Fundamental Theories of Physics

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 u-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 Faxen 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..