From a phenomenological and experimental viewpoint, quantum mechanics is remarkably successful in explaining effects and processes in the microscopic world. Recently, however, a growing body of literature has revisited the theory's own formal foundations, specifically addressing issues of self-consistency and completeness. As a major historical example of these foundational challenges, the Einstein-Podolsky-Rosen (EPR) argument famously claimed that the theory is incomplete. While many valid criticisms and refutations of the EPR logic exist, they generally rely on a combination of technical results and conceptual objections to EPR's interpretive assumptions, thus transcending the plain quantum framework itself. Motivated by these trends of examining the theory's structural limits, here we revisit the EPR argument. Focusing on elements essential to the EPR reasoning, we first analyze general quantum correlations for EPR states: (i) the fact that their observables are always associated with non-commuting operators, and (ii) the nature of the correlated information obtained through measurement. From these two points alone, and relying strictly on the core rules of quantum mechanics, we demonstrate how to overturn the alleged EPR incompleteness without invoking any extraneous propositions. Consequently, we show that the standard formalism alone suffices to resolve such type of skepticism regarding the theory's physical reach. This offers, at least in a paradigmatic instance, a powerful indication of a structurally well-founded, self-consistent quantum theory.

Lévy α-stable distributions have important applications in the study of diverse stochastic phenomena, including anomalous diffusion and long-range correlations. Here we revisit the absence of general elementary closed-form expressions for Lévy α-stable distributions with arbitrary parameters. We address this problem from a new angle, based on a Mellin transform approach to the characteristic function. After expressing the Lévy density function as a complex-plane integral, we analyze the pole structure of the integrand in terms of the distribution parameters α and β. In general, the pole structure rules out elementary representations. However, through a remarkable cancellation of poles for specific combinations of α and β, we show that such representations arise only in the known cases of Gaussian, Cauchy-Lorentz, and Lévy-Smirnov densities. To our knowledge, this is the first time that the Mellin transform has been used to systematically investigate the absence of general elementary expressions for Lévy distributions.

Neurons are notably rich in structure and functioning, so rather diverse in their response to stimuli. Consequently, the proper characterization of their dynamical response to external signals is a crucial step in understanding stimulation mechanisms. In particular, kilohertz (kHz) neuronal electrical stimulation tends to induce comportment and drives unseen in (more conventional) lower frequency ranges. Here, we investigate neuronal response of conductance-based models to kHz frequencies stimulation in a broad and often unexplored parameter space region. First, we show that the time evolution exhibited by the paradigmatic Hodgkin-Huxley model under kilohertz stimulation is highly diverse, ranging from regular spiking to chaotic dynamics, as well as displaying regions of complete activity suppression. However, to unveil all these features, a certain level of technical caution is required. For example, we demonstrate that common simplifications of sodium dynamics become inaccurate under these frequency regimes. Also, based on suitable markers, we propose a method for mapping the mentioned behaviors on a stimulation parameter space. Second, by extending the study to models of mammalian central nervous system regions, a comprehensive dynamical atlas is obtained. It provides a rather systematic way to typify the response of rapidly forced conductance-based neurons. Thus, the present findings seems to point to an useful scheme for stimulation-based computational neuroscience research at kilohertz frequencies.

In physics, with the advent of topological materials, and in chemistry, within the scope of chemical graph theory, topological invariants and/or indices have been considered to successfully characterize innumerous systems. In particular, strong links have been identified (both numerically and analytically) between properties of the Ising model on a lattice L and features of the so-called spanning trees (STs) of L . Nontheless, studies exploring this connection tend to address only a handful of cases given the demands of the necessary calculations. But examining only a few instances prevents one from looking for general trends across numerous L ’s, which could eventually reveal universal traits. In this contribution, we present the most comprehensive investigation to date, analyzing the Ising-ST relation for all the L ’s belonging to the families F K of 1 ⩽ K ⩽ 6 –uniform periodic tiling of the plane, in a total of 1248 lattices. With this goal, we develop optimized protocols (taking advantage of a recently proposed Z 4 representation for F K ) to compute for each L its ST constant λ and the Kac–Ward matrix. The determinant of the latter yields the Ising model free energy and consequently the critical temperature T c . Then, considering the relatively large sample generated, we use machine learning techniques, which disclose a general correlation between the Ising critical temperature and the ST constant, described by a simple quadratic polynomial function P . As a benchmark, we test P for some arbitrary lattices (outside F K ), finding rather satisfactory fittings. These results point to a useful classification scheme for Ising T c in 2D, demonstrating that λ can be a relevant topological concept to investigate lattice models. Finally, as a positive ‘side-effect’ of our computations, for these F K ’s we confirm (and even improve) a conjectured inequality associating λ and the effective coordinator number κ of a lattice.

