Destructive effect of fluctuations on the performance of a Brownian gyrator.

The Brownian gyrator (BG) is often called a minimal model of a nano-engine performing a rotational motion, judging solely upon the fact that in non-equilibrium conditions its torque, specific angular momentum  and specific angular velocity  have non-zero mean values. For a time-discretised (with time-step δt) model we calculate here the previously unknown probability density functions (PDFs) of  and . We show that for finite δt, the PDF of  has exponential tails and all moments are therefore well-defined. At the same time, this PDF appears to be effectively broad - the noise-to-signal ratio is generically bigger than unity meaning that  is strongly not self-averaging. Concurrently, the PDF of  exhibits heavy power-law tails and its mean is the only existing moment. The BG is therefore not an engine in the common sense: it does not exhibit regular rotations on each run and its fluctuations are not only a minor nuisance - on contrary, their effect is completely destructive for the performance. Our theoretical predictions are confirmed by numerical simulations and experimental data. We discuss some plausible improvements of the model which may result in a more systematic rotational motion.

Fluctuation Relation for the Dissipative Flux: The Role of Dynamics, Correlations and Heat Baths.

The fluctuation relation stands as a fundamental result in nonequilibrium statistical physics. Its derivation, particularly in the stationary state, places stringent conditions on the physical systems of interest. On the other hand, numerical analyses usually do not directly reveal any specific connection with such physical properties. This study proposes an investigation of such a connection with the fundamental ingredients of the derivation of the fluctuation relation for the dissipation, which includes the decay of correlations, in the case of heat transport in one-dimensional systems. The role of the heat baths in connection with the system's inherent properties is then highlighted. A crucial discovery of our research is that different lattice models obeying the steady-state fluctuation relation may do so through fundamentally different mechanisms, characterizing their intrinsic nature. Systems with normal heat conduction, such as the lattice ϕ4 model, comply with the theorem after surpassing a certain observational time window, irrespective of lattice size. In contrast, systems characterized by anomalous heat conduction, such as Fermi-Pasta-Ulam-Tsingou-ß and harmonic oscillator chains, require extended observation periods for theoretical alignment, particularly as the lattice size increases. In these systems, the heat bath's fluctuations significantly influence the entire lattice, linking the system's fluctuations with those of the bath. Here, the current autocorrelation function allows us to discern the varying conditions under which different systems satisfy with the fluctuation relation. Our findings significantly expand the understanding of the stationary fluctuation relation and its broader implications in the field of nonequilibrium phenomena.

Probability Turns Material: The Boltzmann Equation.

We review, under a modern light, the conditions that render the Boltzmann equation applicable. These are conditions that permit probability to behave like mass, thereby possessing clear and concrete content, whereas generally, this is not the case. Because science and technology are increasingly interested in small systems that violate the conditions of the Boltzmann equation, probability appears to be the only mathematical tool suitable for treating them. Therefore, Boltzmann's teachings remain relevant, and the present analysis provides a critical perspective useful for accurately interpreting the results of current applications of statistical mechanics.

Chaos and multi-layer attractors in asymmetric neural networks coupled with discrete fractional memristor.

This article introduces a novel model of asymmetric neural networks combined with fractional difference memristors, which has both theoretical and practical implications in the rapidly evolving field of computational intelligence. The proposed model includes two types of fractional difference memristor elements: one with hyperbolic tangent memductance and the other with periodic memductance and memristor state described by sine functions. The authenticity of the constructed memristor is confirmed through fingerprint verification. The research extensively investigates the dynamics of a coupled neural network model, analyzing its stability at equilibrium states, studying bifurcation diagrams, and calculating the largest Lyapunov exponents. The results suggest that when incorporating sine memristors, the model demonstrates coexisting state variables depending on the initial conditions, revealing the emergence of multi-layer attractors. The article further demonstrates how the memristor state shifts through numerical simulations with varying memductance values. Notably, the study emphasizes the crucial role of memductance (synaptic weight) in determining the complex dynamical characteristics of neural network systems. To support the analytical results and demonstrate the chaotic response of state variables, the article includes appropriate numerical simulations. These simulations effectively validate the presented findings and provide concrete evidence of the system's chaotic behavior.

Inertial and geometrical effects of self-propelled elliptical Brownian particles.

