991 research outputs found
Invariant measures for Burgers equation with stochastic forcing
In this paper we study the following Burgers equation
du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t)
where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x
and white noise in t. We prove the existence and uniqueness of an invariant
measure by establishing a ``one force, one solution'' principle, namely that
for almost every realization of the force, there is a unique distinguished
solution that exists for the time interval (-infty, +infty) and this solution
attracts all other solutions with the same forcing. This is done by studying
the so-called one-sided minimizers. We also give a detailed description of the
structure and regularity properties for the stationary solutions. In
particular, we prove, under some non-degeneracy conditions on the forcing, that
almost surely there is a unique main shock and a unique global minimizer for
the stationary solutions. Furthermore the global minimizer is a hyperbolic
trajectory of the underlying system of characteristics.Comment: 84 pages, published version, abstract added in migratio
Non-ergodicity of the motion in three dimensional steep repelling dispersing potentials
It is demonstrated numerically that smooth three degrees of freedom
Hamiltonian systems which are arbitrarily close to three dimensional strictly
dispersing billiards (Sinai billiards) have islands of effective stability, and
hence are non-ergodic. The mechanism for creating the islands are corners of
the billiard domain.Comment: 6 pages, 8 figures, submitted to Chao
Turbulence without pressure
We develop exact field theoretic methods to treat turbulence when the effect
of pressure is negligible. We find explicit forms of certain probability
distributions, demonstrate that the breakdown of Galilean invariance is
responsible for intermittency and establish the operator product expansion. We
also indicate how the effects of pressure can be turned on perturbatively.Comment: 12 page
Nonequilibrium, thermostats and thermodynamic limit
The relation between thermostats of "isoenergetic" and "frictionless" kind is
studied and their equivalence in the thermodynamic limit is proved in space
dimension and, for special geometries, .Comment: 22 pages PRA 2-columns format v3: Typos corrected as acknowledge
Proving The Ergodic Hypothesis for Billiards With Disjoint Cylindric Scatterers
In this paper we study the ergodic properties of mathematical billiards
describing the uniform motion of a point in a flat torus from which finitely
many, pairwise disjoint, tubular neighborhoods of translated subtori (the so
called cylindric scatterers) have been removed. We prove that every such system
is ergodic (actually, a Bernoulli flow), unless a simple geometric obstacle for
the ergodicity is present.Comment: 24 pages, AMS-TeX fil
Bifractality of the Devil's staircase appearing in the Burgers equation with Brownian initial velocity
It is shown that the inverse Lagrangian map for the solution of the Burgers
equation (in the inviscid limit) with Brownian initial velocity presents a
bifractality (phase transition) similar to that of the Devil's staircase for
the standard triadic Cantor set. Both heuristic and rigorous derivations are
given. It is explained why artifacts can easily mask this phenomenon in
numerical simulations.Comment: 12 pages, LaTe
Survival of a Diffusing Particle in a Transverse Shear Flow: A First-Passage Problem with Continuously Varying Persistence Exponent
We consider a particle diffusing in the y-direction, dy/dt=\eta(t), subject
to a transverse shear flow in the x-direction, dx/dt=f(y), where x \ge 0 and
x=0 is an absorbing boundary. We treat the class of models defined by f(y) =
\pm v_{\pm}(\pm y)^\alpha where the upper (lower) sign refers to y>0 (y<0). We
show that the particle survives with probability Q(t) \sim t^{-\theta} with
\theta = 1/4, independent of \alpha, if v_{+}=v_{-}. If v_{+} \ne v_{-},
however, we show that \theta depends on both \alpha and the ratio v_{+}/v_{-},
and we determine this dependence.Comment: 4 page
Random walks in a random environment on a strip: a renormalization group approach
We present a real space renormalization group scheme for the problem of
random walks in a random environment on a strip, which includes one-dimensional
random walk in random environment with bounded non-nearest-neighbor jumps. We
show that the model renormalizes to an effective one-dimensional random walk
problem with nearest-neighbor jumps and conclude that Sinai scaling is valid in
the recurrent case, while in the sub-linear transient phase, the displacement
grows as a power of the time.Comment: 9 page
Critical density of a soliton gas
We quantify the notion of a dense soliton gas by establishing an upper bound
for the integrated density of states of the quantum-mechanical Schr\"odinger
operator associated with the KdV soliton gas dynamics. As a by-product of our
derivation we find the speed of sound in the soliton gas with Gaussian spectral
distribution function.Comment: 7 page
On transition to bursting via deterministic chaos
We study statistical properties of the irregular bursting arising in a class
of neuronal models close to the transition from spiking to bursting. Prior to
the transition to bursting, the systems in this class develop chaotic
attractors, which generate irregular spiking. The chaotic spiking gives rise to
irregular bursting. The duration of bursts near the transition can be very
long. We describe the statistics of the number of spikes and the interspike
interval distributions within one burst as functions of the distance from
criticality.Comment: 8 pages, 6 figure
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