arXiv subjects

A. Plastino

Publications and source records attributed to A. Plastino.

At least 163 records · Page 9Linked to original sources

arXiv2005

Density operators that extremize Tsallis entropy and thermal stability effects

Quite general, analytical (both exact and approximate) forms for discrete probability distributions (PD's) that maximize Tsallis entropy for a fixed variance are here investigated. They apply, for instance, in a wide variety of scenarios in which the system is characterized by a series of discrete eigenstates of the Hamiltonian. Using these discrete PD's as "weights" leads to density operators of a rather general character. The present study allows one to vividly exhibit the effects of non-extensivity. Varying Tsallis' non-extensivity index $q$ one is seen to pass from unstable to stable systems and even to unphysical situations of infinite energy.

cond-mat.stat-mech↗

arXiv2005

The $p-$sphere and the geometric substratum of power law probability distributions

Links between power law probability distributions and marginal distributions of uniform laws on $p$-spheres in $\mathbb{R}^{n}$ show that a mathematical derivation of the Boltzmann-Gibbs distribution necessarily passes through power law ones. Results are also given that link parameters $p$ and $n$ to the value of the non-extensivity parameter $q$ that characterizes these power laws in the context of non-extensive statistics.

cond-mat.stat-mech↗

arXiv2005

Landau-Ginzburg method applied to finite fermion systems: Pairing in Nuclei

Given the spectrum of a Hamiltonian, a methodology is developed which employs the Landau-Ginsburg method for characterizing phase transitions in infinite systems to identify phase transition remnants in finite fermion systems. As a first application of our appproach we discuss pairing in finite nuclei.

nucl-th↗

arXiv2005

Stochastic invertible mappings between power law and Gaussian probability distributions

We construct "stochastic mappings" between power law probability distributions (PD's) and Gaussian ones. To a given vector $N$, Gaussian distributed (respectively $Z$, exponentially distributed), one can associate a vector $X$, "power law distributed", by multiplying $X$ by a random scalar variable $a$, $N= a X$. This mapping is "invertible": one can go via multiplication by another random variable $b$ from $X$ to $N$ (resp. from $X$ to $Z$), i.e., $X=b N$ (resp. $X=b Z$). Note that all the above equalities mean "is distributed as". As an application of this stochastic mapping we revisit the so-called "zero-th law of thermodynamics problem" that bedevils the practitioners of nonextensive thermostatistics.

cond-mat.stat-mech↗

arXiv2005

Equivalence of the four versions of Tsallis statistics

In spite of its undeniable success, there are still open questions regarding Tsallis non-extensive statistical formalism, whose founding stone was laid in 1988 in JSTAT. Some of them are concerned with the so-called normalization problem of just how to evaluate expectation values. The Jaynes MaxEnt approach for deriving statistical mechanics is based on the adoption of (1) a specific entropic functional form S and (2) physically appropriate constraints. The literature on non-extensive thermostatistics has considered, in its historical evolution, four possible choices for the evaluation of expectation values: (i) 1988 Tsallis-original (TO), (ii) Curado-Tsallis (CT), (iii) Tsallis-Mendes- Plastino (TMP), and (iv) the same as (iii), but using centered operators as constraints (OLM). The 1988 was promptly abandoned and replaced, mostly with versions ii) and iii). We will here (a) show that the 1988 is as good as any of the others, (b) demonstrate that the four cases can be easily derived from just one (any) of them, i.e., the probability distribution function in each of these four instances may be evaluated with a unique formula, and (c) numerically analyze some consequences that emerge from these four choices.

cond-mat.stat-mech↗

arXiv2005

Geometric origin of probabilistic distributions in statistical mechanics

A finite dimensional-system whose physics is governed by a Gaussian distribution can be regarded as a subsystem of an infinite dimensional-underlying system described by a uniform distribution on the (infinite dimensional) sphere. In turn, a finite dimensional-system ruled by a power-law probability distribution can be considered as a subsystem of a finite underlying system whose physics is governed by a uniform distribution on a finite dimensional sphere. These two uniform distributions (on finite and infinite dimensional spheres) are the most "natural" distributions since they maximize both Tsallis and Shannon entropies without any extra constraint but bounded support. The main advantage of power-law distributions could thus be assigned to the fact that they correspond to a still more "natural" underlying model, namely a finite dimensional sphere. In many physical situations, the underlying system is so large that the distinction between finite and infinite dimension is difficult to see, so that the Boltzmann-Gibbs distribution acquires a great degree of versatility. Moreover, an explicit value of the power-law exponent can be given in terms of the dimension of the underlying model and a geometric interpretation of the temperature-concept is provided.

cond-mat.stat-mech↗

arXiv2004

The mutual co-implication of thermodynamics' first and second laws

In classical phenomenological thermodynamics the first and second laws can be regarded as independent statements. Statistical mechanics provides a microscopic substratum that explains thermodynamics in probabilistic terms via a microstate probability distribution ${p_i}$. We study here a hitherto unexplored microscopic connection between the two laws. Given an information measure (or entropic form), each of the two laws implies the other through the process $p_i \to p_i+dp_i$.

cond-mat.stat-mech↗

arXiv2004

Wootters' distance revisited: a new distinguishability criterium

The notion of distinguishability between quantum states has shown to be fundamental in the frame of quantum information theory. In this paper we present a new distinguishability criterium by using a information theoretic quantity: the Jensen-Shannon divergence (JSD). This quantity has several interesting properties, both from a conceptual and a formal point of view. Previous to define this distinguishability criterium, we review some of the most frequently used distances defined over quantum mechanics' Hilbert space. In this point our main claim is that the JSD can be taken as a unifying distance between quantum states.