The convergence toward asymptotic states at bifurcation points (BPs) r=rb of 1D mappings of a free parameter r presents scaling laws whose characteristic exponents in principle should depend on the maps non-linear features. Aiming to better understand such comportment, we investigated the logistic-like and sine-like family of maps by studying transcritical, pitchfork, period-doubling, and tangent BPs. For this, we employed the supertracks framework, where continuous functions of r are generated, having the 1D map critical point as the initial condition. Analyzing these functions we obtained, from numerical and analytical procedures, four exponents to describe the asymptotic behavior when r=rb as well as another exponent typifying the case of r>rb. Moreover, we confirmed the universality classes of transcritical and pitchfork BPs proposed in the literature and unveiled novel universality results for period-doubling and tangent BPs. Our findings highlighted the usefulness of the supertracks method, for instance, helping to uncover universality in dynamical systems and allowing to establish parallels with critical phenomena.

We study sums of independent and identically distributed random velocities in special relativity. We show that the resulting one-dimensional velocity distributions are not only stable under relativistic velocity addition, but define a genuinely new class of stochastic processes, namely relativistic Lévy processes. Given a system, this allows identifying distinct relativistic regimes in terms of the distribution's concavity at the origin and the probability of measuring relativistic velocities. These features provide a protocol to assess the relevance of stochastic relativistic effects in actual experiments. As supporting evidence, we find agreement with previous results about heavy-ion diffusion and show that our findings are consistent with the distribution of momentum deviations observed in measurements of antiproton cooling.

In the present contribution, we discuss quantum scattering in 1D periodic finite lattices of N localized potentials by means of an exact Green's function approach. By considering continued fraction techniques, we solve the resulting recurrence relations, thus being able to derive the full structure reflections RN and transmission TN amplitudes in a closed analytic form. The framework allows for dealing with extremely large arrays, in some examples for N up to 1010 cells (or building blocks). For so great N's, in practice the protocol can unveil most of the basic features of the energy band structures of the corresponding infinite systems, demanding relatively little computational effort. We further investigate general scattering properties of distinct lattices, e.g., when their cells are spatially asymmetric or composed by two or more elementary shapes, each shape commonly modeled in the literature in terms of Dirac's delta, rectangular, trapezoidal, and triangular barriers. As concrete applications, we address the problem of parameter optimization of heterostructures used to build solar cells and the identification of some transmission resonance modes, relevant in the study of band-pass transmission in superlattices.

In this contribution, we investigate how to correctly describe sums of independent and identically distributed random velocities in the theory of special relativity. We derive a one-dimensional probability distribution of velocities stable under relativistic velocity addition. In a given system, this allows identifying distinct physical regimes in terms of the distribution's concavity at the origin and the probability of measuring relativistic velocities. These features provide a protocol to assess the relevance of stochastic relativistic effects in actual experiments. As examples, we find agreement with previous results about heavy-ion diffusion and show that our findings are consistent with the distribution of momentum deviations observed in measurements of antiproton cooling.

The numerical hailstone sequences, or orbits, generated by the Collatz map have been disclosed to present relevant features commonly associated with complex systems. It is so despite the extreme simplicity of the arithmetic dynamical system iteration rule. Indeed, for a positive integer n , the Collatz map f reads $f(n) = n/2$ ( $f(n) = 3 n + 1$ ) for n even (odd). Seeking to elucidate this surprising fact, here we unveil distinct characteristics of stochastic-like behavior for collections of Collatz orbits by considering methods commonly employed to temporal series, as cryptography tests, power-spectrum, detrended fluctuation, auto-correlation and entropy measure. Besides confirming previous predictions that the Collatz orbits display some global properties of geometric Brownian motion, our results are likewise able to explain, at least heuristically, the reasons for so. In special, we show by means of comprehensive analysis that our findings cannot be ascribed to standard chaotic evolution. Moreover, we identify novel short- and mid-range correlations in the Collatz orbits. The Collatz map is hence a paradigmatic example of an arithmetic dynamical system which could also be regarded as displaying key characteristics of an arithmetic statistical physics system, explaining its dynamical richness.

