52 research outputs found
Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations
Quasi-stationary, or metastable, states play an important role in
two-dimensional turbulent fluid flows where they often emerge on time-scales
much shorter than the viscous time scale, and then dominate the dynamics for
very long time intervals. In this paper we propose a dynamical systems
explanation of the metastability of an explicit family of solutions, referred
to as bar states, of the two-dimensional incompressible Navier-Stokes equation
on the torus. These states are physically relevant because they are associated
with certain maximum entropy solutions of the Euler equations, and they have
been observed as one type of metastable state in numerical studies of
two-dimensional turbulence. For small viscosity (high Reynolds number), these
states are quasi-stationary in the sense that they decay on the slow, viscous
timescale. Linearization about these states leads to a time-dependent operator.
We show that if we approximate this operator by dropping a higher-order,
non-local term, it produces a decay rate much faster than the viscous decay
rate. We also provide numerical evidence that the same result holds for the
full linear operator, and that our theoretical results give the optimal decay
rate in this setting.Comment: 21 pages, 2 figures. Version 3: minor error from version 2 correcte
Extinction of multiple shocks in the modular Burgers’ equation
We consider multiple shock waves in the Burgers’ equation with a modular advection term. It was previously shown that the modular Burgers’ equation admits a traveling viscous shock with a single interface, which is stable against smooth and exponentially localized perturbations. In contrast, we suggest in the present work with the help of energy estimates and numerical simulations that the evolution of shock waves with multiple interfaces leads to finite-time coalescence of two consecutive interfaces. We formulate a precise scaling law of the finite-time extinction supported by the interface equations and by numerical simulation
Statistical Physics of Vehicular Traffic and Some Related Systems
In the so-called "microscopic" models of vehicular traffic, attention is paid
explicitly to each individual vehicle each of which is represented by a
"particle"; the nature of the "interactions" among these particles is
determined by the way the vehicles influence each others' movement. Therefore,
vehicular traffic, modeled as a system of interacting "particles" driven far
from equilibrium, offers the possibility to study various fundamental aspects
of truly nonequilibrium systems which are of current interest in statistical
physics. Analytical as well as numerical techniques of statistical physics are
being used to study these models to understand rich variety of physical
phenomena exhibited by vehicular traffic. Some of these phenomena, observed in
vehicular traffic under different circumstances, include transitions from one
dynamical phase to another, criticality and self-organized criticality,
metastability and hysteresis, phase-segregation, etc. In this critical review,
written from the perspective of statistical physics, we explain the guiding
principles behind all the main theoretical approaches. But we present detailed
discussions on the results obtained mainly from the so-called
"particle-hopping" models, particularly emphasizing those which have been
formulated in recent years using the language of cellular automata.Comment: 170 pages, Latex, figures include
Large Scale Stochastic Dynamics
In focus are interacting stochastic systems with many components, ranging from stochastic partial differential equations to discrete systems as interacting particles on a lattice moving through random jumps.
More specifically one wants to understand the large scale behavior, large in spatial extent but also over long time spans, as entailed by the characterization of stationary measures, effective macroscopic evolution laws, transport of conserved fields, homogenization, self-similar structure and scaling, critical dynamics, aging, dynamical phase transitions, large deviations, to mention only a few key items
Interplay of Analysis and Probability in Physics
The main purpose of this workshop was to foster interaction between researchers in the fields of analysis and probability with the aim of joining forces to understand difficult problems from physics rigorously. 52 researchers of all age groups and from many parts of Europe and overseas attended. The talks and discussions evolved around five topics on the interface between analysis and probability. The main goal of the workshop, the systematic encouragement of intense discussions between the two communities, was achieved to a high extent
Cascades and transitions in turbulent flows
Turbulence is characterized by the non-linear cascades of energy and other
inviscid invariants across a huge range of scales, from where they are injected
to where they are dissipated. Recently, new experimental, numerical and
theoretical works have revealed that many turbulent configurations deviate from
the ideal 3D/2D isotropic cases characterized by the presence of a strictly
direct/inverse energy cascade, respectively. We review recent works from a
unified point of view and we present a classification of all known transfer
mechanisms. Beside the classical cases of direct and inverse cascades, the
different scenarios include: split cascades to small and large scales
simultaneously, multiple/dual cascades of different quantities, bi-directional
cascades where direct and inverse transfers of the same invariant coexist in
the same scale-range and finally equilibrium states where no cascades are
present, including the case when a condensate is formed. We classify all
transitions as the control parameters are changed and we analyse when and why
different configurations are observed. Our discussion is based on a set of
paradigmatic applications: helical turbulence, rotating and/or stratified
flows, MHD and passive/active scalars where the transfer properties are altered
as one changes the embedding dimensions, the thickness of the domain or other
relevant control parameters, as the Reynolds, Rossby, Froude, Peclet, or Alfven
numbers. We discuss the presence of anomalous scaling laws in connection with
the intermittent nature of the energy dissipation in configuration space. An
overview is also provided concerning cascades in other applications such as
bounded flows, quantum, relativistic and compressible turbulence, and active
matter, together with implications for turbulent modelling. Finally, we present
a series of open problems and challenges that future work needs to address.Comment: accepted for publication on Physics Reports 201
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