Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits...Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.展开更多
A key strategy for raising students’humanistic aptitude is through music education.Culture is the soul of music and the source of innovation in music education.Under the current social education concept,it is advocat...A key strategy for raising students’humanistic aptitude is through music education.Culture is the soul of music and the source of innovation in music education.Under the current social education concept,it is advocated to carry forward excellent traditional and red culture in teaching,hence this paper further explores the path of integrating red culture into music education.The incorporation of the Taihang spirit,the most representative cultural resource in Shanxi,into music education in colleges and universities will promote the concept of Lide Shuren,further enhance students’training,and enable students to fulfill the dual education requirements.展开更多
In this paper,the path integral solutions for a general n-dimensional stochastic differential equa-tions(SDEs)withα-stable Lévy noise are derived and verified.Firstly,the governing equations for the solutions of...In this paper,the path integral solutions for a general n-dimensional stochastic differential equa-tions(SDEs)withα-stable Lévy noise are derived and verified.Firstly,the governing equations for the solutions of n-dimensional SDEs under the excitation ofα-stable Lévy noise are obtained through the characteristic function of stochastic processes.Then,the short-time transition probability density func-tion of the path integral solution is derived based on the Chapman-Kolmogorov-Smoluchowski(CKS)equation and the characteristic function,and its correctness is demonstrated by proving that it satis-fies the governing equation of the solution of the SDE,which is also called the Fokker-Planck-Kolmogorov equation.Besides,illustrative examples are numerically considered for highlighting the feasibility of the proposed path integral method,and the pertinent Monte Carlo solution is also calculated to show its correctness and effectiveness.展开更多
Path integral technique is discussed using Hamilton Jacobi method. The Hamilton Jacobi function of non-natural Lagrangian is obtained using separation of variables method. This function makes an important role in path...Path integral technique is discussed using Hamilton Jacobi method. The Hamilton Jacobi function of non-natural Lagrangian is obtained using separation of variables method. This function makes an important role in path integral quantization. The path integral is obtained as integration over the canonical phase space coordinates, which contains the generalized coordinate q and the generalized momentum p. One illustrative example is considered to explain the application of our formalism.展开更多
We outline a proposal for an experimental test of Everett’s many-worlds interpretation of quantum mechanics that could potentially verify the existence of a multiverse. This proposal is based on a quantum field theor...We outline a proposal for an experimental test of Everett’s many-worlds interpretation of quantum mechanics that could potentially verify the existence of a multiverse. This proposal is based on a quantum field theory formulation of many-worlds through the path integral formalism and a careful choice of the vacuum state.展开更多
This study focuses on the master of arts education in higher education institutions in Guangxi Zhuang Autonomous Region of China,explores the path of integrating Guangxi Zhuang’s intangible cultural heritage with the...This study focuses on the master of arts education in higher education institutions in Guangxi Zhuang Autonomous Region of China,explores the path of integrating Guangxi Zhuang’s intangible cultural heritage with the teaching of master of arts,and puts forward the teaching mode of“thinking guidance-autonomous judgement-program construction.”A theoretical model of innovative transformation of intangible cultural heritage is also summarized.Through the development of this study,it is expected to further enrich the practical teaching mechanism of master of arts education in Chinese universities and form a master of arts teaching model with strong local cultural characteristics.At the same time,the teaching reform based on the integration of Guangxi Zhuang’s intangible cultural heritage and master of arts education also has strong practical significance for promoting the inheritance and innovation of Chinese intangible cultural heritage,promoting the development of cultural and creative industries,and serving the economic and social development of Guangxi.展开更多