Wave confinement, e.g., in waveguides, gives rise to a huge number of distinct phenomena. Among them, amplitude gain is a recurrent and relevant effect in undulatory processes. Using a general purpose protocol to solve wave equations, the boundary wall method, we demonstrate that for relatively simple geometries, namely, a few leaky or opaque obstacles inside a theta wedge waveguide (described by the Helmholtz equation), one can obtain a considerable wave amplification in certain spatially localized regions of the system. The approach relies on an expression for the wedge waveguide exact Green's function in the case of theta = pi /M (M = 1, 2, ...), derived through the method of images allied to group theory concepts. The formula is particularly amenable to numerical calculations, greatly facilitating simulations. As an interesting by-product of the present framework, we are able to obtain the eigenstates of certain closed shapes (billiards) placed within the waveguide, as demonstrated for triangular structures. Finally, we briefly discuss possible concrete realizations for our setups in the context of matter and electromagnetic (for some particular modes and conditions) waves.

A wide class of the nonlinear Langevin equations driven by a generic multiplicative Ornstein–Uhlenbeck noise is investigated, with time–space-dependent drift and diffusion coefficients. Solutions for the probability density function and generalized n-moment (averages of a generic function) are obtained. Also, a generalized second Einstein relation is attained. These generalizations lead to novel associations among certain characteristics of the stochastic systems. Given the broad applicability of these models, the present general theoretical results might be useful in the discussion of different processes.

Bidimensional crystals display unique properties of both fundamental and applied interest, with a good part of these properties being related to the topological aspects of 2D materials. Discrete quantum walks models, commonly used in the area of quantum information, are mathematical constructions in which the underlying network topology plays a fundamental role in determining the systems behavior. Here we present a complete scattering quantum walks approach to study 2D honeycomb lattice problems, the structure of paradigmatic 2D Dirac materials like graphene, germanene and silicene. The framework great flexibility relies on considering two arbitrary 3 x 3 unitary scattering matrices (Gamma) over cap ((+/-)) to describe the local dynamics in the lattice fundamental cell. From a simple analytic choice for (Gamma) over cap ((+/-)), we address important aspects of 2D materials like transport characteristics. We also readily obtain analytic formulas for quantities which are commonly derived from a tight-binding approximation. Most importantly, we derive a rather general equation for the system energy bands based on the determinant of products of (Gamma) over cap ((+/-)). We show that by properly setting these matrices (numerically), we get good agreements between our calculations for the pi and pi* energy bands of the graphene, germanene and siliciene with accurate ab initio methods in the literature. We finally briefly discuss how the (Gamma) over cap ((+/-)) could be computed from first principles, making the present an useful protocol to investigate 2D materials.

An essential action in quantum information processing is the manipulation (control) of a single qubit, ideally a closed two-level system. However, in realistic applications, quantum processes are often under the influence of the external environment, e.g., presenting some degree of dissipation and decoherence. In this paper we address the emerging difficulties in the (tracking) quantum control of a two-level system under the influence of both Markovian and non-Markovian noise. We employ a same framework to treat both situations, a Lindblad-type equation, but considering that for the former (latter) case, the decay rate ⠂ is time independent (dependent). We discuss the conditions leading to a breakdown of the quantum control and eventual ways to overcome the problem, like employing a fast control scheme or controlling the off-diagonal terms of the system density matrix. Surprisingly, for Markovian noise such breakdown time decreases with ⠂ not as an exponential but as a power law. This indicates that the quantum control should be possible for a coupling between the system and the environment stronger than previously expected. Moreover, we find that for non-Markovian noise, the breakdown time is longer when there is backflow, i.e., ⠂(t) can be negative. The present theoretical results point to certain favorable scenarios to operate qubits even in a noisy medium.

There is demand in diverse fields for a reliable method of estimating the entropy associated with correlations. The estimation of a unique entropy directly from the Pearson correlation matrix has remained an open problem for more than half a century. All existing approaches lack generality insofar as they require thresholding choices that arbitrarily remove possibly important information. Here we propose an objective procedure for directly estimating a unique entropy of a general Pearson matrix. We show that upon rescaling the Pearson matrix satisfies all necessary conditions for an analog of the von Neumann entropy to be well defined. No thresholding is required. We demonstrate the method by estimating the entropy from neuroimaging time series of the human brain under the influence of a psychedelic.

