Spatial distribution order parameter prediction of collective system using graph network

In the past few decades, the study of collective motion phase transition process has made great progress. It is also important for the description of the spatial distribution of particles. In this work, we propose a new order parameter φ to quantify the degree of order in the spatial distribution of particles. The results show that the spatial distribution order parameter can effectively describe the transition from a disorderly moving phase to a phase with a coherent motion of the particle distribution and the same conclusion could be obtained for systems with different sizes. Furthermore, we develop a powerful molecular dynamic graph network(MDGNet) model to realize the long-term prediction of the self-propelled collective system solely from the initial particle positions and movement angles. Employing this model, we successfully predict the order parameters of the specified time step. And the model can also be applied to analyze other types of complex systems with local interactions.

A tentative model for the explanation of B?th law using the order parameter of seismicity in natural time

Using the order parameter of seismicity defined in natural time, we suggest a simple model for the expla- nation of Bath law, according to which a mainshock differs in magnitude from its largest aftershock by approximately 1.2 regardless of the mainshock magnitude. In addition, the validity of Bath law is studied in the Global Centroid Moment Tensor catalogue by using two different aftershock definitions. It is found that the mean of this difference, when considering all the pairs mainshock-largest aftershock, does not markedly differ from 1.2 and the corresponding distributions do not depend on the mainshock＇s magnitude threshold in a statistically significant manner. Finally, the analysis of the cumulative distribution functions provides evidence in favour of the proposed model.

A novel approach to topological defects in a vector order parameter system

Based on Duan＇s topological current theory,we propose a novel approach to study the topological properties of topological defects in a two-dimensional complex vector order parameter system.This method shows explicitly the fine topological structure of defects.The branch processes of defects in the vector order parameter system have also been investigated with this method.

Temperature dependence of ratio between dielectric anisotropy and order parameter in fluorinated nematic liquid crystals

Temperature dependence of ratio between dielectric anisotropy and order parameter of fluorinated nematic liquid crystal is investigated by using a semi-empirical molecular orbital package that can accurately calculate an angle between molecular dipole moment and long axis. We optimize the molecular conformations with three semi-empirical Hamiltonians AM1, PM3 and PM5, and then make a comparison between computational results and experimental measurements. It is shown that the results obtained from AM1 method are in good agreement with the measurements. The present study offers an applicable method to predict the dielectric properties of liquid crystal material.

Order parameters and synchronization of FitzHugh-Nagumo small-world networks

This paper numerically investigates the order parameter and synchronisation in the small world connected FitzHugh-Nagumo excitable systems. The simulations show that the order parameter continuously decreases with increasing D, the quality of the synchronisation worsens for large noise intensity. As the coupling intensity goes up, the quality of the synchronisation worsens, and it finds that the larger rewiring probability becomes the larger order parameter. It obtains the complete phase diagram for a wide range of values of noise intensity D and control parameter g.

Polarizabilities and Orientational Order Parameter in N-(p-n-Ethoxybenzylidene)- p-n-Alkoxy Anilines, 2O.Om LC Compounds

n.m, n.Om, nO.m and nO.Om compounds play an important role in the fundamental and applied aspects. As a part of our systematic studies of the above homologous series regarding the synthesis, characterization and phase transition studies, here the authors report the polarizabilities and orientational order parameter, S in N-(p-n-ethoxybenzylidene)-p-n-alkoxy anilines, 2O.Om with m = 3 to 10 liquid crystalline compounds except with m = 5 which has been already published. The rest of the compounds exhibit the nematic LC phase except with m = 10 which shows the smectic-C phase along the nematic phase which is originally not detected by Godzwon et al. But the author’s observations with TM and differential scanning calorimeter have shown the existence on SmC phase in addition to nematic phase. Further, SmC is clarified using the histogram technique whose details are given below and are published elsewhere. The orientstonal order parameter, S is estimated using different methods and compared with the value obtained from birefringence, δn = (ne – no) where no field model is used. It is observed in these compounds case as in other LC compounds Vuks isotropic model is favored compared to that of anisotropic model proposed by Neugebauer.