As a sequel to our recent work [1], in which a control framework was developed for large-scale joint swarms of unmanned ground (UGV) and aerial (UAV) vehicles, the present paper proposes cognitive and meta-cognitive s...As a sequel to our recent work [1], in which a control framework was developed for large-scale joint swarms of unmanned ground (UGV) and aerial (UAV) vehicles, the present paper proposes cognitive and meta-cognitive supervisor models for this kind of distributed robotic system. The cognitive supervisor model is a formalization of the recently Nobel-awarded research in brain science on mammalian and human path integration and navigation, performed by the hippocampus. This is formalized here as an adaptive Hamiltonian path integral, and efficiently simulated for implementation on robotic vehicles as a pair of coupled nonlinear Schr?dinger equations. The meta-cognitive supervisor model is a modal logic of actions and plans that hinges on a weak causality relation that specifies when atoms may change their values without specifying that they must change. This relatively simple logic is decidable yet sufficiently expressive to support the level of inference needed in our application. The atoms and action primitives of the logic framework also provide a straight-forward way of connecting the meta-cognitive supervisor with the cognitive supervisor, with other modules, and to the meta-cognitive supervisors of other robotic platforms in the swarm.展开更多
A path-integral representation of central spin system immersed in an antiferromagnetic environment was investigated. To carry out this study, we made use of the discrete-time propagator method associated with a basic ...A path-integral representation of central spin system immersed in an antiferromagnetic environment was investigated. To carry out this study, we made use of the discrete-time propagator method associated with a basic set involving coherent states of Grassmann variables which made it possible to obtain the analytical propagator which is the centerpiece of the study. In this study, we considered that the environment was in the low-temperature and low-excitation limit and was split into 2 subnets that do not interact with each other. The evaluation of our system was made by considering the first neighbor approximation. From the formalism of the path integrals, it is easy to evaluate the partition function and thermodynamic properties followed from an appropriate tracing over Grassmann variables in the imaginary time domain. We show that the energy of the system depends on the number of sites <em>n</em> when <em>β </em><em></em><span></span>→ 0.展开更多
The major difficulty for the Feynman Path Integral Monte Carlo (PIMC) simulations of the quantum systems of particles is the so called “sign problem”, arising due to the fast oscillations of the path integral integr...The major difficulty for the Feynman Path Integral Monte Carlo (PIMC) simulations of the quantum systems of particles is the so called “sign problem”, arising due to the fast oscillations of the path integral integrand depending on the complex-valued action. Our aim is to find universal techniques being able to solve this problem. The new method combines the basic ideas of the Metropolis and Hasting algorithms and is based on the Picard-Lefschetz theory and complex-valued version of Morse theory. The basic idea is to choose the Lefschetz thimbles as manifolds approaching the saddle point of the integrand. On this thimble the imaginary part of the complex-valued action remains constant. As a result the integrand on each thimble does not oscillate, so the “sign problem” disappears and the integral can be calculated much more effectively. The developed approach allows also finding saddle points in the complexified space of path integral integration. Some simple test calculations and comparisons with available analytical results have been carried out.展开更多
We have investigated the effects of compression and quantization on atomic distribution in ice Ic and in its compressed states at 77 K and 10 K, using the path integral molecular dynamics (PIMD) simulations over wide ...We have investigated the effects of compression and quantization on atomic distribution in ice Ic and in its compressed states at 77 K and 10 K, using the path integral molecular dynamics (PIMD) simulations over wide range of volume. It has been found that the high density amorphous ice (HDA) is attained by compression but volume range to retain ice structure is wider at 10 K than 77 K. We have discovered that quantum dispersion of atoms in ice Ic at 10 K induces non-zero probability that hydrogen-bonded H<sub>2</sub>O molecular molecules are oriented nonlinearly in the crystal structure, which was believed to contain exclusively linear orientation of hydrogen-bonded molecular pairs in this ice. It has been found that for HDA there is each non-zero probability of orientational disorder of hydrogen-bonded H<sub>2</sub>O pairs, of such uniform distribution of H atoms as observed in supercritical fluids in general, and of H atoms located at the O-O midpoint. The present PIMD simulations have revealed that these observed anomalous characteristics of atomic distribution in HDA are caused by both quantization of atoms and compression of the system.展开更多
The quantum probability theory of fuzzy event is suggested by using the idea and method of fuzzy mathematics, giving the form of fuzzy event path integral, membership degree amplitude, fuzzy field function, Green func...The quantum probability theory of fuzzy event is suggested by using the idea and method of fuzzy mathematics, giving the form of fuzzy event path integral, membership degree amplitude, fuzzy field function, Green function, physical quantity and fuzzy diagram. This theory reforms quantum mechanics, making the later become its special case. This theory breaks unitarity, gauge invariance, probability conservation and information conservation, making these principles become approximate ones under certain conditions. This new theory, which needs no renormalization and can naturally give meaningful results which are in accordance with the experiments, is the proper theory to describe microscopic high-speed phenomenon, whereas quantum mechanics is only a proper theory to describe microscopic low-speed phenomenon. This theory is not divergent under the condition of there being no renormalization and infinitely many offsetting terms, thereby it can become the theoretical framework required for the quantization of gravity.展开更多
The stochastic response of a multi‐degree‐of‐freedom nonlinear dynamical system is determined based on the recently developed Wiener path integral(WPI)technique.The system can be construed as a representative model...The stochastic response of a multi‐degree‐of‐freedom nonlinear dynamical system is determined based on the recently developed Wiener path integral(WPI)technique.The system can be construed as a representative model of electrostatically coupled arrays of micromechanical oscillators,and relates to an experiment performed by Buks and Roukes.Compared to alternative modeling and solution treatments in the literature,the paper exhibits the following novelties.First,typically adopted linear,or higher‐order polynomial,approximations of the nonlinear electrostatic forces are circumvented.Second,for the first time,stochastic modeling is employed by considering a random excitation component representing the effect of diverse noise sources on the system dynamics.Third,the resulting high‐dimensional,nonlinear system of coupled stochastic differential equations governing the dynamics of the micromechanical array is solved based on the WPI technique for determining the response joint probability density function.Comparisons with pertinent Monte Carlo simulation data demonstrate a quite high degree of accuracy and computational efficiency exhibited by the WPI technique.Further,it is shown that the proposed model can capture,at least in a qualitative manner,the salient aspects of the frequency domain response of the associated experimental setup.展开更多
This paper intends to study the stochastic response and reliability of the roll motion under the action of wind and wave excitation.The roll motion in random beam seas is described by a four-dimensional(4D)Markov dyna...This paper intends to study the stochastic response and reliability of the roll motion under the action of wind and wave excitation.The roll motion in random beam seas is described by a four-dimensional(4D)Markov dynamic system whose probabilistic properties are governed by the Fokker-Planck(FP)equation.The 4D path integration(PI)method,an efficient numerical technique based on the Markov property of the 4D system,is applied in order to solve the high dimensional FP equation and then the stochastic statistics of the roll motion are derived.Based on the obtained response statistics,the reliability evaluation of the ship stability is performed and the effect of wind action is studied.The accuracy of the 4D PI method and the reliability evaluation is assessed by the versatile Monte Carlo simulation(MCS)method.展开更多
In this work,we study the dissipation mechanism and frictional force of a nanometer-sized tip scanning a metal surface via a path integral approach.The metal,with internal degrees of freedom(c,c^(†))and a tip with an ...In this work,we study the dissipation mechanism and frictional force of a nanometer-sized tip scanning a metal surface via a path integral approach.The metal,with internal degrees of freedom(c,c^(†))and a tip with an internal degree of freedom(d,d^(†))couple with one another by means of an exchanged potential,V.Having integrated out all internal degrees of freedom,we obtain the in-out amplitude.Moreover,we calculate the imaginary part of the in-out amplitude and the frictional force.We find the imaginary part of the in-out amplitude to be positive,and correlated to the sliding velocity in most cases.The frictional force is proportional to the sliding velocity for the case where v<0.01.However,for cases where v>0.01,the frictional force demonstrates nonlinear dependence on sliding velocity.展开更多
Applicability of Feynman path integral approach to numerical simulations of quantum dynamics of an electron in real time domain is examined.Coherent quantum dynamics is demonstrated with one dimensional test cases(qua...Applicability of Feynman path integral approach to numerical simulations of quantum dynamics of an electron in real time domain is examined.Coherent quantum dynamics is demonstrated with one dimensional test cases(quantum dot models)and performance of the Trotter kernel as compared with the exact kernels is tested.Also,a novel approach for finding the ground state and other stationary sates is presented.This is based on the incoherent propagation in real time.For both approaches the Monte Carlo grid and sampling are tested and compared with regular grids and sampling.We asses the numerical prerequisites for all of the above.展开更多
