Maria Groppi

Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas

A kinetic approach is presented for the analysis of a gas mixture with two kinds of nonconservative interactions. In a bimolecular chemical reaction, mass transfer and energy of chemical link arise, and in inelastic mechanical encounters, molecules get excited or de‐excited due to their quantized structure. Molecules undergo transitions between energy levels also by absorption and emission of photons of the self‐consistent radiation field. From the kinetic Boltzmann‐type equations, the problem of equilibria and of their stability is addressed. A detailed balance principle is proved and a Lyapunov functional is constructed; mass action law and Plancks law of radiation are recovered.

A Bhatnagar-Gross-Krook-type approach for chemically reacting gas mixtures

A recently proposed consistent approach for elastically scattering gas mixtures, of the type introduced by Bhatnagar, Gross, and Krook (BGK), has been extended to deal with a four species gas undergoing reversible bimolecular chemical reactions. The single BGK collision operator introduced for each species must take into account also transfer of mass and of energy of chemical bond. Suitable auxiliary fields have then to be introduced not only for temperatures and velocities, but also for densities, in order to fulfill correctly balance equations for mass, momentum, and total energy. The exact collision equilibrium, satisfying the mass action law of chemistry, is also recovered, and the proper choice of collision frequencies is discussed. Preliminary numerical results for the relaxation problem in space-homogeneous conditions are reported and briefly commented on.

How different two almost identical action potentials can be: A model study on cardiac repolarization

Spatial heterogeneity in the properties of ion channels generates spatial dispersion of ventricular repolarization, which is modulated by gap junctional coupling. However, it is possible to simulate conditions in which local differences in excitation properties are electrophysiologically silent and only play a role in pathological states. We use a numerical procedure on the Luo-Rudy phase 1 model of the ventricular action potential (AP1) in order to find a modified set of model parameters which generates an action potential profile (AP2) almost identical to AP1. We show that, although the two waveforms elicited from resting conditions as a single AP are very similar and belong to membranes sharing similar passive electrical properties, the modified membrane generating AP2 is a weaker current source than the one generating AP1, has different sensitivity to up/down-regulation of ion channels and to extracellular potassium, and a different electrical restitution profile. We study electrotonic interaction of AP1- and AP2- type membranes in cell pairs and in cable conduction, and find differences in source-sink properties which are masked in physiological conditions and become manifest during intercellular uncoupling or partial block of ion channels, leading to unidirectional block and spatial repolarization gradients. We provide contour plot representations that summarize differences and similarities. The present report characterizes an inverse problem in cardiac cells, and strengthen the recently emergent notion that a comprehensive characterization and validation of cell models and their components are necessary in order to correctly understand simulation results at higher levels of complexity.

Shock structure analysis in chemically reacting gas mixtures by a relaxation-time kinetic model

The shock structure in a gas mixture undergoing a bimolecular chemical reaction is studied by means of a reactive kinetic relaxation model. The relevant nonlinear integrodifferential equations are numerically solved in one space dimension with upstream and downstream asymptotic equilibrium conditions satisfying the reactive Rankine–Hugoniot relations and entropy condition. Numerical results are presented, emphasizing the role of Mach number, upstream concentration fractions, and change in the chemical composition across the shock.

Kinetic theory of a diatomic gas with reactions of dissociation and recombination through a transition state

Extended kinetic equations, according to the scattering kernel formulation of the Boltzmann equation, are derived for a chemical reaction of dissociation and recombination in the frame of the transition-state theory. Conservation laws and moment equations are discussed, and, in the spontaneous asymptotic limit induced by the transition species, collision equilibria are determined to leading order. In the spirit of the stationary-state approximation, a closed set of fluid-dynamic equations of the Euler type are then obtained for the main macroscopic fields. Preliminary numerical results are finally presented and briefly commented on.

Approximate solutions to the problem of stationary shear flow of smooth granular materials

Abstract The inelastic Boltzmann equation is used in order to study stationary shear flows in a rarefied granular gas of hard spheres. We resort to a Gaussian moment approximation in order to calculate the pressure tensor in three dimensions. The method is discussed along with previously introduced techniques: asymptotic expansion in the near elastic limit, and pseudo-Maxwellian approximation. Numerical results and approximate analytic formulas for the pressure tensor are presented and briefly commented on. A comparison with the pseudo-Maxwellian solution is discussed in detail.

Modelling of predator–prey trophic interactions. Part I: two trophic levels

Abstract.A class of lumped parameter models to describe the local dynamics in a controlled environment of a two-trophic chain is considered. The class is characterized by a trophic function (functional response of predator to the abundance of prey) depending on the ratio of prey biomass x and a linear function of predator biomass y: f(qx/[(1-ρ)k+ρy]), where q is the efficiency of the predation process, k is a reference biomass, and ρ (0≤ρ≤1) specifies the predation model. The trophic function is defined only by some properties determining its shape. A stability analysis of the models has been performed by taking the parameters q and ρ as bifurcation parameters: the regions in the (ρ,q) plane of existence and stability of nonnegative equilibrium states and limit cycles are determined. This analysis shows that the behaviour of the models is qualitatively similar for 0≤ρ<1 (in particular the null state is always a saddle point), while the value ρ=1 gives rise to some kind of structural instability of the system (in particular the null state becomes an attractor for sufficiently high predation efficiency).

Euler closure of the Boltzmann equations for resonant bimolecular reactions

The problem of a rigorous hydrodynamic closure at the Euler level of the complicated set of integro-differential nonlinear Boltzmann-type equations, describing the evolution of a chemically reactive gas mixture at the kinetic level, is addressed. The case in which the process is driven by both elastic scattering and resonant transitions is worked out and discussed by a formal mathematical procedure based on typical asymptotic methods of kinetic theory.

React or wait: which optimal culling strategy to control infectious diseases in wildlife.

We applied optimal control theory to an SI epidemic model to identify optimal culling strategies for diseases management in wildlife. We focused on different forms of the objective function, including linear control, quadratic control, and control with limited amount of resources. Moreover, we identified optimal solutions under different assumptions on disease-free host dynamics, namely: self-regulating logistic growth, Malthusian growth, and the case of negligible demography. We showed that the correct characterization of the disease-free host growth is crucial for defining optimal disease control strategies. By analytical investigations of the model with negligible demography, we demonstrated that the optimal strategy for the linear control can be either to cull at the maximum rate at the very beginning of the epidemic (reactive culling) when the culling cost is low, or never to cull, when culling cost is high. On the other hand, in the cases of quadratic control or limited resources, we demonstrated that the optimal strategy is always reactive. Numerical analyses for hosts with logistic growth showed that, in the case of linear control, the optimal strategy is always reactive when culling cost is low. In contrast, if the culling cost is high, the optimal strategy is to delay control, i.e. not to cull at the onset of the epidemic. Finally, we showed that for diseases with the same basic reproduction number delayed control can be optimal for acute infections, i.e. characterized by high disease-induced mortality and fast dynamics, while reactive control can be optimal for chronic ones.

Solvability of linear kinetic equations with multi-energetic inelastic scattering

Abstract In this paper we shall analyse the linear Boltzmann equation describing the motion of test particles through a background of heavy field particles that can appear in several energy states. The inelastic scattering process consists in the exchange of quanta of energy between the field and test particles. The well-posedness of the problem is investigated by means of the substochastic semigroup theory and the conditions on the scattering collision frequencies are given for the evolution to keep the collision rate finite and to preserve the total number of particles.