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.
The motivation for this theoretical paper comes from recent experiments of a heat transfer system of two thermally coupled rings rotating in opposite directions with equal angular velocities that present anti-parity-time (APT) symmetry. The theoretical model predicted a rest-to-motion temperature distribution phase transition during the symmetry breaking for a particular rotation speed. In this work we show that the system exhibits a parity-time ($\mathcal{PT}$) phase transition at the exceptional point in which eigenvalues and eigenvectors of the corresponding non-Hermitian Hamiltonian coalesce. We analytically solve the heat diffusive system at the exceptional point and show that one can pass through the phase transition that separates the unbroken and broken phases by changing the radii of the rings. In the case of unbroken $\mathcal{PT}$ symmetry the temperature profiles exhibit damped Rabi oscillations at the exceptional point. Our results unveils the behavior of the system at the exceptional point in heat diffusive systems.
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