arXiv subjects

Ignacio S. Gomez

Publications and source records attributed to Ignacio S. Gomez.

At least 19 recordsLinked to original sources

arXiv2026

Inverse characterization and non-uniqueness of the Gaussian configurational partition function

Given a configurational partition function, in this work we investigate the inverse problem of reconstructing the one-dimensional potential. For the case of the Gaussian partition function, the configurational density of states (CDOS) is univocally determined, being the harmonic potential uniquely recovered due to its symmetric single--well feature. When multiples inverse branches are considered, the uniqueness of the potential is broken and more information is needed in order to obtain a unique potential. The branch topology imposes alternating orientations, thus allowing distinct spatial realizations. Finally, the Gaussian partition function only determines the CDOS but not the potential, manifesting in this way the precise scope and limitations of the inverse characterization of the Gaussian configurational partition function.

math-ph↗

arXiv2026

Finite nonlinear mathematical structures induced by the Tsallis $q$-sum

The Tsallis $q$-sum is one of the fundamental nonlinear composition laws of nonextensive statistical mechanics. Although it has been extensively investigated in continuous settings, its implications for finite mathematical structures remain less explored. Here we investigate this question by replacing ordinary additive laws with the Tsallis $q$-sum. We first introduce an axiomatic $q$-cardinality of finite sets satisfying a nonlinear additivity principle. Existence and uniqueness are established, leading to an explicit expression that continuously recovers the classical cardinality as $q\to1$. We then propose a nonlinear matrix composition induced by the same deformation and establish its principal algebraic properties, including associativity, the neutral element, and a characterization of its noncommutativity in terms of the ordinary matrix commutator. An illustrative example over $M_2(\mathbb Z_6)$ shows how the deformation can modify the center of a finite matrix algebra when $1-q$ is a zero divisor. These results indicate that the Tsallis $q$-sum provides a natural mechanism for constructing nonlinear finite mathematical structures and connects nonextensive statistical mechanics with finite algebra and nonlinear mathematical physics.

math-ph↗

arXiv2025

Mixed dynamics from the classical and quantum ergodic hierarchy

Based on the classical and quantum ergodic hierarchy, a framework for mixed systems with a phase space composed by two uncorrelated integrable and chaotic regions is presented. It provides some features of mixed systems connecting the intuitive notion of a mixed phase space with the mixing level of the ergodic hierarchy. The formalism is illustrated with the kicked rotator.

math-ph↗

arXiv2025

Partition function for position-dependent mass systems from superestatistics

In this work, we show a connection between superstatistics and position-dependent mass (PDM) systems in the context of the canonical ensemble. The key point is to set the fluctuation distribution of the inverse temperature in terms od the system PDM. For PDMs associated to Tsallis and Kaniadakis nonextensive statistics, the pressure and entropy of the ideal gas result lower than the standard case but maintaining monotonic behavior. Gas of non-interacting harmonic oscillators provided with quadratic and exponential PDMs exhibit a behavior of standard ED harmonic oscillator gas and a linear specific heat respectively, the latter being consistent with Nernst's third law of thermodynamics. Thus, a combined PDM-superstatistics scenario offers an alternative way to study the effects of the inhomogeneities of PDM systems in their thermodynamics.

cond-mat.stat-mech↗

arXiv2024

Symplectic Quantization and General Constraint Structure of a Prototypical Second-Class System

We discuss a general prototypical constrained Hamiltonian system with a broad application in quantum field theory and similar contexts where dynamics is defined through a functional action obeying a stationarity principle. The prototypical model amounts to a Dirac-Bergmann singular system, whose constraints restrict the actual dynamics to occur within a differential submanifold, as is the case in the major part of field theoretical models with gauge symmetry. We apply the Dirac-Bergmann algorithm in its full generality unraveling a total of $4m$ second-class constraints and obtain the corresponding Dirac brackets algebra in phase space. We follow with the Faddeev-Jackiw-Barcelos-Wotzasek approach in which the geometric character of the mentioned submanifold is emphasized by means of an internal metric function encoding its symplectic properties. We consider two straightforward examples, applying our general results to constrained motion along a toroidal geometry and to a Lorentz violating toy model in field theory. Since toroidal geometry has been recently used in cosmological models, we suggest how our results could lead to different proposals for the shape of the universe in cosmology.

hep-th↗

arXiv2023

Exact solution and coherent states of an asymmetric oscillator with position-dependent mass

