1,721,122 research outputs found
Self-similarity and power-like tails in nonconservative kinetic models
No full textIn this paper, we discuss the large-time behavior of solution of a simple kinetic model of Boltzmann-Maxwell type, such that the temperature is time decreasing and/or time increasing. We show that, under the combined effects of the nonlinearity
and of the time-monotonicity of the temperature, the kinetic model has non trivial quasi-stationary states with power law tails. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution.
The same idea is applied to investigate the large-time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and/or decreasing of the mean wealth. In this last case, the large time behavior of
the solution shows a Pareto power law tail. Numerical results confirm the previous analysis
Dinamiche sociali ed equazioni alle derivate parziali in ambito epidemiologico
No full textIn questo breve sunto divulgativo discuteremo l’importanza delle dinamiche sociali in ambito epidemico e la loro modellizzazione matematica tramite equazioni alle derivate parziali. Presenteremo inizialmente modelli di interazione tra individui in cui le caratteristiche sociali, come l’età degli individui, il numero di contatti sociali e la loro ricchezza economica, giocano un ruolo chiave nella diffusione di un’epidemia. Successivamente, accenneremo a modelli che tengono conto anche di caratteristiche aggiuntive quali la carica virale e le difese immunitarie dell’individuo. Infine, analizzeremo alcuni modelli alle derivate parziali per la descrizione degli spostamenti degli individui, sia su scala urbana che extra urbana, ed evidenzieremo come le dinamiche di movimento giochino un ruolo chiave sull’avanzamento dell’epidemia
Overpopulated tails in nonconservative kinetic models
No full textIn this paper, we discuss the large-time behavior of the solution of a simple kinetic model of Boltzmann-Maxwell type, such that the temperature is decreases with time or increases with time. We show that, under the combined effects of the nonlinearity and of the time-monotonicity of the temperature, the kinetic model has nontrivial quasi-stationary states with power law tails. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution. The same idea is applied to investigate the large-time behavior of an elementary kinetic model of an economy involving both exchanges between agents and increasing and/or decreasing of the mean wealth. In this last case, the large-time behavior of the solution shows a Pareto power law tail. Numerical results confirm the previous analysis
Interacting Multiagent Systems. Kinetic equations and Monte Carlo methods
No full textThe description of emerging collective phenomena and self-organization in systems composed of large numbers of individuals has gained increasing interest from various research communities in biology, ecology, robotics and control theory, as well as sociology and economics.
Applied mathematics is concerned with the construction, analysis and interpretation of mathematical models that can shed light on significant problems of the natural sciences as well as our daily lives. To this set of problems belongs the description of the collective behaviours of complex systems composed by a large enough number of individuals. Examples of such systems are interacting agents in a financial market, potential voters during political elections, or groups of animals with a tendency to flock or herd. Among other possible approaches, this book provides a step-by-step introduction to the mathematical modelling based on a mesoscopic description and the construction of efficient simulation algorithms by Monte Carlo methods.
The arguments of the book cover various applications, from the analysis of wealth distributions, the formation of opinions and choices, the price dynamics in a financial market, to the description of cell mutations and the swarming of birds and fishes. By means of methods inspired by the kinetic theory of rarefied gases, a robust approach to mathematical modelling and numerical simulation of multi-agent systems is presented in detail. The content is a useful reference text for applied mathematicians, physicists, biologists and economists who want to learn about modelling and approximation of such challenging phenomena.
Readership: Graduate students and researchers in mathematics, physics, economics, and biology. Also those with an interest in econophysics, collective behaviour, and swarming
On a class of Fokker–Planck equations with subcritical confinement
No full textWe study the relaxation to equilibrium for a class of linear one-dimensional Fokker–Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker–Planck equations is that, for any given probability density e(x), the diffusion coefficient can be built to have e(x) as steady state. This representation of the equilibrium density can be fruitfully used to obtain one-dimensional Wirtinger-type inequalities and to recover, for a sufficiently regular density e(x), a polynomial rate of convergence to equilibrium. Numerical results then confirm the theoretical analysis, and allow to conjecture that convergence to equilibrium with positive rate still holds for steady states characterized by a very slow polynomial decay at infinity
Variation in geometries and displacement along thrust faults: a quantitative analysis from sandbox models
No full textThe present work combines theoretical analysis, laboratory experiments, and field examples together with quantitative analysis to understand displacement variation along fault surfaces with varying dip angle. Two sandbox experiments are described to quantitatively analyze the interactions between shortening and sedimentation during late stages of deformation. The two models consisted of sand layers with single and two microbeads layer/s within the sandpack, sieved over horizontal frictional detachment. The models demonstrate that the overlying sediment load bears a major role on kinematic of developing structures. Fault displacement varied along imbricate thrusts due to variation in fault dip angle in the listric geometry. Maximum displacement occurred in the up-dip direction (up-section or near fault tip) along the faults where fault dip angles were shallower and favourable for thrust displacement. In contrast, the displacement was limited in the down-dip direction (down-section or near basal detachment) where fault dip angles were steeper. The variation in fault displacement with fault dip along the fault surfaces made it possible to establish an inverse relationship between the fault dip and related displacement values, i.e. lower is the dip angle, greater the displacement and vice versa. Despite the inherent limitations and assumptions of laboratory experiments, application of results to natural examples sheds new light on the mechanics and geometry of thrust faults from the deep to shallow structural levels as well as structural patterns that could appear as anomalous at a first sight
Kinetic theory for a binary gas-mixture: problems in unbounded domains by the semidiscrete Boltzmann equation
No full textSocial climbing and Amoroso distribution
No full textWe 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
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