Retarded and instantaneous evolution equations of macroobservables in non-equilibrium statistical mechanics

The relations between evolution equations (EE) for macroobservables of the retarded and instantaneous (time-convolutionless) type are discussed from a general point of view. Far from equilibrium processes are included in using linear operator representations of nonlinear EE. Conditions under which a given retarded EE can be memory renormalized are formulated and the renormalized versions of nonequilibrium theories of Robertson and Grabert are derived. Memory kernels containing long time tails which cannot be renormalized are considered explicitly. The corresponding instantaneous EE exhibit the existence of two different regimes of decay, the transition between them taking place at a critical time tK ∼ ln α, where α characterizes the strength of the long time tail contribution. If using a logarithmic time scale, at very large times the relaxation is described by a Markovian equation with a universal transport coefficient which is independent of the microscopic properties of the system.