arXiv subjects

M. Tessarotto

Publications and source records attributed to M. Tessarotto.

At least 19 recordsLinked to original sources

arXiv2017

On the Boltzmann-Grad limit of the Master Kinetic Equation

In this paper the problem is posed of the prescription of the so-called Boltzmann-Grad (BG) limit ($\mathcal{L}_{BG}$) for the $N-$body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous collisions. The statistical description is couched in terms of the Master kinetic equation, i.e., the kinetic equation which realizes the axiomatic "\textit{ab initio}" approach to the classical statistical mechanics of finite hard-sphere systems recently developed (Tessarotto \textit{et al.}, 2013-2017). The issue addressed here concerns the prescription of the BG-limit operator and specifically the non-commutative property of $\mathcal{L}_{BG}$ with the free-streaming operator which enters the same kinetic equation.\ It is shown\ that the form of the resulting limit equation remains in principle non-unique, its precise realization depending critically on the way the action of the same operator is prescribed. Implications for the global prescription of the Boltzmann equation are pointed out.

math-ph↗

arXiv2012

The kinetic theory of quasi-stationary collisionless accretion disc plasmas

Astrophysical plasmas in accretion discs are usually treated in the framework of fluid or MHD approaches but there are some situations where these treatments become inadequate and one needs to revert to the more fundamental underlying kinetic theory. This occurs when the plasma becomes effectively collisionless or weakly-collisional such as, for example, in radiatively inefficient accretion flows onto black holes. In this paper, we lay down the basics of kinetic theory in these contexts. In particular, we formulate the kinetic theory for quasi-stationary collisionless accretion disc plasmas in the framework of a Vlasov-Maxwell description, taking the plasma to be non-relativistic, axisymmetric, gravitationally-bound and subject to electromagnetic fields. Quasi-stationary solutions for the kinetic distribution functions are constructed which are shown to admit temperature anisotropies. The physical implications of the theory are then investigated and the equations of state and angular momentum conservation law are discussed. Analysis of the Ampere equation reveals the existence of a quasi-stationary kinetic dynamo which gives rise to self-generation of poloidal and azimuthal magnetic fields and operates even in the absence of turbulence and/or instability phenomena.

astro-ph.HE↗

arXiv2012

Absolute stability of axisymmetric perturbations in strongly-magnetized collisionless axisymmetric accretion disk plasmas

The physical mechanism responsible for driving accretion flows in astrophysical accretion disks is commonly thought to be related to the development of plasma instabilities and turbulence. A key question is therefore the determination of consistent equilibrium configurations for accretion-disk plasmas and investigation of their stability properties. In the case of collisionless plasmas kinetic theory provides the appropriate theoretical framework. This paper presents a kinetic description of low-frequency and long-wavelength axisymmetric electromagnetic perturbations in non-relativistic, strongly-magnetized and gravitationally-bound axisymmetric accretion-disk plasmas in the collisionless regime. The analysis, carried out within the framework of the Vlasov-Maxwell description, relies on stationary kinetic solutions of the Vlasov equation which allow for the simultaneous treatment of non-uniform fluid fields, stationary accretion flows and temperature anisotropies. It is demonstrated that, for such solutions, no axisymmetric unstable perturbations can exist occurring on characteristic time and space scales which are long compared with the Larmor gyration time and radius. Hence, these stationary configurations are actually stable against axisymmetric kinetic instabilities of this type. As a fundamental consequence, this rules out the possibility of having the axisymmetric magneto-rotational or thermal instabilities to arise in these systems.

astro-ph.HE↗

arXiv2010

Theory of multi-point probability densities for incompressible Navier-Stokes fluids

