arXiv subjects

Massimo Tessarotto

Publications and source records attributed to Massimo Tessarotto.

52 records · Page 3Linked to original sources

arXiv2008

On the Boltzmann-Grad Limit for the classical hard-spheres system

Despite the progress achieved by kinetic theory, the search of possible exact kinetic equations remains elusive to date. This concerns, specifically, the issue of the validity of the conjecture proposed by Grad (Grad, 1972) and developed in a seminal work by Lanford (Lanford, 1974) that kinetic equations - such as the Boltzmann equation for a gas of classical hard spheres - might result exact in an appropriate asymptotic limit, usually denoted as Boltzmann-Grad limit. The Lanford conjecture has actually had a profound influence on the scientific community, giving rise to a whole line of original research in kinetic theory and mathematical physics. Nevertheless, certain aspects of the theory remain to be addressed and clarified. The purpose of this paper is to investigate the possible existence of the strong Boltzmann-Grad limit for the BBGKY hierarchy. Contrary to previous approaches in which the w*-convergence was considered for the definition of the Boltzmann-Grad limit functions, based on their construction in terms of time-series expansions obtained from the BBGKY hierarchy, here we look for the possible existence of strong limit functions in the sense of local convergence in phase space. The result is based on the adoption of the Klimontovich approach to statistical mechanics, permitting the explicit representation of the $s$-body reduced distribution functions in terms of the Klimontovich probability density.

math-ph↗

arXiv2008

On the existence of canonical gyrokinetic variables for chaotic magnetic fields

The gyrokinetic description of particle dynamics faces a basic difficulty when a special type of canonical variables is sought, i.e., the so-called \textit{gyrokinetic canonical variables}. These are defined in such a way that two of them are respectively identified with the gyrophase-angle, describing the fast particle gyration motion around magnetic field lines, and its canonically conjugate momentum. In this paper we intend to discuss the conditions of existence for these variables.

physics.plasm-ph↗

arXiv2008

Lagrangian dynamics of incompressible thermofluids

A key aspect of fluid dynamics is the correct definition of the \textit{% phase-space} Lagrangian dynamics which characterizes arbitrary fluid elements of an incompressible fluid. Apart being an unsolved theoretical problem of fundamental importance, the issue is relevant to exhibit the connection between fluid dynamics and the classical dynamical systems underlying incompressible and non-isothermal fluid, typically founded either on: a) a \textit{configuration-space} Lagrangian description of the dynamics of fluid elements; b) a kinetic description of the molecular dynamics, based on a discrete representation of the fluid. The goal of this paper is to show that the exact Lagrangian dynamics can be established based on the inverse kinetic theory (IKT) for incompressible fluids recently pointed out (Ellero \textit{et al.}, 2004-2006, \cite{Ellero2004}). The result is reached by adopting an IKT approach based on a \textit{restricted phase-space representation} of the fluid, in which the configuration space coincides with the physical fluid domain. The result appears of potential importance in applied fluid dynamics and CFD.

physics.flu-dyn↗

arXiv2008

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 $f(\mathbf{r,v},t),$ being respectively $\mathbf{r}$ and\textbf{\}$\mathbf{v}$ the position an velocity of a test particle with probability density $f(\mathbf{r,v},t)$. 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.

physics.flu-dyn↗

arXiv2007

Lattice Boltzmann inverse kinetic approach for the incompressible Navier-Stokes equations

In spite of the large number of papers appeared in the past which are devoted to the lattice Boltzmann (LB) methods, basic aspects of the theory still remain unchallenged. An unsolved theoretical issue is related to the construction of a discrete kinetic theory which yields \textit{exactly} the fluid equations, i.e., is non-asymptotic (here denoted as \textit{LB inverse kinetic theory}). The purpose of this paper is theoretical and aims at developing an inverse kinetic approach of this type. In principle infinite solutions exist to this problem but the freedom can be exploited in order to meet important requirements. In particular, the discrete kinetic theory can be defined so that it yields exactly the fluid equation also for arbitrary non-equilibrium (but suitably smooth) kinetic distribution functions and arbitrarily close to the boundary of the fluid domain. Unlike previous entropic LB methods the theorem can be obtained without functional constraints on the class of the initial distribution functions. Possible realizations of the theory and asymptotic approximations are provided which permit to determine the fluid equations \textit{with prescribed accuracy.} As a result, asymptotic accuracy estimates of customary LB approaches and comparisons with the Chorin artificial compressibility method are discussed.

physics.flu-dyn↗

arXiv2006

An exact pressure evolution equation for the incompressible Navier-Stokes equations

In this paper the issue of the determination of the fluid pressure in incompressible fluids is addressed, with particular reference to the search of algorithms which permit to advance in time the fluid pressure without actually solving numerically the Poisson equation. Based on an inverse kinetic approach recently proposed for the incompressible Navier-Stokes equations we intend to prove that an exact evolution equation can be obtained which advances in time self-consistently the fluid pressure. The new equation is susceptible of numerical implementation in Lagrangian CFD simulation codes.

physics.flu-dyn↗

arXiv2006

Kinetic description of classical matter infalling in black holes

A popular aspect of black holes physics is the mathematical analogy between their laws, coming from general relativity and the laws of thermodynamics. The analogy is achieved by identifying a suitable set of observables, precisely: \emph{(a)} $E=M$ (being $E$ the thermodynamic free energy and $M$ the mass of the BH), \emph{(b)} $T=ακ$ (with $T$ the absolute temperature, $κ$ the so-called surface gravity on event horizon and $% α$ a suitable dimensional constant) and \emph{(c)} $S=(1/8πα)A $ (where $S$ is the thermodynamic entropy of the black hole and $A$ the surface of the event horizon). However, despite numerous investigations and efforts spent on the subject, the theoretical foundations of such identifications between physical quantities belonging to apparently unrelated frameworks are not yet clear. The goal of this work is to provide the contribution to the black hole entropy, coming from matter in the black hole exterior. We propose a classical solution for the kinetic description of matter falling into a black hole, which permits to evaluate both the kinetic entropy and the entropy production rate of classical infalling matter at the event horizon. The formulation is based on a relativistic kinetic description for classical particles in the presence of an event horizon. An H-theorem is established which holds for arbitrary models of black holes and is valid also in the presence of contracting event horizons.

