Transport Theory and Collective Modes II: Long-Time Tail and Green-Kubo Formalism

Abstract

The long-time tail effect (i.e., a non-Markovian effect) in a velocity autocorrelation function for moderately dense classical gases in d-dimensional space is estimated for arbitray n-mode coupling by superposition of the Markov equations for the collective modes which has been introduced through the complex spectral representation of the Liouville operator in the previous paper. Taking into account intermediate nonhydrodynamic modes in a transition between hydrodynamic states, we found slower decay processes in the long-time tail. These new processes lead to a critical dimension at d = 4 as in the renormalization group, that is, higher modes processes lead to slower decay process in the autocorrelation function for d = 4, while they lead to quicker decay process for d > 4. This conclusion clashes with the traditional point of view, which leads to the critical dimension d = 2. These slower processes invalidate the traditional kinetic equations for bare distribution functions obtained by a truncation of the BBGKY hierarchy for d < 4, as well as the Green-Kubo formalism, as there appear contributions of order t⁻¹, t^−1/2, ... coming from multiple mode-mode couplings even for d = 3.