24 research outputs found
Decoherence produces coherent states: an explicit proof for harmonic chains
We study the behavior of infinite systems of coupled harmonic oscillators as
t->infinity, and generalize the Central Limit Theorem (CLT) to show that their
reduced Wigner distributions become Gaussian under quite general conditions.
This shows that generalized coherent states tend to be produced naturally. A
sufficient condition for this to happen is shown to be that the spectral
function is analytic and nonlinear. For a rectangular lattice of coupled
oscillators, the nonlinearity requirement means that waves must be dispersive,
so that localized wave-packets become suppressed. Virtually all harmonic
heat-bath models in the literature satisfy this constraint, and we have good
reason to believe that coherent states and their generalizations are not merely
a useful analytical tool, but that nature is indeed full of them. Standard
proofs of the CLT rely heavily on the fact that probability densities are
non-negative. Although the CLT generally fails if the probability densities are
allowed to take negative values, we show that a CLT does indeed hold for a
special class of such functions. We find that, intriguingly, nature has
arranged things so that all Wigner functions belong to this class.Comment: Final published version. 17 pages, Plain TeX, no figures. Online at
http://astro.berkeley.edu/~max/gaussians.html (faster from the US), from
http://www.mpa-garching.mpg.de/~max/gaussians.html (faster from Europe) or
from [email protected]
Characterization and stability problems for finite quadratic forms
Sufficient conditions are given under which the distribution of a finite quadratic form in independent identically distributed symmetric random variables defines uniquely the underlying distribution. Moreover, a stability theorem for quadratic forms is proved. (orig.)Available from TIB Hannover: RR 4487(1999,4) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman
Symbolically Quantifying Response Time in Stochastic Models using Moments and Semirings
International audienceWe study quantitative properties of the response time in stochastic models. For instance, we are interested in quantifying bounds such that a high percentage of the runs answers a query within these bounds. To study such problems, computing probabilities on a state-space blown-up by a factor depending on the bound could be used, but this solution is not satisfactory when the bound is large. In this paper, we propose a new symbolic method to quantify bounds on the response time, using the moments of the distribution of simple stochastic systems. We prove that the distribution (and hence the bounds) is uniquely defined given its moments. We provide optimal bounds for the response time over all distributions having a pair of these moments. We explain how to symbolically compute in polynomial time any moment of the distribution of response times using adequately-defined semirings. This allows us to compute optimal bounds in parametric models and to reduce complexity for computing optimal bounds in hierarchical models