arXiv subjects

L. Velazquez

Publications and source records attributed to L. Velazquez.

At least 19 recordsLinked to original sources

arXiv2020

Occupation time for classical and quantum walks

This is a personal tribute to Lance Littlejohn on the occasion of his 70th birthday. It is meant as a present to him for many years of friendship. It is not written in the "Satz-Beweis" style of Edmund Landau or even in the format of a standard mathematics paper. It is rather an invitation to a fairly new, largely unexplored, topic in the hope that Lance will read it some afternoon and enjoy it. If he cares about complete proofs he will have to wait a bit longer; we almost have them but not in time for this volume. We hope that the figures will convince him and other readers that the phenomena displayed here are interesting enough.

quant-ph↗

arXiv2020

A pure non-neutral plasma under an external harmonic field: equilibrium thermodynamics and chaos

Motivated by the precedent study of Ordenes-Huanca and Velazquez [JSTAT \textbf{093303} (2016)], we address the study of a simple model of a pure non-neutral plasma: a system of identical non-relativistic charged particles confined under an external harmonic field with frequency $ω$. We perform the equilibrium thermo-statistical analysis in the framework of continuum approximation. This study reveals the existence of two asymptotic limits: the known Brillouin steady state at zero temperature, and the gas of harmonic oscillators in the limit of high temperatures. The non-extensive character of this model is evidenced by the associated thermodynamic limit, $N\rightarrow+\infty: U/N^{7/3}=const$, which coincides with the thermodynamic limit of a self-gravitating system of non-relativistic point particles in presence of Newtonian gravitation. Afterwards, the dynamics of this model is analyzed through numerical simulations. It is verified the agreement of thermo-statistical estimations and the temporal expectation values of the same macroscopic observables. The system chaoticity is addressed via numerical computation of Lyapunov exponents in the framework of the known \emph{tangent dynamics}. The temperature dependence of Lyapunov exponent $λ$ approaches to zero in the two asymptotic limits of this model, reaching its maximum during the transit between them. The chaos of the present model is very strong, since its rate is faster than the characteristic timescale of the microscopic dynamics $τ_{dyn}=1/ω$. A qualitative analysis suggests that such a strong chaoticity cannot be explained in terms of collision events because of their respective characteristic timescales are quite different, $τ_{ch}\propto τ_{dyn}/N^{1/4}$ and $τ_{coll}\propto τ_{dyn}$.

cond-mat.stat-mech↗

arXiv2016

Thermodynamics of gravitational clustering phenomena: $N$-body self-gravitating gas on the sphere $\mathbb{S}^{3}\subset\mathbb{R}^{4}$

This work is devoted to the thermodynamics of gravitational clustering, a collective phenomenon with a great relevance in the $N$-body cosmological problem. We study a classical self-gravitating gas of identical non-relativistic particles defined on the sphere $\mathbb{S}^{3}\subset \mathbb{R}^{4}$ by considering gravitational interaction that corresponds to this geometric space. The analysis is performed within microcanonical description of an isolated Hamiltonian system by combining continuum approximation and steepest descend method. According to numerical solution of resulting equations, the gravitational clustering can be associated with two microcanonical phase transitions. A first phase transition with a continuous character is associated with breakdown of $SO(4)$ symmetry of this model. The second one is the gravitational collapse, whose continuous or discontinuous character crucially depends on the regularization of short-range divergence of gravitation potential. We also derive the thermodynamic limit of this model system, the astrophysical counterpart of Gibbs-Duhem relation, the order parameters that characterize its phase transitions and the equation of state. Other interesting behavior is the existence of states with negative heat capacities, which appear when the effects of gravitation turn dominant for energies sufficiently low. Finally, we comment the relevance of some of these results in the study of astrophysical and cosmological situations. Special interest deserves the gravitational modification of the equation of state due to the local inhomogeneities of matter distribution. Although this feature is systematically neglected in studies about Universe expansion, the same one is able to mimic an effect that is attributed to the dark energy: a negative pressure.