Study of Molecular Polarizabilities and Orientational Order Parameter in the Nematic Phase of 6.O12O.6 and 7.O12O.7

As a part of our systematic study of the physical properties of the compounds of symmetric liquid crystal dimer homologous series, α,ω-bis-(4-n-alkylaniline benzylidene-4’-oxy) alkanes (m.OnO.m), we report here an optical study of two nematogenic compounds of this series viz., 6.O12O.6 and 7.O12O.7 by carrying out the measurement of extraordinary and ordinary refractive indices using modified spectrometer. The molecular polarizability anisotropies are evaluated using Lippincott δ-function model, the molecular vibration method, Haller’s extrapolation method and scaling factor method. The molecular polrizabilities, αe and αo are calculated using Vuks’ isotropic field model and Neugebauer’s anisotropic field model. The orientational order parameter, S, is estimated by using the molecular polarizability values calculated from refractive index and denstity data as well as polarizability anisotropy values calculated as a function of temperature in the nematic phase of both of the dimers. These values are discussed with reference to the order parameter values obtained directly from the optical birefringence, Δn, data without assuming any internal field. A comparison of the order parameter values from all of these methods was carried out in the light of available literature data.

SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.

Effect of Types and Orders of Electromagnetic Field Finite Element Meshes on Power Communication Harmonic Parameters Calculation Results of Tubular Hydrogenerators

In generator design field,waveform total harmonic distortion(THD)and telephone harmonic factor(THF)are parameters commonly used to measure the impact of generator no-load voltage harmonics on the power communication quality.Tubular hydrogenerators are considered the optimal generator for exploiting low-head,high-flow hydro resources,and they have seen increasingly widespread application in China's power systems recent years.However,owing to the compact and constrained internal space of such generators,their internal magnetic-field harmonics are pronounced.Therefore,accurate calculation of their THD and THF is crucial during the analysis and design stages to ensure the quality of power communication.Especially in the electromagnetic field finite element modeling analysis of such generators,the type and order of the finite element meshes may have a significant impact on the THD and THF calculation results,which warrants in-depth research.To address this,this study takes a real 34 MW large tubular hydrogenerator as an example,and establishes its electromagnetic field finite element model under no-load conditions.Two types of meshes,five mesh densities,and two mesh orders are analyzed to reveal the effect of electromagnetic field finite element mesh types and orders on the calculation results of THD and THF for such generators.

Numerical Simulation and Parameter Estimation of Fractional-Order Dynamic Epidemic Model for COVID-19

The outbreak of COVID-19 in 2019 resulted in numerous infections and deaths. In order to better study the transmission of COVID-19, this article adopts an improved fractional-order SIR model. Firstly, the properties of the model are studied, including the feasible domain and bounded solutions of the system. Secondly, the stability of the system is discussed, among other things. Then, the GMMP method is introduced to obtain numerical solutions for the COVID-19 system and combined with the improved MH-NMSS-PSO parameter estimation method to fit the real data of Delhi, India from April 1, 2020 to June 30, 2020. The results show that the fitting effect is quite ideal. Finally, long-term predictions were made on the number of infections. We accurately estimate that the peak number of infections in Delhi, India, can reach around 2.1 million. This paper also compares the fitting performance of the integer-order COVID-19 model and the fractional-order COVID-19 model using the real data from Delhi. The results indicate that the fractional-order model with different orders, as we proposed, performs the best.

From microscopic theory to macroscopic theory——symmetries and order parameters of rigid molecules

Density functional theory is used to describe the phase behaviors of rigid molecules. The construc- tion of the kernel function is discussed. Excluded-volume potential is calculated for two types of molecules with C2v symmetry. Molecular symmetries lead to the symmetries of the kernel function and the density function, enabling a reduction of configuration space. By approximating the kernel function with a polynomial, the system can be fully characterized by some moments corresponding to the form of the kernel function. The symmetries of the kernel function determine the form of the polynomial, while the coefficients are determined by the tem- perature and molecular parameters. The analysis of the impact of coefficients helps us to choose independent variables in the moments as order parameters. Combining the analysis and some simulation results, we propose a minimal set of order parameters for bent-core molecules.

Optimization of anchorage support parameters for soft rock tunnel based on displacement control theory

In the construction of a soft rock tunnel,it is critical to accurately estimate the pre-stressed anchor support parameters for surrounding rock reinforcement;otherwise,engineering disasters may occur.This paper presents a support parameter selection method that aims to allow deformation as a control objective,which was applied to the tunnel located in Muzailing Highway,Min County,Dingxi City,Gansu Province,China.Through theoretical analysis,we have identified five factors that influence pre-stressing anchorages.The selection of mechanical parameters for the rock mass was carried out using an inverse analysis method.Compared with the measured data,the maximum displacement error of the numerical simulation results was only 0.07 m.The length of anchor cable,circumferential spacing of anchor cable,longitudinal spacing,and pre-stress index are adopted as the input parameters for the support vector machine neural network model based on particle swarm optimization(PSO-LSSVM).Besides,the vault subsidence and the maximum deformation of surrounding rock are considered as output values(performance indices).The goodness of fit between the predicted values and the simulated values exceeds 0.9.Finally,all support parameters within the acceptable deformation range are calculated.The optimal support variables are derived by considering the construction cost and duration.The field application results show that it is feasible to construct the sample database utilizing the numerical simulation approach by taking the displacement as the control target and using the neural network to specify the appropriate support parameters.

Optimizing near-carbon-free nuclear energy systems:advances in reactor operation digital twin through hybrid machine learning algorithms for parameter identification and state estimation

Accurate and efficient online parameter identification and state estimation are crucial for leveraging digital twin simulations to optimize the operation of near-carbon-free nuclear energy systems.In previous studies,we developed a reactor operation digital twin(RODT).However,non-differentiabilities and discontinuities arise when employing machine learning-based surrogate forward models,challenging traditional gradient-based inverse methods and their variants.This study investigated deterministic and metaheuristic algorithms and developed hybrid algorithms to address these issues.An efficient modular RODT software framework that incorporates these methods into its post-evaluation module is presented for comprehensive comparison.The methods were rigorously assessed based on convergence profiles,stability with respect to noise,and computational performance.The numerical results show that the hybrid KNNLHS algorithm excels in real-time online applications,balancing accuracy and efficiency with a prediction error rate of only 1%and processing times of less than 0.1 s.Contrastingly,algorithms such as FSA,DE,and ADE,although slightly slower(approximately 1 s),demonstrated higher accuracy with a 0.3%relative L_2 error,which advances RODT methodologies to harness machine learning and system modeling for improved reactor monitoring,systematic diagnosis of off-normal events,and lifetime management strategies.The developed modular software and novel optimization methods presented offer pathways to realize the full potential of RODT for transforming energy engineering practices.

Parameter Estimation of a Valve-Controlled Cylinder System Model Based on Bench Test and Operating Data Fusion

The accurate estimation of parameters is the premise for establishing a high-fidelity simulation model of a valve-controlled cylinder system.Bench test data are easily obtained,but it is challenging to emulate actual loads in the research on parameter estimation of valve-controlled cylinder system.Despite the actual load information contained in the operating data of the control valve,its acquisition remains challenging.This paper proposes a method that fuses bench test and operating data for parameter estimation to address the aforementioned problems.The proposed method is based on Bayesian theory,and its core is a pool fusion of prior information from bench test and operating data.Firstly,a system model is established,and the parameters in the model are analysed.Secondly,the bench and operating data of the system are collected.Then,the model parameters and weight coefficients are estimated using the data fusion method.Finally,the estimated effects of the data fusion method,Bayesian method,and particle swarm optimisation(PSO)algorithm on system model parameters are compared.The research shows that the weight coefficient represents the contribution of different prior information to the parameter estimation result.The effect of parameter estimation based on the data fusion method is better than that of the Bayesian method and the PSO algorithm.Increasing load complexity leads to a decrease in model accuracy,highlighting the crucial role of the data fusion method in parameter estimation studies.

THEORY OF INTRAMOLECULAR ORIENTATIONAL ORDER OF MESOGENIC UNITS IN MC-LCPS AND ITS APPLICATION

New concepts such as intramolecular orientational order parameter and corresponding model as well as theory were proposed to describe the intramolecular orientation of mesogenic units in the liquid crystalline polymer chains. The relationship between the intramolecular orientational order parameter and the molecular geometrical parameters such as the bond angle, the bond rotational angle and the rotational potential energy of chemical bonds was deduced. A significant even-odd oscillation of the intramolecular orientational order parameter of LCPs with different length of flexible spacer was found and rationally related to even-odd zig-zag manner of transition properties The verification and application of the theory are also discussed. The isotropic transition temperature predicted by the theory is shown to be in favourable agreement with the experiments.

Introducing the nth-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-FASAM-N): II. Illustrative Example

This work highlights the unparalleled efficiency of the “nth-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-FASAM-N) by considering the well-known Nordheim-Fuchs reactor dynamics/safety model. This model describes a short-time self-limiting power excursion in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident. This nonlinear paradigm model is sufficiently complex to model realistically self-limiting power excursions for short times yet admits closed-form exact expressions for the time-dependent neutron flux, temperature distribution and energy released during the transient power burst. The nth-FASAM-N methodology is compared to the extant “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-CASAM-N) showing that: (i) the 1st-FASAM-N and the 1st-CASAM-N methodologies are equally efficient for computing the first-order sensitivities;each methodology requires a single large-scale computation for solving the “First-Level Adjoint Sensitivity System” (1st-LASS);(ii) the 2nd-FASAM-N methodology is considerably more efficient than the 2nd-CASAM-N methodology for computing the second-order sensitivities since the number of feature-functions is much smaller than the number of primary parameters;specifically for the Nordheim-Fuchs model, the 2nd-FASAM-N methodology requires 2 large-scale computations to obtain all of the exact expressions of the 28 distinct second-order response sensitivities with respect to the model parameters while the 2nd-CASAM-N methodology requires 7 large-scale computations for obtaining these 28 second-order sensitivities;(iii) the 3rd-FASAM-N methodology is even more efficient than the 3rd-CASAM-N methodology: only 2 large-scale computations are needed to obtain the exact expressions of the 84 distinct third-order response sensitivities with respect to the Nordheim-Fuchs model’s parameters when applying the 3rd-FASAM-N methodology, while the application of the 3rd-CASAM-N methodology requires at least 22 large-scale computations for computing the same 84 distinct third-order sensitivities. Together, the nth-FASAM-N and the nth-CASAM-N methodologies are the most practical methodologies for computing response sensitivities of any order comprehensively and accurately, overcoming the curse of dimensionality in sensitivity analysis.

Introducing the nth-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-FASAM-N): I. Mathematical Framework

This work presents the “nth-Order Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviated as “nth-FASAM-N”), which will be shown to be the most efficient methodology for computing exact expressions of sensitivities, of any order, of model responses with respect to features of model parameters and, subsequently, with respect to the model’s uncertain parameters, boundaries, and internal interfaces. The unparalleled efficiency and accuracy of the nth-FASAM-N methodology stems from the maximal reduction of the number of adjoint computations (which are considered to be “large-scale” computations) for computing high-order sensitivities. When applying the nth-FASAM-N methodology to compute the second- and higher-order sensitivities, the number of large-scale computations is proportional to the number of “model features” as opposed to being proportional to the number of model parameters (which are considerably more than the number of features).When a model has no “feature” functions of parameters, but only comprises primary parameters, the nth-FASAM-N methodology becomes identical to the extant nth CASAM-N (“nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems”) methodology. Both the nth-FASAM-N and the nth-CASAM-N methodologies are formulated in linearly increasing higher-dimensional Hilbert spaces as opposed to exponentially increasing parameter-dimensional spaces thus overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems. Both the nth-FASAM-N and the nth-CASAM-N are incomparably more efficient and more accurate than any other methods (statistical, finite differences, etc.) for computing exact expressions of response sensitivities of any order with respect to the model’s features and/or primary uncertain parameters, boundaries, and internal interfaces.

Order parameter analysis of synchronization transitions on star networks

The collective behaviors of populations of coupled oscillators have attracted significant attention in recent years. In this paper, an order parameter approach is proposed to study the low-dimensional dynamical mechanism of collective synchronizations, by adopting the star-topology of coupled oscil- lators as a prototype system. The order parameter equation of star-linked phase oscillators can be obtained in terms of the Watanabe--Strogatz transformation, Ott--Antonsen ansatz, and the ensem- ble order parameter approach. Different solutions of the order parameter equation correspond to the diverse collective states, and different bifurcations reveal various transitions among these collective states. The properties of various transitions in the star-network model are revealed by using tools of nonlinear dynamics such as time reversibility analysis and linear stability analysis.

Shape phase transition in the odd Sm nuclei: effective order parameter and odd–even effect

Some binding energy related quantities serving as effective order parameters have been used to analyze the shape phase transition in the odd Sm nuclei. It is found that the signals of phase transition in the odd Sm nuclei are greatly enhanced in contrast to the even Sm nuclei. A further analysis shows that the transitional behaviors related to pairing in the Sm nuclei can be well described by the mean field plus pairing interaction model, with a monotonic decrease in the pairing strength G.

Universal order-parameter and quantum phase transition for two-dimensional q-state quantum Potts model

We investigate quantum phase transitions for q-state quantum Potts models(q=2,3,4)on a square lattice and for the Ising model on a honeycomb lattice by using the infinite projected entangled-pair state algorithm with a simplified updating scheme.We extend the universal order parameter to a two-dimensional lattice system,which allows us to explore quantum phase transitions with symmetry-broken order for any translation-invariant quantum lattice system of the symmetry group G.The universal order parameter is zero in the symmetric phase,and it ranges from zero to unity in the symmetry-broken phase.The ground-state fidelity per lattice site is computed,and a pinch point is identified on the fidelity surface near the critical point.The results offer another example highlighting the connection between(i)critical points for a quantum many-body system undergoing a quantum phase-transition and(ii)pinch points on a fidelity surface.In addition,we discuss three quantum coherence measures:the quantum Jensen–Shannon divergence,the relative entropy of coherence,and the l1norm of coherence,which are singular at the critical point,thereby identifying quantum phase transitions.