The Boltzmann equation and its applications

Carlo Cercignani

This book gives a complete exposition of the present status of the theory of the Boltzmann equation and its applications. The Boltzmann equation, an integrodifferential equation established by Boltzmann in 1872 to describe the state of a dilute gas, still forms the basis for the kinetic theory of gases. It has proved fruitful not only for the study of the classical gases Boltzmann had in mind, but also, properly generalized, for electron transport in nuclear reactors, photon transport in superfluids, and radiative transport in planetary and stellar atmospheres. The text presents a unified approach to the problems arising in these different fields, by exploiting similarities whenever they exist and underlining the differences when necessary. But the main exposition is tied to the classical equation established by Boltzmann. Hence the detailed description of applications refers almost exclusively to monatomic neutral gases. Appropiate references are given to papers dealing with similar problems arising in other fields, with particular concern for neutron transport. A unique feature is the detailed consideration of the boundary conditions to be used in connection with the Boltzmann equation. Other topics covered in detail are the derivation of the Boltzmann equation from first principles, the theory of the linearized Boltzmann equation, the use of model equations, and the various regimes of rarefied gas dynamics. In addition to updating the material to 1987, the main improvement over the previous book of the author, "Theory and Application of the Boltzmann equation" is the detailed survey of the use of the techniques of functional analysis in connection with the nonlinear Boltzmann equation, a subject which has greatly progressed in the last ten years.

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This book gives a complete exposition of the present status of the theory of the Boltzmann equation and its applications. The Boltzmann equation, an integrodifferential equation established by Boltzmann in 1872 to describe the state of a dilute gas, still forms the basis for the kinetic theory of gases. It has proved fruitful not only for the study of the classical gases Boltzmann had in mind, but also, properly generalized, for electron transport in nuclear reactors, photon transport in superfluids, and radiative transport in planetary and stellar atmospheres. The text presents a unified approach to the problems arising in these different fields, by exploiting similarities whenever they exist and underlining the differences when necessary. But the main exposition is tied to the classical equation established by Boltzmann. Hence the detailed description of applications refers almost exclusively to monatomic neutral gases. Appropiate references are given to papers dealing with similar problems arising in other fields, with particular concern for neutron transport. A unique feature is the detailed consideration of the boundary conditions to be used in connection with the Boltzmann equation. Other topics covered in detail are the derivation of the Boltzmann equation from first principles, the theory of the linearized Boltzmann equation, the use of model equations, and the various regimes of rarefied gas dynamics. In addition to updating the material to 1987, the main improvement over the previous book of the author, "Theory and Application of the Boltzmann equation" is the detailed survey of the use of the techniques of functional analysis in connection with the nonlinear Boltzmann equation, a subject which has greatly progressed in the last ten years.

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[目次]

  • 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.

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[目次]

  • 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.

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この本の情報

書名 The Boltzmann equation and its applications
著作者等 Cercignani, Carlo
シリーズ名 Applied mathematical sciences
出版元 Springer-Verlag
刊行年月 c1988
版表示 Softcover reprint of the original 1st ed. 1988
ページ数 xii, 455 p.
大きさ 25 cm
ISBN 3540966374
0387966374
9781461269953
NCID BA01469128
※クリックでCiNii Booksを表示
言語 英語
出版国 アメリカ合衆国
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