cond-mat.stat-mech↗

arXiv2016

Truncated $γ$-exponential models for tidal stellar systems

We introduce a parametric family of models to characterize the properties of astrophysical systems in a quasi-stationary evolution under the incidence evaporation. We start from an one-particle distribution $f_γ\left(\mathbf{q},\mathbf{p}|β,\varepsilon_{s}\right)$ that considers an appropriate deformation of Maxwell-Boltzmann form with inverse temperature $β$, in particular, a power-law truncation at the scape energy $\varepsilon_{s}$ with exponent $γ>0$. This deformation is implemented using a generalized $γ$-exponential function obtained from the \emph{fractional integration} of ordinary exponential. As shown in this work, this proposal generalizes models of tidal stellar systems that predict particles distributions with \emph{isothermal cores and polytropic haloes}, e.g.: Michie-King models. We perform the analysis of thermodynamic features of these models and their associated distribution profiles. A nontrivial consequence of this study is that profiles with isothermal cores and polytropic haloes are only obtained for low energies whenever deformation parameter $γ<γ_{c}\simeq 2.13$.

astro-ph.GA↗

arXiv2016

Remarks about the thermodynamics of astrophysical systems in mutual interaction and related notions

General aspects about the thermodynamics of astrophysical systems are discussed, overall, those concerning to astrophysical systems in mutual interaction (or the called \emph{open astrophysical systems}). A special interest is devoted along the paper to clarify several misconceptions that are still common in the recent literature, such as the direct application to the astrophysical scenario of notions and theoretical frameworks that were originally conceived to deal with extensive systems of the everyday practice (large systems with short-range interactions).

astro-ph.GA↗

arXiv2016

Extended canonical Monte Carlo methods: Improving accuracy of microcanonical calculations using a re-weighting technique

Velazquez and Curilef have proposed a methodology to extend Monte Carlo algorithms that are based on canonical ensemble. According to our previous study, their proposal allows us to overcome slow sampling problems in systems that undergo any type of temperature-driven phase transition. After a comprehensive review about ideas and connections of this framework, we discuss the application a re-weighting technique to improve the accuracy of microcanonical calculations, specifically, the well-known multi-histograms method of Ferrenberg and Swendsen. As example of application, we reconsider the study of four-state Potts model on the square lattice $L\times L$ with periodic boundary conditions. This analysis allows us to detect the existence of a very small latent heat per site $q_{L}$ during the occurrence of temperature-driven phase transition of this model, whose size dependence seems to follow a power-law $q_{L}(L)\propto(1/L)^{z}$ with exponent $z\simeq0$.$26\pm0$.$02$. It is discussed the compatibility of these results with the continuous character of temperature-driven phase transition when $L\rightarrow+\infty$.

cond-mat.stat-mech↗

arXiv2013

Curvature of fluctuation geometry and its implications on Riemannian fluctuation theory

Fluctuation geometry was recently proposed as a counterpart approach of Riemannian geometry of inference theory. This theory describes the geometric features of the statistical manifold $\mathcal{M}$ of random events that are described by a family of continuous distributions $dp(x|θ)$. A main goal of this work is to clarify the statistical relevance of Levi-Civita curvature tensor $R_{ijkl}(x|θ)$ of the statistical manifold $\mathcal{M}$. For this purpose, the notion of \emph{irreducible statistical correlations} is introduced. Specifically, a distribution $dp(x|θ)$ exhibits irreducible statistical correlations if every distribution $dp(\check{x}|θ)$ obtained from $dp(x|θ)$ by considering a coordinate change $\check{x}=ϕ(x)$ cannot be factorized into independent distributions as $dp(\check{x}|θ)=\prod_{i}dp^{(i)}(\check{x}^{i}|θ)$. It is shown that the curvature tensor $R_{ijkl}(x|θ)$ arises as a direct indicator about the existence of irreducible statistical correlations. Moreover, the curvature scalar $R(x|θ)$ allows to introduce a criterium for the applicability of the \emph{gaussian approximation} of a given distribution function. This type of asymptotic result is obtained in the framework of the second-order geometric expansion of the distributions family $dp(x|θ)$, which appears as a counterpart development of the high-order asymptotic theory of statistical estimation. In physics, fluctuation geometry represents the mathematical apparatus of a Riemannian extension for Einstein's fluctuation theory of statistical mechanics. Some exact results of fluctuation geometry are now employed to derive the \emph{invariant fluctuation theorems}.

math-ph↗

arXiv2013

Improving the efficiency of Monte Carlo simulations of systems that undergo temperature-driven phase transitions

