The eigenvalue problem for linear macroscopic equations

The normal mode analysis of systems of linear macroscopic equations in irreversible thermodynamics is extended in several ways. When the characteristics equation has multiple roots, there may appear normal solutions that do not decay purely exponentially, but a closed form for the Green function and the autocorrelation function can still be given. Furthermore, nonexponential decay is associated only with accidental, not with systematic degeneracy. We also discuss the case of external parameters that break microscopic time reversibility. In this case the orthonormality relations between the normal mode vectors are replaced by biorthonormality relations between the normal modes of the system studied and those of the system with reversed external parameters. Finally we discuss systems in which the second order energy is only positive semi-definite.