The Boltzmann Equation and Its ApplicationsSpringer Science & Business Media, 06.12.2012 - 455 Seiten Statistical mechanics may be naturally divided into two branches, one dealing with equilibrium systems, the other with nonequilibrium systems. The equilibrium properties of macroscopic systems are defined in principle by suitable averages in well-defined Gibbs's ensembles. This provides a frame work for both qualitative understanding and quantitative approximations to equilibrium behaviour. Nonequilibrium phenomena are much less understood at the present time. A notable exception is offered by the case of dilute gases. Here a basic equation was established by Ludwig Boltzmann in 1872. The Boltzmann equation still forms the basis for the kinetic theory of gases and has proved fruitful not only for a study of the classical gases Boltzmann had in mind but also, properly generalized, for studying electron transport in solids and plasmas, neutron transport in nuclear reactors, phonon transport in superfluids, and radiative transfer in planetary and stellar atmospheres. Research in both the new fields and the old one has undergone a considerable advance in the last thirty years. |
Inhalt
| 1 | |
THE BOLTZMANN EQUATION | 40 |
GASSURFACE INTERACTION AND THE HTHEOREM | 104 |
Reciprocity | 111 |
A remarkable inequality | 115 |
Maxwells boundary conditions Accommodation coefficients | 118 |
Mathematical models for gassurface interaction | 122 |
Physical models for gassurface interaction | 130 |
Application of the general method to the Kramers and Milne problems | 294 |
Application to the flow between parallel plates and the critical problem of a slab | 299 |
Unsteady solutions of kinetic models with constant collision frequency | 306 |
Analytical solutions of specific problems | 310 |
More general models | 315 |
Some special cases | 319 |
Unsteady solutions of kinetic models with velocity dependent collision frequency | 322 |
Analytic continuation | 330 |
Scattering of molecular beams | 134 |
The Htheorem Irreversibility | 137 |
Equilibrium states and Maxwellian distributions | 142 |
Appendix | 149 |
References | 156 |
LINEAR TRANSPORT 1 The linearized collision operator | 158 |
The linearized Boltzmann equation | 161 |
The linear Boltzmann equation Neutron transport and radiative transfer | 165 |
Uniqueness of the solution for initial and boundary value problems | 172 |
Further investigation of the linearized collision term | 174 |
The decay to equilibrium and the spectrum of the collision operator | 180 |
Steady onedimensional problems Transport coefficients | 189 |
The general case | 200 |
Linearized kinetic models | 205 |
The variational principle | 212 |
Greens function | 215 |
The integral equation approach | 222 |
References | 229 |
SMALL AND LARGE MEAN FREE PATHS 1 The Knudsen number | 232 |
The Hilbert expansion | 234 |
The ChapmanEnskog expansion | 239 |
Criticism of the ChapmanEnskog method | 245 |
Initial boundary and shock layers | 248 |
Further remarks on the ChapmanEnskog method and the computation of transport coefficients | 260 |
Free molecule flow past a convex body | 262 |
Free molecule flow in presence of nonconvex boundaries | 271 |
Nearly freemolecule flows | 278 |
References | 283 |
ANALYTICAL SOLUTIONS OF MODELS 1 The method of elementary solutions | 286 |
Elementary solutions of the simplest transport equation | 288 |
Sound propagation in monatomic gases | 334 |
Twodimensional and threedimensional problems Flow past solid bodies | 338 |
Fluctuations and light scattering | 344 |
Appendix | 345 |
References | 348 |
THE TRANSITION REGIME 1 Introduction | 351 |
The variational method | 355 |
Monte Carlo methods | 359 |
Problems of flow and heat transfer in regions bounded by planes or cylinders | 361 |
Shockwave structure | 369 |
External flows | 377 |
Expansion of a gas into a vacuum | 380 |
References | 385 |
THEOREMS ON THE SOLUTIONS OF THE BOLTZMANN EQUATION 1 Introduction | 392 |
Mollified and other modified versions of the Boltzmann equation | 398 |
Nonstandard analysis approach to the Boltzmann equation | 401 |
Local existence and validity of the Boltzmann equation | 405 |
Global existence near equilibrium | 407 |
Perturbations of vacuum | 412 |
Homoenergetic solutions | 414 |
Boundary value problems The linearized and weakly nonlinear cases | 417 |
Nonlinear boundary value problems | 422 |
Concluding remarks | 425 |
References | 426 |
APPENDIX | 431 |
References | 439 |
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| 450 | |
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