59 research outputs found
Separating Solution of a Quadratic Recurrent Equation
In this paper we consider the recurrent equation
for with and given. We give conditions
on that guarantee the existence of such that the sequence
with tends to a finite positive limit as .Comment: 13 pages, 6 figures, submitted to J. Stat. Phy
First passage time exponent for higher-order random walks:Using Levy flights
We present a heuristic derivation of the first passage time exponent for the
integral of a random walk [Y. G. Sinai, Theor. Math. Phys. {\bf 90}, 219
(1992)]. Building on this derivation, we construct an estimation scheme to
understand the first passage time exponent for the integral of the integral of
a random walk, which is numerically observed to be . We discuss
the implications of this estimation scheme for the integral of a
random walk. For completeness, we also address the case. Finally, we
explore an application of these processes to an extended, elastic object being
pulled through a random potential by a uniform applied force. In so doing, we
demonstrate a time reparameterization freedom in the Langevin equation that
maps nonlinear stochastic processes into linear ones.Comment: 4 figures, submitted to PR
First-passage and extreme-value statistics of a particle subject to a constant force plus a random force
We consider a particle which moves on the x axis and is subject to a constant
force, such as gravity, plus a random force in the form of Gaussian white
noise. We analyze the statistics of first arrival at point of a particle
which starts at with velocity . The probability that the particle
has not yet arrived at after a time , the mean time of first arrival,
and the velocity distribution at first arrival are all considered. We also
study the statistics of the first return of the particle to its starting point.
Finally, we point out that the extreme-value statistics of the particle and the
first-passage statistics are closely related, and we derive the distribution of
the maximum displacement .Comment: Contains an analysis of the extreme-value statistics not included in
first versio
Translation-invariance of two-dimensional Gibbsian point processes
The conservation of translation as a symmetry in two-dimensional systems with
interaction is a classical subject of statistical mechanics. Here we establish
such a result for Gibbsian particle systems with two-body interaction, where
the interesting cases of singular, hard-core and discontinuous interaction are
included. We start with the special case of pure hard core repulsion in order
to show how to treat hard cores in general.Comment: 44 pages, 6 figure
High Temperature Expansions and Dynamical Systems
We develop a resummed high-temperature expansion for lattice spin systems
with long range interactions, in models where the free energy is not, in
general, analytic. We establish uniqueness of the Gibbs state and exponential
decay of the correlation functions. Then, we apply this expansion to the
Perron-Frobenius operator of weakly coupled map lattices.Comment: 33 pages, Latex; [email protected]; [email protected]
Fractal entropy of a chain of nonlinear oscillators
We study the time evolution of a chain of nonlinear oscillators. We focus on
the fractal features of the spectral entropy and analyze its characteristic
intermediate timescales as a function of the nonlinear coupling. A Brownian
motion is recognized, with an analytic power-law dependence of its diffusion
coefficient on the coupling.Comment: 6 pages, 3 figures, revised version to appear in Phys. Rev.
Anomalous Diffusion in Infinite Horizon Billiards
We consider the long time dependence for the moments of displacement < |r|^q
> of infinite horizon billiards, given a bounded initial distribution of
particles. For a variety of billiard models we find ~ t^g(q) (up to
factors of log t). The time exponent, g(q), is piecewise linear and equal to
q/2 for q2. We discuss the lack of dependence of this result
on the initial distribution of particles and resolve apparent discrepancies
between this time dependence and a prior result. The lack of dependence on
initial distribution follows from a remarkable scaling result that we obtain
for the time evolution of the distribution function of the angle of a
particle's velocity vector.Comment: 11 pages, 7 figures Submitted to Physical Review
On the convergence of cluster expansions for polymer gases
We compare the different convergence criteria available for cluster
expansions of polymer gases subjected to hard-core exclusions, with emphasis on
polymers defined as finite subsets of a countable set (e.g. contour expansions
and more generally high- and low-temperature expansions). In order of
increasing strength, these criteria are: (i) Dobrushin criterion, obtained by a
simple inductive argument; (ii) Gruber-Kunz criterion obtained through the use
of Kirkwood-Salzburg equations, and (iii) a criterion obtained by two of us via
a direct combinatorial handling of the terms of the expansion. We show that for
subset polymers our sharper criterion can be proven both by a suitable
adaptation of Dobrushin inductive argument and by an alternative --in fact,
more elementary-- handling of the Kirkwood-Salzburg equations. In addition we
show that for general abstract polymers this alternative treatment leads to the
same convergence region as the inductive Dobrushin argument and, furthermore,
to a systematic way to improve bounds on correlations
Accelerating cycle expansions by dynamical conjugacy
Periodic orbit theory provides two important functions---the dynamical zeta
function and the spectral determinant for the calculation of dynamical averages
in a nonlinear system. Their cycle expansions converge rapidly when the system
is uniformly hyperbolic but greatly slowed down in the presence of
non-hyperbolicity. We find that the slow convergence can be associated with
singularities in the natural measure. A properly designed coordinate
transformation may remove these singularities and results in a dynamically
conjugate system where fast convergence is restored. The technique is
successfully demonstrated on several examples of one-dimensional maps and some
remaining challenges are discussed
Exact Asymptotic Results for Persistence in the Sinai Model with Arbitrary Drift
We obtain exact asymptotic results for the disorder averaged persistence of a
Brownian particle moving in a biased Sinai landscape. We employ a new method
that maps the problem of computing the persistence to the problem of finding
the energy spectrum of a single particle quantum Hamiltonian, which can be
subsequently found. Our method allows us analytical access to arbitrary values
of the drift (bias), thus going beyond the previous methods which provide
results only in the limit of vanishing drift. We show that on varying the
drift, the persistence displays a variety of rich asymptotic behaviors
including, in particular, interesting qualitative changes at some special
values of the drift.Comment: 17 pages, two eps figures (included
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