33 research outputs found
Geometric Rényi divergence: A comparative measure with applications to atomic densities
An alternative one-parameter measure of divergence is proposed, quantifying the discrepancy among general probability densities. Its main mathematical properties include (i) comparison among an arbitrary number of functions, (ii) the possibility of assigning different weights to each function according to its relevance on the comparative procedure, and (iii) ability to modify the relative contribution of different regions within the domain. Applications to the study of atomic density functions, in both conjugated spaces, show the versatility and universality of this divergence
Entanglement and the Born-Oppenheimer approximation in an exactly solvable quantum many-body system
We investigate the correlations between different bipartitions of an exactly
solvable one-dimensional many-body Moshinsky model consisting of Nn "nuclei"
and Ne "electrons". We study the dependence of entanglement on the
inter-particle interaction strength, on the number of particles, and on the
particle masses. Consistent with kinematic intuition, the entanglement between
two subsystems vanishes when the subsystems have very different masses, while
it attains its maximal value for subsystems of comparable mass. We show how
this entanglement feature can be inferred by means of the Born-Oppenheimer
Ansatz, whose validity and breakdown can be understood from a quantum
information point of view.Comment: Accepted in Eur. Phys. J. D (2014
Synchronization and Redundancy: Implications for Robustness of Neural Learning and Decision Making
Learning and decision making in the brain are key processes critical to
survival, and yet are processes implemented by non-ideal biological building
blocks which can impose significant error. We explore quantitatively how the
brain might cope with this inherent source of error by taking advantage of two
ubiquitous mechanisms, redundancy and synchronization. In particular we
consider a neural process whose goal is to learn a decision function by
implementing a nonlinear gradient dynamics. The dynamics, however, are assumed
to be corrupted by perturbations modeling the error which might be incurred due
to limitations of the biology, intrinsic neuronal noise, and imperfect
measurements. We show that error, and the associated uncertainty surrounding a
learned solution, can be controlled in large part by trading off
synchronization strength among multiple redundant neural systems against the
noise amplitude. The impact of the coupling between such redundant systems is
quantified by the spectrum of the network Laplacian, and we discuss the role of
network topology in synchronization and in reducing the effect of noise. A
range of situations in which the mechanisms we model arise in brain science are
discussed, and we draw attention to experimental evidence suggesting that
cortical circuits capable of implementing the computations of interest here can
be found on several scales. Finally, simulations comparing theoretical bounds
to the relevant empirical quantities show that the theoretical estimates we
derive can be tight.Comment: Preprint, accepted for publication in Neural Computatio
Quantum entanglement in exactly soluble atomic models: The Moshinsky model with three electrons, and with two electrons in a uniform magnetic field
We investigate the entanglement-related features of the eigenstates of two exactly soluble atomic models: a one-dimensional three-electron Moshinsky model, and a three-dimensional two-electron Moshinsky system in an external uniform magnetic field. We analytically compute the amount of entanglement exhibited by the wavefunctions corresponding to the ground, first and second excited states of the three-electron model. We found that the amount of entanglement of the system tends to increase with energy, and in the case of excited states we found a finite amount of entanglement in the limit of vanishing interaction. We also analyze the entanglement properties of the ground and first few excited states of the two-electron Moshinsky model in the presence of a magnetic field. The dependence of the eigenstates' entanglement on the energy, as well as its behaviour in the regime of vanishing interaction, are similar to those observed in the three-electron system. On the other hand, the entanglement exhibits a monotonically decreasing behavior with the strength of the external magnetic field. For strong magnetic fields the entanglement approaches a finite asymptotic value that depends on the interaction strength. For both systems studied here we consider a perturbative approach in order to shed some light on the entanglement's dependence on energy and also to clarify the finite entanglement exhibited by excited states in the limit of weak interactions. As far as we know, this is the first work that provides analytical and exact results for the entanglement properties of a three-electron model.Fil: Bouvrie, P. A.. Universidad de Granada; EspañaFil: Majtey, Ana Paula. Universidad de Granada; España. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Plastino, Ángel Ricardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina. Universidad Nacional de La Plata; ArgentinaFil: Sánchez Moreno, P.. Universidad de Granada; EspañaFil: Dehesa, J. S.. Universidad de Granada; Españ
Three strongly correlated charged bosons in a one-dimensional harmonic trap: natural orbital occupancies
We study a one-dimensional system composed of three charged bosons confined
in an external harmonic potential. More precisely, we investigate the
ground-state correlation properties of the system, paying particular attention
to the strong-interaction limit. We explain for the first time the nature of
the degeneracies appearing in this limit in the spectrum of the reduced density
matrix. An explicit representation of the asymptotic natural orbitals and their
occupancies is given in terms of some integral equations.Comment: 6 pages, 4 figures, To appear in European Physical Journal
Testing one-body density functionals on a solvable model
There are several physically motivated density matrix functionals in the
literature, built from the knowledge of the natural orbitals and the occupation
numbers of the one-body reduced density matrix. With the help of the equivalent
phase-space formalism, we thoroughly test some of the most popular of those
functionals on a completely solvable model.Comment: Latex, 16 pages, 4 figure
Interpolated wave functions for nonadiabatic simulations with the fixed-node quantum Monte Carlo method
Simulating nonadiabatic effects with many-body wave function approaches is an
open field with many challenges. Recent interest has been driven by new
algorithmic developments and improved theoretical understanding of properties
unique to electron-ion wave functions. Fixed-node diffusion Monte Caro is one
technique that has shown promising results for simulating electron-ion systems.
In particular, we focus on the CH molecule for which previous results suggested
a relatively significant contribution to the energy from nonadiabatic effects.
We propose a new wave function ansatz for diatomic systems which involves
interpolating the determinant coefficients calculated from configuration
interaction methods. We find this to be an improvement beyond previous wave
function forms that have been considered. The calculated nonadiabatic
contribution to the energy in the CH molecule is reduced compared to our
previous results, but still remains the largest among the molecules under
consideration.Comment: 7 pages, 3 figure
Review on computational methods for Lyapunov functions
Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function