The Kuramoto model is one of the most profound and classical models of coupled phase oscillators.Because of the global couplings between oscillators,its precise critical exponents can be obtained using the mean-field ...The Kuramoto model is one of the most profound and classical models of coupled phase oscillators.Because of the global couplings between oscillators,its precise critical exponents can be obtained using the mean-field approximation(MFA),where the time average of the modulus of the mean-field is defined as the order parameter.Here,we further study the phase fluctuations of oscillators from the mean-field using the eigen microstate theory(EMT),which was recently developed.The synchronization of phase fluctuations is identified by the condensation and criticality of eigen microstates with finite eigenvalues,which follow the finite-size scaling with the same critical exponents as those of the MFA in the critical regime.Then,we obtain the complete critical behaviors of phase oscillators in the Kuramoto model.We anticipate that the critical behaviors of general phase oscillators can be investigated by using the EMT and different critical exponents from those of the MFA will be obtained.展开更多
We propose an eigen microstate approach(EMA)for analyzing quantum phase transitions in quantum many-body systems,introducing a novel framework that does not require prior knowledge of an order parameter.Using the tran...We propose an eigen microstate approach(EMA)for analyzing quantum phase transitions in quantum many-body systems,introducing a novel framework that does not require prior knowledge of an order parameter.Using the transversefield Ising model(TFIM)as a case study,we demonstrate the effectiveness of EMA by identifying key features of the phase transition through the scaling behavior of eigenvalues and the structure of associated eigen microstates.Our results reveal substantial changes in the ground state of the TFIM as it undergoes a phase transition,as reflected in the behavior of specific componentsξ_(i)^((k))within the eigen microstates.This method is expected to be applicable to other quantum systems where predefining an order parameter is challenging.展开更多
Ln this paper, the super-inverse iterative method is proposed to compute the accurate and complete eigen-solutions for anti-plane cracks/notches with multi-materials, arbitrary opening angles and various surface condi...Ln this paper, the super-inverse iterative method is proposed to compute the accurate and complete eigen-solutions for anti-plane cracks/notches with multi-materials, arbitrary opening angles and various surface conditions. Taking the advantage of the knowledge of the variation forms of the eigen-functions, a series of numerical techniques are proposed to simplify the computation and speed up the convergence rare of the inverse iteration. A number of numerical examples are given to demonstrate the excellent accuracy, efficiency and reliability of the proposed approach.展开更多
Emergence refers to the existence or formation of collective behaviors in complex systems.Here,we develop a theoretical framework based on the eigen microstate theory to analyze the emerging phenomena and dynamic evol...Emergence refers to the existence or formation of collective behaviors in complex systems.Here,we develop a theoretical framework based on the eigen microstate theory to analyze the emerging phenomena and dynamic evolution of complex system.In this framework,the statistical ensemble composed of M microstates of a complex system with N agents is defined by the normalized N×M matrix A,whose columns represent microstates and order of row is consist with the time.The ensemble matrix A can be decomposed as■,where r=min(N,M),eigenvalueσIbehaves as the probability amplitude of the eigen microstate U_I so that■and U_I evolves following V_I.In a disorder complex system,there is no dominant eigenvalue and eigen microstate.When a probability amplitudeσIbecomes finite in the thermodynamic limit,there is a condensation of the eigen microstate UIin analogy to the Bose–Einstein condensation of Bose gases.This indicates the emergence of U_I and a phase transition in complex system.Our framework has been applied successfully to equilibrium threedimensional Ising model,climate system and stock markets.We anticipate that our eigen microstate method can be used to study non-equilibrium complex systems with unknown orderparameters,such as phase transitions of collective motion and tipping points in climate systems and ecosystems.展开更多
