Self-consistent Mode-Coupling Formulation of Spectra of Nonlinear (Anharmonic) Systems

Abstract

The calculation of spectra of nonlinearly coupled oscillators is a basic problem in nonequilibrium statistical mechanics which has broad range of implications (e.g. the vibrational spectra of polyatomic molecules). It is well established that classical trajectories of nonlinear systems may be classified as “quasiperiodic” or “stochastic” (chaotic) depending on the nonlinear coupling and the initial conditions.^1–3 The quantum mechanical significance of these concepts is however not at all clear and is the subject of numerous current studies.⁴,⁵ There is obviously a need to develop a theoretical framework which will allow us to study classical and quantum systems along similar lines and to compare their behaviour in detail. In the present paper, an exact reduced equation of motion which allows the calculation of zero temperature correlation functions in nonlinear (anharmonic) quantum systems is derived. A self consistent procedure is subsequently developed which enables us to solve for the correlation functions by mapping the anharmonic problem into a harmonic problem with higher dimensionality. The latter assumes the form of a nonlinear map analogous to those used in classical nonlinear dynamics and may exhibit critical dependence on the anharmonic potential.⁶ The present approach is based on the mode-coupling formalism⁷ and utilizes a new type of reduced equation of motion developed recently.^{8, 9}