Quantitative lower bounds for the full Boltzmann equation, Part I: Periodic boundary conditions

We prove the appearance of an explicit lower bound on the solution to the full Boltzmann equation in the torus for a broad family of collision kernels including in particular long-range interaction models, under the assumption of some uniform bounds on some hydrodynamic quantities. This lower bound is independent of time and space. When the collision kernel satisfies Grad's cutoff assumption, the lower bound is a global Maxwellian and its asymptotic behavior in velocity is optimal, whereas for non-cutoff collision kernels the lower bound we obtain decreases exponentially but faster than the Maxwellian. Our results cover solutions constructed in a spatially homogeneous setting, as well as small-time or close-to-equilibrium solutions to the full Boltzmann equation in the torus. The constants are explicit and depend on the a priori bounds on the solution.Comment: 37 page

On the kinetic systems for simple reacting spheres : modeling and linearized equations

Series: Springer Proceedings in Mathematics & Statistics, Vol. 75In this work we present some results on the kinetic theory of chemically reacting gases, concerning the model of simple reacting spheres (SRS) for a gaseous mixture undergoing a chemical reaction of type A1 +A2 A3 +A4. Starting from the approach developed in paper [11], we provide properties of the SRS system needed in the mathematical and physical analysis of the model. Our main result in this proceedings provides basic properties of the SRS system linearized around the equilibrium, including the explicit representations of the kernels of the linearized SRS operators.Fundação para a Ciência e a Tecnologia (FCT), PEst-C/MAT/UI0013/2011, SFRH/BD/28795/200

A Continuum Description of Rarefied Gas Dynamics (I)--- Derivation From Kinetic Theory

We describe an asymptotic procedure for deriving continuum equations from the kinetic theory of a simple gas. As in the works of Hilbert, of Chapman and of Enskog, we expand in the mean flight time of the constituent particles of the gas, but we do not adopt the Chapman-Enskog device of simplifying the formulae at each order by using results from previous orders. In this way, we are able to derive a new set of fluid dynamical equations from kinetic theory, as we illustrate here for the relaxation model for monatomic gases. We obtain a stress tensor that contains a dynamical pressure term (or bulk viscosity) that is process-dependent and our heat current depends on the gradients of both temperature and density. On account of these features, the equations apply to a greater range of Knudsen number (the ratio of mean free path to macroscopic scale) than do the Navier-Stokes equations, as we see in the accompanying paper. In the limit of vanishing Knudsen number, our equations reduce to the usual Navier-Stokes equations with no bulk viscosity.Comment: 16 page

Towards Rigorous Derivation of Quantum Kinetic Equations

We develop a rigorous formalism for the description of the evolution of states of quantum many-particle systems in terms of a one-particle density operator. For initial states which are specified in terms of a one-particle density operator the equivalence of the description of the evolution of quantum many-particle states by the Cauchy problem of the quantum BBGKY hierarchy and by the Cauchy problem of the generalized quantum kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation is established in the space of trace class operators. The links of the specific quantum kinetic equations with the generalized quantum kinetic equation are discussed.Comment: 25 page

Boltzmann equation and hydrodynamic fluctuations

We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics.Comment: This is a more detailed version of a related paper: I.V. Karlin, M. Colangeli, M. Kroger, PRL 100 (2008) 214503, arXiv:0801.2932. It contains comparison between predictions and experiment, in particular. 11 pages, 6 figures, 2 table

Hydrodynamic modes, Green-Kubo relations, and velocity correlations in dilute granular gases

It is shown that the hydrodynamic modes of a dilute granular gas of inelastic hard spheres can be identified, and calculated in the long wavelength limit. Assuming they dominate at long times, formal expressions for the Navier-Stokes transport coefficients are derived. They can be expressed in a form that generalizes the Green-Kubo relations for molecular systems, and it is shown that they can also be evaluated by means of $N$-particle simulation methods. The form of the hydrodynamic modes to zeroth order in the gradients is used to detect the presence of inherent velocity correlations in the homogeneous cooling state, even in the low density limit. They manifest themselves in the fluctuations of the total energy of the system. The theoretical predictions are shown to be in agreement with molecular dynamics simulations. Relevant related questions deserving further attention are pointed out

