Mahler measures, elliptic curves, and L-functions for the free energy of the Ising model.

This work establishes links between the Ising model and elliptic curves via Mahler measures. First, we reformulate the partition function of the Ising model on the square, triangular, and honeycomb lattices in terms of the Mahler measure of a Laurent polynomial whose variety's projective closure defines an elliptic curve. Next, we obtain hypergeometric formulas for the partition functions on the triangular and honeycomb lattices and review the known series for the square lattice.

Intermediate statistics: Addressing the thermoelectric properties of solids.

We study the thermodynamics of a crystalline solid by applying intermediate statistics obtained by deforming known solid state models using the mathematics of q analogs. We apply the resulting q deformation to both the Einstein and Debye models and study the deformed thermal and electrical conductivities and the deformed Debye specific heat. We find that the q deformation acts in two different ways-but not necessarily as independent mechanisms.

Generalization to d-dimensions of a fermionic path integral for exact enumeration of polygons on hypercubic lattices.

The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectured that the three-dimensional (3D) and higher dimensional problems are not solvable-or, to be more precise, that there are no differentiably finite (D-finite) solutions.

The Statistics of -Statistics.

Almost two decades ago, Ernesto P. Borges and Bruce M. Boghosian embarked on the intricate task of composing a manuscript to honor the profound contributions of Constantino Tsallis to the realm of statistical physics, coupled with a concise exploration of -Statistics.

Reminiscences of Half a Century of Life in the World of Theoretical Physics.

Selma Lagerlöf said that culture is what remains when one has forgotten everything we had learned. Without any warranty, through ongoing research tasks, that I will ever attain this high level of wisdom, I simply share here reminiscences that have played, during my life, an important role in my incursions in science, mainly in theoretical physics. I end by presenting some perspectives for future developments.

Non-additive entropies and statistical mechanics at the edge of chaos: a bridge between natural and social sciences.

The Boltzmann-Gibbs (BG) statistical mechanics constitutes one of the pillars of contemporary theoretical physics. It is constructed upon the other pillars-classical, quantum, relativistic mechanics and Maxwell equations for electromagnetism-and its foundations are grounded on the optimization of the BG (additive) entropic functional [Formula: see text]. Its use in the realm of classical mechanics is legitimate for vast classes of nonlinear dynamical systems under the assumption that the maximal Lyapunov exponent is (currently referred to as ), and its validity has been experimentally verified in countless situations.

Senses along Which the Entropy Is Unique.

The Boltzmann-Gibbs-von Neumann-Shannon entropy SBG=-k∑ipilnpi as well as its continuous and quantum counterparts, constitute the grounding concept on which the BG statistical mechanics is constructed. This magnificent theory has produced, and will most probably keep producing in the future, successes in vast classes of classical and quantum systems. However, recent decades have seen a proliferation of natural, artificial and social complex systems which defy its bases and make it inapplicable.

Medical Applications of Nonadditive Entropies.

The Boltzmann-Gibbs additive entropy SBG=-k∑ipilnpi and associated statistical mechanics were generalized in 1988 into nonadditive entropy Sq=k1-∑ipiqq-1 and nonextensive statistical mechanics, respectively. Since then, a plethora of medical applications have emerged. In the present review, we illustrate them by briefly presenting image and signal processings, tissue radiation responses, and modeling of disease kinetics, such as for the COVID-19 pandemic.

Lévy walkers inside spherical shells with absorbing boundaries: Towards settling the optimal Lévy walk strategy for random searches.

The Lévy flight foraging hypothesis states that organisms must have evolved adaptations to exploit Lévy walk search strategies. Indeed, it is widely accepted that inverse square Lévy walks optimize the search efficiency in foraging with unrestricted revisits (also known as nondestructive foraging). However, a mathematically rigorous demonstration of this for dimensions D≥2 is still lacking.

What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution.

An exact solution of the Ising model on the simple cubic lattice is one of the long-standing open problems in rigorous statistical mechanics. Indeed, it is generally believed that settling it would constitute a methodological breakthrough, fomenting great prospects for further application, similarly to what happened when Lars Onsager solved the two-dimensional model eighty years ago. Hence, there have been many attempts to find analytic expressions for the exact partition function , but all such attempts have failed due to unavoidable conceptual or mathematical obstructions.

Effect of the search space dimensionality for finding close and faraway targets in random searches.

We investigate the dependence on the search space dimension of statistical properties of random searches with Lévy α-stable and power-law distributions of step lengths. We find that the probabilities to return to the last target found (P_{0}) and to encounter faraway targets (P_{L}), as well as the associated Shannon entropy S, behave as a function of α quite differently in one (1D) and two (2D) dimensions, a somewhat surprising result not reported until now. While in 1D one always has P_{0}≥P_{L}, an interesting crossover takes place in 2D that separates the search regimes with P_{0}>P_{L} for higher α and P_{0} View Article and Find Full Text PDF

Spectrum of the tight-binding model on Cayley trees and comparison with Bethe lattices.

