arXiv subjects

Miguel A. Rodriguez

Publications and source records attributed to Miguel A. Rodriguez.

At least 19 recordsLinked to original sources

arXiv2026

Laurent Sequences, Extended Rota Algebras and Categorical Discretization of Dynamical Systems

We introduce a novel integrability-preserving discretization for a broad class of differential equations with variable coefficients, encompassing both linear and nonlinear cases. The construction is achieved via a categorical approach that enables a unified treatment of continuous and discrete dynamical systems. Our theoretical framework is grounded on a generalization of G. C. Rota's finite operator calculus, which enables us to extend the theory of basic sequence of polynomials to the setting of Laurent polynomials. Accordingly, we introduce the notion of an \textit{extended Rota algebra}, defined as a Galois differential algebra in which all difference operators act as derivations on the space of Laurent power series with respect to a suitably defined functional product. The core of our theory relies on the existence of covariant functors between the newly proposed Rota category of Galois differential algebras and suitable categories of abstract dynamical systems. In this setting, under certain regularity assumptions, a differential equation and its discrete analogues are naturally interpreted as objects of the same category. This perspective enables the construction of a vast class of integrable maps that share with their continuous analogues a wide set of exact solutions, \textit{regular} or \textit{singular} and, in the linear case, the Picard-Vessiot group.

math-ph↗

arXiv2022

On higher-dimensional superintegrable systems: A new family of classical and quantum Hamiltonian models

We introduce a family of $n$-dimensional Hamiltonian systems which, contain, as special reductions, several superintegrable systems as the Tremblay-Turbiner-Winternitz system, a generalized Kepler potential and the anisotropic harmonic oscillator with Rosochatius terms. We conjecture that there exist special values in the space of parameters, apart from those leading to known cases, for which this new Hamiltonian family is superintegrable.

math-ph↗

arXiv2022

Energy Consumption Analysis Of Machining Centers Using Bayesian Analysis And Genetic Optimization

Responding to the current urgent need for low carbon emissions and high efficiency in manufacturing processes, the relationships between three different machining factors (depth of cut, feed rate, and spindle rate) on power consumption and surface finish (roughness) were analysed by applying a Bayesian seemingly unrelated regressions (SUR) model. For the analysis, an optimization criterion was established and minimized by using an optimization algorithm that combines evolutionary algorithm methods with a derivative-based (quasi-Newton) method to find the optimal conditions for energy consumption that obtains a good surface finish quality. A Bayesian ANOVA was also performed to identify the most important factors in terms of variance explanation of the observed outcomes. The data were obtained from a factorial experimental design performed in two computerized numerical control (CNC) vertical machining centers (Haas UMC-750 and Leadwell V-40iT). Some results from this study show that the feed rate is the most influential factor in power consumption, and the depth of cut is the factor with the stronger influence on roughness values. An optimal operational point is found for the three factors with a predictive error of less than 0.01% and 0.03% for the Leadwell V-40iT machine and the Haas UMC-750 machine, respectively.

stat.AP↗

arXiv2019

FEniCS Mechanics: A Package for Continuum Mechanics Simulations

FEniCS Mechanics is a Python package to facilitate computational mechanics simulations. The Python library dolfin, from the FEniCS Project, is used to formulate and numerically solve the problem in variational form. The general balance laws from continuum mechanics are used to enable rapid prototyping of different material laws. In addition to its generality, FEniCS Mechanics also checks the input provided by users to ensure that problem definitions are physically consistent. In turn, this code enables simulations of custom mechanics problems to be more accessible to those with limited programming or mechanics knowledge.

cs.CE↗

arXiv2016

On the construction of partial difference schemes II: discrete variables and Schwarzian lattices

In the process of constructing invariant difference schemes which approximate partial differential equations we write down a procedure for discretizing an arbitrary partial differential equation on an arbitrary lattice. An open problem is the meaning of a lattice which does not satisfy the Clairaut--Schwarz--Young theorem. To analyze it we apply the procedure on a simple example, the potential Burgers equation with two different lattices, an orthogonal lattice which is invariant under the symmetries of the equation and satisfies the commutativity of the partial difference operators and an exponential lattice which is not invariant and does not satisfy the Clairaut--Schwarz--Young theorem. A discussion on the numerical results is also presented showing the different behavior of both schemes for two different exact solutions and their numerical approximations.

math-ph↗

arXiv2016

Critical behavior of su(1|1) supersymmetric spin chains with long-range interactions

We introduce a general class of su$(1|1)$ supersymmetric spin chains with long-range interactions which includes as particular cases the su$(1|1)$ Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su$(1|1)$ permutation operator, and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low energy excitations and the low temperature behavior of the free energy, which coincides with that of a $(1+1)$-dimensional conformal field theory (CFT) with central charge $c=1$ when the chemical potential lies in the critical interval $(0,\mathcal E(π))$, $\mathcal E(p)$ being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson $(1+1)$-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge $c=1$. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su$(1|1)$ elliptic chain.