Active particles that self-propel by transforming energy into mechanical motion represent a growing area of research in mathematics, physics, and chemistry. Here we investigate the dynamics of nonspherical inertial active particles moving in a harmonic potential, introducing geometric parameters which take into account the role of eccentricity for nonspherical particles. A comparison between the overdamped and underdamped models for elliptical particles is performed. The model of overdamped active Brownian motion has been used to describe most of the basic aspects of micrometer-sized particles moving in a liquid ("microswimmers"). We consider active particles by extending the active Brownian motion model to incorporate translation and rotation inertia and account for the role of eccentricity. We show how the overdamped and the underdamped models behave in the same way for small values of activity (Brownian case) if eccentricity is equal to zero, but increasing eccentricity leads the two dynamics to substantially depart from each other-in particular, the action of a torque induced by external forces, induced a marked difference close to the walls of the domain if eccentricity is high. Effects induced by inertia include an inertial delay time of the self-propulsion direction from the particle velocity, and the differences between the overdamped and underdamped systems are particularly evident in the first and second moments of the particle velocities. Comparison with the experimental results of vibrated granular particles shows good agreement and corroborates the notion that self-propelling massive particles moving in gaseous media are dominated by inertial effects.

Exact Response Theory for Time-Dependent and Stochastic Perturbations.

The exact, non perturbative, response theory developed within the field of non-equilibrium molecular dynamics, also known as TTCF (transient time correlation function), applies to quite general dynamical systems. Its key element is called the dissipation function because it represents the power dissipated by external fields acting on the particle system of interest, whose coupling with the environment is given by deterministic thermostats. This theory has been initially developed for time-independent external perturbations, and then it has been extended to time-dependent perturbations. It has also been applied to dynamical systems of different nature, and to oscillator models undergoing phase transitions, which cannot be treated with, e.g., linear response theory. The present work includes time-dependent stochastic perturbations in the theory using the Karhunen-Loève theorem. This leads to three different investigations of a given process. In the first, a single realization of the stochastic coefficients is fixed, and averages are taken only over the initial conditions, as in a deterministic process. In the second, the initial condition is fixed, and averages are taken with respect to the distribution of stochastic coefficients. In the last investigation, one averages over both initial conditions and stochastic coefficients. We conclude by illustrating the applicability of the resulting exact response theory with simple examples.

Transport and nonequilibrium phase transitions in polygonal urn models.

We study the deterministic dynamics of N point particles moving at a constant speed in a 2D table made of two polygonal urns connected by an active rectangular channel, which applies a feedback control on the particles, inverting the horizontal component of their velocities when their number in the channel exceeds a fixed threshold. Such a bounce-back mechanism is non-dissipative: it preserves volumes in phase space. An additional passive channel closes the billiard table forming a circuit in which a stationary current may flow. Under specific constraints on the geometry and on the initial conditions, the large N limit allows nonequilibrium phase transitions between homogeneous and inhomogeneous phases. The role of ergodicity in making a probabilistic theory applicable is discussed for both rational and irrational urns. The theoretical predictions are compared with the numerical simulation results. Connections with the dynamics of feedback-controlled biological systems are highlighted.

Work and Thermal Fluctuations in Crystal Indentation under Deterministic and Stochastic Thermostats: The Role of System-Bath Coupling.

The Jarzynski equality (JE) was originally derived under the deterministic Hamiltonian formalism, and later, it was demonstrated that stochastic Langevin dynamics also lead to the JE. However, the JE has been verified mainly in small, low-dimensional systems described by Langevin dynamics. Although the two theoretical derivations apparently lead to the same expression, we illustrate that they describe fundamentally different experimental conditions. While the Hamiltonian framework assumes that the thermal bath producing the initial canonical equilibrium switches off for the duration of the work process, the Langevin bath effectively acts on the system. Moreover, the former considers an environment with which the system may interact, whereas the latter does not. In this study, we investigate the effect of the bath on the measurable quantity of the JE through molecular dynamics simulations of crystal nanoindentation employing deterministic and stochastic thermostats. Our analysis shows that the distributions of the kinetic energy and the mechanical work produced during the indentation processes are affected by the interaction between the system and the thermostat baths. As a result, the type of thermostatting has also a clear effect on the left-hand side of the JE, which enables the estimation of the free-energy difference characterizing the process.

Spectral fingerprints of nonequilibrium dynamics: The case of a Brownian gyrator.

The same system can exhibit a completely different dynamical behavior when it evolves in equilibrium conditions or when it is driven out-of-equilibrium by, e.g., connecting some of its components to heat baths kept at different temperatures. Here we concentrate on an analytically solvable and experimentally relevant model of such a system-the so-called Brownian gyrator-a two-dimensional nanomachine that performs a systematic, on average, rotation around the origin under nonequilibrium conditions, while no net rotation takes place under equilibrium ones. On this example, we discuss a question whether it is possible to distinguish between two types of a behavior judging not upon the statistical properties of the trajectories of components but rather upon their respective spectral densities. The latter are widely used to characterize diverse dynamical systems and are routinely calculated from the data using standard built-in packages. From such a perspective, we inquire whether the power spectral densities possess some "fingerprint" properties specific to the behavior in nonequilibrium. We show that indeed one can conclusively distinguish between equilibrium and nonequilibrium dynamics by analyzing the cross-correlations between the spectral densities of both components in the short frequency limit, or from the spectral densities of both components evaluated at zero frequency. Our analytical predictions, corroborated by experimental and numerical results, open a new direction for the analysis of a nonequilibrium dynamics.

