Critical Behavior Of System Research Articles

Complete critical bound region of the generalized exponential cosine screened Coulomb potential

The complete critical bound region of the generalized exponential cosine screened Coulomb potential (ECSCP) V(r)=−r−1e−(βcos⁡θ)rcos⁡[(βsin⁡θ)r] is investigated by employing the generalized pseudospectral method. Highly accurate critical screening parameters are determined over the entire parameter space, spanning from the exponential screened Coulomb potential (θ = 0), via the oscillatory ECSCP ( θ=π/4 ), to the strongest oscillatory cosine Coulomb potential (CCP) ( θ=π/2 ). With an efficient extrapolation technique, the critical screening parameter for the CCP is estimated at about βc=2.34 . By analyzing the variation of the ground state energy in the critical bound region, we separate the parameter space into two regions, i.e. the exponential region at small and moderate values of θ where the exponential screening term dominates the critical behavior of system, and the cosine region for θ close to π/4 where the oscillatory cosine term dominates.

Generalized scale behavior and renormalization group for data analysis

Some recent results showed that the renormalization group (RG) can be considered as a promising framework to address open issues in data analysis. In this work, we focus on one of these aspects, closely related to principal component analysis (PCA) for the case of large dimensional data sets with covariance having a nearly continuous spectrum. In this case, the distinction between ‘noise-like’ and ‘non-noise’ modes becomes arbitrary and an open challenge for standard methods. Observing that both RG and PCA search for simplification for systems involving many degrees of freedom, we aim to use the RG argument to clarify the turning point between noise and information modes. The analogy between coarse-graining renormalization and PCA has been investigated in Bradde and Bialek (2017 J. Stat. Phys. 167 462–75), from a perturbative framework, and the implementation with real sets of data by the same authors showed that the procedure may reflect more than a simple formal analogy. In particular, the separation of sampling noise modes may be controlled by a non-Gaussian fixed point, reminiscent of the behaviour of critical systems. In our analysis, we go beyond the perturbative framework using nonperturbative techniques to investigate non-Gaussian fixed points and propose a deeper formalism allowing us to go beyond power-law assumptions for explicit computations.

Marginal Effect of Structural Defects on the Nonequilibrium Critical Behavior of the Two-Dimensional Ising Model

The influence of various initial magnetizations m0 and structural defects on the nonequilibrium critical behavior of the two-dimensional Ising model is numerically simulated by Monte Carlo methods. Based on analysis of the time dependence of magnetization and the two-time dependences of autocorrelation function and dynamic susceptibility, we revealed the influence of logarithmic corrections and the crossover phenomena of percolation behavior on the nonequilibrium characteristics and the critical exponents. Violation of the fluctuation–dissipation theorem is studied, and the limiting fluctuation–dissipation ratio is calculated for the case of high-temperature initial state. The influence of various initial states on the limiting fluctuation–dissipation ratio is investigated. The nonequilibrium critical dynamics of weakly disordered systems with spin concentrations p ≥ 0.9 is shown to belong to the universality class of the nonequilibrium critical behavior of the pure model and to be characterized by the same critical exponents and the same limiting fluctuation–dissipation ratios. The nonequilibrium critical behavior of systems with p ≤ 0.85 demonstrates that the universal characteristics of the nonequilibrium critical behavior depend on the defect concentration and the dynamic scaling is violated, which is related to the influence of the crossover effects of percolation behavior.

Requirements-Driven Test Generation for Autonomous Vehicles With Machine Learning Components

Autonomous vehicles are complex systems that are challenging to test and debug. A requirements-driven approach to the development process can decrease the resources required to design and test these systems, while simultaneously increasing the reliability. We present a testing framework that uses signal temporal logic (STL), which is a precise and unambiguous requirements language. Our framework evaluates test cases against the STL formulae and additionally uses the requirements to automatically identify test cases that fail to satisfy the requirements. One of the key features of our tool is the support for machine learning (ML) components in the system design, such as deep neural networks. The framework allows evaluation of the control algorithms, including the ML components, and it also includes models of CCD camera, lidar, and radar sensors, as well as the vehicle environment. We use multiple methods to generate test cases, including covering arrays, which is an efficient method to search discrete variable spaces. The resulting test cases can be used to debug the controller design by identifying controller behaviors that do not satisfy requirements. The test cases can also enhance the testing phase of development by identifying critical corner cases that correspond to the limits of the system's allowed behaviors. We present STL requirements for an autonomous vehicle system, which capture both component-level and system-level behaviors. Additionally, we present three driving scenarios and demonstrate how our requirements-driven testing framework can be used to identify critical system behaviors, which can be used to support the development process.

Confusion scheme in machine learning detects double phase transitions and quasi-long-range order.

