Complex spectral representation of the Liouvillian and kinetic theory in nonequilibrium physics

Abstract

Kinetic theory in nonequilibrium statistical physics is discussed in terms of the complex spectral representation of the Liouvillian for quantum systems. We show that the collision operator appearing in the kinetic equation is the "self-energy part" of the Liouvillian. Solving the dispersion equation associated to the "self-energy part", one can construct the resonance states of the Liouvillian which have complex eigenvalues. The imaginary part of the eigenvalue gives a decay rate in irreversible process in nonequilibrium statistical physics. As illustrations of the resonance states of the Liouvillian we consider two examples; one is a one-dimensional system in which a polaron is weakly interacting with a thermal reservoir consisting of an acoustic phonon field, and the other is a molecular chain that is described by the Davydov Hamiltonian, which has been introduced as a simple model that describes a protein chain. We show that the imaginary part of the spectrum of the Liouvillian in these systems has rich structures showing band structure, an accumulation point, and a fractal structure reminiscent of Hofstadter's butterfly.