We present Path Integral Monte Carlo C code for calculation of quantum mechanical transition amplitudes for 1Dmodels.The SPEEDUP C code is based on the use of higher-order short-time effective actions and implemented ...We present Path Integral Monte Carlo C code for calculation of quantum mechanical transition amplitudes for 1Dmodels.The SPEEDUP C code is based on the use of higher-order short-time effective actions and implemented to themaximal order p=18 in the time of propagation(Monte Carlo time step),which substantially improves the convergence of discretized amplitudes to their exact continuum values.Symbolic derivation of higher-order effective actions is implemented in SPEEDUP Mathematica codes,using the recursive Schrodinger equation approach.In addition to the general 1D quantum theory,developed Mathematica codes are capable of calculating effective actions for specific models,for general 2D and 3D potentials,as well as for a general many-body theory in arbitrary number of spatial dimensions.展开更多
Multidimensional tunneling appears in many problems at nano scale.The high dimensionality of the potential energy surface(e.g.many degrees of freedom)poses a great challenge in both theoretical and numerical descripti...Multidimensional tunneling appears in many problems at nano scale.The high dimensionality of the potential energy surface(e.g.many degrees of freedom)poses a great challenge in both theoretical and numerical description of tunneling.Numerical simulation based on Schrodinger equation is often prohibitively expensive.We propose an accurate,efficient,robust and easy-to-implement numerical method to calculate the ground state tunneling splitting based on imaginary-time path integral(‘instanton’formulation).The method is genuinely multi-dimensional and free from any additional ad hoc assumptions on potential energy surface.It enables us to calculate the effects of all coupling modes on the tunneling degree of freedom without loss.We also review in this paper some theoretical background and survey some recent work from other groups in calculating multidimensional quantum tunneling effects in chemical reactions.展开更多
Feynman’s path integral reformulates the quantum Schrödinger differential equation to be an integral equation.It has been being widely used to compute internuclear quantum-statistical effects on many-body molecu...Feynman’s path integral reformulates the quantum Schrödinger differential equation to be an integral equation.It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems.In this Review,the molecular Schrödinger equation will first be introduced,together with the BornOppenheimer approximation that decouples electronic and internuclear motions.Some effective semiclassical potentials,e.g.,centroid potential,which are all formulated in terms of Feynman’s path integral,will be discussed and compared.These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrödinger’s energy eigenvalues.As a result,path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques.To complement these techniques,we will examine how Kleinert’s variational perturbation(KP)theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential.To enable the powerful KP theory to be practical for many-body molecular systems,we have proposed a new path-integral method:automated integrationfree path-integral(AIF-PI)method.Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method,we have used it to perform ab initio path-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions.The computational procedure of using our AIF-PI method,along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy(AMAZE),are also highlighted in this review.展开更多
文摘Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.
基金Achievements of“Continuing the Red Bloodline and Inheriting the Spirit of Taihang”by Shanxi Province’s characteristic cultural education brand.
文摘A key strategy for raising students’humanistic aptitude is through music education.Culture is the soul of music and the source of innovation in music education.Under the current social education concept,it is advocated to carry forward excellent traditional and red culture in teaching,hence this paper further explores the path of integrating red culture into music education.The incorporation of the Taihang spirit,the most representative cultural resource in Shanxi,into music education in colleges and universities will promote the concept of Lide Shuren,further enhance students’training,and enable students to fulfill the dual education requirements.
基金This work was supported by the Key International(Regional)Joint Research Program of the National Natural Science Foundation of China(No.12120101002).
文摘In this paper,the path integral solutions for a general n-dimensional stochastic differential equa-tions(SDEs)withα-stable Lévy noise are derived and verified.Firstly,the governing equations for the solutions of n-dimensional SDEs under the excitation ofα-stable Lévy noise are obtained through the characteristic function of stochastic processes.Then,the short-time transition probability density func-tion of the path integral solution is derived based on the Chapman-Kolmogorov-Smoluchowski(CKS)equation and the characteristic function,and its correctness is demonstrated by proving that it satis-fies the governing equation of the solution of the SDE,which is also called the Fokker-Planck-Kolmogorov equation.Besides,illustrative examples are numerically considered for highlighting the feasibility of the proposed path integral method,and the pertinent Monte Carlo solution is also calculated to show its correctness and effectiveness.
