4,674 research outputs found
Schroedinger's Interpolating Dynamics and Burgers' Flows
We discuss a connection (and a proper place in this framework) of the
unforced and deterministically forced Burgers equation for local velocity
fields of certain flows, with probabilistic solutions of the so-called
Schr\"{o}dinger interpolation problem. The latter allows to reconstruct the
microscopic dynamics of the system from the available probability density data,
or the input-output statistics in the phenomenological situations. An issue of
deducing the most likely dynamics (and matter transport) scenario from the
given initial and terminal probability density data, appropriate e.g. for
studying chaos in terms of densities, is here exemplified in conjunction with
Born's statistical interpretation postulate in quantum theory, that yields
stochastic processes which are compatible with the Schr\"{o}dinger picture free
quantum evolution.Comment: Latex file, to appear in "Chaos, Solitons and Fractals
Periodic Homogenization for Inertial Particles
We study the problem of homogenization for inertial particles moving in a
periodic velocity field, and subject to molecular diffusion. We show that,
under appropriate assumptions on the velocity field, the large scale, long time
behavior of the inertial particles is governed by an effective diffusion
equation for the position variable alone. To achieve this we use a formal
multiple scale expansion in the scale parameter. This expansion relies on the
hypo-ellipticity of the underlying diffusion. An expression for the diffusivity
tensor is found and various of its properties studied. In particular, an
expansion in terms of the non-dimensional particle relaxation time (the
Stokes number) is shown to co-incide with the known result for passive
(non-inertial) tracers in the singular limit . This requires the
solution of a singular perturbation problem, achieved by means of a formal
multiple scales expansion in Incompressible and potential fields are
studied, as well as fields which are neither, and theoretical findings are
supported by numerical simulations.Comment: 31 pages, 7 figures, accepted for publication in Physica D. Typos
corrected. One reference adde
Burgers' Flows as Markovian Diffusion Processes
We analyze the unforced and deterministically forced Burgers equation in the
framework of the (diffusive) interpolating dynamics that solves the so-called
Schr\"{o}dinger boundary data problem for the random matter transport. This
entails an exploration of the consistency conditions that allow to interpret
dispersion of passive contaminants in the Burgers flow as a Markovian diffusion
process. In general, the usage of a continuity equation , where stands for the
Burgers field and is the density of transported matter, is at variance
with the explicit diffusion scenario. Under these circumstances, we give a
complete characterisation of the diffusive transport that is governed by
Burgers velocity fields. The result extends both to the approximate description
of the transport driven by an incompressible fluid and to motions in an
infinitely compressible medium. Also, in conjunction with the Born statistical
postulate in quantum theory, it pertains to the probabilistic (diffusive)
counterpart of the Schr\"{o}dinger picture quantum dynamics.Comment: Latex fil
Stochastic modelling of nonlinear dynamical systems
We develop a general theory dealing with stochastic models for dynamical
systems that are governed by various nonlinear, ordinary or partial
differential, equations. In particular, we address the problem how flows in the
random medium (related to driving velocity fields which are generically bound
to obey suitable local conservation laws) can be reconciled with the notion of
dispersion due to a Markovian diffusion process.Comment: in D. S. Broomhead, E. A. Luchinskaya, P. V. E. McClintock and T.
Mullin, ed., "Stochaos: Stochastic and Chaotic Dynamics in the Lakes",
American Institute of Physics, Woodbury, Ny, in pres
Burgers velocity fields and dynamical transport processes
We explore a connection of the forced Burgers equation with the
Schr\"{o}dinger (diffusive) interpolating dynamics in the presence of
deterministic external forces. This entails an exploration of the consistency
conditions that allow to interpret dispersion of passive contaminants in the
Burgers flow as a Markovian diffusion process. In general, the usage of a
continuity equation , where
stands for the Burgers field and is the
density of transported matter, is at variance with the explicit diffusion
scenario. Under these circumstances, we give a complete characterisation of the
diffusive matter transport that is governed by Burgers velocity fields. The
result extends both to the approximate description of the transport driven by
an incompressible fluid and to motions in an infinitely compressible medium.Comment: Latex fil
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