Discrete Kinetic Theory Research Articles

The Role of the Table of Games in the Discrete Thermostatted Kinetic Theory

This paper is concerned with the mathematical modeling of complex living systems whose element microscopic state contains variables which can attain discrete values. Specifically, the main mathematical frameworks of the discrete thermostatted kinetic theory for active particles are reviewed and generalized. In the generalized thermostatted frameworks, which are based on nonlinear ordinary or partial differential equations, the elements of the system are viewed as active particles that are able to perform certain strategies modeled by introducing a functional-state variable called activity. Interactions, which are responsible of the evolution of the system, are modeled using the fundamentals of stochastic game theory and may be influenced by the action of an external force field coupled to a Gaussian-type thermostat. In particular, the interaction domain is modeled by introducing a weighted function and different non-homogeneous discrete frameworks are proposed and coupled with a specific thermostat. Two recent models derived within this approach are reviewed and refer to vehicular and pedestrian dynamics. Future research perspectives are discussed in the whole paper from theoretical and modeling viewpoints.

Relaxation-rate formula for the entropic lattice Boltzmann model**Project supported by the National Natural Science Foundation of China (Grant Nos. 11471185, 11801030, and 11861131004).

A relaxation-rate formula is presented for the entropic lattice Boltzmann model (ELBM) — a discrete kinetic theory for hydrodynamics. The simple formula not only guarantees the discrete time H-theorem but also gives full consideration to the consistency with hydrodynamics. The relaxation rate calculated with the formula effectively characterizes the drastic changes of the flow fields. By using this formula, the computational cost of the ELBM is significantly reduced and the model now can be efficiently used for a broad range of applications including high Reynolds number flows.

Essentially Entropic Lattice Boltzmann Model.

The entropic lattice Boltzmann model (ELBM), a discrete space-time kinetic theory for hydrodynamics, ensures nonlinear stability via the discrete time version of the second law of thermodynamics (the H theorem). Compliance with the H theorem is numerically enforced in this methodology and involves a search for the maximal discrete path length corresponding to the zero dissipation state by iteratively solving a nonlinear equation. We demonstrate that an exact solution for the path length can be obtained by assuming a natural criterion of negative entropy change, thereby reducing the problem to solving an inequality. This inequality is solved by creating a new framework for construction of Padé approximants via quadrature on appropriate convex function. This exact solution also resolves the issue of indeterminacy in case of nonexistence of the entropic involution step. Since our formulation is devoid of complex mathematical library functions, the computational cost is drastically reduced. To illustrate this, we have simulated a model setup of flow over the NACA-0012 airfoil at a Reynolds number of 2.88×10^{6}.

Discrete kinetic theory for 2D modeling of a moving crowd: Application to the evacuation of a non-connected bounded domain

This paper concerns the mathematical modeling of the motion of a crowd in a non connected bounded domain, based on kinetic and stochastic game theories. The proposed model is a mesoscopic probabilistic approach that retains features obtained from both micro- and macro-scale representations; pedestrian interactions with various obstacles being managed from a probabilistic perspective. A proof of the existence and uniqueness of the proposed mathematical model’s solution is given for large times. A numerical resolution scheme based on the splitting method is implemented and then applied to crowd evacuation in a non connected bounded domain with one rectangular obstacle. The evacuation time of the room is then calculated by our technique, according to the dimensions and position of a square-shaped obstacle, and finally compared to the time obtained by a deterministic approach by means of randomly varying some of its parameters.

Asymptotic limits of a discrete Kinetic Theory model of vehicular traffic

The paper proposes a rigorous method to construct the hyperbolic asymptotic limit of the discrete Kinetic Theory model of vehicular traffic proposed in [8]. A second-order macroscopic model of the Payne–Whitham type is derived and the coefficients of the equations are obtained from the detailed description of the microscopic interactions developed in the kinetic model.