Eigen characters of the fundamental equations, equilibrium equation of stress and harmony equation of deformation, of the traditional elastic mechanics under geometrical space were testified by means of the concept of...Eigen characters of the fundamental equations, equilibrium equation of stress and harmony equation of deformation, of the traditional elastic mechanics under geometrical space were testified by means of the concept of standard space, and the modal equilibrium equation and the modal harmony equation under mechanical space were obtained. Based on them and the modal Hooke’s law, a new system of the fundamental equation of elastic mechanics is given. The advantages of the theory given here are as following: the form of the fundamental equation is in common for both isotropy and anisotropy, both force method and displacement method, both force boundary and displacement boundary; the number of stress functions is equal to that of the anisotropic subspaces, which avoids the man made mistakes; the solution of stress field or strain field is given in form of the modal superimposition, which makes calculation simplified greatly; no matter how complicated the anisotropy of solids may be, the complete solutions can be obtained.展开更多
Diapycnal mixing plays an important role in the ocean circulation.Internal waves are a kind of bridge relating the diapycnal mixing to external sources of mechanical energy.Difficulty in obtaining eigen solutions of i...Diapycnal mixing plays an important role in the ocean circulation.Internal waves are a kind of bridge relating the diapycnal mixing to external sources of mechanical energy.Difficulty in obtaining eigen solutions of internal waves over curved topography is a limitation for further theoretical study on the generation problem and scattering process.In this study,a kind of transform method is put forward to derive the eigen solutions of internal waves over subcritical topography in twodimensional and linear framework.The transform converts the curved topography in physical space to flat bottom in transform space while the governing equation of internal waves is still hyperbolic if proper transform function is selected.Thus,one can obtain eigen solutions of internal waves in the transform space.Several examples of transform functions,which convert the linear slope,the convex slope,and the concave slope to flat bottom,and the corresponding eigen solutions are illustrated.A method,using a polynomial to approximate the transform function and least squares method to estimate the undetermined coefficients in the polynomial,is introduced to calculate the approximate expression of the transform function for the given subcritical topography.展开更多
The fundamental equations and the corresponding boundary condition of elastic mechanics under mechanical representation are given by using the conception of eigen space and elastic variation principle. It is proved th...The fundamental equations and the corresponding boundary condition of elastic mechanics under mechanical representation are given by using the conception of eigen space and elastic variation principle. It is proved theoretically that the solution of anisotropic elastic mechanics consists of modal ones, which are obtained respectively from the modal equation of the different subspaces. A simple application is also given.展开更多
Anisotropic viscoelastic mechanics is studied under anisotropicsubspace. It is proved that there also exist the eigen properties forviscoelastic medium. The modal Maxwell's equation, modal dynamicalequation (or mo...Anisotropic viscoelastic mechanics is studied under anisotropicsubspace. It is proved that there also exist the eigen properties forviscoelastic medium. The modal Maxwell's equation, modal dynamicalequation (or modal equilibrium equation) and modal compatibilityequation are ob- tained. Based on them, a new theory of anisotropicviscoelastic mechanics is presented. The advan- tages of the theoryare as follows: 1) the equations are all scalar, and independent ofeach other. The number of equations is equal to that of anisotrpicsubspaces, 2) no matter how complicated the anisotropy of solids maybe, the form of the definite equation and the boundary condition arein com- mon and explicit, 3) there is no distinction between theforce method and the displacement method for statics, that is, theequilibrium equation and the compatibility equation areindistinguishable under the mechanical space, 4) each modal equationhas a definite physical meaning, for example, the modal equations oforder one and order two express the value change and sheardeformation respec- tivley for isotropic solids, 5) there also existthe potential functions which are similar to the stress functions ofelastic mechanics for viscoelastic mechanics, but they are notman-made, 6) the final so- lution of stress or strain is given in theform of modal superimposition, which is suitable to the proxi- matecalculation in engineering.展开更多
