Dynamic fluctuation-dissipation theory for generalized Langevin equations: Constructive constraints, stability, and realizability

Using the initial-value formulation, a dynamic theory for systems evolving according to a generalized Langevin equation is developed, providing conditions on the existence of equilibrium behavior and its fluctuation-dissipation implications. For systems fulfilling the property of local realizability, that for all the practical purposes corresponds to the postulate of the existence of a Markovian embedding, physical constraints, expressed in the form of dissipative stability and stochastic realizability are derived. If these two properties are met, Kubo theory is constructively recovered, while if one of these conditions is violated a thermodynamic equilibrium behavior does not exist (and this occurs also for "well-behaved dissipative systems"), with implications in the linear response theory.