28,438 research outputs found
Quantum Trajectories in Random Environment: the Statistical Model for a Heat Bath
In this article, we derive the stochastic master equations corresponding to
the statistical model of a heat bath. These stochastic differential equations
are obtained as continuous time limits of discrete models of quantum repeated
measurements. Physically, they describe the evolution of a small system in
contact with a heat bath undergoing continuous measurement. The equations
obtained in the present work are qualitatively different from the ones derived
in \cite{A1P1}, where the Gibbs model of heat bath has been studied. It is
shown that the statistical model of a heat bath provides clear physical
interpretation in terms of emissions and absorptions of photons. Our approach
yields models of random environment and unravelings of stochastic master
equations. The equations are rigorously obtained as solutions of martingale
problems using the convergence of Markov generators
Dimension reduction for systems with slow relaxation
We develop reduced, stochastic models for high dimensional, dissipative
dynamical systems that relax very slowly to equilibrium and can encode long
term memory. We present a variety of empirical and first principles approaches
for model reduction, and build a mathematical framework for analyzing the
reduced models. We introduce the notions of universal and asymptotic filters to
characterize `optimal' model reductions for sloppy linear models. We illustrate
our methods by applying them to the practically important problem of modeling
evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof
Stochastic Master Equations in Thermal Environment
We derive the stochastic master equations which describe the evolution of
open quantum systems in contact with a heat bath and undergoing indirect
measurements. These equations are obtained as a limit of a quantum repeated
measurement model where we consider a small system in contact with an infinite
chain at positive temperature. At zero temperature it is well-known that one
obtains stochastic differential equations of jump-diffusion type. At strictly
positive temperature, we show that only pure diffusion type equations are
relevant
Markov Chains Approximations of jump-Diffusion Quantum Trajectories
"Quantum trajectories" are solutions of stochastic differential equations
also called Belavkin or Stochastic Schr\"odinger Equations. They describe
random phenomena in quantum measurement theory. Two types of such equations are
usually considered, one is driven by a one-dimensional Brownian motion and the
other is driven by a counting process. In this article, we present a way to
obtain more advanced models which use jump-diffusion stochastic differential
equations. Such models come from solutions of martingale problems for
infinitesimal generators. These generators are obtained from the limit of
generators of classical Markov chains which describe discrete models of quantum
trajectories. Furthermore, stochastic models of jump-diffusion equations are
physically justified by proving that their solutions can be obtained as the
limit of the discrete trajectories
Non Markovian Quantum Repeated Interactions and Measurements
A non-Markovian model of quantum repeated interactions between a small
quantum system and an infinite chain of quantum systems is presented. By
adapting and applying usual pro jection operator techniques in this context,
discrete versions of the integro-differential and time-convolutioness Master
equations for the reduced system are derived. Next, an intuitive and rigorous
description of the indirect quantum measurement principle is developed and a
discrete non Markovian stochastic Master equation for the open system is
obtained. Finally, the question of unravelling in a particular model of
non-Markovian quantum interactions is discussed.Comment: 22 page
Perseus: Randomized Point-based Value Iteration for POMDPs
Partially observable Markov decision processes (POMDPs) form an attractive
and principled framework for agent planning under uncertainty. Point-based
approximate techniques for POMDPs compute a policy based on a finite set of
points collected in advance from the agents belief space. We present a
randomized point-based value iteration algorithm called Perseus. The algorithm
performs approximate value backup stages, ensuring that in each backup stage
the value of each point in the belief set is improved; the key observation is
that a single backup may improve the value of many belief points. Contrary to
other point-based methods, Perseus backs up only a (randomly selected) subset
of points in the belief set, sufficient for improving the value of each belief
point in the set. We show how the same idea can be extended to dealing with
continuous action spaces. Experimental results show the potential of Perseus in
large scale POMDP problems
Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction
Virtually all questions that one can ask about the behavioral and structural
complexity of a stochastic process reduce to a linear algebraic framing of a
time evolution governed by an appropriate hidden-Markov process generator. Each
type of question---correlation, predictability, predictive cost, observer
synchronization, and the like---induces a distinct generator class. Answers are
then functions of the class-appropriate transition dynamic. Unfortunately,
these dynamics are generically nonnormal, nondiagonalizable, singular, and so
on. Tractably analyzing these dynamics relies on adapting the recently
introduced meromorphic functional calculus, which specifies the spectral
decomposition of functions of nondiagonalizable linear operators, even when the
function poles and zeros coincide with the operator's spectrum. Along the way,
we establish special properties of the projection operators that demonstrate
how they capture the organization of subprocesses within a complex system.
Circumventing the spurious infinities of alternative calculi, this leads in the
sequel, Part II, to the first closed-form expressions for complexity measures,
couched either in terms of the Drazin inverse (negative-one power of a singular
operator) or the eigenvalues and projection operators of the appropriate
transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at
http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht
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