Derivation of the linear and nonlinear non-Markov fluctuation-dissipation relations of the first kind for a dynamical model

A dynamical system consisting of a subsystem having the variablesz=(q,p) and of another dynamical system (thermostat) is considered in the nonquantum case. Using a dynamical equation, it is shown that the linear and quadratic non-Markov fluctuation-dissipation relations (FDRs) of the first kind are valid in the first nonvanishing approximation in interaction constants. Applying these FDRs, one can determine the statistical properties of the fluctuations when the form of the nonlinear phenomenological equation is known. The non-Markov FDRs of the first kind are the direct generalization (to the inertial case) of the Markov FDRs that are the consequence of detailed balance.     

The time-local view of nonequilibrium statistical mechanics. I. Linear theory of transport and relaxation

The various approaches to nonequilibrium statistical mechanics may be subdivided into convolution and convolutionless (time-local) ones. While the former, put forward by Zwanzig, Mori, and others, are used most commonly, the latter are less well developed, but have proven very useful in recent applications. The aim of the present series of papers is to develop the time-local picture (TLP) of nonequilibrium statistical mechanics on a new footing and to consider its physical implications for topics such as the formulation of irreversible thermodynamics. The most natural approach to TLP is seen to derive from the Fourier-Laplace transform ) of pertinent time correlation functions, which on the physical sheet typically displays an essential singularity at z= and a number of macroscopic and microscopic poles in the lower half-plane corresponding to long- and short-lived modes, respectively, the former giving rise to the autonomous macrodynamics, whereas the latter are interpreted as doorway modes mediating the transfer of information from relevant to irrelevant channels. Possible implications of this doorway mode concept for socalled extended irreversible thermodynamics are briefly discussed. The pole structure is used for deriving new kinds of generalized Green-Kubo relations expressing macroscopic quantities, transport coefficients, e.g., by contour integrals over current-current correlation functions obeying Hamiltonian dynamics, the contour integration replacing projection. The conventional Green-Kubo relations valid for conserved quantities only are rederived for illustration. Moreover, may be expressed by a Laurent series expansion in positive and negative powers ofz, from which a rigorous, general, and straightforward method is developed for extracting all macroscopic quantities from so-called secularly divergent expansions of as obtained from the application of conventional many-body techniques to the calculation of . The expressions are formulated as time scale expansions, which should rapidly converge if macroscopic and microscopic time scales are sufficiently well separated, i.e., if lifetime (memory) effects are not too large.