Kinetic Theory of Gases and Plasmas

ISBN-13: 9789401056021 / Angielski / Miękka / 2012 / 429 str.

Ppjm Schram

Opis

Kinetic theory is the link between the non--equilibrium statistical mechanics of many particle systems and macroscopic or phenomenological physics. Therefore much attention is paid in this book both to the derivation of kinetic equations with their limitations and generalizations on the one hand, and to the use of kinetic theory for the description of physical phenomena and the calculation of transport coefficients on the other hand. The book is meant for researchers in the field, graduate students and advanced undergraduate students. At the end of each chapter a section of exercises is added not only for the purpose of providing the reader with the opportunity to test his understanding of the theory and his ability to apply it, but also to complete the chapter with relevant additions and examples that otherwise would have overburdened the main text of the preceding sections. The author is indebted to the physicists who taught him Statistical Mechanics, Kinetic Theory, Plasma Physics and Fluid Mechanics. I gratefully acknowledge the fact that much of the inspiration without which this book would not have been possible, originated from what I learned from several outstanding teachers. In particular I want to mention the late Prof. dr. H. C. Brinkman, who directed my first steps in the field of theoretical plasma physics, my thesis advisor Prof. dr. N. G. Van Kampen and Prof. dr. A. N. Kaufman, whose course on Non-Equilibrium Statistical Mechanics in Berkeley I remember with delight.