Thanks to the development of machine learning techniques, it has been shown that the supervised learning can be useful to study critical phenomena of various systems. However, the supervised learning cannot be done without labels which require knowledge about critical behavior of the system. To overcome this barrier, unsupervised machine learning methods have been considered and the confusion scheme has been proposed. In this study, we use the confusion scheme of the unsupervised learning and investigate critical behavior of various systems which exhibit single (double) phase transitions with (without) quasi-long-range order. In detail, we choose the two-color Ashkin-Teller model, the XY model, and the eight-state clock model as such systems and snapshots of the spin configurations at various temperatures are collected via Monte Carlo simulations to be used as input data for the unsupervised machine learning. We also put focus on the size dependence of results and validate the availability of the confusion scheme in thermodynamic limit. Our results indicate that the confusion scheme of the unsupervised learning successfully locates the approximate transition points for all models and becomes more accurate as the system size is increased. We also find a characteristic feature of the result which reflects the presence of a quasi-long-range order. We conclude that regardless of the presence of a quasi-long-range order, single and double phase transitions can be detected via the confusion scheme.

Gaussian statistics as an emergent symmetry of the stochastic scalar Burgers equation.

Symmetries play a conspicuous role in the large-scale behavior of critical systems. In equilibrium they allow us to classify asymptotics into different universality classes, and out of equilibrium, they sometimes emerge as collective properties which are not explicit in the "bare" interactions. Here we elucidate the emergence of an up-down symmetry in the asymptotic behavior of the stochastic scalar Burgers equation in one and two dimensions, manifested by the occurrence of Gaussian fluctuations even within the time regime controlled by nonlinearities. This robustness of Gaussian behavior contradicts naive expectations due to the detailed relation-including the lack of up-down symmetry-between the Burgers equation and the Kardar-Parisi-Zhang equation, which paradigmatically displays non-Gaussian fluctuations described by Tracy-Widom distributions. We reach our conclusions via a dynamic renormalization group study of the field statistics, confirmed by direct evaluation of the field probability distribution function from numerical simulations of the dynamical equation.

Instability of Insulators near Quantum Phase Transitions.

Thin films of amorphous indium oxide undergo a magnetic field driven superconducting to insulator quantum phase transition. In the insulating phase, the current-voltage characteristics show large current discontinuities due to overheating of electrons. We show that the onset voltage for the discontinuities vanishes as we approach the quantum critical point. As a result, the insulating phase becomes unstable with respect to any applied voltage making it, at least experimentally, immeasurable. We emphasize that unlike previous reports of the absence of linear response near quantum phase transitions, in our system, the departure from equilibrium is discontinuous. Because the conditions for these discontinuities are satisfied in most insulators at low temperatures, and due to the decay of all characteristic energy scales near quantum phase transitions, we believe that this instability is general and should occur in various systems while approaching their quantum critical point. Accounting for this instability is crucial for determining the critical behavior of systems near the transition.

Refractory period in network models of excitable nodes: self-sustaining stable dynamics, extended scaling region and oscillatory behavior

Networks of excitable nodes have recently attracted much attention particularly in regards to neuronal dynamics, where criticality has been argued to be a fundamental property. Refractory behavior, which limits the excitability of neurons is thought to be an important dynamical property. We therefore consider a simple model of excitable nodes which is known to exhibit a transition to instability at a critical point (λ = 1), and introduce refractory period into its dynamics. We use mean-field analytical calculations as well as numerical simulations to calculate the activity dependent branching ratio that is useful to characterize the behavior of critical systems. We also define avalanches and calculate probability distribution of their size and duration. We find that in the presence of refractory period the dynamics stabilizes while various parameter regimes become accessible. A sub-critical regime with λ < 1.0, a standard critical behavior with exponents close to critical branching process for λ = 1, a regime with 1 < λ < 2 that exhibits an interesting scaling behavior, and an oscillating regime with λ > 2.0. We have therefore shown that refractory behavior leads to a wide range of scaling as well as periodic behavior which are relevant to real neuronal dynamics.

Lambda phage genetic switch as a system with critical behaviour

Critical behaviour pervades scientific disciplines as diverse as geology, economy or sociology. The critical behaviour of cell control systems is an open issue whose role has not yet been fully explored. The control of the expression of lambda phage DNA in the host cell can be classified as a system with critical behaviour. Lambda phage is a virus that infects Escherichia coli. Its core genes maintain one of two states; lysogeny or lysis. Current knowledge of the lambda phage genetic network allows to build a computational model of transcriptional control of the genes involved in the lytic-lysogenic switch and to simulate the temporal changes of their expression. Here, we focused on the computational simulation of these gene expressions to demonstrate critical behaviour of the system.

