arXiv subjects

L. Velazquez

Publications and source records attributed to L. Velazquez.

At least 37 records · Page 2Linked to original sources

arXiv2009

Geometrical aspects and connections of the energy-temperature fluctuation relation

Recently, we have derived a generalization of the known canonical fluctuation relation $k_{B}C=β^{2}< δU^{2} >$ between heat capacity $C$ and energy fluctuations, which can account for the existence of macrostates with negative heat capacities $C<0$. In this work, we presented a panoramic overview of direct implications and connections of this fluctuation theorem with other developments of statistical mechanics, such as the extension of canonical Monte Carlo methods, the geometric formulations of fluctuation theory and the relevance of a geometric extension of the Gibbs canonical ensemble that has been recently proposed in the literature.

cond-mat.stat-mech↗

arXiv2009

Thermodynamic fluctuation relation for temperature and energy

The present work extends the well-known thermodynamic relation $C=β^{2}< δ{E^{2}}>$ for the canonical ensemble. We start from the general situation of the thermodynamic equilibrium between a large but finite system of interest and a generalized thermostat, which we define in the course of the paper. The resulting identity $< δβδ{E}> =1+< δ{E^{2}}% > \partial ^{2}S(E) /\partial {E^{2}}$ can account for thermodynamic states with a negative heat capacity $C<0$; at the same time, it represents a thermodynamic fluctuation relation that imposes some restrictions on the determination of the microcanonical caloric curve $β(E) =\partial S(E) /\partial E$. Finally, we comment briefly on the implications of the present result for the development of new Monte Carlo methods and an apparent analogy with quantum mechanics.

cond-mat.stat-mech↗

arXiv2009

Uncertainty relations of Statistical Mechanics

Recently, we have presented some simple arguments supporting the existence of certain complementarity between thermodynamic quantities of temperature and energy, an idea suggested by Bohr and Heinsenberg in the early days of Quantum Mechanics. Such a complementarity is expressed as the impossibility of perform an exact simultaneous determination of the system energy and temperature by using an experimental procedure based on the thermal equilibrium with other system regarded as a measure apparatus (thermometer). In this work, we provide a simple generalization of this latter approach with the consideration of a thermodynamic situation with several control parameters.

cond-mat.stat-mech↗

arXiv2009

Polynomial perturbations of hermitian linear functionals and difference equations

This paper is devoted to the study of general (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree. The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters. Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a kind of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional.

math.CA↗

arXiv2009

Matrix valued Szego polynomials and quantum random walks

We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case the study of QRWs on the set of non-negative integers can be handled using scalar valued (Laurent) polynomials and a scalar valued measure on the circle. In the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions. We show how our tools yield results in the well known case of the Hadamard walk, but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods. More precisely we consider QRWs on the set of non- negative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being selfcontained, but we re- frain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 18]. See also the recent notes [19].

quant-ph↗

arXiv2007

Generalized Gibbs canonical ensemble: A possible physical scenario

After reviewing some fundamental results derived from the introduction of the generalized Gibbs canonical ensemble, such as the called thermodynamic uncertainty relation, it is described a physical scenario where such a generalized ensemble naturally appears as a consequence of a modification of the energetic interchange mechanism between the interest system and its surrounding, which could be relevant within the framework of long-range interacting systems.

cond-mat.stat-mech↗

arXiv2006

Enhancing Monte Carlo methods by using a generalized fluctuation theory

According to the recently obtained thermodynamic uncertainty relation, the microcanonical regions with a negative heat capacity can be accessed within a canonical-like description by using a thermostat with a fluctuating inverse temperature. This far-reaching conclusion is used in this Letter for enhancing the potentialities of the well-known Swendsen-Wang cluster algorithm in order to access to the anomalous microcanonical states of the q=10 state Potts model on a square lattice, which exhibits a first-order phase transition in its thermodynamical description.

cond-mat.stat-mech↗

arXiv2006

Generalized Fluctuation Theory based on the reparametrization invariance of the microcanonical ensemble

