Lecture 3

1. Lecture 3

2. Full statistical description of the system of N particles is given by the many particle distribution function: in the phase space of 6N dimensions. One-particle distribution function: Two-particle distribution function, Three-particle distribution function, …

3. Luiville equation Equation for one-particle distribution function: interaction with other particles external fields

4. Mean force in plasma, acting on considered particle from many rather far situated particles, does not depend on their strict location and resembles the external force. Short-range collisions long-range collisions are defined by Maxwell equations, where in right hand sides of equations include not only internal but also external sources.

5. The kinetic description of a plasma consisting of several species of non relativistic charged particles (ions and electrons) is based on the determination of the distribution functions in the 6 dimensional phase space . Such approximation is relevant for the plasma of sufficiently low density, when it can be treated as ideal. The task of a basic kinetic theory is to provide and solve an equation for distribution function related to the dynamics of each individual particle subject to their mutual interaction and the effect of external forces.

6. The Vlasov equation is a differential equation describing time evolution of the distribution function of charged particles with Coulumn interaction.

7. Many-particle distribution function One-particle distribution function Macroscopic description Density Velocity Pressure

8. 1. Continuity equation 2. Momentum equation change due to electromagnetic fields change due to nonhomogeneity of the space change due to interactions with particles of other sorts

9. 3. Energy equation Heat flux due to the thermal conductivity Mean energy of chaotic movement The change of energy due to collisions with the particles of the other type

10. 5 equations for variables Unknown functions In order to solve the obtained system of equations it is necessary to define these unknown functions through variables Laws of conservation of the number of particles, their momentum and energy Infinite number of such equations, called equations for the moments of kinetic equations

11. The full statistical description of the system of N charged particles is given by F and Luiville equation Which is equivalent to the infinite chain of kinetic equations for one-particle, two-particle, and so on distribution functions We take only first equation in this chain and lose information about the behavior of all other particles Equations for moments of kinetic equation

12. Kinetic equation is equivalent to infinite chain of equations for moments We take only 3 first equations for moments and once more loose the information (all velocity space) Maxwell equations in quasistatic approximation

13. The approach to obtain hydrodynamic equations from the first principles was first introduced by Chapman and Enskog, and then used for plasma by Braginskii. When the characteristic scales at which the hydrodynamic parameters of plasma vary significantly, are considerably larger than mean free paths of plasma particles the distribution functions can be expressed as sums of zero order distribution functions, which correspond to the thermodynamic steady-state distribution functions, and a small perturbation. Substituting such expressions into the kinetic equations they can be solved approximately with respect to the perturbations, so that all dissipative fluxes can be expressed in terms of gradients of temperature, density and velocity and in terms of electric field, leading to the closed set of hydrodynamic equations.

14. KINETIC EQUATIONS The plasma consisting of several species of nonrelativistic charged particles is described by: Two-particle elastic electron collisions including screening are taken into account. We use the Landau form for collision terms. Ionization and recombination are not studied. Let us multiply the kinetic equations by and integrate over the velocity space. As the result we obtain the infinite system of equations for the moments of the distribution functions.

15. Balance equations The first three moments of kinetic equation for species , which are related to the laws of conservation of density, momentum and energy determine the system of hydrodynamic equations for each species: conservation of particle density, momentum, energy All the quantities are defined with reference to the mean velocity of the species

16. When the characteristic scales at which the hydrodynamic parameters of plasma vary significantly, are considerably larger than free paths of plasma particles, and when bulk velocities as well as the electric field in comoving reference frame are sufficiently small, then the state of plasma is close locally to the thermodynamic equilibrium. The perturbations should be much smaller than the zero order distribution functions

17. Electron kinetic equation The kinetic equation for the electrons If the parameters are small, then at each point in and space the electron distribution function differs little from Maxwellian distribution function with smoothly varying parameters. This makes it possible to use the expansion in powers of the small parameters to solve the kinetic equation and then calculate in terms of the hydrodynamic parameters of the plasma. As a result we obtain the closed system of hydrodynamic equations.

18. EMHD equations

19. : Electron-ion friction force the dissipative heat flux in the electrons: and the rate of thermal transfer between electrons and ions:

20. In order to complete the derivation of the MHD equations it is necessary to solve the kinetic equations for ions and express in terms of and

21. Ion equations

22. Ion thermal flux

23. MHD equations