A classical and quantum noise model
Item
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Title
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A classical and quantum noise model
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Identifier
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d_2009_2013:22ec669950a3:10256
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identifier
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10433
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Creator
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Yang, Yejun,
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Contributor
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Leon Cohen
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Date
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2009
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Language
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English
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Publisher
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City University of New York.
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Subject
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Theoretical physics | Quantum physics | characteristic | noise | quantum | wigner
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Abstract
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We develop detailed statistics of a noise model that consists of N independent harmonic oscillators where the total force is given by the sum of the individual forces. This model was first proposed in the paper by Ford, Kac, and Mazur that was aimed at deriving the Langevin equation from first principles. We extend the model and calculate relevant probability distributions and other statistical quantities such as the autocorrelation function. In the usual model one assumes that the initial position and momentum values are stochastic variables that determine the statistical features of the force by ensemble averaging over those quantities. We extend that by also treating the mass and frequency as statistical quantities. We consider both the equilibrium case, that is the canonical distribution for the initial positions and momenta, for the but we also consider other initial distributions and show that this leads to non-stationary autocorrelation functions. One of our basic aims is to also develop this model for the quantum case and compare the results with the classical case.;The general approach we use for the calculation of the statistical quantities is by way of the characteristic function. We use the characteristic function approach because the oscillators are independent of each other. However, the quantum characteristic function present unique difficulties because the initial momentum and position operators do not commute. We use the Weyl correspondence to define the quantum characteristic function and we derive explicit expressions for both the pure case and mixtures. We show that many of the statistical quantities can be expressed in terms of the Wigner distribution.;In addition, we consider the time-frequency Wigner spectrum of momentum governed by the Langevin equation when the random driving term is quantum noise. We obtaine an explicit equation. The equation is solved exactly and includes both the transient and the stationary part. The time-dependent Wigner spectrum generalizes the result of Wang and Uhlenbeck wherein they showed that for the white noise driving force the power spectrum in the stationary state regime is Lorenzian. We show that our solution reduces to the classical solution when the parameters of the quantum noise are such that the white noise limit is approached and when the long time limit is taken.
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Type
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dissertation
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Source
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2009_2013.csv
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degree
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Ph.D.
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Program
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Physics
