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Molecular Gas Dynamics : Theory, Techniques, and Applications
This self-contained work is an up-to-date treatment of the basic theory of molecular gas dynamics and its various applications. Recent progress in the field has greatly enhanced the original theory and stimulated interesting and critical gas dynamic phenomena and problems. This book, unique in the literature, presents working knowledge, theory, techniques, and typical phenomena in rarefied gases for theoretical development and applications.
Mathematical Models of Granular Matter
Granular matter displays a variety of peculiarities that distinguish it from other appearances studied in condensed matter physics and renders its overall mathematical modelling somewhat arduous. Prominent directions in the modelling granular flows are analyzed from various points of view. Foundational issues, numerical schemes and experimental results are discussed. The volume furnishes a rather complete overview of the current research trends in the mechanics of granular matter. Various chapters introduce the reader to different points of view and related techniques. New models describing granular bodies as complex bodies are presented. Results on the analysis of the inelastic Boltzmann equations are collected in different chapters. Gallavotti-Cohen symmetry is also discussed.
From Hyperbolic Systems to Kinetic Theory : A Personalized Quest
Equations of state are not always effective in continuum mechanics. Maxwell and Boltzmann created a kinetic theory of gases, using classical mechanics. How could they derive the irreversible Boltzmann equation from a reversible Hamiltonian framework? By using probabilities, which destroy physical reality! Forces at distance are non-physical as we know from Poincaré's theory of relativity. Yet Maxwell and Boltzmann only used trajectories like hyperbolas, reasonable for rarefied gases, but wrong without bound trajectories if the "mean free path between collisions" tends to 0. Tartar relies on his H-measures, a tool created for homogenization, to explain some of the weaknesses, e.g. from quantum mechanics: there are no "particles", so the Boltzmann equation and the second principle, can not apply. He examines modes used by energy, proves which equation governs each mode, and conjectures that the result will not look like the Boltzmann equation, and there will be more modes than those indexed by velocity!
Entropy Methods for the Boltzmann Equation : Lectures from a Special Semester at the Centre Émile Borel, Institut H. Poincaré, Paris, 2001
Entropy and entropy production have recently become mathematical tools for kinetic and hydrodynamic limits, when deriving the macroscopic behaviour of systems from the interaction dynamics of their many microscopic elementary constituents at the atomic or molecular level. During a special semester on Hydrodynamic Limits at the Centre Émile Borel in Paris, 2001 two of the research courses were held by C. Villani and F. Rezakhanlou. Both illustrate the major role of entropy and entropy production in a mutual and complementary manner and have been written up and updated for joint publication. Villani describes the mathematical theory of convergence to equilibrium for the Boltzmann equation and its relation to various problems and fields, including information theory, logarithmic Sobolev inequalities and fluid mechanics. Rezakhanlou discusses four conjectures for the kinetic behaviour of the hard sphere models and formulates four stochastic variations of this model, also reviewing known results for these.
Mathematical Modeling of Complex Biological Systems : A Kinetic Theory Approach
Describes the evolution of several socio-biological systems using mathematical kinetic theory. Specifically, it deals with modeling and simulations of biological systems—comprised of large populations of interacting cells—whose dynamics follow the rules of mechanics as well as rules governed by their own ability to organize movement and biological functions. The authors propose a new biological model for the analysis of competition between cells of an aggressive host and cells of a corresponding immune system.Because the microscopic description of a biological system is far more complex than that of a physical system of inert matter, a higher level of analysis is needed to deal with such complexity. Mathematical models using kinetic theory may represent a way to deal with such complexity, allowing for an understanding of phenomena of nonequilibrium statistical mechanics not described by the traditional macroscopic approach. The proposed models are related to the generalized Boltzmann equation and describe the population dynamics of several interacting elements (kinetic population models).The particular models proposed by the authors are based on a framework related to a system of integro-differential equations, defining the evolution of the distribution function over the microscopic state of each element in a given system. Macroscopic information on the behavior of the system is obtained from suitable moments of the distribution function over the microscopic states of the elements involved.
Mathematical Foundation of Turbulent Viscous Flows : Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1-5, 2003
Five leading specialists reflect on different and complementary approaches to fundamental questions in the study of the Fluid Mechanics and Gas Dynamics equations. Constantin presents the Euler equations of ideal incompressible fluids and discusses the blow-up problem for the Navier-Stokes equations of viscous fluids, describing some of the major mathematical questions of turbulence theory.
Macroscopic Transport Equations for Rarefied Gas Flows : Approximation Methods in Kinetic Theory
This book discusses classical and modern methods to derive macroscopic transport equations for rarefied gases from the Boltzmann equation, for small and moderate Knudsen numbers, i.e.as well as the new order of magnitude method, which avoids the short-comings of the classical methods, but retains their benefits.
