2,083 research outputs found
On the Mixing of Diffusing Particles
We study how the order of N independent random walks in one dimension evolves
with time. Our focus is statistical properties of the inversion number m,
defined as the number of pairs that are out of sort with respect to the initial
configuration. In the steady-state, the distribution of the inversion number is
Gaussian with the average <m>~N^2/4 and the standard deviation sigma N^{3/2}/6.
The survival probability, S_m(t), which measures the likelihood that the
inversion number remains below m until time t, decays algebraically in the
long-time limit, S_m t^{-beta_m}. Interestingly, there is a spectrum of
N(N-1)/2 distinct exponents beta_m(N). We also find that the kinetics of
first-passage in a circular cone provides a good approximation for these
exponents. When N is large, the first-passage exponents are a universal
function of a single scaling variable, beta_m(N)--> beta(z) with
z=(m-<m>)/sigma. In the cone approximation, the scaling function is a root of a
transcendental equation involving the parabolic cylinder equation, D_{2
beta}(-z)=0, and surprisingly, numerical simulations show this prediction to be
exact.Comment: 9 pages, 6 figures, 2 table
Pattern Selection and Super-patterns in the Bounded Confidence Model
We study pattern formation in the bounded confidence model of opinion
dynamics. In this random process, opinion is quantified by a single variable.
Two agents may interact and reach a fair compromise, but only if their
difference of opinion falls below a fixed threshold. Starting from a uniform
distribution of opinions with compact support, a traveling wave forms and it
propagates from the domain boundary into the unstable uniform state.
Consequently, the system reaches a steady state with isolated clusters that are
separated by distance larger than the interaction range. These clusters form a
quasi-periodic pattern where the sizes of the clusters and the separations
between them are nearly constant. We obtain analytically the average separation
between clusters L. Interestingly, there are also very small quasi-periodic
modulations in the size of the clusters. The spatial periods of these
modulations are a series of integers that follow from the continued fraction
representation of the irrational average separation L.Comment: 6 pages, 6 figure
Fragmentation of Random Trees
We study fragmentation of a random recursive tree into a forest by repeated
removal of nodes. The initial tree consists of N nodes and it is generated by
sequential addition of nodes with each new node attaching to a
randomly-selected existing node. As nodes are removed from the tree, one at a
time, the tree dissolves into an ensemble of separate trees, namely, a forest.
We study statistical properties of trees and nodes in this heterogeneous
forest, and find that the fraction of remaining nodes m characterizes the
system in the limit N --> infty. We obtain analytically the size density phi_s
of trees of size s. The size density has power-law tail phi_s ~ s^(-alpha) with
exponent alpha=1+1/m. Therefore, the tail becomes steeper as further nodes are
removed, and the fragmentation process is unusual in that exponent alpha
increases continuously with time. We also extend our analysis to the case where
nodes are added as well as removed, and obtain the asymptotic size density for
growing trees.Comment: 9 pages, 5 figure
Stationary states and energy cascades in inelastic gases
We find a general class of nontrivial stationary states in inelastic gases
where, due to dissipation, energy is transfered from large velocity scales to
small velocity scales. These steady-states exist for arbitrary collision rules
and arbitrary dimension. Their signature is a stationary velocity distribution
f(v) with an algebraic high-energy tail, f(v) ~ v^{-sigma}. The exponent sigma
is obtained analytically and it varies continuously with the spatial dimension,
the homogeneity index characterizing the collision rate, and the restitution
coefficient. We observe these stationary states in numerical simulations in
which energy is injected into the system by infrequently boosting particles to
high velocities. We propose that these states may be realized experimentally in
driven granular systems.Comment: 4 pages, 4 figure
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