Recently, Velazquez and Curilef have proposed a methodology to extend Monte Carlo algorithms based on canonical ensemble, which is aimed to overcome slow sampling problems associated with temperature-driven discontinuous phase transitions. We show in this work that Monte Carlo algorithms extended with this methodology also exhibit a remarkable efficiency near a critical point. Our study is performed for the particular case of 2D four-state Potts model on the square lattice with periodic boundary conditions. This analysis reveals that the extended version of Metropolis importance sample is more efficient than the usual Swendsen-Wang and Wolff cluster algorithms. These results demonstrate the effectiveness of this methodology to improve the efficiency of MC simulations of systems that undergo any type of temperature-driven phase transition.

cond-mat.stat-mech↗

arXiv2012

Principles of classical statistical mechanics: A perspective from the notion of complementarity

Quantum mechanics and classical statistical mechanics are two physical theories that share several analogies in their mathematical apparatus and physical foundations. In particular, classical statistical mechanics is hallmarked by the complementarity between two descriptions that are unified in thermodynamics: (i) the parametrization of the system macrostate in terms of mechanical macroscopic observables $I=\{I^{i}\}$; and (ii) the dynamical description that explains the evolution of a system towards the thermodynamic equilibrium. As expected, such a complementarity is related to the uncertainty relations of classical statistical mechanics $ΔI^{i}Δη_{i}\geq k$. Here, $k$ is the Boltzmann's constant, $η_{i}=\partial \mathcal{S}(I|θ)/\partial I^{i}$ are the restituting generalized forces derived from the entropy $\mathcal{S}(I|θ)$ of a closed system, which is found in an equilibrium situation driven by certain control parameters $θ=\{θ^α\}$. These arguments constitute the central ingredients of a reformulation of classical statistical mechanics from the notion of complementarity. In this new framework, Einstein postulate of classical fluctuation theory $dp(I|θ)\sim\exp[\mathcal{S}(I|θ)/k]dI$ appears as the correspondence principle between classical statistical mechanics and thermodynamics in the limit $k\rightarrow0$, while the existence of uncertainty relations can be associated with the non-commuting character of certain operators.

cond-mat.stat-mech↗

arXiv2012

Equilibrium fluctuation theorems compatible with anomalous response

Previously, we have derived a generalization of the canonical fluctuation relation between heat capacity and energy fluctuations $C=β^{2}<δU^{2}>$, which is able to describe the existence of macrostates with negative heat capacities $C<0$. In this work, we extend our previous results for an equilibrium situation with several control parameters to account for the existence of states with anomalous values in other response functions. Our analysis leads to the derivation of three different equilibrium fluctuation theorems: the \textit{fundamental and the complementary fluctuation theorems}, which represent the generalization of two fluctuation identities already obtained in previous works, and the \textit{associated fluctuation theorem}, a result that has no counterpart in the framework of Boltzmann-Gibbs distributions. These results are applied to study the anomalous susceptibility of a ferromagnetic system, in particular, the case of 2D Ising model.

cond-mat.stat-mech↗

arXiv2012

Understanding critical behavior in the framework of the extended equilibrium fluctuation theorem

Recently (arXiv:0910.2870), we have derived a fluctuation theorem for systems in thermodynamic equilibrium compatible with anomalous response functions, e.g. the existence of states with \textit{negative heat capacities} $C<0$. In this work, we show that the present approach of the fluctuation theory introduces new insights in the understanding of \textit{critical phenomena}. Specifically, the new theorem predicts that the environmental influence can radically affect critical behavior of systems, e.g. to provoke a suppression of the divergence of correlation length $ξ$ and some of its associated phenomena as spontaneous symmetry breaking. Our analysis reveals that while response functions and state equations are \emph{intrinsic properties} for a given system, critical behaviors are always \emph{relative phenomena}, that is, their existence crucially depend on the underlying environmental influence.

cond-mat.stat-mech↗

arXiv2011

The Quantum Walk of F. Riesz

We exhibit a way to associate a quantum walk (QW) on the non-negative integers to any probability measure on the unit circle. This forces us to consider one step transitions that are not traditionally allowed. We illustrate this in the case of a very interesting measure, originally proposed by F. Riesz for a different purpose.

