Carlo Cercignani

Bio: Carlo Cercignani is an academic researcher from Polytechnic University of Milan. The author has contributed to research in topics: Boltzmann equation & Boundary value problem. The author has an hindex of 46, co-authored 251 publications receiving 13637 citations.

The Boltzmann equation and its applications

Abstract: I. Basic Principles of The Kinetic Theory of Gases.- 1. Introduction.- 2. Probability.- 3. Phase space and Liouville's theorem.- 4. Hard spheres and rigid walls. Mean free path.- 5. Scattering of a volume element in phase space.- 6. Time averages, ergodic hypothesis and equilibrium states.- References.- II. The Boltzmann Equation.- 1. The problem of nonequilibrium states.- 2. Equations for the many particle distribution functions for a gas of rigid spheres.- 3. The Boltzmann equation for rigid spheres.- 4. Generalizations.- 5. Details of the collision term.- 6. Elementary properties of the collision operator. Collision invariants.- 7. Solution of the equation Q(f,f) = 0.- 8. Connection between the microscopic description and the macroscopic description of gas dynamics.- 9. Non-cutoff potentials and grazing collisions. Fokker-Planck equation.- 10. Model equations.- References.- III. Gas-Surface Interaction and the H-Theorem.- 1. Boundary conditions and the gas-surface interaction.- 2. Computation of scattering kernels.- 3. Reciprocity.- 4. A remarkable inequality.- 5. Maxwell's boundary conditions. Accommodation coefficients.- 6. Mathematical models for gas-surface interaction.- 7. Physical models for gas-surface interaction.- 8. Scattering of molecular beams.- 9. The H-theorem. Irreversibility.- 10. Equilibrium states and Maxwellian distributions.- References.- IV, Linear Transport.- 1. The linearized collision operator.- 2. The linearized Boltzmann equation.- 3. The linear Boltzmann equation. Neutron transport and radiative transfer.- 4. Uniqueness of the solution for initial and boundary value problems.- 5. Further investigation of the linearized collision term.- 6. The decay to equilibrium and the spectrum of the collision operator.- 7. Steady one-dimensional problems. Transport coefficients.- 8. The general case.- 9. Linearized kinetic models.- 10. The variational principle.- 11. Green's function.- 12. The integral equation approach.- References.- V. Small and Large Mean Free Paths.- 1. The Knudsen number.- 2. The Hilbert expansion.- 3. The Chapman-Enskog expansion.- 4. Criticism of the Chapman-Enskog method.- 5. Initial, boundary and shock layers.- 6. Further remarks on the Chapman-Enskog method and the computation of transport coefficients.- 7. Free molecule flow past a convex body.- 8. Free molecule flow in presence of nonconvex boundaries.- 9. Nearly free-molecule flows.- References.- VI. Analytical Solutions of Models.- 1. The method of elementary solutions.- 2. Splitting of a one-dimensional model equation.- 3. Elementary solutions of the simplest transport equation.- 4. Application of the general method to the Kramers and Milne problems.- 5. Application to the flow between parallel plates and the critical problem of a slab.- 6. Unsteady solutions of kinetic models with constant collision frequency.- 7. Analytical solutions of specific problems.- 8. More general models.- 9. Some special cases.- 10. Unsteady solutions of kinetic models with velocity dependent collision frequency.- 11. Analytic continuation.- 12. Sound propagation in monatomic gases.- 13. Two-dimensional and three-dimensional problems. Flow past solid bodies.- 14. Fluctuations and light scattering.- References.- VII. The Transition Regime.- 1. Introduction.- 2. Moment and discrete ordinate methods.- 3. The variational method.- 4. Monte Carlo methods.- 5. Problems of flow and heat transfer in regions bounded by planes or cylinders.- 6. Shock-wave structure.- 7. External flows.- 8. Expansion of a gas into a vacuum.- References.- VIII. Theorems on the Solutions of the Boltzmann Equation.- 1. Introduction.- 2. The space homogeneous case.- 3. Mollified and other modified versions of the Boltzmann equation.- 4. Nonstandard analysis approach to the Boltzmann equation.- 5. Local existence and validity of the Boltzmann equation.- 6. Global existence near equilibrium.- 7. Perturbations of vacuum.- 8. Homoenergetic solutions.- 9. Boundary value problems. The linearized and weakly nonlinear cases.- 10. Nonlinear boundary value problems.- 11. Concluding remarks.- References.- References.- Author Index.

The mathematical theory of dilute gases

Abstract: This book is devoted to the presentation of rigorous mathematical results in the kinetic theory of a gas of hard spheres. Recent developments as well as classical results are presented in a unified way, such that the book should become the standard reference on the subject. There is no such book available at present. The reader will find a systematic treatment of the main mathematical results, a discussion of open problems, and a guide to the existing literature. There is a rigorous and comprehensive presentation of strict validation of the Boltzmann equations, global existence theory, and the fluid-dynamical limits. The authors also review and discuss classical derivation and properties of the Boltzmann equation, particle simulation methods, and boundary conditions.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Traffic and related self-driven many-particle systems

Abstract: Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws

Abstract: In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics.