arXiv subjects

Luis Benet

Publications and source records attributed to Luis Benet.

At least 19 recordsLinked to original sources

arXiv2023

A framework to test interval arithmetic libraries and their IEEE 1788-2015 compliance

As developers of libraries implementing interval arithmetic, we faced the same difficulties when it comes to testing our libraries. What must be tested? How can we devise relevant test cases for unit testing? How can we ensure a high (and possibly 100%) test coverage? Before considering these questions, we briefly recall the main features of interval arithmetic and of the IEEE 1788-2015 standard for interval arithmetic. After listing the different aspects that, in our opinion, must be tested, we contribute a first step towards offering a test suite for an interval arithmetic library. First we define a format that enables the exchange of test cases, so that they can be read and tried easily. Then we offer a first set of test cases, for a selected set of mathematical functions. Next, we examine how the Julia interval arithmetic library, IntervalArithmetic.jl, actually performs to these tests. As this is an ongoing work, we list extra tests that we deem important to perform.

cs.MS↗

arXiv2022

Quantum-classical correspondence of strongly chaotic many-body spin models

We study the quantum-classical correspondence for systems with interacting spin-particles that are strongly chaotic in the classical limit. This is done in the presence of constants of motion associated with the fixed angular momenta of individual spins. Our analysis of the Lyapunov spectra reveals that the largest Lyapunov exponent agrees with the Lyapunov exponent that determines the local instability of each individual spin moving under the influence of all other spins. Within this picture, we introduce a rigorous and simple test of ergodicity for the spin motion, and use it to identify when classical chaos is both strong and global in phase space. In the quantum domain, our analysis of the Hamiltonian matrix in a proper representation allows us to obtain the conditions for the onset of quantum chaos as a function of the model parameters. From the comparison between the quantum and classical domains, we demonstrate that quantum quantities, such as the local density of states (LDOS) and the shape of the chaotic eigenfunctions written in the non-interacting many-body basis, have well-defined classical counterparts. Another central finding is the relationship between the Kolmogorov-Sinai entropy and the width of the LDOS, which is useful for studies of many-body dynamics.

cond-mat.stat-mech↗

arXiv2022

A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting $k$-body fermionic Embedded Gaussian Ensembles

In spite of its simplicity, the central limit theorem captures one of the most outstanding phenomena in mathematical physics, that of universality. While this classical result is well understood it is still not very clear what happens to this universal behaviour when the random variables become correlated. A fruitful mathematically laboratory to investigate the rising of new universal properties is offered by the set of eigenvalues of random matrices. In this regard a lot of work has been done using the standard random matrix ensembles and focusing on the distribution of extreme eigenvalues. In this case, the distribution of the largest -- or smallest -- eigenvalue departs from the Fisher-Tippett-Gnedenko theorem yielding the celebrated Tracy-Widom distribution. One may wonder, yet again, how robust is this new universal behaviour captured by the Tracy-Widom distribution when the correlation among eigenvalues changes. Few answers have been provided to this poignant question and our intention in the present work is to contribute to this interesting unexplored territory. Thus, we study numerically the probability distribution for the normalized largest eigenvalue of the interacting $k$-body fermionic orthogonal and unitary Embedded Gaussian Ensembles in the diluted limit. We find a smooth transition from a slightly asymmetric Gaussian-like distribution, for small $k/m$, to the Tracy-Widom distribution as $k/m\to 1$, where $k$ is the rank of the interaction and $m$ is the number of fermions. Correlations at the edge of the spectrum are stronger for small values of $k/m$, and are independent of the number of particles considered. Our results indicate that subtle correlations towards the edge of the spectrum distinguish the statistical properties of the spectrum of interacting many-body systems in the dilute limit, from those expected for the standard random matrix ensembles.

cond-mat.dis-nn↗

arXiv2022

Testing interval arithmetic libraries, including their IEEE-1788 compliance

As developers of libraries implementing interval arithmetic, we faced the same difficulties when it came to testing our libraries. What must be tested? How can we devise relevant test cases for unit testing? How can we ensure a high (and possibly 100%) test coverage? In this paper we list the different aspects that, in our opinion, must be tested, giving indications on the choice of test cases. Then we examine how several interval arithmetic libraries actually perform tests. Next, we introduce two frameworks developed specifically to gather test cases and to incorporate easily new libraries in order to test them, namely JInterval and ITF1788. Not every important aspects of our libraries fit in these frameworks and we list extra tests that we deem important, but not easy, to perform.

