update, a randomly selected group adopts the majority opin
ion if all group members belong to the same class; if not, majority rule is applied with probability $epsilon$. Consensus is achieved in a time that scales logarithmically with population size if $epsilongeq epsilon_c=frac{1}{9}$. For $epsilon <epsilon_c$, the population can get trapped in a polarized state, with one class preferring the $+1$ state and the other preferring $-1$. The time to escape this polarized state and reach consensus scales exponentially with population size.
A gas composed of a large number of atoms evolving according to Newtonian dynamics is often described by continuum hydrodynamics. Proving this rigorously is an outstanding open problem, and precise numerical demonstrations of the equivalence of the h
ydrodynamic and microscopic descriptions are rare. We test this equivalence in the context of the evolution of a blast wave, a problem that is expected to be at the limit where hydrodynamics could work. We study a one-dimensional gas at rest with instantaneous localized release of energy for which the hydrodynamic Euler equations admit a self-similar scaling solution. Our microscopic model consists of hard point particles with alternating masses, which is a nonintegrable system with strong mixing dynamics. Our extensive microscopic simulations find a remarkable agreement with Euler hydrodynamics, with deviations in a small core region that are understood as arising due to heat conduction.
We introduce and study the dynamics of an emph{immortal} critical branching process. In the classic, critical branching process, particles give birth to a single offspring or die at the same rates. Even though the average population is constant in ti
me, the ultimate fate of the population is extinction. We augment this branching process with immortality by positing that either: (a) a single particle cannot die, or (b) there exists an immortal stem cell that gives birth to ordinary cells that can subsequently undergo critical branching. We discuss the new dynamical aspects of this immortal branching process.
We investigate parking in a one-dimensional lot, where cars enter at a rate $lambda$ and each attempts to park close to a target at the origin. Parked cars also depart at rate 1. An entering driver cannot see beyond the parked cars for more desirable
open spots. We analyze a class of strategies in which a driver ignores open spots beyond $tau L$, where $tau$ is a risk threshold and $L$ is the location of the most distant parked car, and attempts to park at the first available spot encountered closer than $tau L$. When all drivers use this strategy, the probability to park at the best available spot is maximal when $tau=frac{1}{2}$, and parking at the best available spot occurs with probability $frac{1}{4}$.
The time which a diffusing particle spends in a certain region of space is known as the occupation time, or the residence time. Recently the joint occupation time statistics of an ensemble of non-interacting particles was addressed using the single-p
article statistics. Here we employ the Macroscopic Fluctuation Theory (MFT) to study the occupation time statistics of many emph{interacting} particles. We find that interactions can significantly change the statistics and, in some models, even cause a singularity of the large-deviation function describing these statistics. This singularity can be interpreted as a dynamical phase transition. We also point out to a close relation between the MFT description of the occupation-time statistics of non-interacting particles and the level 2 large deviation formalism which describes the occupation-time statistics of a single particle.
We introduce and study a simple and natural class of solvable stochastic lattice gases. This is the class of emph{Strong Particles}. The name is due to the fact that when they try to jump to an occupied site they succeed pushing away a pile of partic
les. For this class of models we explicitly compute the transport coefficients. We also discuss some generalizations and the relations with other classes of solvable models.
We analyze the joint distributions and temporal correlations between the partial maximum $m$ and the global maximum $M$ achieved by a Brownian Bridge on the subinterval $[0,t_1]$ and on the entire interval $[0,t]$, respectively. We determine three pr
obability distribution functions: The joint distribution $P(m,M)$ of both maxima; the distribution $P(m)$ of the partial maximum; and the distribution $Pi(G)$ of the gap between the maxima, $G = M-m$. We present exact results for the moments of these distributions and quantify the temporal correlations between $m$ and $M$ by calculating the Pearson correlation coefficient.
We study the correlations between the maxima $m$ and $M$ of a Brownian motion (BM) on the time intervals $[0,t_1]$ and $[0,t_2]$, with $t_2>t_1$. We determine exact forms of the distribution functions $P(m,M)$ and $P(G = M - m)$, and calculate the mo
ments $mathbb{E}{left(M - mright)^k}$ and the cross-moments $mathbb{E}{m^l M^k}$ with arbitrary integers $l$ and $k$. We show that correlations between $m$ and $M$ decay as $sqrt{t_1/t_2}$ when $t_2/t_1 to infty$, revealing strong memory effects in the statistics of the BM maxima. We also compute the Pearson correlation coefficient $rho(m,M)$, the power spectrum of $M_t$, and we discuss a possibility of extracting the ensemble-averaged diffusion coefficient in single-trajectory experiments using a single realization of the maximum process.
We investigate statistics of lead changes of the maxima of two discrete-time random walks in one dimension. We show that the average number of lead changes grows as $pi^{-1}ln(t)$ in the long-time limit. We present theoretical and numerical evidence
that this asymptotic behavior is universal. Specifically, this behavior is independent of the jump distribution: the same asymptotic underlies standard Brownian motion and symmetric Levy flights. We also show that the probability to have at most n lead changes behaves as $t^{-1/4}[ln t]^n$ for Brownian motion and as $t^{-beta(mu)}[ln t]^n$ for symmetric Levy flights with index $mu$. The decay exponent $beta(mu)$ varies continuously with the Levy index when $0<mu<2$, while $beta=1/4$ for $mu>2$.
We study the statistics of a tagged particle in single-file diffusion, a one-dimensional interacting infinite-particle system in which the order of particles never changes. We compute the two-time correlation function for the displacement of the tagg
ed particle for an arbitrary single-file system. We also discuss single-file analogs of the arcsine law and the law of the iterated logarithm characterizing the behavior of Brownian motion. Using a macroscopic fluctuation theory we devise a formalism giving the cumulant generating functional. In principle, this functional contains the full statistics of the tagged particle trajectory---the full single-time statistics, all multiple-time correlation functions, etc. are merely special cases.
