Conservative Force Fields in Nonextensive Kinetic Theory

We investigate the nonextensive $q$-distribution function for a gas in the presence of an external field of force possessing a potential $U({\bf r})$. In the case of a dilute gas, we show that the power law distribution including the potential energy factor term can rigorously be deduced based on kinetic theoretical arguments. This result is significant as a preliminary to the discussion of long range interactions according to nonextensive thermostatistics and the underlying kinetic theory. As an application, the historical problem of the unbounded isothermal planetary atmospheres is rediscussed. It is found that the maximum height for the equilibrium atmosphere is $z_{max} = K_{B}T/mg(1-q)$. In the extensive limit, the exponential Boltzmann factor is recovered and the length of the atmosphere becomes infinite.