math.NA↗

arXiv2021

Spectra, eigenstates and transport properties of a $\mathcal{PT}$-symmetric ring

We study, analytically and numerically, a simple $\mathcal{PT}$-symmetric tight-binding ring with an onsite energy $a$ at the gain and loss sites. We show that if $a\neq 0$, the system generically exhibits an unbroken $\mathcal{PT}$-symmetric phase. We study the nature of the spectrum in terms of the singularities in the complex parameter space as well as the behavior of the eigenstates at large values of the gain and loss strength. We find that in addition to the usual exceptional points, there are "diabolical points", and inverse exceptional points at which complex eigenvalues reconvert into real eigenvalues. We also study the transport through the system. We calculate the total flux from the source to the drain, and how it splits along the branches of the ring. We find that while usually the density flows from the source to the drain, for certain eigenstates a stationary "backflow" of density from the drain to the source along one of the branches can occur. We also identify two types of singular eigenstates, i.e. states that do not depend on the strength of the gain and loss, and classify them in terms of their transport properties.

quant-ph↗

arXiv2020

Using posterior predictive distributions to analyse epidemic models: COVID-19 in Mexico City

Epidemiological models contain a set of parameters that must be adjusted based on available observations. Once a model has been calibrated, it can be used as a forecasting tool to make predictions and to evaluate contingency plans. It is customary to employ only point estimators for such predictions. However, some models may fit the same data reasonably well for a broad range of parameter values, and this flexibility means that predictions stemming from such models will vary widely, depending on the particular parameter values employed within the range that give a good fit. When data are poor or incomplete, model uncertainty widens further. A way to circumvent this problem is to use Bayesian statistics to incorporate observations and use the full range of parameter estimates contained in the posterior distribution to adjust for uncertainties in model predictions. Specifically, given the epidemiological model and a probability distribution for observations, we use the posterior distribution of model parameters to generate all possible epidemiological curves via the posterior predictive distribution. From the envelope of all curves one can extract the worst-case scenario and study the impact of implementing contingency plans according to this assessment. We apply this approach to the potential evolution of COVID-19 in Mexico City and assess whether contingency plans are being successful and whether the epidemiological curve has flattened.

q-bio.PE↗

arXiv2019

Spectral and transport properties of a $\mathcal{PT}$-symmetric tight-binding chain with gain and loss

We derive a continuity equation to study transport properties in a $\mathcal{PT}$-symmetric tight-binding chain with gain and loss in symmetric configurations. This allows us to identify the density fluxes in the system, and to define a transport coefficient to characterize the efficiency of transport of each state. These quantities are studied explicitly using analytical expressions for the eigenvalues and eigenvectors of the system. We find that in states with broken $\mathcal{PT}$-symmetry, transport is inefficient, in the sense that either inflow exceeds outflow and density accumulates within the system, or outflow exceeds inflow, and the system becomes depleted. We also report the appearance of two subsets of interesting eigenstates whose eigenvalues are independent on the strength of the coupling to gain and loss. We call these opaque and transparent states. Opaque states are decoupled from the contacts and there is no transport; transparent states exhibit always efficient transport. Interestingly, the appearance of such eigenstates is connected with the divisors of the length of the system plus one and the position of the contacts. Thus the number of opaque and transparent states varies very irregularly.

quant-ph↗

arXiv2019

On the dynamics of Comet 1P/Halley: Lyapunov and power spectra

Using a purely Newtonian model for the Solar System, we investigate the dynamics of comet 1P/Halley considering in particular the Lyapunov and power spectra of its orbit, using the nominal initial conditions of JPL's Horizons system. We carry out precise numerical integrations of the $(N+1)$-restricted problem and the first variational equations, considering a time span of $2\times10^5$~yr. The power spectra are dominated by a broadband component, with peaks located at the current planetary frequencies, including contributions from Jupiter, Venus, the Earth and Saturn, as well as the $1:6$ resonance among Halley and Jupiter and higher harmonics. From the average value of the maximum Lyapunov exponent we estimate the Lyapunov time of the comet's nominal orbit, obtaining $\tau_L \simeq 562$~yr; the remaining independent Lyapunov exponents (not related by time-reversal symmetry) tend asymptotically to zero as $t^{-1/2}$. Yet, our results do not display convergence of the maximum Lyapunov exponent. We argue that the lack of convergence of the maximum Lyapunov exponent is a signature of transient chaos which will lead to an eventual ejection of the comet from the Solar System.

astro-ph.EP↗

arXiv2019

Two-particle quantum correlations in stochastically-coupled networks

Quantum walks in dynamically-disordered networks have become an invaluable tool for understanding the physics of open quantum systems. In this work, we introduce a novel approach to describe the dynamics of indistinguishable particles in noisy quantum networks. By making use of stochastic calculus, we derive a master equation for the propagation of two non-interacting correlated particles in tight-binding networks affected by off-diagonal dynamical disorder. We show that the presence of noise in the couplings of a quantum network creates a pure-dephasing-like process that destroys all coherences in the single-particle Hilbert subspace. Remarkably, we find that when two or more correlated particles propagate in the network, coherences accounting for particle indistinguishability are robust against the impact of noise, thus showing that it is possible, in principle, to find specific conditions for which many indistinguishable particles can traverse dynamically-disordered systems without losing their ability to interfere. These results shed light on the role of particle indistinguishability in the preservation of quantum coherence in dynamically-disordered quantum networks.