We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. {\bf 62}, 092101 (2021)] in the classical and quantum formalisms, by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space $(x, p)$ to a deformed one $(x_γ, Π_γ)$. Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.

quant-ph↗

arXiv2022

Studies of transport coefficients in charged AdS$_{4}$ black holes on $κ$-deformed space

In this work, we study the effect of $κ$-deformed space on the thermodynamic quantities, this are find through the holographic renormalization that provide the free energy, which is fundamental to derive the another thermodynamic quantities. For this scenario we consider an charged AdS$_{4}$ black hole for an Einstein-Maxwell model where the derivative quadrivector is replaced by a deformed version inspired in Kaniadakis statistics. Besides, we extract the transport coefficient know as electrical conductivity.

hep-th↗

arXiv2021

Supersymmetric quantum mechanics and coherent states for a deformed oscillator with position-dependent effective mass

We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator, associated with a position-dependent effective mass, by means of the supersymmetric formalism. From the supersymmetric partner Hamiltonians and the shape invariance technique we obtain the eigenstates and the eigenvalues along with the ladders operators, thus showing a preservation of the supersymmetric structure in terms of the deformed counterpartners. The deformed space in supersymmetry allows to characterize position-dependent effective mass, uniform field interactions and to obtain a generalized uncertainty relation (GUP) that behaves as a distinguishability measure for the coherent states, these latter satisfying a periodic evolution of the GUP corrections.

quant-ph↗

arXiv2020

Deformed Fokker-Planck equation: inhomogeneous medium with a position-dependent mass

We present the Fokker-Planck equation (FPE) for an inhomogeneous medium with a position-dependent mass particle by making use of the Langevin equation, in the context of a generalized deformed derivative for an arbitrary deformation space where the linear (nonlinear) character of the FPE is associated with the employed deformed linear (nonlinear) derivative. The FPE for an inhomogeneous medium with a position-dependent diffusion coefficient is equivalent to a deformed FPE within a deformed space, described by generalized derivatives, and constant diffusion coefficient. The deformed FPE is consistent with the diffusion equation for inhomogeneous media when the temperature and the mobility have the same position-dependent functional form as well as with the nonlinear Langevin approach. The deformed version of the H-theorem permits to express the Boltzmann-Gibbs entropic functional as a sum of two contributions, one from the particles and the other from the inhomogeneous medium. The formalism is illustrated with the infinite square well and the confining potential with linear drift coefficient. Connections between superstatistics and position-dependent Langevin equations are also discussed.

cond-mat.stat-mech↗

arXiv2020

Morse potential in relativistic contexts from generalized momentum operator, Pekeris approximation revisited and mapping

In this work we explore a generalization of the Dirac and Klein-Gordon (KG) oscillators, provided with a deformed linear momentum inspired in nonextensive statistics, that gives place to the Morse potential in relativistic contexts by first principles. In the (1+1)-dimensional case the relativistic oscillators are mapped into the quantum Morse potential. Using the Pekeris approximation, in the (3+1)-dimensional case we study the thermodynamics of the S-waves states (l=0) of the H2, LiH, HCl and CO molecules (in the non-relativistic limit) and of a relativistic electron, where Schottky anomalies (due to the finiteness of the Morse spectrum) and spin contributions to the heat capacity are reported. By revisiting a generalized Pekeris approximation, we provide a mapping from (3+1)-dimensional Dirac and KG equations with a spherical potential to an associated one-dimensional Schrödinger-like equation, and we obtain the family of potentials for which this mapping corresponds to a Schrödinger equation with non-minimal coupling.

math-ph↗

arXiv2020

$κ$-Deformed quantum and classical mechanics for a system with position-dependent effective mass

We present the quantum and classical mechanics formalisms for a particle with position-dependent mass in the context of a deformed algebraic structure (named $κ$-algebra), motivated by the Kappa-statistics. From this structure we obtain deformed versions of the position and momentum operators, which allow to define a point canonical transformation that maps a particle with constant mass in a deformed space into a particle with position-dependent mass in the standard space. We illustrate the formalism with a particle confined in an infinite potential well and the Mathews-Lakshmanan oscillator, exhibiting uncertainty relations depending on the deformation.

quant-ph↗

arXiv2020

Splitting frequency of the (2+1)-dimensional Duffin-Kemmer-Petiau oscillator in an external magnetic field