The main interest of the present work is the generalization of the Boltzmann-Gibbs distributions and the fluctuation theory based on the consideration of the reparametrization invariance of the microcanonical ensemble. This approach allows a novel interpretation of some anomalous phenomena observed in the non extensive systems like the existence of the negative specific heats as well as possibilities the enhancing of some Monte Carlo methods based on the Statistical Mechanics.

cond-mat.stat-mech↗

arXiv2006

Metropolis Monte Carlo algorithm based on the reparametrization invariance

We introduce a modification of the well-known Metropolis importance sampling algorithm by using a methodology inspired on the consideration of the reparametrization invariance of the microcanonical ensemble. The most important feature of the present proposal is the possibility of performing a suitable description of microcanonical thermodynamic states during the first-order phase transitions by using this local Monte Carlo algorithm.

cond-mat.stat-mech↗

arXiv2006

Basis of a non Riemannian Geometry within the Equilibrium Thermodynamics

Microcanonical description is characterized by the presence of an internal symmetry closely related with the dynamical origin of this ensemble: the reparametrization invariance. Such symmetry possibilities the development of a non Riemannian geometric formulation within the microcanonical description of an isolated system, which leads to an unexpected generalization of the Gibbs canonical ensemble and the classical fluctuation theory for the open systems (where the inverse temperature and the total energy E behave as complementary thermodynamical quantities), the improvement of Monte Carlo simulations based on the canonical ensemble, as well as a reconsideration of any classification scheme of the phase transitions based on the concavity of the microcanonical entropy.

cond-mat.stat-mech↗

arXiv2006

Remarks about the Phase Transitions within the Microcanonical description

According to the reparametrization invariance of the microcanonical ensemble, the only microcanonically relevant phase transitions are those involving an ergodicity breaking in the thermodynamic limit: the zero-order phase transitions and the continuous phase transitions. We suggest that the microcanonically relevant phase transitions are not associated directly with topological changes in the configurational space as the Topological Hypothesis claims, instead, they could be related with topological changes of certain subset A of the configurational space in which the system dynamics is effectively trapped in the thermodynamic limit N→∞.

cond-mat.stat-mech↗

arXiv2006

Thermo-Statistical description of the Hamiltonian non extensive systems: The selfsimilarity scaling laws

The foundations for a thermo-statistical description of the called non extensive Hamiltonian systems are reconsidered. The relevance of the parametric resonance as a fundamental mechanism of the Hamiltonian chaoticity in those systems with bound motions in the configurational space is discussed. The universality of this mechanism suggests us the possibility of performing a thermo-statistical description with microcanonical basis in the context of the long-range interacting Hamiltonian systems. The concept of selfsimilarity is proposed as an appropriate generalization of the well-known extensive conditions exhibited by the traditional systems, which is used to justify a given generalized thermodynamic formalism starting from the consideration of the microcanonical ensemble, i.e. the nonextensive Statistics of Tsallis. These ideas are illustrated by considering a recent proposed astrophysical model based on the quasi-ergodic character of the microscopic dynamics of these paradigmatic examples of real long-range interacting systems.

cond-mat.stat-mech↗

arXiv2006

Thermo-Statistical description of the Hamiltonian non extensive systems: The reparametrization invariance

In the present paper we continue our reconsideration about the foundations for a thermostatistical description of the called Hamiltonian nonextensive systems (see in cond-mat/0604290). After reviewing the selfsimilarity concept and the necessary conditions for the ensemble equivalence, we introduce the reparametrization invariance of the microcanonical description as an internal symmetry associated with the dynamical origin of this ensemble. Possibility of developing a geometrical formulation of the thermodynamic formalism based on this symmetry is discussed, with a consequent revision about the classification of phase-transitions based on the concavity of the Boltzmann entropy. The relevance of such conceptions are analyzed by considering the called Antonov isothermal model.

