Claudio Asci

The Lagrangian dynamics of thermal tracer particles in Navier-Stokes fluids

A basic issue for Navier-Stokes (NS) fluids is their characterization in terms of the so-called NS phase-space classical dynamical system, which provides a mathematical model for the description of the dynamics of infinitesimal (or ideal) tracer particles in these fluids. The goal of this paper is to analyze the properties of a particular subset of solutions of the NS dynamical system, denoted as thermal tracer particles (TTPs), whose states are determined uniquely by the NS fluid fields. Applications concerning both deterministic and stochastic NS fluids are pointed out. In particular, in both cases it is shown that in terms of the ensemble of TTPs a statistical description of NS fluids can be formulated. In the case of stochastic fluids this feature permits to uniquely establish the corresponding Langevin and Fokker-Planck dynamics. Finally, the relationship with the customary statistical treatment of hydrodynamic turbulence (HT) is analyzed and a solution to the closure problem for the statistical description of HT is proposed.

Generating Uniform Random Vectors

In this paper, we study the rate of convergence of the Markov chain Xn+1=AXn+bn mod p, where A is an integer matrix with nonzero integer eigenvalues and {bn}n is a sequence of independent and identically distributed integer vectors. If λi≠±1 for all eigenvalues λi of A, then n=O((log p)2) steps are sufficient and n=O(log p) steps are necessary to have Xn sampling from a nearly uniform distribution. Conversely, if A has the eigenvalue λ1=±1, and λi≠±1 for all i≠1, n=O(p2) steps are necessary and sufficient.

Asymptotic orderings and approximations of the Master kinetic equation for large hard spheres systems

In this paper the problem is posed of determining the physically-meaningful asymptotic orderings holding for the statistical description of a large N-body system of hard spheres, i.e., formed by N equivalent to 1/epsilon >> 1 particles, which are allowed to undergo instantaneous and purely elastic unary, binary or multiple collisions. Starting point is the axiomatic treatment recently developed [Tessarotto et al., 2013-2016] and the related discovery of an exact kinetic equation realized by Master equation which advances in time the 1-body probability density function (PDF) for such a system. As shown in the paper the task involves introducing appropriate asymptotic orderings in terms of epsilon for all the physically-relevant parameters. The goal is that of identifying the relevant physically-meaningful asymptotic approximations applicable for the Master kinetic equation, together with their possible relationships with the Boltzmann and Enskog kinetic equations, and holding in appropriate asymptotic regimes. These correspond either to dilute or dense systems and are formed either by small-size or finite-size identical hard spheres, the distinction between the various cases depending on suitable asymptotic orderings in terms of epsilon.

Generating Uniform Random Vectors in Zpk: The General Case

AbstractWe consider the rate of convergence of the Markov chain Xn+1=AXn+Bn (mod p), where A is an integer matrix with nonzero eigenvalues, and {Bn}n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Qk invariant under A. If

The Mathematical Modelling of Thermal Tracer Particles in Navier-Stokes Fluids

|\lambda_{i}|\not =1

Beta-hypergeometric distributions and random continued fractions

for all eigenvalues λi of A, then n=O((ln p)2) steps are sufficient and n=O(ln p) steps are necessary to have Xn sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λi that are roots of positive integer numbers, |λ1|=1 and |λi|>1 for all

On the behavior of homogeneous, isotropic and stationary turbulence

i\not =1

Microscopic statistical description of incompressible Navier-Stokes granular fluids

, then O(p2) steps are necessary and sufficient.

Global validity of the Master kinetic equation for hard-sphere systems

In this paper we report the discovery of a subset of socalled ideal tracer particles (ITP (4)) belonging to NavierStokes (NS) fluids which are denoted as thermal tracer particles (TTP). Their states are found to be uniquely dependent on the local state of the fluid. This means that a suitable statistical ensemble of TTPs should reproduce exactly the dynamics of the fluid. In other words, it should be possible to determine the fluid fields characterizing the fluid state by means of suitable statistical averages on the ensemble of TTPs so that they satisfy identically the required set of fluid equations. The result applies to NS fluids described as mesoscopic, i.e., continuous fluids, which can be either viscous or inviscid, compressible or incompressible, thermal or isothermal, isentropic or nonisentropic. We shall assume that the state of these fluids is represented by an ensemble of observables {Z(r, t)} ≡ {Zi(r, t), i = 1, .., n} (with n an integer ≥ 1), i.e., fluid fields, which can be unambiguously prescribed as continuous and suitably smooth functions, respectively, in Ω × I and in the open set Ω × I (in the following to be identified with a bounded subset of R). We intend to show that, as a remarkable consequence, the phasespace dynamical system which advances in time (the states of) these particles can be uniquely prescribed in such a way to determines self-consistently the time evolution of the complete set of fluid equations characterizing the fluid. This implies that TTPs must reproduce exactly the dynamics of the fluid. In other words, by means of an appropriate statistical averages on the ensemble of the TTPs, it is possible to determine the time-evolution of the fluid state, in such a way that it satisfies identically the required set of fluid equations. In this paper we point out that the existence of TTPs can be reached by means of a Gedanken experiment on the NS fluid, i.e., by inspecting the properties of the underlying clsssical dynamical system which advances in time the state of a NS fluid.