The Metropolis algorithm is widely used in Monte Carlo (MC) simulations in diverse areas of science and technology, especially for problems formulated in terms of lattice models. A common situation is the necessity to perform long sequential processing, e.g., when looking for equilibrium states of distinct physical systems. Hence, even marginal increases in efficiency of the algorithm individual steps can lead to significant reductions in absolute execution runtimes. Usual speedup procedures include hardware updates, parallelization (when possible) and sampling methods. Here we follow a different direction in trying to decrease the full execution times of MC approaches: algorithmic optimization. We show that the algorithms can be improved by implementing relatively few and simple changes in their organization and structure. First, we discuss some refinements for the pseudo-random number generator, addressing the broadly employed Mersenne-Twister algorithm (MT19937-64). Second, we develop a protocol to precalculate the Boltzmann factor, thereby avoiding the high cost of repeatedly calls to this exponential function (indeed, a very recurring step in the standard Metropolis method). To benchmark our proposals we choose the Ising model since it is one of the best known and more extensively studied problems in statistical physics. We consider the mentioned optimizations and different computational elements, like compilers and Hamiltonian variables ranges, testing the efficiency to obtain the system solutions. Our results suggest that the present set of improvement schemes—namely; decreasing the processing time for both, to generate a random number and to implement the one-flip Metropolis step; systematically enforcing optimization for the maximum quantity of algorithm structures accessing random numbers in a code; and considerably reducing the amount of required computations of the MC probabilistic actualization term—might constitute a relevant addition to the existing collection of expediting techniques in MC computational routines.

In many instances, the dynamical richness and complexity observed in natural phenomena can be related to stochastic drives influencing their temporal evolution. For example, random noise allied to spatial asymmetries may induce stabilization of otherwise diverging trajectories in dynamical systems. However, to identify how exactly this takes place in actual processes usually is not a simple task. Here we unveil a few trends leading to dynamical stabilization and diversity of behavior by introducing Gaussian white noise to a class of exactly solvable non-linear deterministic models displaying space-dependent drifts. For the resulting nonlinear Langevin equations, the associated Fokker-Planck equations can be solved through the similarity method or the Fourier transform technique. By comparing the cases with and without noise, we discuss the changes in the systems dynamical characteristics. Simple examples of drift and diffusion coefficients are explicitly analyzed and comparisons with some other models in the literature are made. Our study illustrates the rich phenomenology originated from spatially heterogeneous dynamical systems under the influence of white noise.

Extracting reliable information on certain physical properties of materials, such as thermal transport, can be computationally very demanding. Aiming to overcome such difficulties in the particular case of lattice thermal conductivity (LTC) of 2D nanomaterials, we propose a simple, fast, and accurate semi-empirical approach for LTC calculation. The approach is based on parameterized thermochemical equations and Arrhenius-like fitting procedures, thus avoiding molecular dynamics or ab initio protocols, which frequently require computationally expensive simulations. As a proof of concept, we obtain the LTC of some prototypical physical systems, such as graphene (and other 2D carbon allotropes), hexagonal boron nitride (hBN), silicene, germanene, binary, and ternary BNC lattices and two examples of the fullerene network family. Our obtained values are in good agreement with other theoretical and experimental estimations, nonetheless, being derived in a rather straightforward way, at a fraction of the usual computational cost.

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}<P_{L} for lower α, depending on the initial distance to the last target found. We also obtain in 2D a maximum in the entropy S for α∈(0,2], not observed in 1D apart from the trivial α→0 ballistic limit. Improving the understanding of the role of dimensionality in random searches is relevant in diverse contexts, as in the problem of encounter rates in biology and ecology.

The boundary wall method (BWM) is a general purpose protocol to treat boundary value problems for wave equations, specially Helmholtz’s (the case addressed here). Similarly to most approaches, the BWM may be computationally demanding for large borders C , at which the wave function must satisfy specified boundary conditions. Also, despite the fact the BWM is an exact procedure, usually it is not amenable to closed form solutions. The BWM relies on the Green’s function G 0 of the embedding domain V of C . However, in many instances—like for C modeling a billiard—the specific V is not really fundamental and thus one has a certain freedom to choose distinct domains and so G 0 ’s. Here we consider this characteristic of the BWM and show how to obtain some analytical results and solve numerically semi-infinite waveguides by exploring proper Green’s functions. As examples, we discuss rectangular, triangular and trapezoidal structures with both Dirichlet and leaking boundaries as well as scattering states within semi-infinite rectangular waveguides.

We devise a simple heuristic method for obtaining the relaxation time and electrical conductivity dependence on the temperature of carriers in 2D semiconductors. The approach is computationally straightforward. It relies on the BoltzTraP algorithm (from the Boltzmann transport equation), on a direct fitting procedure, and on a proper scaling at a reference temperature. The approach provides a good estimate for the figure of merit ZT, an important characterization of thermoelectricity in materials. We employ our approach to analyze promising 2D systems for thermoelectric applications, namely, nitrogenated holey graphene (NHG), boron-doped NHG, and tungsten disulfide 2D-WS2. In all these cases, our results agree with computationally expensive calculations available in the literature at a fraction of the computing time.