Narrow Results

In this work, a general formulation, which is based on steady boundary layer problems for the Boltzmann equation, of a half-space problem is considered. The number of conditions on the indata at the interface needed to obtain well-posedness is investigated. The solutions will converge exponentially fast “far away” from the interface. For linearized kinetic half-space problems similar to the one of evaporation and condensation in kinetic theory, slowly varying modes might occur near regime transitions where the number of conditions needed to obtain well-posedness changes (corresponding to transition between evaporation and condensation, or subsonic and supersonic evaporation/condensation), preventing uniform exponential speed of convergence. However, those modes might be eliminated by imposing extra conditions on the indata at the interface. Flow velocities at the far end for which regime transitions occur are presented for Boltzmann equations: for monatomic and polyatomic single species and mixtures; as well as bosons and fermions.

We study a kinetic model for a system of two species of particles interacting via a repulsive long range potential and with a reservoir at fixed temperature. The interaction between the particles is modeled by a Vlasov term and the thermal bath by a Fokker–Planck term. We show that in the diffusive and sharp interface limit the motion of the interfaces at low temperature is described by a Stefan problem or a Mullins–Sekerka motion, depending on the time scale.

Starting from a mesoscopic principle of moment conservation, discrete Boltzmann collision operators J^h are constructed, which both converge to bounded collision operators J_Ω and have the same collision invariants as the original Boltzmann collision operator J. The crucial point of this construction is the application of a weak formulation of the gain operator to remove the post-collision velocities from it as well as the development of moment conserving integration formulas for the approximation of surface integrals over the unit sphere. Finally two applications for the discrete operators are presented.

A temperature jump in water traveling through a pipe is delayed by the absorption of heat into the pipe wall. The convective transport of heat by the water and the conductive exchange with the interior pipe wall are described by an exact but highly singular micro-model. The limiting form of this parabolic initial-boundary-value problem is a distributed microstructure model, which can successively be better approximated by simpler first- and second-order kinetic models. This model provides a means to validate and calibrate these classical multi-temperature kinetic models which are used to describe the delay.

We construct and investigate a new nonlocal kinetic model for the formation and movement of animal groups in two dimensions. The model generalizes to two dimensions, the one-dimensional hyperbolic model from (R. Eftimie, G. de Vries, M. A. Lewis and F. Lutscher, Modeling group formation and activity patterns in self-organizing collectives of individuals, Bull. Math. Biol.69 (2007) 1537–1566). The main modeling aspect in the present approach concerns the assumptions we make on the turning rates, to include, in a nonlocal fashion, the three types of social interactions that act among individuals of a group: attraction, repulsion and alignment. We show that solutions to the new mathematical model are bounded, along with their gradients. We also present numerical results to illustrate three types of group formations that we obtained with the new model, starting from random initial conditions: (i) swarms (aggregation into a group, with no preferred direction of motion), (ii) parallel/translational motion (uniform spatial density, movement in a certain preferred direction) and (iii) parallel groups (aggregation into a group, with movement in a preferred direction).

We present and analyze kinetic models for the size evolution of a huge number of populations subject to interactions which describe both birth and death in a single population, and migration between them. If the mean size of the population is preserved, we prove that the solution to the underlying kinetic equation converges to equilibrium as time goes to infinity, and in various relevant cases we recover its main properties. In addition, by considering a suitable asymptotic procedure (the limit of quasi-invariant interactions) a simpler kinetic description of the model is derived. This procedure allows to describe the evolution process in terms of a linear kinetic transport-type equation. Among the various processes that can be described in this way, one recognizes a process which is closely related to the Lea–Coulson model of mutation processes in bacteria, a variation of the original model proposed by Luria and Delbrück, and a model recently proposed to describe evolution of the cross-genomic family abundance (i.e. the number of genes of a given family found in different genomes).

We present and discuss various one-dimensional linear Fokker–Planck-type equations that have been recently considered in connection with the study of interacting multi-agent systems. In general, these Fokker–Planck equations describe the evolution in time of some probability density of the population of agents, typically the distribution of the personal wealth or of the personal opinion, and are mostly obtained by linear or bilinear kinetic models of Boltzmann type via some limit procedure. The main feature of these equations is the presence of variable diffusion, drift coefficients and boundaries, which introduce new challenging mathematical problems in the study of their long-time behavior.

Call centers are service networks in which agents provide telephone-based services. An important part of call center operations is represented by service durations. In recent statistical analysis of real data, it has been noted that the distribution of service times reveals a remarkable fit to the lognormal distribution. In this paper, we discuss a possible source of this behavior by resorting to classical methods of statistical mechanics of multi-agent systems. The microscopic service time variation leading to a linear kinetic equation with lognormal equilibrium density is built up introducing as main criterion for decision a suitable value function in the spirit of the prospect theory of Kahneman and Twersky.

In recent years, it has been increasing evidence that lognormal distributions are widespread in physical and biological sciences, as well as in various phenomena of economics and social sciences. In social sciences, the appearance of lognormal distribution has been noticed, among others, when looking at body weight, and at women’s age at first marriage. Likewise, in economics, lognormal distribution appears when looking at consumption in a western society, at call-center service times, and others. The common feature of these situations, which describe the distribution of a certain people’s hallmark, is the presence of a desired target to be reached by repeated choices. In this paper, we discuss a possible explanation of lognormal distribution forming in human activities by resorting to classical methods of statistical mechanics of multi-agent systems. The microscopic variation of the hallmark around its target value, leading to a linear Fokker–Planck-type equation with lognormal equilibrium density, is built up introducing as main criterion for decision a suitable value function in the spirit of the prospect theory of Kahneman and Twersky.

We introduce a class of new one-dimensional linear Fokker–Planck-type equations describing the dynamics of the distribution of wealth in a multi-agent society. The equations are obtained, via a standard limiting procedure, by introducing an economically relevant variant to the kinetic model introduced in 2005 by Cordier, Pareschi and Toscani according to previous studies by Bouchaud and Mézard. The steady state of wealth predicted by these new Fokker–Planck equations remains unchanged with respect to the steady state of the original Fokker–Planck equation. However, unlike the original equation, it is proven by a new logarithmic Sobolev inequality with weight and classical entropy methods that the solution converges exponentially fast to equilibrium.

We introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker–Planck type, describing the dynamics of individuals of a multi-agent society questing for high status in the social hierarchy. At the Boltzmann level, the microscopic variation of the status of agents around a universal desired target, is built up introducing as main criterion for the change of status a suitable value function in the spirit of the prospect theory of Kahneman and Twersky. In the asymptotics of grazing interactions, the solution density of the Boltzmann-type kinetic equation is shown to converge towards the solution of a Fokker–Planck type equation with variable coefficients of diffusion and drift, characterized by the mathematical properties of the value function. The steady states of the statistical distribution of the social status predicted by the Fokker–Planck equations belong to the class of Amoroso distributions with Pareto tails, which correspond to the emergence of a social elite. The details of the microscopic kinetic interaction allow to clarify the meaning of the various parameters characterizing the resulting equilibrium. Numerical results then show that the steady state of the underlying kinetic equation is close to Amoroso distribution even in an intermediate regime in which interactions are not grazing.

We introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker–Planck type, describing the dynamics of the amount of consumer spending transactions of a multi-agent society, that, in the presence of social distancing laws consequent to an epidemic spreading, can reduce the volume of a consistent part of economic commercial activities requiring the presence. At the Boltzmann level, the microscopic variation of the number of transactions around a desired commercial target, is built up by joining a random risk with a deterministic change. Both the risk and target value parameters are here dependent of the number of social contacts of the multi-agent system, which can vary in time in reason of the governments decisions to control the epidemic spreading by introducing restrictions on the social activities. In the asymptotics of grazing interactions, the solution density of the Boltzmann-type kinetic equation is shown to converge towards the solution of a Fokker–Planck-type equation with variable coefficients of diffusion and drift, characterized by the details of the elementary interaction. The economic description is then coupled with the evolution equations of a new SIR-type compartmental epidemic system suitable to describe both the classical epidemic spreading and the social contacts evolution in dependence of the multi-agent social heterogeneity. The (local in time) steady states of the statistical distribution of the amount of economic transactions predicted by the Fokker–Planck equation are shown to be inverse Gamma densities, with time-dependent polynomial tails at infinity which characterize the consequences of the control policies on the time evolution of the Pareto index.