quant-ph↗

arXiv2018

Robustness of optimal transport in disordered interacting many-body networks

The robustness of quantum transport under various perturbations is analyzed in disordered interacting many-body systems, which are constructed from the embedded Gaussian random matrix ensembles (EGEs). The transport efficiency can be enhanced drastically, if centrosymmetry (csEGE) is imposed. When the csEGE is perturbed with an ordinary EGE, the transport efficiency in the optimal cases is reduced significantly, while in the suboptimal cases the changes are less pronounced. Qualitatively the same behavior is observed, when parity and centrosymmetry are broken by block perturbations. Analyzing the influence of the environment coupling, optimal transport is observed at a certain coupling strength, while too weak and too strong coupling reduce the transport. Taking into account the effects of decoherence, in the EGE the transport efficiency approaches its maximum at a finite nonzero decoherence strength (environment-assisted transport). In the csEGE the efficiency decays monotonically with the decoherence but is always larger than in the EGE.

quant-ph↗

arXiv2016

A simple model for the location of Saturn's F ring

In this paper, we introduce a simplified model to understand the location of Saturn's F ring. The model is a planar restricted five-body problem defined by the gravitational field of Saturn, including its second zonal harmonic $J_2$, the shepherd moons Prometheus and Pandora, and Titan. We compute accurate long-time numerical integrations of (about 2.5 million) non-interacting test-particles initially located in the region between the orbits of Prometheus and Pandora, and address whether they escape or remain trapped in this region. We obtain a wide region of initial conditions of the test particles that remain confined. We consider a dynamical stability indicator for the test particles' motion defined by computing the ratio of the standard deviation over the average value of relevant dynamical quantities, in particular, for the mean-motion and the semi-major axis. This indicator separates clearly a subset of trapped initial conditions that appear as very localized stripes in the initial semi-major axis and eccentricity space for the most stable orbits. Retaining only these test particles, we obtain a narrow eccentric ring which displays sharp edges and collective alignment. The semi-major axis of the accumulation stripes of the stable ring-particles can be associated with resonances, mostly involving Prometheus' outer Lindblad and co-rotation resonances, but not exclusively. Pandora's inner Lindblad and co-rotation resonances as well as low-order three-body resonances typically coincide with gaps, i.e., regions of instabilities. Comparison of our results with the nominal data for the F ring shows some correspondence.

astro-ph.EP↗

arXiv2016

Efficient quantum transport in disordered interacting many-body networks

The coherent transport of $n$ fermions in disordered networks of $l$ single-particle states connected by $k$-body interactions is studied. These networks are modeled by embedded Gaussian random matrix ensemble (EGE). The conductance bandwidth as well as the ensemble-averaged total current attain their maximal values if the system is highly filled $n \sim l-1$ and $k\sim n/2$. For the cases $k=1$ and $k=n$ the bandwidth is minimal. We show that for all parameters the transport is enhanced significantly whenever centrosymmetric ensemble (csEGE) are considered. In this case the transmission shows numerous resonances of perfect transport. Analyzing the transmission by spectral decomposition, we find that centrosymmetry induces strong correlations and enhances the extrema of the distributions. This suppresses destructive interference effects in the system and thus, causes backscattering-free transmission resonances which enhance the overall transport. The distribution of the total current for the csEGE has a very large dominating peak for $n=l-1$, close to the highest observed currents.

quant-ph↗

arXiv2015

Probing two-particle exchange processes in two-mode Bose-Einstein condensates

We study the fidelity decay and its freeze for an initial coherent state of two-mode Bose-Einstein condensates in the Fock regime considering a Bose-Hubbard model that includes two-particle tunneling terms. By using linear-response theory we find scaling properties of the fidelity as a function of the particle number that prove the existence of two-particle mode-exchange when a non-degeneracy condition is fulfilled. Tuning the energy difference of the two modes serves to distinguish the presence of two-particle mode-exchange terms through the appearance of certain singularities. Numerical results confirm our findings. Experimental verification of our findings could improve cold atom interferometry.

cond-mat.quant-gas↗

arXiv2015

Quantum efficiencies in finite disordered networks connected by many-body interactions