文摘Path integral technique is discussed using Hamilton Jacobi method. The Hamilton Jacobi function of non-natural Lagrangian is obtained using separation of variables method. This function makes an important role in path integral quantization. The path integral is obtained as integration over the canonical phase space coordinates, which contains the generalized coordinate q and the generalized momentum p. One illustrative example is considered to explain the application of our formalism.
文摘We outline a proposal for an experimental test of Everett’s many-worlds interpretation of quantum mechanics that could potentially verify the existence of a multiverse. This proposal is based on a quantum field theory formulation of many-worlds through the path integral formalism and a careful choice of the vacuum state.
基金2023 Innovation Project of Guangxi Graduate Education“Innovation Transformation·Integration of Industry and Education-Research on the Integration Path of Zhuang Intangible Cultural Heritage and Master of Arts Course Teaching”(Project number:JGY2023052)2023 Special Project of Guangxi 14th Five-Year Plan for Educational Science“Revitalisation of Non-Heritage-Integration of Industry and Education-Research on the Service of Regional Economic Development of Design Professional Innovation and Entrepreneurship Education in Guangxi Colleges and Universities”(Project number:2023ZJY1836)。
文摘This study focuses on the master of arts education in higher education institutions in Guangxi Zhuang Autonomous Region of China,explores the path of integrating Guangxi Zhuang’s intangible cultural heritage with the teaching of master of arts,and puts forward the teaching mode of“thinking guidance-autonomous judgement-program construction.”A theoretical model of innovative transformation of intangible cultural heritage is also summarized.Through the development of this study,it is expected to further enrich the practical teaching mechanism of master of arts education in Chinese universities and form a master of arts teaching model with strong local cultural characteristics.At the same time,the teaching reform based on the integration of Guangxi Zhuang’s intangible cultural heritage and master of arts education also has strong practical significance for promoting the inheritance and innovation of Chinese intangible cultural heritage,promoting the development of cultural and creative industries,and serving the economic and social development of Guangxi.
文摘As a sequel to our recent work [1], in which a control framework was developed for large-scale joint swarms of unmanned ground (UGV) and aerial (UAV) vehicles, the present paper proposes cognitive and meta-cognitive supervisor models for this kind of distributed robotic system. The cognitive supervisor model is a formalization of the recently Nobel-awarded research in brain science on mammalian and human path integration and navigation, performed by the hippocampus. This is formalized here as an adaptive Hamiltonian path integral, and efficiently simulated for implementation on robotic vehicles as a pair of coupled nonlinear Schr?dinger equations. The meta-cognitive supervisor model is a modal logic of actions and plans that hinges on a weak causality relation that specifies when atoms may change their values without specifying that they must change. This relatively simple logic is decidable yet sufficiently expressive to support the level of inference needed in our application. The atoms and action primitives of the logic framework also provide a straight-forward way of connecting the meta-cognitive supervisor with the cognitive supervisor, with other modules, and to the meta-cognitive supervisors of other robotic platforms in the swarm.
文摘A path-integral representation of central spin system immersed in an antiferromagnetic environment was investigated. To carry out this study, we made use of the discrete-time propagator method associated with a basic set involving coherent states of Grassmann variables which made it possible to obtain the analytical propagator which is the centerpiece of the study. In this study, we considered that the environment was in the low-temperature and low-excitation limit and was split into 2 subnets that do not interact with each other. The evaluation of our system was made by considering the first neighbor approximation. From the formalism of the path integrals, it is easy to evaluate the partition function and thermodynamic properties followed from an appropriate tracing over Grassmann variables in the imaginary time domain. We show that the energy of the system depends on the number of sites <em>n</em> when <em>β </em><em></em><span></span>→ 0.
文摘The major difficulty for the Feynman Path Integral Monte Carlo (PIMC) simulations of the quantum systems of particles is the so called “sign problem”, arising due to the fast oscillations of the path integral integrand depending on the complex-valued action. Our aim is to find universal techniques being able to solve this problem. The new method combines the basic ideas of the Metropolis and Hasting algorithms and is based on the Picard-Lefschetz theory and complex-valued version of Morse theory. The basic idea is to choose the Lefschetz thimbles as manifolds approaching the saddle point of the integrand. On this thimble the imaginary part of the complex-valued action remains constant. As a result the integrand on each thimble does not oscillate, so the “sign problem” disappears and the integral can be calculated much more effectively. The developed approach allows also finding saddle points in the complexified space of path integral integration. Some simple test calculations and comparisons with available analytical results have been carried out.