quant-ph↗

arXiv2004

Fisher information, Wehrl entropy, and Landau Diamagnetism

Using information theoretic quantities like the Wehrl entropy and Fisher's information measure we study the thermodynamics of the problem leading to Landau's diamagnetism, namely, a free spinless electron in a uniform magnetic field. It is shown that such a problem can be "translated" into that of the thermal harmonic oscillator. We discover a new Fisher-uncertainty relation, derived via the Cramer-Rao inequality, that involves phase space localization and energy fluctuations.

cond-mat.stat-mech↗

arXiv2004

Thermodynamics' first law: what information theory tells us

Thermodynamics, and in particular its first law, is of fundamental importance to Science, and therefore of great general interest to all physicists. The first law, although undoubtedly true, and believed by everyone to be true because of its many verified consequences, rests on a rather weak experimental foundation as its path independent aspect has never been directly verified, and rests on a somewhat weak foundation apropos the need for invoking the so-called adiabatic theorem (AT) to prove it from first principles. We provide here a more direct and convincing theoretical demonstration, without the AT and some other usually employed axioms.

cond-mat.stat-mech↗

arXiv2004

Jensen-Shannon divergence, Fisher information, and Wootters' hypothesis

We discuss different statistical distances in probability space, with emphasis on the Jensen-Shannon divergence, vis-a-vis {\it metrics} in Hilbert space and their relationship with Fisher's information measure. This study provides further reconfirmation of Wootters' hypothesis concerning the possibility that statistical fluctuations in the outcomes of measurements be regarded (at least partly) as responsible for the Hilbert-space structure of quantum mechanics.

quant-ph↗

arXiv2004

Reciprocity relations between ordinary temperature and the Frieden-Soffer's Fisher-temperature

Frieden and Soffer conjectured some years ago the existence of a ``Fisher temperature" T_F that would play, with regards to Fisher's information measure I, the same role that the ordinary temperature T plays vis-a-vis Shannon's logarithmic measure. Here we exhibit the existence of reciprocity relations between T_F and T and provide an interpretation with reference to the meaning of T_F for the canonical ensemble.

cond-mat.stat-mech↗

arXiv2004

Constructive approximations of the $q=1/2$ maximum entropy distribution from redundant and noisy data

The problem of constructing the $q=1/2$ non-extensive maximum entropy distributions from redundant and noisy data is considered. A strategy is proposed, which evolves through the following steps: i)independent constraints are first pre-selected by recourse to a data-independent technique to be discussed here. ii)the data are a posteriori used to determine the parameters of the distribution by a previously introduced forward approach. iii) A backward approach is proposed for reducing the parameters of such distribution. The previously introduced forward approach is generalised here in order to make it suitable for dealing with very noisy data.

math-ph↗

arXiv2004

Fisher information and Hamilton's canonical equations

We show that the mathematical form of the information measure of Fisher's I for a Gibbs' canonical probability distribution (the most important one in statistical mechanics) incorporates important features of the intrinsic structure of classical mechanics and has a universal form in terms of "forces" and "accelerations", i.e., one that is valid for all Hamiltonian of the form T+V. If the system of differential equations associated to Hamilton's canonical equations of motion is linear, one can easily ascertain that the Fisher information per degree of freedom is proportional to the inverse temperature and to the number of these degrees. This equipartition of I is also seen to hold in a simple example involving a non-linear system of differential equations.

cond-mat.stat-mech↗

arXiv2004

Escort--Husimi distributions, Fisher information and nonextensivity

We evaluate generalized information measures constructed with Husimi distributions and connect them with the Wehrl entropy, on the one hand, and with thermal uncertainty relations, on the other one. The concept of escort distribution plays a central role in such a study. A new interpretation concerning the meaning of the nonextensivity index $q$ is thereby provided. A physical lower bound for $q$ is also established, together with a ``state equation" for $q$ that transforms the escort-Cramer--Rao bound into a thermal uncertainty relation.

cond-mat.stat-mech↗

arXiv2004

Heisenberg-Fisher thermal uncertainty measure

With the help of the coherent states' basis we establish an interesting connection among i) the so-called Wehrl entropy, ii) Fisher's information measure $I$, and iii) the canonical ensemble entropy for the one-dimensional quantum harmonic oscillator (HO). We show that the contribution of the excited HO spectrum to the mean thermal energy is given by $I$, while the pertinent canonical partition function is given by another Fisher measure: the so-called shift invariant one, minus the HO's ground state energy. Our findings should be of interest in view of the fact that it has been shown that the Legendre transform structure of thermodynamics can be replicated without any change if one replaces the Boltzmann-Gibbs-Shannon entropy by Fisher's information measure [{\it Physical Review E} {\bf 60}, 48 (1999)]. New Fisher-related uncertainty relations are also advanced.

cond-mat.stat-mech↗

arXiv2003

Power-Law distributions and Fisher's information measure

We show that thermodynamic uncertainties (TU) it preserve their form in passing from Boltzmann-Gibbs' statistics to Tsallis' one provided that we express these TU in terms of the appropriate variable conjugate to the temperature in a nonextensive context.

cond-mat.stat-mech↗

arXiv2003

q-Thermodynamics: First law for quasi-stationary states

We discuss peculiar aspects of the first law of thermodynamics for systems characterized by the presence of meta-equilibrium quasi-stationary states for which the pertinent phase/configuration spaces is generally inhomogeneous. As a consequence, the naive additivity requirement for thermodynamic quantities ceases to be satisfied.

cond-mat.stat-mech↗