This function must be a continuous function of its arguments, positive and finite at any instant of time, and must tend to zero as the velocity becomes infinitely large. In the hot plasma model, all the macroscopic variables of physical interest are deduced from distribution function. This is possible because the moments of the function are related to these variables. The first four moments are related to
The dependence of the distribution function on the independent variables r, v, and t is governed by an equation known as the Boltzmann equation. In the absense of collisions, Liouville's theorem is formed. In addition, a very useful approximate way to describe the dynamics of a plasma is to consider that the motions of the plasma particles are governed by the applied external fields plus the macroscopic average internal fields, smoothed in space and time, due to presence and motion of all plasma particles. The Vlasov equation is derived from the Boltzmann equation using this kind of reasoning. The Vlasov equation, together with the Maxwell equations for the internal electromagnetic fields and equations for the plasma charge density and plasma current density, constitute a complete set of self-consistent equations to be solved simultaneously.
An equilibrium distribution function, like the Maxwell-Boltzmann function, is the time- independent solution of the Bolzmann equation in the absence of external forces. It represents the most probable distribution function satisfying the macroscopic conditions imposed on the system.
The time-dependent Bolzmann equation is diffucult to solve, and some approximate methods to derive macroscopic variables are needed. This is possible by deriving differential equations governing the temporal and spatial variations of the macroscopic variables directly from the Bolzmann equation without solving it. These differential equations are called the macroscopic transport equations, and they can be obtained by taking the moments of the Boltzmann equation. The first three moments are obtained by multiplying the equation by m, mv, and mv^2/2, and integrating over all the velocity space, results being, respectively,
The simplest closed system is known as the cold plasma model, and it contains only the equations of conservations of mass and momentum (macroscopic variables being the number density and the mean velocity). The highest moment of this system, the kinetic pressure dyad, is taken to be zero in the truncation process. Thus the cold plasma model assumes a zero plasma temperature, and the corresponding distribution function is a Dirac delta function centered at the macroscopic flow velocity! This model can be used in the study of small amplitude electromagnetic waves propagating in the plasmas, with phase velocities much larger than the thermal velocity of the particles (magnetoionic theory).
In the warm plasma model, also the energy equation is included, but the term involving the heat flux vector is neglected. Because the thermal conductivity is thus zero, the plasma is non-viscous and, consequently, the non-diagonal terms of the pressure dyad are all equal to zero. When the diagonal terms of the dyad are also assumed to be equal, the third macroscopic variable in this approximation is the scalar pressure. The energy equation reduces to the adiabatic equation (warm plasma model is also called the adiabatic approximation).
Above we have considered equations that are written separately for each particle species considered important. However, plasma can also be considered as a conducting fluid (one fluid theory) without specifying its various individual species. In this case, each macroscopic variable is formed by adding the contributions of the various particle species in the plasma. Using simplified forms of the transport equations (conservations of mass, momentum, and energy) and additional electrodynamic equations (Maxwell curl equations, conservations of electric charge, and generalized Ohm's law) the so called magnetohydrodynamic (MHD) theory is created. Note that the truncation of the equations can be done either in cold or warm plasma levels.
To summarize, the level of complication in the starting equation set determines three basic plasma theories: