0.07

0.06

0.5

# Distribution (arbitrary units)

0.05

0.04

0.03

# Distribution (arbitrary units)

0.4

0.3

0.2

0.02

0.01

0.1

0

0

0

1

2

3

4

0

1

2

3

4

vx

v

Figure 1: The Maxwell-Boltzmann distribution for a velocity component (left) and for the speed (right). The most probable velocity along any given axis is zero, as seen in the left plot, while the most probable speed is non-zero (right plot).

3.5

Fluid description of plasmas

3.5.1

Fluid parameters

By summing over all velocities, we get a fluid description of a plasma. Here, each particle species is described by what is known as fluid parameters: density, flow speed, temperature and so on. We get the number density by simply summing the distribution function for all velocities:

n_{α}(r, t) =

f_{α}(r, v, t) d^{3}v.

(12)

This is the number of particles of species α per unit volume (SI unit: m^{−3}). We get the mass density by multiplying the number of particles per unit volume by the mass of each particle, so

ρ_{m }

=

m_{α}n_{α},

(13)

α

and similarly the charge density is given by

ρ=

q_{α}n_{α}.

(14)

α

# By defining the mean velocity of the particles of species α,

v_{α}(r, t) =

1 n_{α }

v f_{α}(r, v, t) d^{3}v,

(15)

we can also calculate the current density in the plasma as

j=

q α n α v α .

(16)

α

In this course, we will normally assume that the plasmas we study consist of two particle species: electrons (e) and protons (i). Such a plasma is called a two-component plasma. In this case, we get

ρ_{m }

= m e n e + m i n i ≈ m i n i ( b e c a u s e m e

m i ) , ρ = e ( n i − n e ) , a n d j = e ( n i v i − n e v e ) .

8