Using the eigen theory of solid mechanics, the eigen properties of anisotropic viscoelastic bodies with Kelvin-Voigt model were studied, and the generalized Stokes equation of anisotropic viscoelastic dynamics was obt...Using the eigen theory of solid mechanics, the eigen properties of anisotropic viscoelastic bodies with Kelvin-Voigt model were studied, and the generalized Stokes equation of anisotropic viscoelastic dynamics was obtained, which gives the three-dimensional pattern of viscoelastical waves. The laws of viscoelastical waves of different anisotropical bodies were discussed.Several new conclusiones are given.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.12135003,71731002,and 12471141)the Postdoctoral Fellowship Program of CPSF(Grant No.GZC20231179)+1 种基金the China Postdoctoral Science Foundation-Tianjin Joint Support Program(Grant No.2023T001TJ)the Tianjin Education Commission scientific Research Project(Grant No.2023SK070)。
文摘The Kuramoto model is one of the most profound and classical models of coupled phase oscillators.Because of the global couplings between oscillators,its precise critical exponents can be obtained using the mean-field approximation(MFA),where the time average of the modulus of the mean-field is defined as the order parameter.Here,we further study the phase fluctuations of oscillators from the mean-field using the eigen microstate theory(EMT),which was recently developed.The synchronization of phase fluctuations is identified by the condensation and criticality of eigen microstates with finite eigenvalues,which follow the finite-size scaling with the same critical exponents as those of the MFA in the critical regime.Then,we obtain the complete critical behaviors of phase oscillators in the Kuramoto model.We anticipate that the critical behaviors of general phase oscillators can be investigated by using the EMT and different critical exponents from those of the MFA will be obtained.
基金supported by the National Natural Science Foundation of China(Grant Nos.12475033,12135003,12174194,and 12405032)the National Key Research and Development Program of China(Grant No.2023YFE0109000)+1 种基金supported by the Fundamental Research Funds for the Central Universitiessupport from the China Postdoctoral Science Foundation(Grant No.2023M730299).
文摘We propose an eigen microstate approach(EMA)for analyzing quantum phase transitions in quantum many-body systems,introducing a novel framework that does not require prior knowledge of an order parameter.Using the transversefield Ising model(TFIM)as a case study,we demonstrate the effectiveness of EMA by identifying key features of the phase transition through the scaling behavior of eigenvalues and the structure of associated eigen microstates.Our results reveal substantial changes in the ground state of the TFIM as it undergoes a phase transition,as reflected in the behavior of specific componentsξ_(i)^((k))within the eigen microstates.This method is expected to be applicable to other quantum systems where predefining an order parameter is challenging.
基金The project is supported by the Natural Science Foundation of China.
文摘Ln this paper, the super-inverse iterative method is proposed to compute the accurate and complete eigen-solutions for anti-plane cracks/notches with multi-materials, arbitrary opening angles and various surface conditions. Taking the advantage of the knowledge of the variation forms of the eigen-functions, a series of numerical techniques are proposed to simplify the computation and speed up the convergence rare of the inverse iteration. A number of numerical examples are given to demonstrate the excellent accuracy, efficiency and reliability of the proposed approach.
基金supported by the Key Research Program of Frontier Sciences,Chinese Academy of Sciences(Grant No.QYZD-SSW-SYS019)。
文摘Emergence refers to the existence or formation of collective behaviors in complex systems.Here,we develop a theoretical framework based on the eigen microstate theory to analyze the emerging phenomena and dynamic evolution of complex system.In this framework,the statistical ensemble composed of M microstates of a complex system with N agents is defined by the normalized N×M matrix A,whose columns represent microstates and order of row is consist with the time.The ensemble matrix A can be decomposed as■,where r=min(N,M),eigenvalueσIbehaves as the probability amplitude of the eigen microstate U_I so that■and U_I evolves following V_I.In a disorder complex system,there is no dominant eigenvalue and eigen microstate.When a probability amplitudeσIbecomes finite in the thermodynamic limit,there is a condensation of the eigen microstate UIin analogy to the Bose–Einstein condensation of Bose gases.This indicates the emergence of U_I and a phase transition in complex system.Our framework has been applied successfully to equilibrium threedimensional Ising model,climate system and stock markets.We anticipate that our eigen microstate method can be used to study non-equilibrium complex systems with unknown orderparameters,such as phase transitions of collective motion and tipping points in climate systems and ecosystems.