An investigation of the diversity of strains of enteroaggregative Escherichia coli isolated from cases associated with a large multi-pathogen foodborne outbreak in the UK

Following a large outbreak of foodborne gastrointestinal (GI) disease, a multiplex PCR approach was used retrospectively to investigate faecal specimens from 88 of the 413 reported cases. Gene targets from a range of bacterial GI pathogens were detected, including Salmonella species, Shigella species and Shiga toxin-producing Escherichia coli, with the majority (75%) of faecal specimens being PCR positive for aggR associated with the Enteroaggregative E. coli (EAEC) group. The 20 isolates of EAEC recovered from the outbreak specimens exhibited a range of serotypes, the most frequent being O104:H4 and O131:H27. None of the EAEC isolates had the Shiga toxin (stx) genes. Multilocus sequence typing and single nucleotide polymorphism analysis of the core genome confirmed the diverse phylogeny of the strains. The analysis also revealed a close phylogenetic relationship between the EAEC O104:H4 strains in this outbreak and the strain of E. coli O104:H4 associated with a large outbreak of haemolytic ureamic syndrome in Germany in 2011. Further analysis of the EAEC plasmids, encoding the key enteroaggregative virulence genes, showed diversity with respect to FIB/FII type, gene content and genomic architecture. Known EAEC virulence genes, such as aggR, aat and aap, were present in all but one of the strains. A variety of fimbrial genes were observed, including genes encoding all five known fimbrial types, AAF/1 to AAF/V. The AAI operon was present in its entirety in 15 of the EAEC strains, absent in three and present, but incomplete, in two isolates. EAEC is known to be a diverse pathotype and this study demonstrates that a high level of diversity in strains recovered from cases associated with a single outbreak. Although the EAEC in this study did not carry the stx genes, this outbreak provides further evidence of the pathogenic potential of the EAEC O104:H4 serotype

Hydrodynamic theory for granular gases

A granular gas subjected to a permanent injection of energy is described by means of hydrodynamic equations derived from a moment expansion method. The method uses as reference function not a Maxwellian distribution $f_{\sf M}$ but a distribution $f_0 = \Phi f_{\sf M}$, such that $\Phi$ adds a fourth cumulant $\kappa$ to the velocity distribution. The formalism is applied to a stationary conductive case showing that the theory fits extraordinarily well the results coming from our molecular dynamic simulations once we determine $\kappa$ as a function of the inelasticity of the particle-particle collisions. The shape of $\kappa$ is independent of the size $N$ of the system.Comment: 10 pages, 9 figures, more about our research in http://www.cec.uchile.cl/cinetica

Three-dimensional lattice-Boltzmann simulations of critical spinodal decomposition in binary immiscible fluids

We use a modified Shan-Chen, noiseless lattice-BGK model for binary immiscible, incompressible, athermal fluids in three dimensions to simulate the coarsening of domains following a deep quench below the spinodal point from a symmetric and homogeneous mixture into a two-phase configuration. We find the average domain size growing with time as $t^\gamma$, where $\gamma$ increases in the range $0.545 < \gamma < 0.717$, consistent with a crossover between diffusive $t^{1/3}$ and hydrodynamic viscous, $t^{1.0}$, behaviour. We find good collapse onto a single scaling function, yet the domain growth exponents differ from others' works' for similar values of the unique characteristic length and time that can be constructed out of the fluid's parameters. This rebuts claims of universality for the dynamical scaling hypothesis. At early times, we also find a crossover from $q^2$ to $q^4$ in the scaled structure function, which disappears when the dynamical scaling reasonably improves at later times. This excludes noise as the cause for a $q^2$ behaviour, as proposed by others. We also observe exponential temporal growth of the structure function during the initial stages of the dynamics and for wavenumbers less than a threshold value.Comment: 45 pages, 18 figures. Accepted for publication in Physical Review