Overdamping is a regime in which friction is sufficiently large that the motion either decays to its equilibrium position or it crosses the equilibrium position exactly once before returning monotonically towards the equilibrium position. The phenomena of overdamping has been studied classically and quantum mechanically only for the case of the linear damped harmonic oscillator. Here we study the classical and quantum dynamics of a family of over-damped non linear systems. The main objective of this paper is to find a Lagrangian and Hamiltonian framework to study over-damped non linear systems and to show that a quantum mechanical description can be developed in the momentum representation. Our results reduce to the well known solution of the linear damped harmonic oscillator when the non linear part is set to zero.
In this dissertation, we studied the generation of squeezed states induced by a timedependent interaction and the in uence of temperature on the strength of the squeezing in a condensed matter model. The model studied comprised two quantum harmonic oscillators, with a time-dependent, non-linear coupling between them. The in uence of the thermal bath on the non-equilibrium dynamics of the model was represented in terms of non-Hamiltonian thermostats and a collection of independent harmonic oscillators with Ohmic spectral density. The equations of motion were studied in the Wigner representation, which introduces a phase space description for the model. The representation of the system in quantum phase space allowed us to investigate the di erence between purely classical evolution and the relative importance of quantum corrections with respect to the dynamics. The dynamics was studied by means of computer simulation techniques. The numerical simulation of the non-equilibrium statistical mechanics of both time-dependent and non-linear interactions allowed us to investigate conditions beyond those in recent literature [1, 2].