795 research outputs found

    A Second-Order Stochastic Leap-Frog Algorithm for Langevin Simulation

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    Langevin simulation provides an effective way to study collisional effects in beams by reducing the six-dimensional Fokker-Planck equation to a group of stochastic ordinary differential equations. These resulting equations usually have multiplicative noise since the diffusion coefficients in these equations are functions of position and time. Conventional algorithms, e.g. Euler and Heun, give only first order convergence of moments in a finite time interval. In this paper, a stochastic leap-frog algorithm for the numerical integration of Langevin stochastic differential equations with multiplicative noise is proposed and tested. The algorithm has a second-order convergence of moments in a finite time interval and requires the sampling of only one uniformly distributed random variable per time step. As an example, we apply the new algorithm to the study of a mechanical oscillator with multiplicative noise.Comment: 3 pages, 4 figures, to submit to XX International LINAC conferenc

    Classical Dynamics for Linear Systems: The Case of Quantum Brownian Motion

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    It has long been recognized that the dynamics of linear quantum systems is classical in the Wigner representation. Yet many conceptually important linear problems are typically analyzed using such generally applicable techniques as influence functionals and Bogoliubov transformations. In this Letter we point out that the classical equations of motion provide a simpler and more intuitive formalism for linear quantum systems. We examine the important problem of Brownian motion in the independent oscillator model, and show that the quantum dynamics is described directly and completely by a c-number Langevin equation. We are also able to apply recent insights into quantum Brownian motion to show that the classical Fokker-Planck equation is always local in time, regardless of the spectral density of the environment.Comment: 9 pages, LaTe

    How Wigner Functions Transform Under Symplectic Maps

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    It is shown that, while Wigner and Liouville functions transform in an identical way under linear symplectic maps, in general they do not transform identically for nonlinear symplectic maps. Instead there are ``quantum corrections'' whose hbar tending to zero limit may be very complicated. Examples of the behavior of Wigner functions in this limit are given in order to examine to what extent the corresponding Liouville densities are recovered.Comment: 8 pages, 6 figures [RevTeX/epsfig, macro included]. To appear in Proceedings of the Advanced Beam Dynamics Workshop on Quantum Aspects of Beam Physics (Monterey, CA 1998
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