We revisit the (2+1)-dimensional DKP oscillator in an external magnetic field by means of 4x4 and 6x6 representations of the DKP field, thus obtaining several cases studied in the literature. We found an splitting in the frequency of the DKP oscillator according to the spin projection that arises as an interplay between the oscillator, the external field and the spin, from which the energies and the eigenfunctions are expressed in a unified way. For certain critical values of the magnetic field the oscillation in the components of spin projections -1 and 1 is cancelled. We study the thermodynamics of the canonical ensemble of the vectorial sector, where a phase transition is reported when the cancellation of the oscillation occurs. Thermodynamic potentials converge rapidly to their asymptotical expressions in the high temperature limit with the partition function symmetric under the reversion of the magnetic field.

math-ph↗

arXiv2019

Algebraic structures and deformed Schrödinger equations from groups entropies

Motivated by the group entropy theory, in this work we generalize the algebra of real numbers (that we called G-algebra), from which we develop an associated G-differential calculus. Thus, the algebraic structures corresponding to the Tsallis and Kappa statistics are obtained as special cases when the Tsallis and Kappa group classes are chosen. We employ the G-algebra to formulate a generalized G-deformed Schrödinger equation and we illustrate it with the infinite potential well, where the effective mass is related with the G-algebra structure and the $q$-deformed (standard) Schrödinger equation results an special case for the Tsallis (Boltzmann-Gibbs) group class. The non-uniform zeros spacing of the G-deformed eigenfunctions is expressed in terms of the generalized sum of the G-algebra.

math-ph↗

arXiv2019

Information-theoretic measures for a position-dependent mass system in an infinite potential well

In this work we calculate the Cramér-Rao, the Fisher-Shannon and the López-Ruiz-Mancini-Calbert (LMC) complexity measures for eigenstates of a deformed Schrödinger equation, being this intrinsically linked with position-dependent mass (PDM) systems. The formalism presented is illustrated with a particle confined in an infinite potential well. Abrupt variation of the complexity near to the asymptotic value of the PDM-function $m(x)$ and erasure of its asymmetry along with negative values of the entropy density in the position space, are reported as a consequence of the interplay between the deformation and the complexity.

quant-ph↗

arXiv2018

Majorization and dynamics of continuous distributions

In this work we show how the concept of majorization in continuous distributions can be employed to characterize chaotic, diffusive and quantum dynamics. The key point lies in that majorization allows to define an intuitive arrow of time, within a continuous dynamics, along with an associated majorized Second Law which implies the standard Second Law of thermodynamics but not viceversa. Moreover, mixing dynamics, generalized Fokker-Planck equations and quantum evolutions are explored as majorized ordered chains along the time evolution, being the stationary states the infimum elements.

math-ph↗

arXiv2018

Hermite-Gaussian model for quantum states

In order to characterize quantum states within the context of information geometry, we propose a generalization of the Gaussian model, which we called the Hermite-Gaussian model. We obtain the Fisher-Rao metric and the scalar curvature for this model, and we show its relation with the one-dimensional quantum harmonic oscillator. Moreover, using this model we characterize some failies of states of the quantum harmonic oscillator. We find that for the eigenstates of the Hamiltonian, mixtures of eigenstates and even or odd superpositions of eienstates the associated Fisher-Rao metrics are diagonal.

math-ph↗

arXiv2018

A unified time scale for quantum chaotic regimes

We present a generalised time scale for quantum chaos dynamics, motivated by nonextensive statistical mechanics. It recovers, as particular cases, the relaxation (Heisenberg) and the random (Ehrenfest) time scales. Moreover, we show that the generalised time scale can also be obtained from a nonextensive version of the Kolmogorov-Sinai entropy by considering the graininess of quantum phase space and a generalised uncorrelation between subsets of the phase space. Lyapunov and regular regimes for the fidelity decay are obtained as a consequence of a nonextensive generalisation of the $m$th point correlation function for a uniformly distributed perturbation in the classical limit.

quant-ph↗

arXiv2018

Fisher metric from relative entropy group

In this work we consider the Fisher metric which results from the Hessian of the relative entropy group, that we called Fisher metric group, and we obtain the corresponding ones to the Boltzmann-Gibbs, Tsallis, Kaniadakis and Abe-Borges-Roditi classes. We prove that the scalar curvature of the Fisher metric group results a multiple of the standard Fisher one, with the factor of proportionality given by the local properties of the entropy group. For the Tsallis class, the softening and strengthening of the scalar curvature is illustrated with the $2D$ correlated model, from which their associated indexes for the canonical ensemble of a pair of interacting harmonic oscillators, are obtained.

math-ph↗