Study of Classical and Quantum Open Systems

Details Study of Classical and Quantum Open Systems Study of Classical and Quantum Open Systems

2010-08-03

arxiv.org/abs/1008.0491v1

Physics

Condensed Matter

Statistical Mechanics

honors thesis, 58 pp, 40 figs

Scientific paper

This thesis covers various aspects of open systems in classical and quantum mechanics. In the first part, we deal with classical systems. The bath-of-oscillators formalism is used to describe an open system, and the phenomenological Langevin equation is recovered. The Fokker-Planck equation is derived from its corresponding Langevin equation. The Fokker-Planck equation for a particle in a periodic potential in the high-friction limit is solved using the continued-fraction method. The equilibrium and time-dependent solutions are obtained. Under strong periodic driving, we observe significant non-linear effects in the dynamical hysteresis loops. Shapiro steps appear in the time-average of the drift velocity curves. Similar study is carried out for a dipole in an electric field. In the second part of the thesis, we begin the study of open quantum systems by re-deriving the quantum master equation using perturbation theory. The master equation is then applied to the bath-of-oscillators model. The subtleties and approximations of the master equation are discussed. We then use the master equation to solve the damped harmonic oscillator. The equilibrium solution coincides with the canonical distribution. The steady state response to DC and AC forces is also studied. Driven systems are more challenging in quantum open systems, and we manage to solve the quantum master equation with the continued-fraction method. We obtain the frequency-dependent susceptibility curves, which exhibit typical absorption and dispersion profiles.

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