cond-mat.stat-mech↗

arXiv2006

Matrix orthogonal polynomials whose derivatives are also orthogonal

In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with an hermitian functional. However, we give several examples of non-diagonalizable positive definite weight matrices satisfying a Pearson-type equation, which show that the previous results are non-trivial even in the positive definite case. A detailed analysis is made for the class of matrix functionals which satisfy a Pearson-type equation whose polynomial of degree not greater than 2 is scalar. We characterize the Pearson-type equations of this kind that yield a sequence of matrix orthogonal polynomials, and we prove that these matrix orthogonal polynomials satisfy a second order differential equation even in the non-hermitian case. Finally, we prove and improve a conjecture of Duran and Grunbaum concerning the triviality of this class in the positive definite case, while some examples show the non-triviality for hermitian functionals which are not positive definite.

math.CA↗

arXiv2005

Quasi-stationary evolution of systems driven by particle evaporation

We study the quasi-stationary evolution of systems where an energetic confinement is unable to completely retain their constituents. It is performed an extensive numerical study of a gas whose dynamics is driven by binary encounters and its particles are able to escape from the container when their kinetic energies overcome a given cutou Uc .We use a parametric family of differential cross sections in order to modify the effectiveness of this equilibration mechanism. It is verified that when the binary encounters favor an effective exploration of all accessible velocities, the quasi-stationary evolution is reached when the detailed balance is imposed for all those binary collisions which do not provoke particle evaporation. However, the weakening of this effectiveness leads to energy distribution functions which could be very well fitted by using a Michie-King-like profile. We perform a theoretical analysis, in the context of Hamiltonian systems driven by a strong chaotic dynamics and particle evaporation, in order to take into account the effect of the nonhomogeneous character of the confining potential.

cond-mat.stat-mech↗

arXiv2004

Minimal representations of unitary operators and orthogonal polynomials on the unit circle

In this paper we prove that the simplest band representations of unitary operators on a Hilbert space are five-diagonal. Orthogonal polynomials on the unit circle play an essential role in the development of this result, and also provide a parametrization of such five-diagonal representations which shows specially simple and interesting decomposition and factorization properties. As an application we get the reduction of the spectral problem of any unitary Hessenberg matrix to the spectral problem of a five-diagonal one. Two applications of these results to the study of orthogonal polynomials on the unit circle are presented: the first one concerns Krein's Theorem; the second one deals with the movement of mass points of the orthogonality measure under monoparametric perturbations of the Schur parameters.

math.CA↗

arXiv2003

Remarks about the thermodynamic limit in selfgravitating systems

The present effort addresses the question about the existence of a well-defined thermodynamic limit for the astrophysical systems with the following power law form: to tend the number of particles, N, the total energy, E, and the characteristic linear dimension of the system, L, to infinity, keeping constant E/N^{Λ_{E}} and L/N^{Λ_{L}}, being Λ_{E} and Λ_{L} certain scaling exponent constant. This study is carried out for a system constituted by a non-rotating fluid under the influence of its own Newtonian gravitational interaction. The analysis yields that a thermodynamic limit of the above form will only appear when the local pressure depends on the energy density of fluid as $p=γε$, being $γ$ certain constant. Therefore, a thermodynamic limit with a power law form can be only satisfied by a reduced set of models, such as the selfgravitating gas of fermions and the Antonov isothermal model.

astro-ph↗

arXiv2003

Alternative model of the Antonov problem

Astrophysical systems will never be in a real Thermodynamic equilibrium: they undergo an evaporation process due to the fact that the gravity is not able to confine the particles. Ordinarily, this difficulty is overcome by enclosing the system in a rigid container which avoids the evaporation. We proposed an energetic prescription which is able to confine the particles, leading in this way to an alternative version of the Antonov isothermal model which unifies the well-known isothermal and polytropic profiles. Besides of the main features of the isothermal sphere model: the existence of the gravitational collapse and the energetic region with a negative specific heat, this alternative model has the advantage that the system size naturally appears as a consequence of the particles evaporation.

cond-mat.stat-mech↗