In this paper, by resorting to classical methods of statistical mechanics, we build a kinetic model able to reproduce the observed statistical weight distribution of many diverse species. The kinetic description of the time variations of the weight distribution is based on elementary interactions that describe in a qualitative and quantitative way successive evolutionary updates, and determine explicit equilibrium distributions. Numerical fittings on mammalian eutherians of the order Chiroptera population illustrates the effectiveness of the approach.

Polyamines act as dual modulators on electric eel acetylcholinesterase, modifying both the apparent Km and Ki, depending on substrate levels. A kinetic model was developed to explain the results, based on two-step catalysis, a peripheral site for substrate inhibition apart from the catalytic site, and one binding site for polyamine. This model presented the best fittings to data, when compared with a simpler one considering one catalytic step. A fitting equation built up with sixteen independent parameters let us calculate the kinetic constants. In this way, we were able to solve the parameter identifiability problem arising from model uncertainty when only substrate was used in acetylcholinesterase kinetics. Besides, fitting parameters directly provide information about the binding constants of the different complexes, the modulatory strength of substrate and polyamines, and the effect on the standard activation free energy for acetylcholinesterase.

Substrate inhibition operates mainly on the first catalytic step with an affinity constant of 5.2 mM^-1, which is reduced to one third for the acetylated enzyme. The interaction factor between substrate binding at both sites is about 12. The modulatory strength of polyamines is spermine > spermidine > putrescine. This order is directly related to the number of amino groups in the molecule, and to the calculated free interaction energy. The effect of the number of amino groups on the binding energy is significantly increased in acetylated acetylcholinesterase. It is also inferred that the formation of a quaternary complex enzyme-substrate-substrate-polyamine would not be possible. Some relations between polyamine structure and acetylcholinesterase activity are suggested from estimated constants. Due to the distal amino group distances, it is possible for spermine and spermidine to span along the catalytic gorge of acetylcholinesterase, binding to the catalytic and peripheral sites in a way similar to bisquaternary ammonium inhibitors.

This paper deals with a kinetic modelling of the cellular dynamics of tumors interacting with an active immune defence system. The analysis starts from a detailed modelling of the cellular interactions and follows with the derivation of evolution equations in a framework similar to the one of nonlinear statistical mechanics. A discussion about the qualitative properties of the model and on the possibility of its application in immunology is proposed in the last part of the paper.

We consider a non-local scalar conservation law in two space dimensions which arises as the formal hydrodynamic limit of a Fokker–Planck equation. This Fokker–Planck equation is, in turn, the kinetic description of an individual-based model describing the navigation of self-propelled particles in a pheromone landscape. The pheromone may be linked to the agent distribution itself, leading to a nonlinear, non-local scalar conservation law where the effective velocity vector depends on the pheromone field in a small region around each point, and thus, on the solution itself. After presenting and motivating the problem, we present some numerical simulations of a closely related problem, and then prove a well-posedness and stability result for the conservation law.

We discuss, both from the analytical and the numerical point of view, a kinetic model for wealth distribution in a simple market economy which models, besides binary trade interactions, also taxation and redistribution of collected wealth.

We study the dynamics of groups of undistinguished agents, which, while interacting according to their relative positions, dissipate energy. These models are developed to mimic the collective motion of groups of living individuals such as bird flocks, fish schools or bacteria colonies. According to the Cucker and Smale model,⁷ binary interactions between agents are modelled by dissipative collisions in which the coefficient of restitution depends on their relative distance. Under the assumption of weak dissipation, it is shown that the consequent dynamics can be described at a fluid dynamic level by the Euler equation for compressible fluids, in which the equations for momentum and energy present a dissipative correction.