quant-ph↗

arXiv2016

Generalized isotropic Lipkin-Meshkov-Glick models: ground state entanglement and quantum entropies

We introduce a new class of generalized isotropic Lipkin-Meshkov-Glick models with su$(m+1)$ spin and long-range non-constant interactions, whose non-degenerate ground state is a Dicke state of su$(m+1)$ type. We evaluate in closed form the reduced density matrix of a block of $L$ spins when the whole system is in its ground state, and study the corresponding von Neumann and Rényi entanglement entropies in the thermodynamic limit. We show that both of these entropies scale as $a\log L$ when $L$ tends to infinity, where the coefficient $a$ is equal to $(m-k)/2$ in the ground state phase with $k$ vanishing su$(m+1)$ magnon densities. In particular, our results show that none of these generalized Lipkin-Meshkov-Glick models are critical, since when $L\to\infty$ their Rényi entropy $R_q$ becomes independent of the parameter $q$. We have also computed the Tsallis entanglement entropy of the ground state of these generalized su$(m+1)$ Lipkin-Meshkov-Glick models, finding that it can be made extensive by an appropriate choice of its parameter only when $m-k\ge3$. Finally, in the su$(3)$ case we construct in detail the phase diagram of the ground state in parameter space, showing that it is determined in a simple way by the weights of the fundamental representation of su$(3)$. This is also true in the su$(m+1)$ case; for instance, we prove that the region for which all the magnon densities are non-vanishing is an $(m+1)$-simplex in $\mathbf R^m$ whose vertices are the weights of the fundamental representation of su$(m+1)$.

quant-ph↗

arXiv2013

Observation of Pull-in Instability in Graphene Membranes under Interfacial Forces

We present a unique experimental configuration that allows us to determine the interfacial forces on nearly parallel plates made from the thinnest possible mechanical structures, single and few layer graphene membranes. Our approach consists of using a pressure difference across a graphene membrane to bring the membrane to within ~ 10-20 nm above a circular post covered with SiOx or Au until a critical point is reached whereby the membrane snaps into adhesive contact with the post. Continuous measurements of the deforming membrane with an AFM coupled with a theoretical model allow us to deduce the magnitude of the interfacial forces between graphene and SiOx and graphene and Au. The nature of the interfacial forces at ~ 10 - 20 nm separations is consistent with an inverse fourth power distance dependence, implying that the interfacial forces are dominated by van der Waals interactions. Furthermore, the strength of the interactions is found to increase linearly with the number of graphene layers. The experimental approach can be used to measure the strength of the interfacial forces for other atomically thin two-dimensional materials, and help guide the development of nanomechanical devices such as switches, resonators, and sensors.

cond-mat.mes-hall↗

arXiv2008

Reduction of superintegrable systems: the anisotropic harmonic oscillator

We introduce a new 2N--parametric family of maximally superintegrable systems in N dimensions, obtained as a reduction of an anisotropic harmonic oscillator in a 2N--dimensional configuration space. These systems possess closed bounded orbits and integrals of motion which are polynomial in the momenta. They generalize known examples of superintegrable models in the Euclidean plane.

math-ph↗

arXiv2008

Dynamics of perturbations in disordered chaotic systems

We study the time evolution of perturbations in spatially extended chaotic systems in the presence of quenched disorder. We find that initially random perturbations tend to exponentially localize in space around static pinning centers that are selected by the particular configuration of disorder. The spatial structure of typical perturbations, $δu(x,t)$, is analyzed in terms of the Hopf-Cole transform, $h(x,t) \equiv\ln|δu(x,t)|$. Our analysis shows that the associated surface $h(x,t)$ self-organizes into a faceted structure with scale-invariant correlations. Scaling analysis of critical roughening exponents reveals that there are three different universality classes for error propagation in disordered chaotic systems that correspond to different symmetries of the underlying disorder. Our conclusions are based on numerical simulations of disordered lattices of coupled chaotic elements and equations for diffusion in random potentials. We propose a phenomenological stochastic field theory that gives some insights on the path for a generalization of these results for a broad class of disordered extended systems exhibiting space-time chaos.

cond-mat.stat-mech↗

arXiv2007

Noise effects in extended chaotic system: study on the Lorenz'96 model

We investigate the effects of a time-correlated noise on an extended chaotic system. The chosen model is the Lorenz'96, a kind of toy model used for climate studies. The system is subjected to both temporal and spatiotemporal perturbations. Through the analysis of the system's time evolution and its time correlations, we have obtained numerical evidence for two stochastic resonance-like behaviors. Such behavior is seen when a generalized signal-to-noise ratio function are depicted as a function of the external noise intensity or as function of the system size. The underlying mechanism seems to be associated to a noise-induced chaos reduction. The possible relevance of those findings for an optimal climate prediction are discussed, using an analysis of the noise effects on the evolution of finite perturbations and errors.

