Transport theory and collective modes. I. The case of moderately dense gases

The complex spectral representation of the Liouville operator introduced by Prigogine and others is applied to moderately dense gases interacting thro ugh hard-core potentials in arbitrary d-dimensional spaces. Kinetic equatio ns near equilibrium are constructed in each subspace as introduced in the s pectral decomposition for collective, renormalized reduced distribution fun ctions. Our renormalization is a nonequilibrium effect, as the renormalizat ion effect disappears at equilibrium. It is remarkable that our renormalize d functions strictly obey well-defined Markovian kinetic equations for all d, even though the ordinary distribution functions obey nonMarkovian equati ons with memory effects. One can now define transport coefficients associat ed to the collective modes for all dimensional systems including d = 2. Our formulation hence provides a microscopic meaning of the macroscopic transp ort theory. Moreover, this gives an answer to the long-standing question wh ether or not transport equations exist in two-dimensional systems. The non- Markovian effects for the ordinary distribution function, such as the long- time tails for arbitrary n-mode coupling, are estimated by superposition of the Markovian evolutions of the dressed distribution functions.