Quantifying population dynamics via a geometric mean predator-prey model.

An integrable Hamiltonian variant of the two species Lotka-Volterra (LV) predator-prey model, shortly referred to as geometric mean (GM) predator-prey model, has been recently introduced. Here, we perform a systematic comparison of the dynamics underlying the GM and LV models. Though the two models share several common features, the geometric mean dynamics exhibits a few peculiarities of interest. The structure of the scaled-population variables reduces to the simple harmonic oscillator with dimensionless natural time TGM varying as ωGMt with ωGM=c12c21. We found that the natural timescales of the evolution dynamics are amplified in the GM model compared to the LV one. Since the GM dynamics is ruled by the inter-species rather than the intra-species coefficients, the proposed model might be of interest when the interactions among the species, rather than the individual demography, rule the evolution of the ecosystems.

Novel predator-prey model admitting exact analytical solution.

The Lotka-Volterra predator-prey model still represents the paradigm for the description of the competition in population dynamics. Despite its extreme simplicity, it does not admit an analytical solution, and for this reason, numerical integration methods are usually adopted to apply it to various fields of science. The aim of the present work is to investigate the existence of new predator-prey models sharing the broad features of the standard Lotka-Volterra model and, at the same time, offer the advantage of possessing exact analytical solutions. To this purpose, a general Hamiltonian formalism, which is suitable for treating a large class of predator-prey models in population dynamics within the same framework, has been developed as a first step. The only existing model having the property of admitting a simple exact analytical solution, is identified within the above class of models. The solution of this special predator-prey model is obtained explicitly, in terms of known elementary functions, and its main properties are studied. Finally, the generalization of this model, based on the concept of power-law competition, as well as its extension to the case of N-component competition systems, are considered.

κ-statistics approach to optimal transport waveform inversion.

Extracting physical parameters that cannot be directly measured from an observed data set remains a great challenge in several fields of science and physics. In many of these problems, the construction of a physical model from waveforms is hampered by the phase ambiguity of the recorded wave fronts. In this work, we present an approach for mitigating the effect of phase ambiguity in waveform-driven issues. Our proposal combines the optimal transport theory with the κ-statistical thermodynamics approach. We construct an energy function from the most probable state of a system described by a finite-variance κ-Gaussian distribution to introduce an optimal transport metric. We demonstrate that our proposal outperforms the classical frameworks by considering a nonlinear geophysical data-driven problem based on a wave equation numerical solution. The κ-generalized optimal transport metric is easily adapted to various inverse problems, from estimating power-law exponents to machine learning approaches in quantum mechanics.

Robust parameter estimation based on the generalized log-likelihood in the context of Sharma-Taneja-Mittal measure.

The problem of obtaining physical parameters that cannot be directly measured from observed data arises in several scientific fields. In the classic approach, the well-known maximum likelihood estimation associated with a Gaussian distribution is employed to obtain the model parameters of a complex system. Although this approach is quite popular in statistical physics, only a handful of spurious observations (outliers) make this approach ineffective, violating the Gauss-Markov theorem. In this work, starting from the generalized logarithmic function associated to the Sharma-Taneja-Mittal (STM) information measure, we propose an outlier-resistant approach based on the generalized log-likelihood estimation. In particular, our proposal deforms the Gaussian distribution based on a two-parameter generalization of the ordinary logarithmic function. We have tested the effectiveness of our proposal considering a classic geophysical inverse problem with a very noisy data set. The results show that the task of obtaining physical parameters based on the STM measure from noisy data with several outliers outperforms the classic approach, and therefore, our proposal is a useful tool for statistical physics, information theory, and statistical inference problems.

Composition law of κ-entropy for statistically independent systems.

The intriguing and still open question concerning the composition law of κ-entropy S_{κ}(f)=1/2κ∑_{i}(f_{i}^{1-κ}-f_{i}^{1+κ}) with 0<κ<1 and ∑_{i}f_{i}=1 is here reconsidered and solved. It is shown that, for a statistical system described by the probability distribution f={f_{ij}}, made up of two statistically independent subsystems, described through the probability distributions p={p_{i}} and q={q_{j}}, respectively, with f_{ij}=p_{i}q_{j}, the joint entropy S_{κ}(pq) can be obtained starting from the S_{κ}(p) and S_{κ}(q) entropies, and additionally from the entropic functionals S_{κ}(p/e_{κ}) and S_{κ}(q/e_{κ}),e_{κ} being the κ-Napier number. The composition law of the κ-entropy is given in closed form and emerges as a one-parameter generalization of the ordinary additivity law of Boltzmann-Shannon entropy recovered in the κ→0 limit.

Statistical mechanics in the context of special relativity. II.