An open problem arising in the statistical description of turbulence is related to the \textit{theoretical prediction based on first principles} of the so-called multi-point velocity probability density functions (PDFs) characterizing a Navier-Stokes fluid. In this paper it will be shown that - based on a suitable axiomatic approach - a solution to this problem can actually be achieved based on the so-called inverse kinetic theory (IKT), recently developed for incompressible fluids. More precisely, we intend to show, based on the requirement that \textit{the Boltzmann-Shannon entropy for the s-point velocity PDF ($f_s$) is independent of the order $s$ and is also maximal at all times}, that all multi-point PDFs are \textit{necessarily factorized in terms of the corresponding 1-point velocity PDF} ($f_1$). As a consequence the multi-point PDFs usually considered for the phenomenological description of turbulence can be theoretically predicted \textit{based on the knowledge of $% f_1$ achieved by means of IKT.

physics.flu-dyn↗

arXiv2010

On the initial conditions of the 1-point PDF for incompressible Navier-Stokes fluids

An aspect of fluid dynamics lies in the search of possible statistical models for Navier-Stokes (NS) fluids described by classical solutions of the incompressible Navier-Stokes equations (INSE). This refers in particular to statistical models based on the so-called inverse kinetic theory (IKT) . This approach allows the description of fluid systems by means a suitable 1-point velocity probability density function (PDF) which determines, in terms of suitable "moments", the complete set of fluid fields which define the fluid state. A fundamental related issue lies in the problem of the unique construction of the initial PDF. The goal of this paper is to propose a solution holding for NS fluids. Our claim is that the initial PDF can be uniquely determined by imposing a suitable set of physical realizability constraints.

physics.flu-dyn↗

arXiv2010

On the behavior of homogeneous, isotropic and stationary turbulence

The recent development of a statistical model for incompressible Navier-Stokes (NS) fluids based on inverse kinetic theory (IKT, 2004-2008) poses the problem of searching for particular realizations of the theory which may be relevant for \textit{the statistical description of turbulence} and in particular for the so-called \textit{homogeneous, isotropic}and \textit{stationary} turbulence (HIST). Here the problem is set in terms of the $1-$point velocity probability density function (PDF) which determines a complete IKT-statistical model for NS fluids. This raises the interesting question of identifying the statistical assumptions under which a Gaussian PDF can be achieved in such a context. In this paper it is proven that for the IKT statistical model, HIST requires necessarily that $f_{1}$ must be SIED (namely \textit{stationary}, \textit{isotropic} and \textit{% everywhere-defined}). This implies, in turn, that the functional form of the PDF is uniquely prescribed at all times. In particular, it is found that necessarily the PDF must coincide with an isotropic Gaussian distribution. The conclusion is relevant for the investigation of the so-called homogenous, isotropic and stationary turbulence.

physics.flu-dyn↗

arXiv2009

Theory of the 1-point PDF for incompressible Navier-Stokes fluids

Fundamental aspects of fluid dynamics are related to construction of statistical models for incompressible Navier-Stokes fluids. The latter can be considered either \textit{deterministic} or \textit{stochastic,} respectively for \textit{regular} or \textit{turbulent flows.} In this work we claim that a possible statistical formulation of this type can be achieved by means of the 1-point (local) velocity-space probability density function (PDF, $f_{1}$) to be determined in the framework of the so-called inverse kinetic theory (IKT). There are several important consequences of the theory. These include, in particular, the characterization of the initial PDF [for the statistical model ${f_{1},Γ} ]$ . This is found to be generally non-Gaussian PDF, even in the case of flows which are regular at the initial time. Moreover, both for regular and turbulent flows, its time evolution is provided by a Liouville equation, while the corresponding Liouville operator is found to depend only on a finite number of velocity moments of the same PDF. Hence, its time evolution depends (functionally) solely on the same PDF. In addition, the statistical model here developed determines uniquely both the initial condition and the time evolution of $f_{1}.$ As a basic implication, the theory allows the \textit{exact construction of the corresponding statistical equation for the stochastic-averaged PDF}and the \textit{unique representation of the multi-point PDF}'s in solely in terms of the 1-point PDF. As an example, the case of the reduced 2-point PDF's, usually adopted for the statistical description of NS turbulence, is considered.

