19 research outputs found
Phase Diagram for Turbulent Transport: Sampling Drift, Eddy Diffusivity and Variational Principles
We study the long-time, large scale transport in a three-parameter family of
isotropic, incompressible velocity fields with power-law spectra. Scaling law
for transport is characterized by the scaling exponent and the Hurst
exponent , as functions of the parameters. The parameter space is divided
into regimes of scaling laws of different {\em functional forms} of the scaling
exponent and the Hurst exponent. We present the full three-dimensional phase
diagram.
The limiting process is one of three kinds: Brownian motion (),
persistent fractional Brownian motions () and regular (or smooth)
motion (H=1).
We discover that a critical wave number divides the infrared cutoffs into
three categories, critical, subcritical and supercritical; they give rise to
different scaling laws and phase diagrams. We introduce the notions of sampling
drift and eddy diffusivity, and formulate variational principles to estimate
the eddy diffusivity. We show that fractional Brownian motions result from a
dominant sampling drift
Relativistic diffusion with friction on a pseudoriemannian manifold
We study a relativistic diffusion equation on the Riemannian phase space
defined by Franchi and Le Jan. We discuss stochastic Ito (Langevin)
differential equations (defining the diffusion) as a perturbation by noise of
the geodesic equation. We show that the expectation value of the angular
momentum and the energy grow exponentially fast. We discuss drifts leading to
an equilibrium. It is shown that the diffusion process corresponding to the
Juettner or quantum equilibrium distributions has a bounded expectation value
of angular momentum and energy. The energy and the angular momentum tend
exponentially fast to their equilibrium values. As examples we discuss a
particle in a plane fronted gravitational wave and a particle in de Sitter
universe. It is shown that the relativistic diffusion of momentum in de Sitter
space is the same as the relativistic diffusion on the Minkowski mass-shell
with the temperature proportional to the de Sitter radius.Comment: the version published in CQ
Time-averaging for weakly nonlinear CGL equations with arbitrary potentials
Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the
form: under the periodic boundary conditions, where and
is a smooth function. Let be
the -basis formed by eigenfunctions of the operator . For a
complex function , write it as and
set . Then for any solution of the linear
equation we have . In this work it is
proved that if equation with a sufficiently smooth real potential
is well posed on time-intervals , then for any its
solution , the limiting behavior of the curve
on time intervals of order , as
, can be uniquely characterized by a solution of a certain
well-posed effective equation:
where is a resonant averaging of the nonlinearity . We
also prove a similar results for the stochastically perturbed equation, when a
white in time and smooth in random force of order is added
to the right-hand side of the equation.
The approach of this work is rather general. In particular, it applies to
equations in bounded domains in under Dirichlet boundary conditions