There are few exactly solvable lattice models and even fewer solvable quantum lattice models. Here we address the problem of finding the spectrum of the tight-binding model (equivalently, the spectrum of the adjacency matrix) on Cayley trees. Recent approaches to the problem have relied on the similarity between the Cayley tree and the Bethe lattice.

Along the Lines of Nonadditive Entropies: -Prime Numbers and -Zeta Functions.

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function ζ(s)≡∑n=1∞n-s=∏pprime11-p-s, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended ζ(s) to the complex plane and conjectured that all nontrivial zeros are in the R(z)=1/2 axis. The nonadditive entropy Sq=k∑ipilnq(1/pi)(q∈R;S1=SBG≡-k∑ipilnpi, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function lnqz≡z1-q-11-q(ln1z=lnz). It is already known that this function paves the way for the emergence of a -generalized algebra, using -numbers defined as ⟨x⟩q≡elnqx, which recover the number for q=1.

Reply to Pessoa, P.; Arderucio Costa, B. Comment on "Tsallis, C. Black Hole Entropy: A Closer Look. 2020, , 17".

In the present Reply we restrict our focus only onto the main erroneous claims by Pessoa and Costa in their recent Comment ( , , 1110).

Landscape-scaled strategies can outperform Lévy random searches.

Information on the relevant global scales of the search space, even if partial, should conceivably enhance the performance of random searches. Here we show numerically and analytically that the paradigmatic uninformed optimal Lévy searches can be outperformed by informed multiple-scale random searches in one (1D) and two (2D) dimensions, even when the knowledge about the relevant landscape scales is incomplete. We show in the low-density nondestructive regime that the optimal efficiency of biexponential searches that incorporate all key scales of the 1D landscape of size L decays asymptotically as η_{opt}∼1/sqrt[L], overcoming the result η_{opt}∼1/(sqrt[L]lnL) of optimal Lévy searches.

Connecting complex networks to nonadditive entropies.

Boltzmann-Gibbs statistical mechanics applies satisfactorily to a plethora of systems. It fails however for complex systems generically involving nonlocal space-time entanglement. Its generalization based on nonadditive q-entropies adequately handles a wide class of such systems.

Exploring the Neighborhood of -Exponentials.

The -exponential form eqx≡[1+(1-q)x]1/(1-q)(e1x=ex) is obtained by optimizing the nonadditive entropy Sq≡k1-∑ipiqq-1 (with S1=SBG≡-k∑ipilnpi, where BG stands for Boltzmann-Gibbs) under simple constraints, and emerges in wide classes of natural, artificial and social complex systems. However, in experiments, observations and numerical calculations, it rarely appears in its pure mathematical form. It appears instead exhibiting crossovers to, or mixed with, other similar forms.

Black Hole Entropy: A Closer Look.

In many papers in the literature, author(s) express their perplexity concerning the fact that the ( 3 + 1 ) black-hole 'thermodynamical' entropy appears to be proportional to its area and not to its volume, and would therefore seemingly be nonextensive, or, to be more precise, subextensive. To discuss this question on more clear terms, a non-Boltzmannian entropic functional noted S δ was applied [Tsallis and Cirto, Eur. Phys.

Transient dynamics in a nonequilibrium superdiffusive reaction-diffusion process: Nonequilibrium random search as a case study.

Transient regimes, often difficult to characterize, can be fundamental in establishing final steady states features of reaction-diffusion phenomena. This is particularly true in ecological problems. Here, through both numerical simulations and an analytic approximation, we analyze the transient of a nonequilibrium superdiffusive random search when the targets are created at a certain rate and annihilated upon encounters (a key dynamics, e.

Why Lévy α-stable distributions lack general closed-form expressions for arbitrary α.

The ubiquitous Lévy α-stable distributions lack general closed-form expressions in terms of elementary functions-Gaussian and Cauchy cases being notable exceptions. To better understand this 80-year-old conundrum, we study the complex analytic continuation p_{α}(z), z∈C, of the symmetric Lévy α-stable distribution family p_{α}(x), x∈R, parametrized by 0<α≤2. We first extend known but obscure results, and give a new proof that p_{α}(z) is holomorphic on the entire complex plane for 1<α≤2, whereas p_{α}(z) is not even meromorphic on C for 0<α<1.

Nonadditive Entropies and Complex Systems.

An entropic functional is said if it satisfies, for any two probabilistically independent systems and , that S ( A + B ) = S ( A ) + S ( B ) [...

Scaling properties of d-dimensional complex networks.

The area of networks is very interdisciplinary and exhibits many applications in several fields of science. Nevertheless, there are few studies focusing on geographically located d-dimensional networks. In this paper, we study the scaling properties of a wide class of d-dimensional geographically located networks which grow with preferential attachment involving Euclidean distances through r_{ij}^{-α_{A}} (α_{A}≥0).

Characterizing Complex Networks Using Entropy-Degree Diagrams: Unveiling Changes in Functional Brain Connectivity Induced by Ayahuasca.

With the aim of further advancing the understanding of the human brain's functional connectivity, we propose a network metric which we term the . This metric quantifies the Shannon entropy of the distance distribution to a specific node from all other nodes. It allows to characterize the influence exerted on a specific node considering statistics of the overall network structure.