Star-unitary transformation and stochasticity: emergence of white, 1/f noise through resonances

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Kim, Sungyun

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In this thesis we consider the problem of stochasticity in Hamiltonian dynamics. It was shown by Poincaré that nonintegrable systems do not have constants of motion due to resonances. Divergences due to resonances appear when we try to solve the Hamiltonian by perturbation. In recent years, Prigogine’s group showed that there may exist a new way of solving the Hamiltonian by introducing a non-unitary transformation Λ which removes the divergences systematically. In this thesis we apply this Λ transformation to the problem of stochasticity. To this end, first we study classical Friedrichs model, which describes the interaction between a particle and field. For this model we derive the Λ transformation for general functions of particle modes, and show that the Langevin and Fokker-Planck equations can be derived through the transformed particle density function. It is also shown that the Gaussian white noise structure can be derived through the removal of divergences due to resonances. We extend this to the quantum case, and show that the same structure can be preserved if we keep the normal order of creation and destruction operators. We also study the extended Friedrichs model. This model can be mapped from the case in which a small system is weakly interacting with a reservoir. In this model we show that low frequency 1/f noise is derived due to the sum of resonances effect.