795 research outputs found
A Second-Order Stochastic Leap-Frog Algorithm for Langevin Simulation
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
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
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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