The maxwell model in kinetic theory of weakly ionized gases in electrostatic fields: Ion velocity distribution and moments

Through a multiple series expansion, the stationary velocity distribution of ions in a gas in an electrostatic field is derived. Its final form is a Burnett function series times the (maxwellian) equilibrium distribution. The following assumptions are made: (1) the ions are dilutely and uniformly dispersed in the gas (in thermal equilibrium), and (2) the ion-neutral interaction law is the maxwellian one. The conditions under which the obtained distribution agrees with well-known results valid for the extreme values of the ion-neutral mass ratio are discussed. Moreover, it is proved that the above distribution yields the exact values of the ion velocity moments if the uniform convergence of the series is assumed. However, it is shown that in certain cases the series does not converge at all even at low fields. This result raises the general problem of the limits of validity of the kinetic-theory methods based on Burnett function expansions of the ion velocity distribution.