Kinetic equations for autocorrelation functions in dilute gases

We show that in the case of a dilute gas of neutral particles kinetic equations for autocorrelation functions such as $$leftlangle {hat fleft( {r,v,t} right)hat fleft( {rprime vprime ,tprime } right)} rightrangle ,wherehat fleft( {r,v,t} right) = sum {_{i = 1}^N } delta left( {r - r_i left( t right)} right)delta left( {v - v_i left( {tt} right)} right)$$ , can be obtained in a very simple manner by the use of the truncated BBGKY hierarchy. The resulting equations correspond to the low-density limit of the results of van Leeuwen and Yip. Moreover, the derivation does not make use of the Bogoliubov adiabatic approximation, and therefore includes non-Markovian effects which can be important in describing light scattering from gases and the collisional narrowing of atomic dipole radiation. The resulting equations in the long-wavelength limit correspond to the non-Markovian Boltzmann equation for the self-correlation part and the non-Markovian, linearized Boltzmann equation for the total autocorrelation function.