Greenhouse gas emissions: A rapid submerge of the world.

The investigation of worldwide climate change is a noticeable exploration topic in the field of sciences. Outflow of greenhouse gases in the environment is the main reason behind the worldwide environmental change. Greenhouse gases retain heat from the sun and prompt the earth to become more sultry, resulting in global warming. In this article, a model based technique is proposed to forecast the future climate dynamics globally. Using past data on annual greenhouse gas emissions and per capita greenhouse gas emissions, the fractal curves are generated and a forecast model called the autoregressive integrated moving average model has been employed to anticipate the future scenario in relation to climate change and its impact on sea-level rise. It is necessary to forecast the climate conditions before the situations become acute. Policy measures aimed at lowering CO and other greenhouse gas emissions, or at least slowing down their development, will have a substantial effect on future warming of the earth.

Jarzynski equality on work and free energy: Crystal indentation as a case study.

Mathematical relations concerning particle systems require knowledge of the applicability conditions to become physically relevant and not merely formal. We illustrate this fact through the analysis of the Jarzynski equality (JE), whose derivation for Hamiltonian systems suggests that the equilibrium free-energy variations can be computational or experimentally determined in almost any kind of non-equilibrium processes. This apparent generality is surprising in a mechanical theory. Analytically, we show that the quantity called "work" in the Hamiltonian derivation of the JE is neither a thermodynamic quantity nor mechanical work, except in special circumstances to be singularly assessed. Through molecular dynamics simulations of elastic and plastic deformations induced via nano-indentation of crystalline surfaces that fall within the formal framework of the JE, we illustrate that the JE cannot be verified and that the results of this verification are process dependent.

An exploration of fractal-based prognostic model and comparative analysis for second wave of COVID-19 diffusion.

The coronavirus disease 2019 (COVID-19) pandemic has fatalized 216 countries across the world and has claimed the lives of millions of people globally. Researches are being carried out worldwide by scientists to understand the nature of this catastrophic virus and find a potential vaccine for it. The most possible efforts have been taken to present this paper as a form of contribution to the understanding of this lethal virus in the first and second wave. This paper presents a unique technique for the methodical comparison of disastrous virus dissemination in two waves amid five most infested countries and the death rate of the virus in order to attain a clear view on the behaviour of the spread of the disease. For this study, the data set of the number of deaths per day and the number of infected cases per day of the most affected countries, the USA, Brazil, Russia, India, and the UK, have been considered in the first and second waves. The correlation fractal dimension has been estimated for the prescribed data sets of COVID-19, and the rate of death has been compared based on the correlation fractal dimension estimate curve. The statistical tool, analysis of variance, has also been used to support the performance of the proposed method. Further, the prediction of the daily death rate has been demonstrated through the autoregressive moving average model. In addition, this study also emphasis a feasible reconstruction of the death rate based on the fractal interpolation function. Subsequently, the normal probability plot is portrayed for the original data and the predicted data, derived through the fractal interpolation function to estimate the accuracy of the prediction. Finally, this paper neatly summarized with the comparison and prediction of epidemic curve of the first and second waves of COVID-19 pandemic to visualize the transmission rate in the both times.

Deterministic model of battery, uphill currents, and nonequilibrium phase transitions.

We consider point particles in a table made of two circular cavities connected by two rectangular channels, forming a closed loop under periodic boundary conditions. In the first channel, a bounce-back mechanism acts when the number of particles flowing in one direction exceeds a given threshold T. In that case, the particles invert their horizontal velocity, as if colliding with vertical walls. The second channel is divided in two halves parallel to the first but located in the opposite sides of the cavities. In the second channel, motion is free. We show that, suitably tuning the sizes of cavities of the channels and of T, nonequilibrium phase transitions take place in the N→∞ limit. This induces a stationary current in the circuit, thus modeling a kind of battery, although our model is deterministic, conservative, and time reversal invariant.

Erratum: Fluctuation relations for systems in a constant magnetic field [Phys. Rev. E 102, 030101(R) (2020)].

This corrects the article DOI: 10.1103/PhysRevE.102.030101.

Fluctuation Relations for Dissipative Systems in Constant External Magnetic Field: Theory and Molecular Dynamics Simulations.

We illustrate how, contrary to common belief, transient Fluctuation Relations (FRs) for systems in constant external magnetic field hold without the inversion of the field. Building on previous work providing generalized time-reversal symmetries for systems in parallel external magnetic and electric fields, we observe that the standard proof of these important nonequilibrium properties can be fully reinstated in the presence of net dissipation. This generalizes recent results for the FRs in orthogonal fields-an interesting but less commonly investigated geometry-and enables direct comparison with existing literature. We also present for the first time a numerical demonstration of the validity of the transient FRs with nonzero magnetic field via nonequilibrium molecular dynamics simulations of a realistic model of liquid NaCl.

Dissipation Function: Nonequilibrium Physics and Dynamical Systems.

An exact response theory has recently been developed within the field of Nonequilibrium Molecular Dynamics. Its main ingredient is known as the Dissipation Function, Ω. This quantity determines nonequilbrium properties like thermodynamic potentials do with equilibrium states. In particular, Ω can be used to determine the exact response of particle systems obeying classical mechanical laws, subjected to perturbations of arbitrary size. Under certain conditions, it can also be used to express the response of a single system, in contrast to the standard response theory, which concerns ensembles of identical systems. The dimensions of Ω are those of a rate, hence Ω can be associated with the entropy production rate, provided local thermodynamic equilibrium holds. When this is not the case for a particle system, or generic dynamical systems are considered, Ω can equally be defined, and it yields formal, thermodynamic-like, relations. While such relations may have no physical content, they may still constitute interesting characterizations of the relevant dynamics. Moreover, such a formal approach turns physically relevant, because it allows a deeper analysis of Ω and of response theory than possible in case of fully fledged physical models. Here, we investigate the relation between linear and exact response, pointing out conditions for the validity of the response theory, as well as difficulties and opportunities for the physical interpretation of certain formal results.

Fluctuation relations for systems in a constant magnetic field.

The validity of the fluctuation relations (FRs) for systems in a constant magnetic field is investigated. Recently introduced time-reversal symmetries that hold in the presence of static electric and magnetic fields and of deterministic thermostats are used to prove the transient FRs without invoking, as commonly done, inversion of the magnetic field. Steady-state FRs are also derived, under the t-mixing condition. These results extend the predictive power of important statistical mechanics relations. We illustrate this via the nonlinear response for the cumulants of the dissipation, showing how the alternative FRs enable one to determine analytically null cumulants also for systems in a single magnetic field.

Can India develop herd immunity against COVID-19?

World Health Organization declared the novel coronavirus disease 2019 (COVID-19) outbreak to be a public health crisis of international concern. Further, it provided advice to the global community that countries should place strong measures to detect disease early, isolate and treat cases, trace contacts and promote "social distancing" measures commensurate with the risk. This study analyses the COVID-19 infection data from the top 15 affected countries in which we observed heterogeneous growth patterns of the virus. Hence, this paper applies multifractal formalism on COVID-19 data with the notion that country-specific infection rates follow a power law growth behaviour. According to the estimated generalized fractal dimension curves, the effects of drastic containment measures on the pandemic in India indicate that a significant reduction of the infection rate as its population is concern. Also, comparison results with other countries demonstrate that India has less death rate or more immunity against COVID-19.

Coexistence of Ballistic and Fourier Regimes in the ß Fermi-Pasta-Ulam-Tsingou Lattice.

Commonly, thermal transport properties of one-dimensional systems are found to be anomalous. Here, we perform a numerical and theoretical study of the ß-Fermi-Pasta-Ulam-Tsingou chain, considered a prototypical model for one-dimensional anharmonic crystals, in contact with thermostats at different temperatures. We give evidence that, in steady state conditions, the local wave energy spectrum can be naturally split into modes that are essentially ballistic (noninteracting or scarcely interacting) and kinetic modes (interacting enough to relax to local thermodynamic equilibrium). We show numerically that the well-known divergence of the energy conductivity is related to how the transition region between these two sets of modes shifts in k space with the system size L, due to properties of the collision integral of the system. Moreover, we show that the kinetic modes are responsible for a macroscopic behavior compatible with Fourier's law. Our work sheds light on the long-standing problem of the applicability of standard thermodynamics in one-dimensional nonlinear chains, testbed for understanding the thermal properties of nanotubes and nanowires.

Jump processes with deterministic and stochastic controls.

We consider the dynamics of a one-dimensional system evolving according to a deterministic drift and randomly forced by two types of jump processes, one representing an external, uncontrolled forcing and the other one a control that instantaneously resets the system according to specified protocols (either deterministic or stochastic). We develop a general theory, which includes a different formulation of the master equation using antecedent and posterior jump states, and obtain an analytical solution for steady state. The relevance of the theory is illustrated with reference to stochastic irrigation to assess crop-failure risk, a problem of interest for environmental geophysics.