Knudsen type group for time in ℝ and related Boltzmann type equations

We consider certain Boltzmann type equations on a bounded physical and a bounded velocity space under the presence of both reflective as well as diffusive boundary conditions. We introduce conditions on the shape of the physical space and on the relation between the reflective and the diffusive part in the boundary conditions such that the associated Knudsen type semigroup can be extended to time [Formula: see text]. Furthermore, we provide conditions under which there exists a unique global solution to a Boltzmann type equation for time [Formula: see text] or for time [Formula: see text] for some [Formula: see text] which is independent of the initial value at time 0. Depending on the collision kernel, [Formula: see text] can be arbitrarily small.

The Study of a Wealth Distribution Model with a Linear Collision Kernel

The wealth substitution rate, which describes the substitution relationship between agents’ investment in wealth, is introduced into the collision kernel of the Boltzmann equation to study wealth distribution. Using the continuous trading limit, the Fokker–Planck equation is derived and the steady-state solution is obtained. The results show that the inequality of wealth distribution decreases as the wealth substitution rate increases under certain assumptions. The wealth distribution has a bimodal shape if the wealth substitution rate does not equal one.

The Development and Application of a Kinetic Theory for Modelling Dispersed Particle Flows

This Freeman Scholar article reviews the formulation and application of a kinetic theory for modeling the transport and dispersion of small particles in turbulent gas-flows, highlighting the insights and understanding it has provided and some of the long standing problems in the modeling of dispersed flows it has resolved. The theory has been developed and refined by numerous authors and now forms a rational basis for modeling complex particle laden flows. The formalism and methodology of this approach are discussed and the choice of closure of the kinetic equations involved which ensures realizability and well posedness with exact closure for Gaussian carrier flow fields. The historical development is presented and how single particle kinetic equations resolve the problem of closure of the transport equations for particle mass, momentum and kinetic energy /stress (the so called continuum equations) and the treatment of the dispersed phase as a fluid. The mass fluxes associated with the turbulent aerodynamic driving forces and interfacial stresses are shown to be both dispersive and convective in inhomogeneous turbulence with implications for the build up of particles concentration in near wall turbulent boundary layers and particle pair concentration at small separation. It is shown how this approach deals with the natural wall boundary conditions for a flowing particle suspension and examples are given of partially absorbing surfaces with particle scattering, and gravitational settling; how this approach has revealed the existence of contra gradient diffusion in a developing shear flow and the influence of the turbulence on gravitational settling (the Maxey effect). Particular consideration is given to the general problem of particle transport and deposition in turbulent boundary layers and near wall behavior including particle resuspension. Finally the application of a particle pair formulation for both monodisperse and bidisperse particle flows is reviewed where the differences between the two are compared through the influence of collisions on the particle continuum equations and on the particle collision kernel for the clustering of particles and the degree of random uncorrelated motion (RUM) at the small scales of the turbulence. The inclusion of bidisperse particle suspensions implies the application to polydisperse flows and the evolution of particle size distribution.

On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation

In this paper the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel is studied. Well-posedness and long-time behavior – exponential decay to the global equilibrium – of the analytical solution, and spectral gap estimate for the corresponding linearized gPC-based stochastic Galerkin system are obtained, by using and extending the analytical tools provided in [M. Briant and E. S. Daus, Arch. Ration. Mech. Anal., 3, 1367–1443, 2016] for the deterministic problem in the perturbative regime, and in [E. S. Daus, S. Jin and L. Liu, Kinet. Relat. Models, 12, 909–922, 2019] for the single-species problem with uncertainty. The well-posedness result of the sensitivity system presented here has not been obtained so far even for the single-species case.

Broad excitations in a 2+1D overoccupied gluon plasma

Motivated by the initial stages of high-energy heavy-ion collisions, we study excitations of far-from-equilibrium 2+1 dimensional gauge theories using classical-statistical lattice simulations. We evolve field perturbations over a strongly overoccupied background undergoing self-similar evolution. While in 3+1D the excitations are described by hard-thermal loop theory, their structure in 2+1D is nontrivial and nonperturbative. These nonperturbative interactions lead to broad excitation peaks in spectral and statistical correlation functions. Their width is comparable to the frequency of soft excitations, demonstrating the absence of soft quasiparticles in these theories. Our results also suggest that excitations at higher momenta are sufficiently long-lived, such that an effective kinetic theory description for 2+1 dimensional Glasma-like systems may exist, but its collision kernel must be nonperturbatively determined.

