Advanced plasma theories are extremely important when one tries to explain, for example, the various waves and instabilities found in the plasma environment. Since plasma consist of a very large number of interacting particles, in order to provide a macroscopic description of plasma phenomena it is appropriate to adopt a **statistical** approach. This leads to a great reduction in the amount of information to be handled. In the **kinetic theory** it is necessary to know only the **distribution function** for the system of particles. The distribution function for a specific particle species, f(**r**,**v**,t), is defined as the density of the particles in phase space,

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\begin{eqnarray} f(r,v,t) &=& dn(r,v,t)\over drdv \end{eqnarray} |

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

- number density
- average velocity
- momentum flow dyad
- energy flow triad

For example, the number density n(r,t) is the integral of f(**r**,**v**,t) calculated over the velocity space.

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, m**v**, and mv^2/2, and integrating over all the velocity space, results being, respectively,

- the equation of conservations of mass (continuity equation),
- the equation of conservations of momentum (equation of motion), and
- the equation of conservation of energy (energy equation).

However, at each stage of the hierarchy of moments, the resulting set of transport equations is not complete. Each time a higher moment of the Bolzmann equation is calculated in an attempt to obtain a complete set of transport equations, a new macroscopic variable appears. For example, the continuity equation relates the number density and the mean velocity. The inclusion of momentum equation adds the kinetic pressure dyad, and the inclusion of energy equation adds the heat flow vector. It is thus necessary to **truncate the system of equations at some stage of hierarchy by introducing a simplifying assumption** concerning the highest moment of the distribution function that appears in the system. This truncation creates a **closed system** of transport equations.

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:

- Hot plasma theory
- Warm plasma theory
- Cold plasma theory

There are also other possibilities for approximations:

- Isotropic or anisotropic plasma
- if an external magnetic field is present, plasma is anisotropic, otherwise it is isotropic
- external magnetic field has profound efect in things like plasma wave modes and wave-particle interaction

- Plasma particles
- in simplest case, only electron gas is considered
- in more general case, one or more ion species can be included in the theory
- whole plasma can be treated as a conducting fluid

- Collisional or collisionless plasma
- the additional complications introduced in collisional plasmas include things like wave damping