Critical Behaviors in Contagion Dynamics.

We study the critical behavior of a general contagion model where nodes are either active (e.g., with opinion , or functioning) or inactive (e.g., with opinion , or damaged). The transitions between these two states are determined by (i) spontaneous transitions independent of the neighborhood, (ii) transitions induced by neighboring nodes, and (iii) spontaneous reverse transitions. The resulting dynamics is extremely rich including limit cycles and random phase switching. We derive a unifying mean-field theory. Specifically, we analytically show that the critical behavior of systems whose dynamics is governed by processes (i)–(iii) can only exhibit three distinct regimes: (a) uncorrelated spontaneous transition dynamics, (b) contact process dynamics, and (c) cusp catastrophes. This ends a long-standing debate on the universality classes of complex contagion dynamics in mean field and substantially deepens its mathematical understanding.

Computer simulation of the critical behavior of magnetic systems with competition between the short- and long-range interactions

Critical behavior of Ising ferromagnetic systems with the competition between short- and long-range interactions near the point of a second-order phase transition has been investigated. The effects of long-range interaction lead to the need to calculate the influence of next-nearest spins, which lie inside the sphere of a specific radius. The study was carried out for a power law of decreaseing the exchange integral with distance for a three-dimensional Ising model with a cubic lattice. A comparison with the results of a field-theoretical approach and data of real experiments has been carried out. It has been shown that the critical temperature grows linearly with an increase in the parameters of long-range interaction. Critical exponents have been calculated at different parameters of long-range interaction. It is shown that at small dimensions of a system it is the short-range interaction that exerts the dominant effect on the critical behavior of the system. With an increase in the linear dimension of the system, long-range forces begin to dominate.

P–V criticality and geometrical thermodynamics of black holes with Born–Infeld type nonlinear electrodynamics

In this paper, we take into account the black-hole solutions of Einstein gravity in the presence of logarithmic and exponential forms of nonlinear electrodynamics. At first, we consider the cosmological constant as a dynamical pressure to study the phase transitions and analogy of the black holes with the Van der Waals liquid–gas system in the extended phase space. We make a comparison between linear and nonlinear electrodynamics and show that the lowest critical temperature belongs to Maxwell theory. Also, we make some arguments regarding how power of nonlinearity brings the system to Schwarzschild-like and Reissner–Nordström-like limitations. Next, we study the critical behavior of the system in the context of heat capacity. We show that critical behavior of system is similar to the one in phase diagrams of extended phase space. We also extend the study of phase transition points through geometrical thermodynamics (GTs). We introduce two new thermodynamical metrics for extended phase space and show that divergencies of thermodynamical Ricci scalar (TRS) of the new metrics coincide with phase transition points of the system. Then, we introduce a new method for obtaining critical pressure and horizon radius by considering denominator of the heat capacity.

Аналітичний опис критичної поведінки тривимірного одновісного магнетика в зовнішньому полі з виділенням системи відліку

The critical behavior of systems belonging to the universality class of the three-dimensional Ising model has been studied theoretically. A three-dimensional Ising-like system with exponentially decreasing interaction potential and in the presence of a homogeneous external field was considered in the framework of the collective variables method. A specific feature in the calculation of the partition function and the free energy of a uniaxial magnet consists in singling out a reference system. The role of the latter is played by the molecular-field Hamiltonian. A method to describe the critical behavior with the use of a singled out reference system is developed on the basis of a non-Gaussian (quartic) distribution of order-parameter fluctuations (the p4 model).

Critical behavior of a 3D Ising-like system in the higher non-Gaussian approximation: Inclusion of the critical exponent of the correlation function

A microscopic description of the critical behavior of systems belonging to the universality class of the three-dimensional (3D) Ising model is developed within the collective variables (CV) approach. The higher non-Gaussian approximation (the sextic distribution for the modes of spin-moment density oscillations or the ρ6model) is used. A specific feature of the partition function calculation for an Ising-like system is the inclusion of the correction for the potential averaging. This correction leads to the modified recurrence relations (RR) for the ρ6model and a nonzero critical exponent of the correlation function η. The RR between the coefficients of the effective sextic distributions are written and analyzed. A technique for determining the small critical exponent η is elaborated in the higher non-Gaussian approximation. It is shown that the renormalized critical exponent of the correlation length has a tendency to a reduction in the case when the exponent η is taken into account.