Robust thermal boundary conditions applicable to a wall along which temperature varies in lattice-gas cellular automata

We show that the heat exchange between fluid particles and boundary walls can be achieved by controlling the velocity change rate following the particles' collision with a wall in discrete kinetic theory, such as the lattice-gas cellular automata and the lattice Boltzmann method. We derive a relation between the velocity change rate and temperature so that we can control the velocity change rate according to a given temperature boundary condition. This relation enables us to deal with the thermal boundary whose temperature varies along a wall in contrast to the previous works of the lattice-gas cellular automata. In addition, we present simulation results to compare our method to the existing and give an example in a microchannel with a high temperature gradient boundary condition by the lattice-gas cellular automata.

Moment equations for magnetohydrodynamics

In kinetic treatments of hydrodynamics the macroscopic variables such as density andmomentum are given by moments of distribution functions. In discrete kinetic theory it ispossible to construct a complete set of moments whose evolution provides a completedescription of the dynamics of the underlying kinetic equation. Moreover, the collisionoperator is most easily specified by its action upon moments. This paper presents theequivalent moment system for a kinetic formulation of magnetohydrodynamics that usesvector-valued distribution functions to represent the magnetic field. Besides leading tomodest improvements in numerical stability, this moment system is invaluable for designingnew kinetic equations to model more realistic plasma processes such as current-dependentresistivity.

From generalized kinetic theory to discrete velocity modeling of vehicular traffic. A stochastic game approach

This work reports on vehicular traffic modeling by methods of the discrete kinetic theory. The purpose is to detail a reference mathematical framework for some discrete velocity kinetic models recently introduced in the literature, which proved capable of reproducing interesting traffic phenomena without using experimental information as modeling assumptions. To this end, we firstly derive a general discrete velocity kinetic framework with binary nonlocal interactions. Then, resorting to some ideas of stochastic game theory, we outline specific modeling guidelines for vehicular traffic, and finally we discuss the derivation of the above-mentioned vehicular traffic models from these mathematical structures.

On the modeling of granular traffic flow by the kinetic theory for active particles trend to equilibrium and macroscopic behavior

The modeling of vehicular traffic flow is developed by methods of the discrete mathematical kinetic theory for active particles. The discretization refers to the velocity variable in the case of spatially homogeneity. The discretization overcomes, at least in part, some technical difficulties related to the selection of the correct representation scale. Moreover, the modeling approach includes in the state equation of the vehicle an activity variable suitable to model the quality (low or high) of the vehicle–driver system. This paper aims to be the first of a project concerning traffic flow by active particles methods.

Transition of unsteady flows of evaporation to steady state

We investigate the half-space problem of evaporation and condensation in the scope of discrete kinetic theory. Exact solutions are found to the boundary value problem and the initial boundary value problems of the flow in the half space for a discrete velocity model. The results are used to analyze the transition of the unsteady solutions towards steady states. To cite this article: A. d'Almeida, C. R. Mecanique 336 (2008).

From the discrete kinetic theory of vehicular traffic flow to computing the velocity distribution at equilibrium

This paper deals with the identification of the parameters of a model of vehicular traffic flow using experimental data obtained on the highway Padova–Venezia. Subsequently, simulations of the model are developed corresponding to steady uniform flow conditions. These simulations allow us to describe the velocity distribution in the above conditions corresponding to the parameters obtained by identification on the basis of experimental data.

A note on a Discrete Boltzmann Equation with multiple collisions

We compute a non-trivial explicit solution for the one-dimensional plane 6-velocity discrete Boltzmann model with multiple collisions introduced in [E. Longo, R. Monaco, On the discrete kinetic theory with multiple collisions: Plane six-velocity and unsteady Couette flow, in: Muntz, et al. (Eds.), The Proceedings of Rarefied Gas Dynamics, in: AIAA Publ., vol. 118, 1989, pp. 118–130] which asymptotically connects two particular equilibrium states. We prove that such a solution exists provided that a suitable condition on the differential elastic cross sections holds.

MATHEMATICAL MODELING OF VEHICULAR TRAFFIC: A DISCRETE KINETIC THEORY APPROACH

Following some general ideas on the discrete kinetic and stochastic game theory proposed by one of the authors in a previous work, this paper develops a discrete velocity mathematical model for vehicular traffic along a one-way road. The kinetic scale is chosen because, unlike the macroscopic one, it allows to capture the probabilistic essence of the interactions among the vehicles, and offers at the same time, unlike the microscopic one, the opportunity of a profitable analytical investigation of the relevant global features of the system. The discretization of the velocity variable, rather than being a pure mathematical technicality, plays a role in including the intrinsic granular nature of the flow of vehicles in the mathematical theory of traffic. Other important characteristics of the model concern the gain and loss terms of the kinetic equations, namely the construction of a density-dependent table of games to model velocity transitions and the introduction of a visibility length to account for nonlocal interactions among the vehicles.

