The Establishment of Thermal Equilibrium

Abstract

In a tenuous plasma the velocities and energies of the particles can sustain a nonequilibrium distribution for a considerable period of time. Any difference between the average energies of the electrons and the ions can exist for an especially long time, so that it is often necessary to describe a plasma by two temperatures, the ion temperature T_i and the electron temperature T_e. The establishment of thermal equilibrium at each point in space occurs as a result of random velocity fluctuations. The time required to establish equilibrium is known as the relaxation time. It is necessary to distinguish between three different relaxation times. The establishment of an equilibrium in the electron velocity distribution occurs first, i.e., this process has the shortest relaxation time τ_e. A much longer time is required for the ion velocities to reach an equilibrium distribution. We shall denote this time by τ_i. An even longer time is required for complete energy exchange between the electrons and ions so that the two temperatures become equal. The time required is τ_ei. For identical final temperatures these times are related in the following way:

$$\frac{{{\tau _e}}}{{{\tau _l}}} = {Z^4}\sqrt {\frac{m}{M}} ;\frac{{{\tau _l}}}{{{\tau _{el}}}} = \frac{1}{{{Z^2}}}\sqrt {\frac{m}{M}} .$$