physics.flu-dyn↗

arXiv2009

IKT-approach to MHD turbulence

An open issue in turbulence theory is related to the determination of the exact evolution equation for the probability density associated to the relevant (stochastic) fluid fields. Such an equation in the usual approaches to turbulence reproduces, at most in an approximate sense, the correct fluid equations. In this paper we present a statistical model which applies to an incompressible, resistive and quasi-neutral magnetofluid. The approach is based on the formulation of an inverse kinetic theory (IKT) for the full set of MHD equations appropriate for an incompressible, viscous, quasi-neutral, isentropic, isothermal and resistive magnetofluid. Basic feature of the new approach is that it relies on first principles - including in particular the exact validity of the fluid equations - and thus permits the determination of the correct evolution equation for the probability density. Specific application of the theory here considered concerns the case of statistically homogeneous and stationary MHD turbulence.

physics.flu-dyn↗

arXiv2008

Strong solutions of the incompressible Navier-Stokes equations in external domains: local existence and uniqueness

In this paper the problem of strong solvability of the incompressible Navier-Stokes equations (INSE) is revisited, with the goal of determining the minimal assumptions for the validity of a local existence and uniqueness theorem for the Navier-Stokes fluid fields (solutions of INSE). Emphasis is placed on fluid fields which, together with suitable derivatives, do not necessarily decay at infinity and hence do not belong to Sobolev spaces. For this purpose a novel approach based on a so-called inverse kinetic theory, recently developed by Tessarotto and Ellero, is adopted. This requires the construction of a suitable kinetic equation, advancing in time a suitably smooth kinetic distribution function and providing exactly, as its moment equations, the complete set of fluid equations. In turn, by proper definition of the kinetic equation, this permits the introduction of the so-called \textit{Navier-Stokes dynamical system}, i.e., the dynamical system which advances in time self-consistently the Navier-Stokes fluid fields. Investigation of the properties of this dynamical system is crucial for the establishment of an existence and uniqueness theorem for strong solutions of INSE. The new theorem applies both to bounded and unbounded domains and in the presence of generalized boundaries, represented by surfaces, curves or even sets of isolated points. In particular, for unbounded domains, solutions are considered, which do not necessarily vanish at infinity. Basic consequences for the functional setting of classical solutions are analyzed. \keywords{Navier-Stokes equations \and Kinetic theory \and Dynamical systems} PACS 47.10.ad,47.10.Fg,PACS 47.10.A- MSC 76D03,76D06

math-ph↗

arXiv2008

Variational formulation of the electromagnetic radiation-reaction problem

A fundamental issue in classical electrodynamics is represented by the search of the exact equation of motion for a classical charged particle under the action of its electromagnetic (EM) self-field - the so-called radiation-reaction equation of motion (RR equation). In the past, several attempts have been made assuming that the particle electric charge is localized point-wise (point-charge). These involve the search of possible so-called "regularization" approaches able to deal with the intrinsic divergences characterizing point-particle descriptions in classical electrodynamics. In this paper we intend to propose a new solution to this problem based on the adoption of a variational approach and the treatment of finite-size spherical-shell charges. The approach is based on three key elements: 1) the adoption of the relativistic synchronous Hamilton variational principle recently pointed out (Tessarotto et al, 2006); 2) the variational treatment of the EM self-field, for finite-size charges, taking into account the exact particle dynamics; 3) the adoption of the axioms of classical mechanics and electrodynamics. The new RR equation proposed in this paper, departing significantly from previous approaches, exhibits several interesting properties. In particular: a) unlike the LAD (Lorentz-Abraham-Dirac) equation, it recovers a second-order ordinary differential equation which is fully consistent with the law of inertia, Newton principle of determinacy and Einstein causality principle and b) unlike the LL (Landau-Lifschitz) equation, it holds also in the case of sudden forces. In addition, it is found that the new equation recovers the customary LAD equation in a suitable asymptotic approximation.