nlin.CD↗

arXiv2007

Resonant phenomena in extended chaotic systems subject to external noise: the Lorenz'96 model case

We investigate the effects of a time-correlated noise on an extended chaotic system. The chosen model is the Lorenz'96, a kind of "toy" model used for climate studies. Through the analysis of the system's time evolution and its time and space correlations, we have obtained numerical evidence for two stochastic resonance-like behavior. Such behavior is seen when both, the usual and a generalized signal-to-noise ratio function are depicted as a function of the external noise intensity or the system size. The underlying mechanism seems to be associated to a "noise-induced chaos reduction". The possible relevance of these and other findings for an "optimal" climate prediction are discussed.

nlin.CD↗

arXiv2006

Glassy dynamics, aging and thermally activated avalanches in interface pinning at finite temperatures

We study numerically the out-of-equilibrium dynamics of interfaces at finite temperatures when driven well below the zero-temperature depinning threshold. We go further than previous analysis by including the most relevant non-equilibrium correction to the elastic Hamiltonian. We find that the relaxation dynamics towards the steady-state shows glassy behavior, aging and violation of the fluctuation-dissipation theorem. The interface roughness exponent alpha approx 0.7 is found to be robust to temperature changes. We also study the instantaneous velocity signal in the low temperature regime and find long-range temporal correlations. We argue 1/f-noise arises from the merging of local thermally-activated avalanches of depinning events.

cond-mat.stat-mech↗

arXiv2004

Scaling properties of growing noninfinitesimal perturbations in space-time chaos

We study the spatiotemporal dynamics of random spatially distributed noninfinitesimal perturbations in one-dimensional chaotic extended systems. We find that an initial perturbation of finite size $ε_0$ grows in time obeying the tangent space dynamic equations (Lyapunov vectors) up to a characteristic time $t_{\times}(ε_0) \sim b - (1/λ_{max}) \ln (ε_0)$, where $λ_{max}$ is the largest Lyapunov exponent and $b$ is a constant. For times $t < t_{\times}$ perturbations exhibit spatial correlations up to a typical distance $ξ\sim t^z$. For times larger than $t_{\times}$ finite perturbations are no longer described by tangent space equations, memory of spatial correlations is progressively destroyed and perturbations become spatiotemporal white noise. We are able to explain these results by mapping the problem to the Kardar-Parisi-Zhang universality class of surface growth.

nlin.CD↗

arXiv2004

Lie Symmetries and Exact Solutions of First Order Difference Schemes

We show that any first order ordinary differential equation with a known Lie point symmetry group can be discretized into a difference scheme with the same symmetry group. In general, the lattices are not regular ones, but must be adapted to the symmetries considered. The invariant difference schemes can be so chosen that their solutions coincide exactly with those of the original differential equation.

nlin.SI↗

arXiv2004

Rare Events and Scale--Invariant Dynamics of Perturbations in Delayed Dynamical Systems

We study the dynamics of perturbations in time delayed dynamical systems. Using a suitable space-time coordinate transformation, we find that the time evolution of the linearized perturbations (Lyapunov vector) can be mapped to the linear Zhang surface growth model [Y.-C. Zhang, J. Phys. France {\bf 51}, 2129 (1990)], which is known to describe surface roughening driven by power-law distributed noise. As a consequence, Lyapunov vector dynamics is dominated by rare random events that lead to non-Gaussian fluctuations and multiscaling properties.

cond-mat.stat-mech↗

arXiv2003

Dynamic Scaling of Bred Vectors in Chaotic Extended Systems

We argue that the spatiotemporal dynamics of bred vectors in chaotic extended systems are related to a kinetic roughening process in the Kardar-Parisi-Zhang universality class. This implies that there exists a characteristic length scale corresponding to the typical extend over which the finite-size perturbation is actually correlated in space. This can be used as a quantitative parameter to characterize the degree of projection of the bred vectors into the dynamical attractor.

nlin.CD↗

arXiv2001

Nonequilibrium Phase Transitions in Directed Small-World Networks

Many social, biological, and economic systems can be approached by complex networks of interacting units. The behaviour of several models on small-world networks has recently been studied. These models are expected to capture the essential features of the complex processes taking place on real networks like disease spreading, formation of public opinion, distribution of wealth, etc. In many of these systems relations are directed, in the sense that links only act in one direction (outwards or inwards). We investigate the effect of directed links on the behaviour of a simple spin-like model evolving on a small-world network. We show that directed networks may lead to a highly nontrivial phase diagram including first and second-order phase transitions out of equilibrium.

cond-mat.stat-mech↗