67 research outputs found
Phase-space Lagrangian dynamics of incompressible thermofluids
Phase-space Lagrangian dynamics in ideal fluids (i.e, continua) is usually
related to the so-called {\it ideal tracer particles}. The latter, which can in
principle be permitted to have arbitrary initial velocities, are understood as
particles of infinitesimal size which do not produce significant perturbations
of the fluid and do not interact among themselves. An unsolved theoretical
problem is the correct definition of their dynamics in ideal fluids. The issue
is relevant in order to exhibit the connection between fluid dynamics and the
classical dynamical system, underlying a prescribed fluid system, which
uniquely generates its time-evolution. \
The goal of this paper is to show that the tracer-particle dynamics can be
{\it exactly} established for an arbitrary incompressible fluid uniquely based
on the construction of an inverse kinetic theory (IKT) (Tessarotto \textit{et
al.}, 2000-2008). As an example, the case of an incompressible Newtonian
thermofluid is here considered.Comment: submitted to Physica
Modelling of anthropogenic pollutant diffusion in the atmosphere and applications to civil protection monitoring
A basic feature of fluid mechanics concerns the frictionless phase-space
dynamics of particles in an incompressible fluid. The issue, besides its
theoretical interest in turbulence theory, is important in many applications,
such as the pollutant dynamics in the atmosphere, a problem relevant for civil
protection monitoring of air quality. Actually, both the numerical simulation
of the ABL (atmospheric boundary layer) portion of the atmosphere and that of
pollutant dynamics may generally require the correct definition of the
Lagrangian dynamics which characterizes arbitrary fluid elements of
incompressible thermofluids. We claim that particularly important for
applications would be to consider these trajectories as phase-space
trajectories. This involves, however, the unfolding of a fundamental
theoretical problem up to now substantially unsolved: {\it namely the
determination of the exact frictionless dynamics of tracer particles in an
incompressible fluid, treated either as a deterministic or a turbulent (i.e.,
stochastic) continuum.} In this paper we intend to formulate the necessary
theoretical framework to construct such a type of description. This is based on
a phase-space inverse kinetic theory (IKT) approach recently developed for
incompressible fluids (Ellero \textit{et al.}, 2004-2008). {\it Our claim is
that the conditional frictionless dynamics of a tracer particles - which
corresponds to a prescribed velocity probability density and an arbitrary
choice of the relevant fluid fields - can be exactly specified}.Comment: Contributed paper at RGD26 (Kyoto, Japan, July 2008
Hamiltonian approach to GR - Part 1: covariant theory of classical gravity
A challenging issue in General Relativity concerns the determination of the
manifestly-covariant continuum Hamiltonian structure underlying the Einstein
field equations and the related formulation of the corresponding covariant
Hamilton-Jacobi theory. The task is achieved by adopting a synchronous
variational principle requiring distinction between the prescribed
deterministic metric tensor solution of the Einstein field equations which determines the
geometry of the background space-time and suitable variational fields obeying an appropriate set of continuum Hamilton
equations, referred to here as GR-Hamilton equations It is shown that a
prerequisite for reaching such a goal is that of casting the same equations in
evolutionary form by means of a Lagrangian parametrization for a
suitably-reduced canonical state. As a result, the corresponding
Hamilton-Jacobi theory is established in manifestly-covariant form. Physical
implications of the theory are discussed. These include the investigation of
the structural stability of the GR-Hamilton equations with respect to vacuum
solutions of the Einstein equations, assuming that wave-like perturbations are
governed by the canonical evolution equations
Hamiltonian approach to GR - Part 2: covariant theory of quantum gravity
A non-perturbative quantum field theory of General Relativity is presented
which leads to a new realization of the theory of Covariant Quantum-Gravity
(CQG-theory). The treatment is founded on the recently-identified Hamiltonian
structure associated with the classical space-time, i.e., the corresponding
manifestly-covariant Hamilton equations and the related Hamilton-Jacobi theory.
The quantum Hamiltonian operator and the CQG-wave equation for the
corresponding CQG-state and wave-function are realized in scalar form.
The new quantum wave equation is shown to be equivalent to a set of quantum
hydrodynamic equations which warrant the consistency with the classical GR
Hamilton-Jacobi equation in the semiclassical limit. A perturbative
approximation scheme is developed, which permits the adoption of the harmonic
oscillator approximation for the treatment of the Hamiltonian potential. As an
application of the theory, the stationary vacuum CQG-wave equation is studied,
yielding a stationary equation for the CQG-state in terms of the scalar
invariant-energy eigenvalue associated with the corresponding approximate
quantum Hamiltonian operator. The conditions for the existence of a discrete
invariant-energy spectrum are pointed out. This yields a possible estimate for
the graviton mass together with a new interpretation about the quantum origin
of the cosmological constant
The computational complexity of traditional Lattice-Boltzmann methods for incompressible fluids
It is well-known that in fluid dynamics an alternative to customary direct
solution methods (based on the discretization of the fluid fields) is provided
by so-called \emph{particle simulation methods}. Particle simulation methods
rely typically on appropriate \emph{kinetic models} for the fluid equations
which permit the evaluation of the fluid fields in terms of suitable
expectation values (or \emph{momenta}) of the kinetic distribution function
being respectively and\textbf{\}
the position an velocity of a test particle with probability density
. These kinetic models can be continuous or discrete in
phase space, yielding respectively \emph{continuous} or \emph{discrete kinetic
models} for the fluids. However, also particle simulation methods may be biased
by an undesirable computational complexity. In particular, a fundamental issue
is to estimate the algorithmic complexity of numerical simulations based on
traditional LBM's (Lattice-Boltzmann methods; for review see Succi, 2001
\cite{Succi}). These methods, based on a discrete kinetic approach, represent
currently an interesting alternative to direct solution methods. Here we intend
to prove that for incompressible fluids fluids LBM's may present a high
complexity. The goal of the investigation is to present a detailed account of
the origin of the various complexity sources appearing in customary LBM's. The
result is relevant to establish possible strategies for improving the numerical
efficiency of existing numerical methods.Comment: Contributed paper at RGD26 (Kyoto, Japan, July 2008
Theory of stochastic transitions in area preserving maps
A famous aspect of discrete dynamical systems defined by area-preserving maps
is the physical interpretation of stochastic transitions occurring locally
which manifest themselves through the destruction of invariant KAM curves and
the local or global onset of chaos. Despite numerous previous investigations
(see in particular Chirikov, Greene, Percival, Escande and Doveil and MacKay)
based on different approaches, several aspects of the phenomenon still escape a
complete understanding and a rigorous description. In particular Greene's
approach is based on several conjectures, one of which is that the stochastic
transition leading to the destruction of the last KAM curve in the standard map
is due the linear destabilization of the elliptic points belonging to a
peculiar family of invariants sets {I(m,n)}
(rational iterates) having rational winding numbers and associated to the
last KAM curve. Purpose of this work is to analyze the nonlinear phenomena
leading to the stochastic transition in the standard map and their effect on
the destabilization of the invariant sets associated to the KAM curves,
leading, ultimately, to the destruction of the KAM curves themselves.Comment: 6 pages, 1 figure. Contributed to the Proceedings of the 24th
International Symposium on Rarefied Gas Dynamics, July 10-16, 2004 Porto
Giardino Monopoli (Bari), Ital
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