physics.class-ph↗

arXiv2008

The exact radiation-reaction equation for a classical charged particle

An unsolved problem of classical mechanics and classical electrodynamics is the search of the exact relativistic equations of motion for a classical charged point-particle subject to the force produced by the action of its EM self-field. The problem is related to the conjecture that for a classical charged point-particle there should exist a relativistic equation of motion (RR equation) which results both non-perturbative, in the sense that it does not rely on a perturbative expansion on the electromagnetic field generated by the charged particle and non-asymptotic, i.e., it does not depend on any infinitesimal parameter. In this paper we intend to propose a novel solution to this well known problem, and in particular to point out that the RR equation is necessarily variational. The approach is based on two key elements: 1) the adoption of the relativistic hybrid synchronous Hamilton variational principle recently pointed out (Tessarotto et al, 2006). Its basic feature is that it can be expressed in principle in terms of arbitrary "hybrid" variables (i.e., generally non-Lagrangian and non-Hamiltonian variables); 2) the variational treatment of the EM self-field, taking into account the exact particle dynamics.

physics.class-ph↗

arXiv2008

On the validity of the LAD and LL classical radiation-reaction equations

The search of the correct equation of motion for a classical charged particle under the action of its electromagnetic (EM) self-field, the so-called \textit{radiation-reaction equation of motion}, remains elusive to date. In this paper we intend to point out why this is so. The discussion is based on the direct construction of the EM self-potentials produced by a charged spherical particle under the action of an external EM force. In particular we intend to analyze basic features of the LAD (Lorentz-Abraham-Dirac) and the LL (Landau-Lifschitz) equations. Both are shown to lead to incorrect or incomplete results.

physics.class-ph↗

arXiv2008

Axisymmetric gravitational MHD equilibria in the presence of plasma rotation

In this paper, extending the investigation developed in an earlier paper (Cremaschini et al., 2008), we pose the problem of the kinetic description of gravitational Hall-MHD equilibria which may arise in accretion disks (AD) plasmas close to compact objects. When intense EM and gravitational fields, generated by the central object, are present, a convenient approach can be achieved in the context of the Vlasov-Maxwell description. In this paper the investigation is focused primarily on the following two aspects: 1) the formulation of the kinetic treatment of G-Hall-MHD equilibria. Based on the identification of the relevant first integrals of motion, we show that an explicit representation can be given for the equilibrium kinetic distribution function. For each species this is represented as a superposition of suitable generalized Maxwellian distributions; 2) the determination of the constraints to be placed on the fluid fields for the existence of the kinetic equilibria. In particular, this permits a unique determination of the functional form of the species number densities and of the fluid partial pressures, in terms of suitably prescribed flux functions.

astro-ph↗

arXiv2008

Fokker-Planck Kinetic description of small-scale fluid turbulence for classical incompressible fluids

Extending the statistical approach proposed in a parallel paper \cite% {Tessarotto2008-aa}, purpose of this work is to propose a stochastic inverse kinetic theory for small-scale hydrodynamic turbulence based on the introduction of a suitable \textit{local phase-space probability density function} (pdf). In particular, we pose the problem of the construction of Fokker-Planck kinetic models of hydrodynamic turbulence. The approach here adopted is based on the so-called IKT approach (inverse kinetic theory), developed by Ellero et al. (2004-2008) which permits an exact phase-space description of incompressible fluids based on the adoption of a local pdf. We intend to show that for prescribed models of stochasticity the present approach permits to determine uniquely the time evolution of the stochastic fluid fields. The stochastic-averaged local pdf is shown to obey a kinetic equation which, although generally non-Markovian, locally in velocity-space can be approximated by means of a suitable Fokker-planck kinetic equation. As a side result, the same pdf is proven to have generally a non-Gaussian behavior.

physics.flu-dyn↗

arXiv2008

On the existence of the Boltzmann-Grad limit for a system of hard smooth spheres

Despite the progress achieved by kinetic theory, its rigorous theoretical foundations still remain unsolved to date. This concerns in particular the search of possible exact kinetic equations and, specifically, 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, several aspects of the theory remain to be addressed and clarified. In fact, its validity has been proven for the Boltzmann equation only at most in a weak sense, i.e., if the Boltzmann-Grad limit is defined according to the weak * convergence. While it is doubtful whether the result applies for arbitrary times and for general situations (and in particular more generally for classical systems of particles interacting via binary forces), it remains completely unsolved the issue whether the conjecture might be valid also in a stronger sense (\textit{strong Lanford conjecture}). This paper will point out a physical model providing a counter-example to the strong Lanford conjecture, representing a straightforward generalization of the classical model based on a gas of hard-smooth spheres. In particular we claim that that the one-particle limit function, defined in the sense of the strong Boltzmann-Grad limit, does not generally satisfy the BBGKY (or Boltzmann) hierarchy. The result is important for the theoretical foundations of kinetic theory.

