∇ × B = µ_{0}j + µ_{0 0 }

# ∂E

∂t

(6)

To solve these, we must know the charge density ρ(r, t) and the current density j(r, t). However, these are given by the particle positions and motions by

ρ(r, t) =

q_{k }δ(r − r_{k}(t))

(7)

k

j(r, t) =

q_{k }v_{k }δ(r − r_{k}(t))

(8)

k

# Hence, we have an infinite chain

r, v ⇐ E, B ⇐ ρ, j ⇐ r, v ⇐ ...

(9)

The equations above must therefore be solved simultaneously. Note that the number of equations is very large: there is one of equation (2) for each particle, so in any interesting portion of space, the number of equations will be enormous. However, this description of a plasma is used for doing computer simulations of plasma dynamics, where indeed the above equations or some simplifica- tions of them are solved to find the behaviour of the plasma. Such simulations usually includes a few thousand particles, and are normally not fully three dimensional. However, as the computer capacity increases, the simulations will be more and more realistic, and the importance of numerical simulations of plasmas is likely to increase.

# 3.4 Statistical description of plasmas

Even if we could solve all the equations above, we wouldn’t learn much, since we would get far too much information – we are not interested in the positions and motion of every single particle. Rather, we are interested in statistical averages, for example the density of the plasma. Instead of calculating the motion and position of every particle and then averaging the results, we may try to find equations for the statistical quantities themselves. This is the essence of a statistical description of a plasma.

The fundamental statistical quantity is the distribution function f(r, v, t). This tells us how many particles with velocity near v that are present near the location r at time t. More specifically, the number of particles of species α in the volume d^{3}r = dx dy dz that have velocities in the intervals

[ v x , v x + d v x ] , [ v y , v y + d v y ] , [ v z , v z + d v z ] i s f α ( r , v , t ) d 3 r d 3 v = f ( x , y , z , v x , v y , v z , t ) d x d y d z d v x d v y d v z .

(10)

Hence, the SI unit of f is s^{3}/m^{6}. There is one distribution function for each particle species. In a plasma, there will thus be one distribution function for the electrons, and one for each ion species.

It is possible to construct equations for how the distribution function evolves in time and space due to the influence of electromagnetic fields and other forces. This is the basis for the most advanced plasma theory, called kinetic theory. We will not discuss kinetic theory in this course.

# For a gas or plasma in thermodynamic equilibrium, the distribution function is the Maxwell-

# Boltzmann distribution

f(r, v) =

m 2πKT

3/2

exp

−

1 2

mv^{2 }+ U(r) KT

(11)

where U(r) is the potential energy. The distribution is shown in Figure 1. Plasmas in space are often far from equilibrium, and non-Maxwellian distributions are frequently encountered. Nevertheless, the Maxwell-Boltzmann distribution is often a good approximation.

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