astro-ph↗

arXiv2006

On the uniqueness of continuous inverse kinetic theory for incompressible fluids

Fundamental aspects of inverse kinetic theories for incompressible Navier-Stokes equations concern the possibility of defining uniquely the kinetic equation underlying such models and furthermore, the construction of a kinetic theory implying also the energy equation. The latter condition is consistent with the requirement that fluid fields result classical solutions of the fluid equations. These issues appear of potential relevance both from the mathematical viewpoint and for the physical interpretation of the theory. In this paper we intend to prove that the non-uniqueness feature can be resolved by imposing suitable assumptions. These include, in particular, the requirement that the kinetic equation be equivalent, in a suitable sense, to a Fokker-Planck kinetic equation. Its Fokker-Planck coefficients are proven to be uniquely determined by means of appropriate prescriptions. In addition, as a further result, it is proven that the inverse kinetic equation satisfies both an entropy principle and the energy equation for the fluid fields.

physics.flu-dyn↗

arXiv2006

Non-linear gyrokinetic theory of magnetoplasmas

A crucial issue in relativistic plasma, particularly relevant in the astrophysical context, is the description of highly magnetized plasmas based on a covariant formulation of gyrokinetic dynamics. An interesting case in question is that in which the background electric field (produced either by the same plasma of by other sources) results suitably small (or vanishing) with respect to the magnetic field, while at the same time short-wavelength EM perturbations can be present. The purpose of this work is to extend the relativistic gyrokinetic theory developed by Beklemishev \textit{et al.} [1999-2005] to include, in particular, also the treatment of such a case. We intend to show that this requires the development of a perturbative expansion involving simultaneously both the particle 4-position vector and the corresponding \textit{4-velocity vector}. For this purpose a synchronous form of the relativistic Hamilton variational principle is adopted.

physics.plasm-ph↗

arXiv2006

Inverse kinetic theory for quantum hydrodynamic equations

We propose a solution for the inverse kinetic theory for quantum hydrodynamic equations associated to the non-relativistic Schrödinger equation. It is shown that an inverse kinetic equation of the form of the Vlasov equation can be non-uniquely determined under suitable mathematical prescriptions.

quant-ph↗

arXiv2005

Microscopic statistical description of classical matter around black holes

The problem of statistical description of classical matter, represented by a N-body system of relativistic point particles falling into a black hole, is investigated, adopting a classical framework. A covariant microscopic statistical description is developed for the classical particle system by introducing the definition of observer-related N-particle microscopic distribution function and by suitably extending to general relativity the concept of Gibbs microscopic entropy. The corresponding entropy production rate is calculated taking into account the presence of an event horizon and proven to obey, for N>>1, an exact microscopic H-theorem which holds for arbitrary geometries.

physics.class-ph↗

arXiv2005

H-theorem for classical matter around a black hole

We propose a classical solution for the kinetic description of matter falling into a black hole, which permits to evaluate both the kinetic entropy and the entropy production rate of classical infalling matter at the event horizon. The formulation is based on a relativistic kinetic description for classical particles in the presence of an event horizon. An H-theorem is established which holds for arbitrary models of black holes and is valid also in the presence of contracting event horizons.

physics.class-ph↗

arXiv2004

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.

math-ph↗

arXiv2004

Relativistic kinetic theory of magnetoplasmas

Recently, an increasing interest in astrophysical as well as laboratory plasmas has been manifested in reference to the existence of relativistic flows, related in turn to the production of intense electric fields in magnetized systems. Such phenomena require their description in the framework of a consistent relativistic kinetic theory, rather than on relativistic MHD equations, subject to specific closure conditions. The purpose of this work is to apply the relativistic single-particle guiding-center theory developed by Beklemishev and Tessarotto, including the nonlinear treatment of small-wavelength EM perturbations which may naturally arise in such systems. As a result, a closed set of relativistic gyrokinetic equations, consisting of the collisionless relativistic kinetic equation, expressed in hybrid gyrokinetic variables, and the averaged Maxwell's equations, is derived for an arbitrary four-dimensional coordinate system.

physics.plasm-ph↗

arXiv2004

Canonical Lie-transform method in Hamiltonian gyrokinetics: a new approach

The well-known gyrokinetic problem regards the perturbative expansion related to the dynamics of a charged particle subject to fast gyration motion due to the presence of a strong magnetic field. Although a variety of approaches have been formulated in the past to this well known problem, surprisingly a purely canonical approach based on Lie transform methods is still missing. This paper aims to fill in this gap and provide at the same time new insight in Lie-transform approaches.

physics.plasm-ph↗

arXiv2004

Generalized canonical approach to gyrokinetic theory

We face the well-known gyrokinetic problem, which arises in the description of the dynamics of a charged particle subject to fast gyration for the presence of a strong electromagnetic field. The customary approach to gyrokinetic theory, using canonical variables or identifying them ``a posteriori'' by means of Darboux theorem, leads to potential complications and ambiguities due to the fact that canonical coordinates are field-related. Here we propose an innovative formulation to construct gyrokinetic canonical variables based on the introduction of a new definition of canonical transformation. The new approach permits to shed light on this often controversial issue.

math-ph↗