The quantum efficiency in the transfer of an initial excitation in disordered finite networks, modeled by the $k$-body embedded Gaussian ensembles of random matrices, is studied for bosons and fermions. The influence of the presence or absence of time-reversal symmetry and centrosymmetry/centrohermiticity are addressed. For bosons and fermions, the best efficiencies of the realizations of the ensemble are dramatically enhanced when centrosymmetry (centrohermiticity) is imposed. For few bosons distributed in two single-particle levels this permits perfect state transfer for almost all realizations when one-particle interactions are considered. For fermionic systems the enhancement is found to be maximal for cases when all but one single particle levels are occupied.

quant-ph↗

arXiv2014

Spectral Domain of Large Nonsymmetric Correlated Wishart Matrices

We study {the} complex eigenvalues of the Wishart model defined for nonsymmetric correlation matrices. The model is defined for two statistically equivalent but different Gaussian real matrices, as $\mathsf{C}=\mathsf{AB}^{t}/T$, where $\mathsf{B}^{t}$ is the transpose of $\mathsf{B}$ and both matrices $\mathsf{A}$ and $\mathsf{B}$ are of dimension $N\times T$. We consider {\it actual} correlations between the matrices so that on the ensemble average $\mathsf{C}$ does not vanish. We derive a loop equation for the spectral density of $\mathsf{C}$ in {the} large $N$ and $T$ limit where the ratio $N/T$ is finite. The actual correlations changes the complex eigenvalues of $\mathsf{C}$, and hence their domain from the results known for the vanishing $\mathsf{C}$ or for the uncorrelated $\mathsf{A}$ and $\mathsf{B}$. Using the loop equation we derive {a} result for the contour describing the domain of {the} bulk of the eigenvalues of $\mathsf{C}$. If the nonvanishing-correlation matrix is diagonal with the same element $c\ne0$, the contour is no longer a circle centered at origin but a shifted ellipse. In this case, the loop equation is analytically solvable and we explicitly derive {a} result for the spectral density. For more general cases, our analytical result implies that the contour depends on its symmetric and anti-symmetric parts if the nonvanishing-correlation matrix is nonsymmetric. On the other hand, if it is symmetric then the contour depends only on the spectrum of the correlation matrix. We also provide numerics to justify our analytics.

math-ph↗

arXiv2012

Fractional exclusion statistics and the Random Matrix Boson Ensemble

The k-body Gaussian Embedded Ensemble of Random Matrices is considered for N bosons distributed on two single-particle levels. When k = N, the ensemble is equivalent to the Gaussian Orthogonal Ensemble (GOE), and when k = 2 it corresponds to the Two-body Random Ensemble (TBRE) for bosons. It is shown that the energy spectrum leads to a rank function which is of the form of a discrete generalized beta distribution. The same distribution is obtained assuming N non-interacting quasiparticles that obey the fractional exclusion statistics introduced by Haldane two decades ago.

physics.atom-ph↗

arXiv2011

Fidelity decay in interacting two-level boson systems: Freezing and revivals

We study the fidelity decay in the $k$-body embedded ensembles of random matrices for bosons distributed in two single-particle states, considering the reference or unperturbed Hamiltonian as the one-body terms and the diagonal part of the $k$-body embedded ensemble of random matrices, and the perturbation as the residual off-diagonal part of the interaction. We calculate the ensemble-averaged fidelity with respect to an initial random state within linear response theory to second order on the perturbation strength, and demonstrate that it displays the freeze of the fidelity. During the freeze, the average fidelity exhibits periodic revivals at integer values of the Heisenberg time $t_H$. By selecting specific $k$-body terms of the residual interaction, we find that the periodicity of the revivals during the freeze of fidelity is an integer fraction of $t_H$, thus relating the period of the revivals with the range of the interaction $k$ of the perturbing terms. Numerical calculations confirm the analytical results.

nlin.CD↗

arXiv2010

Fidelity decay of the two-level bosonic embedded ensembles of Random Matrices

We study the fidelity decay of the $k$-body embedded ensembles of random matrices for bosons distributed over two single-particle states. Fidelity is defined in terms of a reference Hamiltonian, which is a purely diagonal matrix consisting of a fixed one-body term and includes the diagonal of the perturbing $k$-body embedded ensemble matrix, and the perturbed Hamiltonian which includes the residual off-diagonal elements of the $k$-body interaction. This choice mimics the typical mean-field basis used in many calculations. We study separately the cases $k=2$ and $3$. We compute the ensemble-averaged fidelity decay as well as the fidelity of typical members with respect to an initial random state. Average fidelity displays a revival at the Heisenberg time, $t=t_H=1$, and a freeze in the fidelity decay, during which periodic revivals of period $t_H$ are observed. We obtain the relevant scaling properties with respect to the number of bosons and the strength of the perturbation. For certain members of the ensemble, we find that the period of the revivals during the freeze of fidelity occurs at fractional times of $t_H$. These fractional periodic revivals are related to the dominance of specific $k$-body terms in the perturbation.

quant-ph↗