文摘Eigen characters of the fundamental equations, equilibrium equation of stress and harmony equation of deformation, of the traditional elastic mechanics under geometrical space were testified by means of the concept of standard space, and the modal equilibrium equation and the modal harmony equation under mechanical space were obtained. Based on them and the modal Hooke’s law, a new system of the fundamental equation of elastic mechanics is given. The advantages of the theory given here are as following: the form of the fundamental equation is in common for both isotropy and anisotropy, both force method and displacement method, both force boundary and displacement boundary; the number of stress functions is equal to that of the anisotropic subspaces, which avoids the man made mistakes; the solution of stress field or strain field is given in form of the modal superimposition, which makes calculation simplified greatly; no matter how complicated the anisotropy of solids may be, the complete solutions can be obtained.
基金the National Nature Science Foundation of China under contract No. 40876015the National High Technology Research and Development Program of China (863 Program) under contract No. 2008AA09A402
文摘Diapycnal mixing plays an important role in the ocean circulation.Internal waves are a kind of bridge relating the diapycnal mixing to external sources of mechanical energy.Difficulty in obtaining eigen solutions of internal waves over curved topography is a limitation for further theoretical study on the generation problem and scattering process.In this study,a kind of transform method is put forward to derive the eigen solutions of internal waves over subcritical topography in twodimensional and linear framework.The transform converts the curved topography in physical space to flat bottom in transform space while the governing equation of internal waves is still hyperbolic if proper transform function is selected.Thus,one can obtain eigen solutions of internal waves in the transform space.Several examples of transform functions,which convert the linear slope,the convex slope,and the concave slope to flat bottom,and the corresponding eigen solutions are illustrated.A method,using a polynomial to approximate the transform function and least squares method to estimate the undetermined coefficients in the polynomial,is introduced to calculate the approximate expression of the transform function for the given subcritical topography.
文摘The fundamental equations and the corresponding boundary condition of elastic mechanics under mechanical representation are given by using the conception of eigen space and elastic variation principle. It is proved theoretically that the solution of anisotropic elastic mechanics consists of modal ones, which are obtained respectively from the modal equation of the different subspaces. A simple application is also given.
文摘Anisotropic viscoelastic mechanics is studied under anisotropicsubspace. It is proved that there also exist the eigen properties forviscoelastic medium. The modal Maxwell's equation, modal dynamicalequation (or modal equilibrium equation) and modal compatibilityequation are ob- tained. Based on them, a new theory of anisotropicviscoelastic mechanics is presented. The advan- tages of the theoryare as follows: 1) the equations are all scalar, and independent ofeach other. The number of equations is equal to that of anisotrpicsubspaces, 2) no matter how complicated the anisotropy of solids maybe, the form of the definite equation and the boundary condition arein com- mon and explicit, 3) there is no distinction between theforce method and the displacement method for statics, that is, theequilibrium equation and the compatibility equation areindistinguishable under the mechanical space, 4) each modal equationhas a definite physical meaning, for example, the modal equations oforder one and order two express the value change and sheardeformation respec- tivley for isotropic solids, 5) there also existthe potential functions which are similar to the stress functions ofelastic mechanics for viscoelastic mechanics, but they are notman-made, 6) the final so- lution of stress or strain is given in theform of modal superimposition, which is suitable to the proxi- matecalculation in engineering.
文摘Using the eigen theory of solid mechanics, the eigen properties of anisotropic viscoelastic bodies with Kelvin-Voigt model were studied, and the generalized Stokes equation of anisotropic viscoelastic dynamics was obtained, which gives the three-dimensional pattern of viscoelastical waves. The laws of viscoelastical waves of different anisotropical bodies were discussed.Several new conclusiones are given.
