157 research outputs found
The Relation between Approximation in Distribution and Shadowing in Molecular Dynamics
Molecular dynamics refers to the computer simulation of a material at the
atomic level. An open problem in numerical analysis is to explain the apparent
reliability of molecular dynamics simulations. The difficulty is that
individual trajectories computed in molecular dynamics are accurate for only
short time intervals, whereas apparently reliable information can be extracted
from very long-time simulations. It has been conjectured that long molecular
dynamics trajectories have low-dimensional statistical features that accurately
approximate those of the original system. Another conjecture is that numerical
trajectories satisfy the shadowing property: that they are close over long time
intervals to exact trajectories but with different initial conditions. We prove
that these two views are actually equivalent to each other, after we suitably
modify the concept of shadowing. A key ingredient of our result is a general
theorem that allows us to take random elements of a metric space that are close
in distribution and embed them in the same probability space so that they are
close in a strong sense. This result is similar to the Strassen-Dudley Theorem
except that a mapping is provided between the two random elements. Our results
on shadowing are motivated by molecular dynamics but apply to the approximation
of any dynamical system when initial conditions are selected according to a
probability measure.Comment: 21 pages, final version accepted in SIAM Dyn Sy
On the Hausdorff dimension of invariant measures of weakly contracting on average measurable IFS
We consider measures which are invariant under a measurable iterated function
system with positive, place-dependent probabilities in a separable metric
space. We provide an upper bound of the Hausdorff dimension of such a measure
if it is ergodic. We also prove that it is ergodic iff the related skew product
is.Comment: 16 pages; to appear in Journal of Stat. Phy
Positive contraction mappings for classical and quantum Schrodinger systems
The classical Schrodinger bridge seeks the most likely probability law for a
diffusion process, in path space, that matches marginals at two end points in
time; the likelihood is quantified by the relative entropy between the sought
law and a prior, and the law dictates a controlled path that abides by the
specified marginals. Schrodinger proved that the optimal steering of the
density between the two end points is effected by a multiplicative functional
transformation of the prior; this transformation represents an automorphism on
the space of probability measures and has since been studied by Fortet,
Beurling and others. A similar question can be raised for processes evolving in
a discrete time and space as well as for processes defined over non-commutative
probability spaces. The present paper builds on earlier work by Pavon and
Ticozzi and begins with the problem of steering a Markov chain between given
marginals. Our approach is based on the Hilbert metric and leads to an
alternative proof which, however, is constructive. More specifically, we show
that the solution to the Schrodinger bridge is provided by the fixed point of a
contractive map. We approach in a similar manner the steering of a quantum
system across a quantum channel. We are able to establish existence of quantum
transitions that are multiplicative functional transformations of a given Kraus
map, but only for the case of uniform marginals. As in the Markov chain case,
and for uniform density matrices, the solution of the quantum bridge can be
constructed from the fixed point of a certain contractive map. For arbitrary
marginal densities, extensive numerical simulations indicate that iteration of
a similar map leads to fixed points from which we can construct a quantum
bridge. For this general case, however, a proof of convergence remains elusive.Comment: 27 page
Quantum Computing and Hidden Variables I: Mapping Unitary to Stochastic Matrices
This paper initiates the study of hidden variables from the discrete,
abstract perspective of quantum computing. For us, a hidden-variable theory is
simply a way to convert a unitary matrix that maps one quantum state to
another, into a stochastic matrix that maps the initial probability
distribution to the final one in some fixed basis. We list seven axioms that we
might want such a theory to satisfy, and then investigate which of the axioms
can be satisfied simultaneously. Toward this end, we construct a new
hidden-variable theory that is both robust to small perturbations and
indifferent to the identity operation, by exploiting an unexpected connection
between unitary matrices and network flows. We also analyze previous
hidden-variable theories of Dieks and Schrodinger in terms of our axioms. In a
companion paper, we will show that actually sampling the history of a hidden
variable under reasonable axioms is at least as hard as solving the Graph
Isomorphism problem; and indeed is probably intractable even for quantum
computers.Comment: 19 pages, 1 figure. Together with a companion paper to appear,
subsumes the earlier paper "Quantum Computing and Dynamical Quantum Models"
(quant-ph/0205059
Discrete-time classical and quantum Markovian evolutions: Maximum entropy problems on path space
The theory of Schroedinger bridges for diffusion processes is extended to
classical and quantum discrete-time Markovian evolutions. The solution of the
path space maximum entropy problems is obtained from the a priori model in both
cases via a suitable multiplicative functional transformation. In the quantum
case, nonequilibrium time reversal of quantum channels is discussed and
space-time harmonic processes are introduced.Comment: 34 page
Maximum-Entropy Weighting of Multi-Component Earth Climate Models
A maximum entropy-based framework is presented for the synthesis of
projections from multiple Earth climate models. This identifies the most
representative (most probable) model from a set of climate models -- as defined
by specified constraints -- eliminating the need to calculate the entire set.
Two approaches are developed, based on individual climate models or ensembles
of models, subject to a single cost (energy) constraint or competing
cost-benefit constraints. A finite-time limit on the minimum cost of modifying
a model synthesis framework, at finite rates of change, is also reported.Comment: Inspired by discussions at the Mathematical and Statistical
Approaches to Climate Modelling and Prediction workshop, Isaac Newton
Institute for Mathematical Sciences, Cambridge, UK, 11 Aug. to 22 Dec. 2010.
Accepted for publication in Climate Dynamics, 8 August 201
Extremal flows in Wasserstein space
We develop an intrinsic geometric approach to the calculus of variations in theWasserstein
space. We show that the flows associated with the Schr\ua8odinger bridge with
general prior, with optimal mass transport, and with the Madelung fluid can all be
characterized as annihilating the first variation of a suitable action. We then discuss
the implications of this unified framework for stochastic mechanics: It entails, in particular,
a sort of fluid-dynamic reconciliation between Bohm\u2019s and Nelson\u2019s stochastic
mechanics
The Central Limit Theorem for Random Dynamical Systems
We consider random dynamical systems with randomly chosen jumps. The choice of deterministic dynamical system and jumps depends on a position. The Central Limit Theorem for random dynamical systems is established
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