Stability Analysis for Smart Homes Energy Management System with Delay Consideration

There is a tendency to consume electrical energy simultaneously by home appliances (HAs) leading to frequent demand peaks. Nowadays the number of (HAs) in modern residences has significantly increased as compared to the past. If the power demand is not met with the limited available power supply, this could lead to major power blackout. Our smart power management system consists of smart electric sensor (SES) and power provisioning controller (PPC). The PPC and SES are used to control and maintain stability with complete power waveform behavior examination. In this paper, we focus on achieving optimal system model for smart homes to reduce the risks of power blackout by applying two different criteria of power assignment for the HAs. A comparative study has been conducted which show the drastic change in the system behavior by changing the power assignment criteria. We also carry out the stability analysis for the proposed system model of smart homes. The stability analysis verifies that the system stability is dependent on some time delay. Simulation results also help us to observe critical system behaviors such as stability and overshoot of energy waveform.

Critical behaviour of a 3D Ising-like system in the ρ^6 model approximation: Role of the correction for the potential averaging

The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method. The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the \rho^6 model). A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector. The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function \eta and the renormalization of the values of other critical exponents. The contributions from this correction to the recurrence relations for the \rho^6 model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out. The expression for a small critical exponent \eta is obtained in a higher non-Gaussian approximation.

Understanding critical behavior in the framework of the extended equilibrium fluctuationtheorem

Recently, we have derived a fluctuation theorem for systems in thermodynamic equilibrium compatiblewith anomalous response functions, e.g. the existence of states with negative heat capacitiesC < 0. In this work, we show that the present approach of the fluctuation theory introduces newinsights in the understanding of critical phenomena. Specifically, the new theorem predictsthat the influence of the environment can radically affect the critical behaviorof systems, e.g. to provoke a suppression of the divergence of correlation lengthξ and some of its associated phenomena such as spontaneous symmetry breaking. Ouranalysis reveals that while response functions and state equations are intrinsic properties ofa given system, critical behaviors are always relative phenomena, that is, their existencecrucially depends on the underlying environmental influence.

A combined molecular dynamics and Monte Carlo study of the approach towards phaseseparation in colloid–polymer mixtures

A coarse-grained model for colloid–polymer mixtures is investigated whereboth colloids and polymer coils are represented as point-like particlesinteracting with spherically symmetric effective potentials. Colloid–colloidand colloid–polymer interactions are described by Weeks–Chandler–Andersenpotentials, while the polymer–polymer interaction is very soft, of strengthkBT/2 for maximum polymer–polymer overlap. This model can be efficiently simulated both byMonte Carlo and molecular dynamics methods, and its phase diagram closely resemblesthat of the well-known Asakura–Oosawa model. The static and dynamic properties of themodel are presented for systems at critical colloid density, varying the polymer density inthe one-phase region. Applying Lees-Edwards boundary conditions, colloid–polymermixtures exposed to shear deformation are considered, and the resulting anisotropy ofcorrelations is studied. Whereas for the considered shear rate, , radial distribution functions and static structure factors indicate only small structuralchanges under shear, an appropriate projection of these correlation functions onto sphericalharmonics is presented that allows us to directly quantify the structural anisotropies.However, the considered shear rate is probably not high enough to see anisotropies in staticstructure factors at small wavenumbers that have been predicted by Onuki and Kawasaki(1979 Ann. Phys. 121 456) for the critical behavior of systems under shear. Theanomalous dependence of the polymer’s self-diffusion constant on polymer density isreferred to the clustering of the colloid particles when approaching the criticalpoint.

Field theory of bicritical and tetracritical points. III. Relaxational dynamics including conservation of magnetization (model C)

We calculate the relaxational dynamical critical behavior of systems of O(n_{ parallel}) plus sign in circleO(n_{ perpendicular}) symmetry including conservation of magnetization by renormalization group theory within the minimal subtraction scheme in two-loop order. Within the stability region of the Heisenberg fixed point and the biconical fixed point, strong dynamical scaling holds, with the asymptotic dynamical critical exponent z=2varphinu-1 , where varphi is the crossover exponent and nu the exponent of the correlation length. The critical dynamics at n_{ parallel}=1 and n_{ perpendicular}=2 is governed by a small dynamical transient exponent leading to nonuniversal nonasymptotic dynamical behavior. This may be seen, e.g., in the temperature dependence of the magnetic transport coefficients.

Field theory of bicritical and tetracritical points. I. Statics

We calculate the static critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by the renormalization group method within the minimal subtraction scheme in two-loop order. Summation methods lead to fixed points describing multicritical behavior. Their stability border lines in the space of the order parameter components n_|| and n_perpendicular and spatial dimension d are calculated. The essential features obtained already in two-loop order for the interesting case of an antiferromagnet in a magnetic field ( n_|| =1, n_perpendicular =2 ) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points, leading to very small transient exponents. We are also able to calculate the flow of static couplings, which allows us to consider the attraction region. Depending on the nonuniversal background parameters, the existence of different multicritical behavior (bicritical or tetracritical) is possible, including a triple point.