math-ph↗

arXiv2008

Exact pressure evolution equation for incompressible fluids

An important aspect of computational fluid dynamics is related to the determination of the fluid pressure in isothermal incompressible fluids. In particular this concerns the construction of an exact evolution equation for the fluid pressure which replaces the Poisson equation and yields an algorithm which is a Poisson solver, i.e., it permits to time-advance exactly the same fluid pressure \textit{without solving the Poisson equation}% . In fact, the incompressible Navier-Stokes equations represent a mixture of hyperbolic and elliptic pde's, which are extremely hard to study both analytically and numerically. In this paper we intend to show that an exact solution to this problem can be achieved adopting the approach based on inverse kinetic theory (IKT) recently developed for incompressible fluids by Ellero and Tessarotto (2004-2007). In particular we intend to prove that the evolution of the fluid fields can be achieved by means of a suitable dynamical system, to be identified with the so-called Navier-Stokes (N-S) dynamical system. As a consequence it is found that the fluid pressure obeys a well-defined evolution equation. The result appears relevant for the construction of Lagrangian approaches to fluid dynamics.

physics.flu-dyn↗

arXiv2008

Generalized covariant gyrokinetic dynamics of magnetoplasmas

A basic prerequisite for the investigation of relativistic astrophysical magnetoplasmas, occurring typically in the vicinity of massive stellar objects (black holes, neutron stars, active galactic nuclei, etc.), is the accurate description of single-particle covariant dynamics, based on gyrokinetic theory (Beklemishev et al.,1999-2005). Provided radiation-reaction effects are negligible, this is usually based on the assumption that both the space-time metric and the EM fields (in particular the magnetic field) are suitably prescribed and are considered independent of single-particle dynamics, while allowing for the possible presence of gravitational/EM perturbations driven by plasma collective interactions which may naturally arise in such systems. The purpose of this work is the formulation of a generalized gyrokinetic theory based on the synchronous variational principle recently pointed out (Tessarotto et al., 2007) which permits to satisfy exactly the physical realizability condition for the four-velocity. The theory here developed includes the treatment of nonlinear perturbations (gravitational and/or EM) characterized locally, i.e., in the rest frame of a test particle, by short wavelength and high frequency. Basic feature of the approach is to ensure the validity of the theory both for large and vanishing parallel electric field. It is shown that the correct treatment of EM perturbations occurring in the presence of an intense background magnetic field generally implies the appearance of appropriate four-velocity corrections, which are essential for the description of single-particle gyrokinetic dynamics.

physics.plasm-ph↗

arXiv2008

IKT approach for quantum hydrodynamic equations

A striking feature of standard quantum mechanics is its analogy with classical fluid dynamics. In particular it is well known the Schrödinger equation can be viewed as describing a classical compressible and non-viscous fluid, described by two (quantum) fluid fields ${ρ,% \mathbf{V}} $, to be identified with the quantum probability density and velocity field. This feature has suggested the construction of a phase-space hidden-variable description based on a suitable inverse kinetic theory (IKT; Tessarotto et al., 2007). The discovery of this approach has potentially important consequences since it permits to identify the classical dynamical system which advances in time the quantum fluid fields. This type of approach, however requires the identification of additional fluid fields. These can be generally identified with suitable directional fluid temperatures $T_{QM,i}$ (for $i=1,2,3$), to be related to the expectation values of momentum fluctuations appearing in the Heisenberg inequalities. Nevertheless the definition given previously for them (Tessarotto et al., 2007) is non-unique. In this paper we intend to propose a criterion, based on the validity of a constant H-theorem, which provides an unique definition for the quantum temperatures.

quant-ph↗