From the discrete kinetic theory to modelling open systems of active particles

This work deals with a methodological development of the kinetic theory for open systems of active particles with discrete states. It essentially refers to the derivation of mathematical tools which provide the guidelines for modelling open systems in different fields of applied sciences. After a description of closed systems, mathematical frameworks suitable for depicting the evolution of open systems are proposed. Finally, some research perspectives towards modelling are outlined.

On the mathematical theory of vehicular traffic flow II: Discrete velocity kinetic models

This paper deals with the modelling of vehicular traffic flow by methods of the discrete mathematical kinetic theory. The discretization is developed in the velocity space by a grid adapted to the local density. The discretization overcomes, at least in part, some technical difficulties related to the selection of the correct representation scale, while the adaptative grid allows an improved description of various phenomena related to vehicular traffic flow. Specific models are proposed and a qualitative and computational analysis is developed to show the properties of the model and their ability to describe real flow conditions. A critical analysis, proposed in the last part of the paper, outlines suitable research perspectives.

Unsteady half-space evaporation and condensation problems on the basis of the discrete kinetic theory

Gas flow is considered on the basis of the discrete models for the Boltzmann equation. The formation and propagation of disturbances, in an initially uniform gas bounded by its plane condensed phase in nonequilibrium with the gas, are investigated in detail when the evaporation or the condensation is taking place from the condensed phase. The behavior of the solutions is clarified, the different types of solutions are investigated, and the profiles of typical solutions are presented. By a numerical approach, we can yield the conditions relating macroscopic variables at infinity and on the condensed phase which allow a steady solution. Our results with simple discrete models are in very good agreement with the ones given by continuous Boltzmann models.

Conservation laws and asymptotic behavior of a model of social dynamics

A conservative social dynamics model is developed within a discrete kinetic framework for active particles, which has been proposed in [M.L. Bertotti, L. Delitala, From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences, Math. Mod. Meth. Appl. Sci. 14 (2004) 1061–1084]. The model concerns a society in which individuals, distinguished by a scalar variable (the activity) which expresses their social state, undergo competitive and/or cooperative interactions. The evolution of the discrete probability distribution over the social state is described by a system of nonlinear ordinary differential equations. The asymptotic trend of their solutions is investigated both analytically and computationally. Existence, stability and attractivity of certain equilibria are proved.

Stationary ‘V’ States for Preferred Motions of Many Particles

We use the discrete kinetic theory with the free-orientation parameter being fixed (\(\pi/4\)) to derive the macroscopic velocity field for many particles flowing through a microdomain. Our results resemble qualitatively other hydrodynamical solutions. The V-shaped velocity field changes as the dominant physical parameter (Knudsen number) varies. We also briefly discuss the possible mechanism due to the entropy production along the boundaries.

On the discrete kinetic theory for active particles. Mathematical tools

This paper deals with the development of a mathematical discrete kinetic theory to model the dynamics of large systems of interacting active particles whose microscopic state includes not only geometrical and mechanical variables (typically position and velocity), but also peculiar functions, called activities, which are able to modify laws of classical mechanics. The number of the above particles is sufficiently large to describe the overall state of the system by a suitable probability distribution over the microscopic state, while the microscopic state is discrete. This paper deals with a methodological approach suitable to derive the mathematical tools and structures which can be properly used to model a variety of models in different fields of applied sciences. The last part of the paper outlines some research perspectives towards modelling.

On the qualitative analysis of the solutions of a mathematical model of social dynamics

This work deals with a family of dynamical systems which were introduced in [M.L. Bertotti, M. Delitala, From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences, Math. Models Methods Appl. Sci. 7 (2004) 1061–1084], modelling the evolution of a population of interacting individuals, distinguished by their social state. The existence of certain uniform distribution equilibria is proved and the asymptotic trend is investigated.