Particle size dynamics in abrading pebble populations

Abrasion of sedimentary particles in fluvial and eolian environments is widely associated with collisions encountered by the particle. Although the physics of abrasion is complex, purely geometric models recover the course of mass and shape evolution of individual particles in low- and middle-energy environments (in the absence of fragmentation) remarkably well. In this paper, we introduce the first model for the collision-driven collective mass evolution of sedimentary particles. The model utilizes results of the individual, geometric abrasion theory as a collision kernel; following techniques adopted in the statistical theory of coagulation and fragmentation, the corresponding Fokker–Planck equation is derived. Our model uncovers a startling fundamental feature of collective particle size dynamics: collisional abrasion may, depending on the energy level, either focus size distributions, thus enhancing the effects of size-selective transport, or it may act in the opposite direction by dispersing the distribution.

Efficient mate finding in planktonic copepods swimming in turbulence

Zooplankton live in dynamic environments where turbulence may challenge their limited swimming abilities. How this interferes with fundamental behavioral processes remains elusive. We reconstruct simultaneously the trajectories of flow tracers and calanoid copepods and we quantify their ability to find mates when ambient flow imposes physical constrains on their motion and impairs their olfactory orientation. We show that copepods achieve high encounter rates in turbulence due to the contribution of advection and vigorous swimming. Males further convert encounters within the perception radius to contacts and then to mating via directed motion toward nearby organisms within the short time frame of the encounter. Inertial effects do not result in preferential concentration, reducing the geometric collision kernel to the clearance rate, which we model accurately by superposing turbulent velocity and organism motion. This behavioral and physical coupling mechanism may account for the ability of copepods to reproduce in turbulent environments.

Particle size dynamics in abrading pebble populations

Abrasion of sedimentary particles in fluvial and aeolian environments is widely associated with collisions encountered by the particle. Although the physics of abrasion is complex, purely geometric models recover the course of mass and shape evolution of individual particles in low and middle energy environments (in the absence of fragmentation) remarkably well. In this paper, utilizing results of this individual, geometric abrasion theory as a collision kernel, following techniques adopted in the statistical theory of coagulation and fragmentation, we construct the corresponding Fokker-Planck equation as the first model for the collision-driven collective mass evolution of sedimentary particles. Our model uncovers a startling fundamental feature of collective particle size dynamics: collisional abrasion may, depending on the energy level, either focus size distributions, thus enhancing the effects of size selective transport or it may act in the opposite direction by dispersing the distribution. This complex behaviour fits geological observations on mass distributions.

Modeling volcanic ash aggregation processes and related impacts on the April–May 2010 eruptions of Eyjafjallajökull volcano with WRF-Chem

Volcanic eruptions eject ash and gases into the atmosphere that can contribute to significant hazards to aviation, public and environment health, and the economy. Several volcanic ash transport and dispersion (VATD) models are in use to simulate volcanic ash transport operationally, but none include a treatment of volcanic ash aggregation processes. Volcanic ash aggregation can greatly reduce the atmospheric budget, dispersion and lifetime of ash particles, and therefore its impacts. To enhance our understanding and modeling capabilities of the ash aggregation process, a volcanic ash aggregation scheme was integrated into the Weather Research Forecasting with online Chemistry (WRF-Chem) model. Aggregation rates and ash mass loss in this modified code are calculated in line with the meteorological conditions, providing a fully coupled treatment of aggregation processes. The updated-model results were compared to field measurements of tephra fallout and in situ airborne measurements of ash particles from the April–May 2010 eruptions of Eyjafjallajökull volcano, Iceland. WRF-Chem, coupled with the newly added aggregation code, modeled ash clouds that agreed spatially and temporally with these in situ and field measurements. A sensitivity study provided insights into the mechanics of the aggregation code by analyzing each aggregation process (collision kernel) independently, as well as by varying the fractal dimension of the newly formed aggregates. In addition, the airborne lifetime (e-folding) of total domain ash mass was analyzed for a range of fractal dimensions, and a maximum reduction of 79.5 % of the airborne ash lifetime was noted.