Spis treści

1. Introduction.- 1.1. The nature and the goals of kinetic theory. Summary and related books..- 1.2. Some concepts from probability theory..- 1.3. Some properties of the Dirac delta function..- 1.4. Phase spaces, conservation of probability and the Liouville equation..- 1.5 Microscopic and macroscopic quantities..- 1.6. Exercises..- 2. Balance Equations.- 2.1. Conservation of particles.- 2.2. Momentum equation..- 2.2.1 Short range interaction forces..- 2.2.2 Long range interaction forces.- 2.2.3 Approximations: Boltzmann gas, Landau gas and electron plasma..- 2.3. Energy equation..- 2.4. Exercises..- 3. Klimontovich Equation, b.b.g.k.y.-hierarchy and vlasov-maxwell Equations.- 3.2. Densities in µ-space..- 3.2. Klimontovich equation..- 3.3. Vlasov-Maxwell equations..- 3.4. The first equation of the B.B.G.K.Y.-hierarchy..- 3.5. The complete hierarchy..- 3.6. Derivation of the B.B.G.K.Y.-hierarchy..- 3.7. Exercises..- 4. Derivation and Properties of the boltzmann equation.- 4.1. The small parameter of the Boltzmann gas..- 4.2. Multiple-time-scales formalism..- 4.2.1 The Van de Pol oscillator..- 4.3. Derivation of the Boltzmann equation..- 4.3.1 First Order theory and Bogoliubov boundary condition..- 4.3.2 Discussion of the kinetic equation. Limitations of Bogoliubov’s approach..- 4.3.3 Bogoliubov’s cylindrical integration..- 4.4. Dynamics of binary collisions..- 4.4.1 An explicit form of the Boltzmann equation..- 4.4.2 Cross-sections.- 4.5. Boltzmann equation and Markov processes..- 4.6. Properties of the Boltzmann equation..- 4.6.1 Positivity of the distribution function and invariance under time reversal..- 4.6.2 H-theorem for a uniform gas..- 4.6.3 H-theorem for a non-uniform gas.- 4.6.4 The pair distribution function in equilibrium..- 4.7. Discussion of irreversibility..- 4.8. Exercises..- 5. Chapman-enskog theory: Asymptotic solution to the boltzmann equation; transport Coefficients.- 5.1. Introduction and table of characteristic quantities..- 5.2. Balance equations..- 5.3. Power series in the Knudsen number and the multiple time scale formalism revisited..- 5.3.1 Zeroth and first Order theory, the Euler equations of hydrodynamics and the Chapman-Enskog integral equation..- 5.3.2 Derivation of the Navier-Stokes equations and the constitutive equations..- 5.4. The role of entropy and the thermodynamic identity..- 5.5. The eigenvalues of the linearized collision Operator and transport coefficients..- 5.5.1 Properties of irreducible tensors..- 5.6. The Maxwell gas..- 5.7. Non-Maxwellian intermolecular interaction..- 5.8. Exercises..- 6. Kinetic theory of Plasmas in the binary collision Approximation..- 6.1. Kinetic theory of gas mixtures. Lorentz gas..- 6.1.1 Expansion of the collision integral J12 in powers of the Square root of the mass ratio..- 6.1.2 Expansion in powers of the Knudsen number. Zeroth and first Order theory..- 6.1.3 Second order theory. Diffusion, thermodiffusion, thermal conductivity and Dufour effect. Onsager symmetry..- 6.2. The electrons in a very weakly ionized gas..- 6.2.1 Transport properties..- 6.2.2 The isotropic part of the distribution function. The Davydov distribution..- 6.2.3 Relaxation towards the Davydov distribution..- 6.3. The Landau equation for a fully ionized plasma..- 6.3.1 Derivation from the Boltzmann equation. Impulse approximation..- 6.3.2 Discussion of the validity of the Landau equation for a plasma..- 6.3.3 The Landau equations for electrons and ions..- 6.4. Calculation of the electrical conductivity..- 6.4.1 Simplifying assumptions..- 6.4.2 Electrical conductivity and velocity dependent collision frequency..- 6.4.3 DC-conductivity and conductivity at rather high frequencies..- 6.4.4 Validity of the Lorentz approximation..- 6.4.5 DC-conductivity and electron-electron collisions.- 6.5. Exercises..- 7. B.G.K.-Models and the slip problem..- 7.1. Linear B.G.K.-model. Its relation to the Boltzmann equation..- 7.2. The non-linear B.G.K.-model. Linearization..- 7.3. The slip problem of Kramers..- 7.4. Solution to the B.G.K. integro-differential equation..- 7.5. The singular integral equation and hydrodynamic slip..- 7.6. The microscopic slip velocity..- 7.7. Exercises..- 8. Kinetic theory of Plasmas, including dynamical screening..- 8.1. Collisions and screening in plasmas. The Lenard approach..- 8.1.1 Equations for the distribution function and the binary correlation function..- 8.1.2. Derivation of the Lenard-Balescu equation..- 8.2. The interaction between two charged particles in a dielectric medium..- 8.2.1 The dynamically screened interaction and the impulse approximation..- 8.2.2 Heuristic derivation of the Lenard-Balescu equation..- 8.3. Properties of the Lenard-Balescu equation..- 8.4. The Landau equation as an approximation to the Lenard-Balescu equation..- 8.5. Completely convergent collision integrals..- 8.5.1 The quantum-mechanical version of the Lenard-Balescu equation..- 8.5.2 Completely convergent classical collision integral..- 8.6. The electrical conductivity at rather high frequencies..- 8.6.1 Calculation of the quantum-mechanical conductivity..- 8.6.2 Calculation of the completely convergent classical conductivity..- 8.7. Excercises..- 9. Linear Response Theory.- 9.1. Linearized Liouville equation..- 9.2. Kubo formulae..- 9.2.1 Derivation..- 9.2.2 Symmetries..- 9.2.3 Time reversal..- 9.3. Electrical conductivity..- 9.3.1 The Kubo-formula..- 9.3.2 Fluctuation-dissipation theorem, Nyquist theorem..- 9.4. Internal agencies..- 9.4.1 Mori method: linearization in small gradients..- 9.4.2 Fluctuations and entropy..- 9.5. Longtime tail of autocorrelation functions..- 9.5.1 Kinetic approximation to the velocity autocorrelation function..- 9.5.2 Asymptotic behaviour for large time..- 9.6. Exercises..- 10. Brownian Motion.- 10.1. Statistical description. Markov processes..- 10.1.1 Fokker-Planck equation for the position. Diffusion..- 10.1.2 Rayleigh particle. Fokker-Planck equation for the velocity..- 10.1.3 Autocorrelation functions of velocity and position..- 10.1.4 Langevin equation..- 10.2. Generalized theory of the velocity autocorrelation function..- 10.2.1 Hydrodynamical forces on a Brownian particle..- 10.2.2 An equation for the velocity autocorrelation function derived from linear response theory, and its solution..- 10.2.3 Long time tales..- 10.3. Hydrodynamic fluctuations and the generalized Langevin equation..- 10.3.1 Induced forces..- 10.3.2 The generalized Faxén theorem..- 10.3.3 Stochastic hydrodynamic equations..- 10.3.4 Generalized Langevin equation and a fluctuation-dissipation theorem..- 10.4. Discussion of the velocity autocorrelation function..- 10.4.1 Solution to the generalized Langevin equation..- 10.4.2 Long time tails..- 10.4.3 Some remaining difficulties..- 10.5. Exercises..- Appendix..- 11. Dense Gases, Renormalized kinetic theory.- 11.1. The Enskog equation for hard sphere dense gases..- 11.1.1 Determination of Y(n)..- 11.1.2 Transport coefficients..- 11.1.3 Self-diffusion. Lorentz-Enskog equation..- 11.2. Limitations of Bogoliubov approach revisited. Hard-sphere gases..- 11.2.1 The binary collision expansion..- 11.2.2 Hard-sphere dynamics. Pseudo-Liouville Equation and -Hierarchy..- 11.3. Renormalization of collisional effects..- 11.3.1 The Choh-Uhlenbeck collision term. The ring operator..- 11.3.2 The diffusion coefficient of a Lorentz gas..- 11.3.3 Self-diffusion..- 11.4. Memory effects in hard-sphere gases and self-diffusion..- 11.4.1 Dynamic cluster expansion..- 11.4.2 Independent particle approximation. Non-Markovian kinetic equation..- 11.4.3 Some results obtainable from the Non-Markovian kinetic equation..- 11.5. Exercises..- 12. Theory of (Slightly) nonideal Plasmas.- 12.1. The Klimontovich equation revisited..- 12.1.1 Fourier transforms.- 12.2. The expansion scheme..- 12.2.1 Initial conditions..- 12.2.2 Derivation of the Lenard-Balescu equation..- 12.2.3 Corrections to the Lenard-Balescu equation..- 12.3. The electrical conductivity at frequencies much lower than the plasma frequency..- 12.3.1 Outline of the method..- 12.3.2 Calculation of the conductivity by means of a completely convergent collision integral..- 12.3.3 Discussion of the results..- 12.4. The electrical conductivity at high frequencies.- 12.4.1 The zeroth and first order conductivity..- 12.4.2 Second order conductivity..- 12.4.3 The conductivity in case of a homogeneous electric field..- 12.4.4 Comparison with Kubo’s formalism..- 12.5. The dispersion relation for plasma waves..- 12.5.1 The dispersion relation in zeroth order..- 12.5.2 The dispersion relation in second order..- 12.6. Remarks about strongly non-ideal plasmas..- 12.6.1 Classification of plasmas, n-T diagram..- 12.6.2 Quantum-statistical methods..- 12.6.3 Some results for thermodynamic equilibrium..- 12.6.4 Some results for the electrical conductivity..- 12.7. Exercises..- References..- Index..