math-ph↗

arXiv2010

One-dimensional quantum walks with one defect

The CGMV method allows for the general discussion of localization properties for the states of a one-dimensional quantum walk, both in the case of the integers and in the case of the non negative integers. Using this method we classify, according to such localization properties, all the quantum walks with one defect at the origin, providing explicit expressions for the asymptotic return probabilities at the origin.

quant-ph↗

arXiv2010

An extension of the associated rational functions on the unit circle

A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the previous ones, the coefficients of this linear combination being self-reciprocal rational functions. We show that, under very general conditions on the self-reciprocal coefficients, this new sequence satisfies orthogonality conditions as well as a recurrence relation. Further, we identify the Caratheodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients. The new class under study includes the associated rational functions as a particular case. As a consequence of the previous general analysis, we obtain explicit representations for the associated rational functions of arbitrary order, as well as for the related Caratheodory function. Such representations are used to find new properties of the associated rational functions.

math.CA↗

arXiv2010

Extending canonical Monte Carlo methods

In this work, we discuss the implications of a recently obtained equilibrium fluctuation-dissipation relation on the extension of the available Monte Carlo methods based on the consideration of the Gibbs canonical ensemble to account for the existence of an anomalous regime with negative heat capacities $C<0$. The resulting framework appears as a suitable generalization of the methodology associated with the so-called \textit{dynamical ensemble}, which is applied to the extension of two well-known Monte Carlo methods: the Metropolis importance sample and the Swendsen-Wang clusters algorithm. These Monte Carlo algorithms are employed to study the anomalous thermodynamic behavior of the Potts models with many spin states $q$ defined on a $d$-dimensional hypercubic lattice with periodic boundary conditions, which successfully reduce the exponential divergence of decorrelation time $τ$ with the increase of the system size $N$ to a weak power-law divergence $τ\propto N^α$ with $α\approx0.2$ for the particular case of the 2D 10-state Potts model.

cond-mat.stat-mech↗

arXiv2010

Extending canonical Monte Carlo methods II

Previously, we have presented a methodology to extend canonical Monte Carlo methods inspired on a suitable extension of the canonical fluctuation relation $C=β^{2}<δE^{2}>$ compatible with negative heat capacities $C<0$. Now, we improve this methodology by introducing a better treatment of finite size effects affecting the precision of a direct determination of the microcanonical caloric curve $β(E) =\partial S(E) /\partial E$, as well as a better implementation of MC schemes. We shall show that despite the modifications considered, the extended canonical MC methods possibility an impressive overcome of the so-called \textit{super-critical slowing down} observed close to the region of a temperature driven first-order phase transition. In this case, the dependence of the decorrelation time $τ$ with the system size $N$ is reduced from an exponential growth to a weak power-law behavior $τ(N)\propto N^α$, which is shown in the particular case of the 2D seven-state Potts model where the exponent $α=0.14-0.18$.

cond-mat.stat-mech↗

arXiv2009

Remarks about the thermostatistical description of the HMF model Part I: Equilibrium Thermodynamics

In this series of papers we shall carry out a reconsideration of the thermodynamical behavior of the called HMF model, a paradigmatic ferromagnetic toy model exhibiting many features of the real long-range interacting systems. This first work is devoted to perform the microcanonical description of this model system: the calculation of microcanonical entropy and some fundamental thermodynamic observables, the distribution and correlation functions, as well as the analysis of the thermodynamical stability and the relevant thermodynamic limit.

cond-mat.stat-mech↗

arXiv2009

Remarks about the thermostatistical description of the HMF model Part II: Phenomenology of Relaxation Dynamics

After a general overview of some features of the relaxation dynamics of the Hamiltonian Mean Field model, its equilibrium thermodynamic properties are used to rephrase the out-of-equilibrium regime for energies below the critical point $u_{c}=0.75$ in terms of an effective dynamical coexistence between a clustered and a gaseous phases, whose existence could be associated to the large relaxation times observed when $u_{1}<u<u_{c}$, with $u_{1}=0.5$. Starting from the hypothesis that the \textit{parametric resonance} is the microscopic mechanism allowing the energetic interchange between the particles during the collisional regime, a phenomenological Fokker-Planck equation based on a Langevin equation with a multiplicative noise is proposed in order to describe the collisional relaxation of this system towards its final equilibrium, which supports the following dependence of the collisional relaxation timescale $τ_{cr}=τ_{0}N\equiv\sqrt{IN/g}$.

cond-mat.stat-mech↗