The special relativity laws emerge as one-parameter (light speed) generalizations of the corresponding laws of classical physics. These generalizations, imposed by the Lorentz transformations, affect both the definition of the various physical observables (e.g., momentum, energy, etc.), as well as the mathematical apparatus of the theory. Here, following the general lines of [Phys. Rev. E 66, 056125 (2002)], we show that the Lorentz transformations impose also a proper one-parameter generalization of the classical Boltzmann-Gibbs-Shannon entropy. The obtained relativistic entropy permits us to construct a coherent and self-consistent relativistic statistical theory, preserving the main features of the ordinary statistical theory, which is recovered in the classical limit. The predicted distribution function is a one-parameter continuous deformation of the classical Maxwell-Boltzmann distribution and has a simple analytic form, showing power law tails in accordance with the experimental evidence. Furthermore, this statistical mechanics can be obtained as the stationary case of a generalized kinetic theory governed by an evolution equation obeying the H theorem and reproducing the Boltzmann equation of the ordinary kinetics in the classical limit.

Two-parameter deformations of logarithm, exponential, and entropy: a consistent framework for generalized statistical mechanics.

A consistent generalization of statistical mechanics is obtained by applying the maximum entropy principle to a trace-form entropy and by requiring that physically motivated mathematical properties are preserved. The emerging differential-functional equation yields a two-parameter class of generalized logarithms, from which entropies and power-law distributions follow: these distributions could be relevant in many anomalous systems. Within the specified range of parameters, these entropies possess positivity, continuity, symmetry, expansibility, decisivity, maximality, concavity, and are Lesche stable. The Boltzmann-Shannon entropy and some one-parameter generalized entropies already known belong to this class. These entropies and their distribution functions are compared, and the corresponding deformed algebras are discussed.

Statistical mechanics in the context of special relativity.

In Ref. [Physica A 296, 405 (2001)], starting from the one parameter deformation of the exponential function exp(kappa)(x)=(sqrt[1+kappa(2)x(2)]+kappax)(1/kappa), a statistical mechanics has been constructed which reduces to the ordinary Boltzmann-Gibbs statistical mechanics as the deformation parameter kappa approaches to zero. The distribution f=exp(kappa)(-beta E+betamu) obtained within this statistical mechanics shows a power law tail and depends on the nonspecified parameter beta, containing all the information about the temperature of the system. On the other hand, the entropic form S(kappa)= integral d(3)p(c(kappa) f(1+kappa)+c(-kappa) f(1-kappa)), which after maximization produces the distribution f and reduces to the standard Boltzmann-Shannon entropy S0 as kappa-->0, contains the coefficient c(kappa) whose expression involves, beside the Boltzmann constant, another nonspecified parameter alpha. In the present effort we show that S(kappa) is the unique existing entropy obtained by a continuous deformation of S0 and preserving unaltered its fundamental properties of concavity, additivity, and extensivity. These properties of S(kappa) permit to determine unequivocally the values of the above mentioned parameters beta and alpha. Subsequently, we explain the origin of the deformation mechanism introduced by kappa and show that this deformation emerges naturally within the Einstein special relativity. Furthermore, we extend the theory in order to treat statistical systems in a time dependent and relativistic context. Then, we show that it is possible to determine in a self consistent scheme within the special relativity the values of the free parameter kappa which results to depend on the light speed c and reduces to zero as c--> infinity recovering in this way the ordinary statistical mechanics and thermodynamics. The statistical mechanics here presented, does not contain free parameters, preserves unaltered the mathematical and epistemological structure of the ordinary statistical mechanics and is suitable to describe a very large class of experimentally observed phenomena in low and high energy physics and in natural, economic, and social sciences. Finally, in order to test the correctness and predictability of the theory, as working example we consider the cosmic rays spectrum, which spans 13 decades in energy and 33 decades in flux, finding a high quality agreement between our predictions and observed data.

Quantum vortices in systems obeying a generalized exclusion principle.

The paper deals with a planar particle system obeying a generalized exclusion principle (EP) and governed, in the mean field approximation, by a nonlinear Schrödinger equation. We show that the EP involves a mathematically simple and physically transparent mechanism, which allows the genesis of quantum vortices in the system. We obtain in a closed form the shape of the vortices and investigate its main physical properties.

New approach to optical analysis of absorbing thin solid films.

A powerful new technique is reported which enables realistic calculation of the optical energy gap of absorbing thin solid films by an analysis of measured transmittance and reflectance spectra in the fundamental absorption region. At the same time a new analytical method allows the thickness of films to be evaluated by measurements of transmittance only.

Microscopic dynamics underlying anomalous diffusion

The time-dependent Tsallis statistical distribution describing anomalous diffusion is usually obtained in the literature as the solution of a nonlinear Fokker-Planck (FP) equation [A.R. Plastino and A. Plastino, Physica A 222, 347 (1995)]. The scope of the present paper is twofold. First, we show that this distribution can be obtained also as a solution of the nonlinear porous media equation. Second, we prove that the time-dependent Tsallis distribution can be obtained also as a solution of a linear FP equation [G. Kaniadakis and P. Quarati, Physica A 237, 229 (1997)] with coefficients depending on the velocity, which describes a generalized Brownian motion. This linear FP equation is shown to arise from a microscopic dynamics governed by a standard Langevin equation in the presence of multiplicative noise.