文摘We have investigated the effects of compression and quantization on atomic distribution in ice Ic and in its compressed states at 77 K and 10 K, using the path integral molecular dynamics (PIMD) simulations over wide range of volume. It has been found that the high density amorphous ice (HDA) is attained by compression but volume range to retain ice structure is wider at 10 K than 77 K. We have discovered that quantum dispersion of atoms in ice Ic at 10 K induces non-zero probability that hydrogen-bonded H<sub>2</sub>O molecular molecules are oriented nonlinearly in the crystal structure, which was believed to contain exclusively linear orientation of hydrogen-bonded molecular pairs in this ice. It has been found that for HDA there is each non-zero probability of orientational disorder of hydrogen-bonded H<sub>2</sub>O pairs, of such uniform distribution of H atoms as observed in supercritical fluids in general, and of H atoms located at the O-O midpoint. The present PIMD simulations have revealed that these observed anomalous characteristics of atomic distribution in HDA are caused by both quantization of atoms and compression of the system.
文摘The quantum probability theory of fuzzy event is suggested by using the idea and method of fuzzy mathematics, giving the form of fuzzy event path integral, membership degree amplitude, fuzzy field function, Green function, physical quantity and fuzzy diagram. This theory reforms quantum mechanics, making the later become its special case. This theory breaks unitarity, gauge invariance, probability conservation and information conservation, making these principles become approximate ones under certain conditions. This new theory, which needs no renormalization and can naturally give meaningful results which are in accordance with the experiments, is the proper theory to describe microscopic high-speed phenomenon, whereas quantum mechanics is only a proper theory to describe microscopic low-speed phenomenon. This theory is not divergent under the condition of there being no renormalization and infinitely many offsetting terms, thereby it can become the theoretical framework required for the quantization of gravity.
文摘The stochastic response of a multi‐degree‐of‐freedom nonlinear dynamical system is determined based on the recently developed Wiener path integral(WPI)technique.The system can be construed as a representative model of electrostatically coupled arrays of micromechanical oscillators,and relates to an experiment performed by Buks and Roukes.Compared to alternative modeling and solution treatments in the literature,the paper exhibits the following novelties.First,typically adopted linear,or higher‐order polynomial,approximations of the nonlinear electrostatic forces are circumvented.Second,for the first time,stochastic modeling is employed by considering a random excitation component representing the effect of diverse noise sources on the system dynamics.Third,the resulting high‐dimensional,nonlinear system of coupled stochastic differential equations governing the dynamics of the micromechanical array is solved based on the WPI technique for determining the response joint probability density function.Comparisons with pertinent Monte Carlo simulation data demonstrate a quite high degree of accuracy and computational efficiency exhibited by the WPI technique.Further,it is shown that the proposed model can capture,at least in a qualitative manner,the salient aspects of the frequency domain response of the associated experimental setup.
文摘This paper intends to study the stochastic response and reliability of the roll motion under the action of wind and wave excitation.The roll motion in random beam seas is described by a four-dimensional(4D)Markov dynamic system whose probabilistic properties are governed by the Fokker-Planck(FP)equation.The 4D path integration(PI)method,an efficient numerical technique based on the Markov property of the 4D system,is applied in order to solve the high dimensional FP equation and then the stochastic statistics of the roll motion are derived.Based on the obtained response statistics,the reliability evaluation of the ship stability is performed and the effect of wind action is studied.The accuracy of the 4D PI method and the reliability evaluation is assessed by the versatile Monte Carlo simulation(MCS)method.
文摘In this work,we study the dissipation mechanism and frictional force of a nanometer-sized tip scanning a metal surface via a path integral approach.The metal,with internal degrees of freedom(c,c^(†))and a tip with an internal degree of freedom(d,d^(†))couple with one another by means of an exchanged potential,V.Having integrated out all internal degrees of freedom,we obtain the in-out amplitude.Moreover,we calculate the imaginary part of the in-out amplitude and the frictional force.We find the imaginary part of the in-out amplitude to be positive,and correlated to the sliding velocity in most cases.The frictional force is proportional to the sliding velocity for the case where v<0.01.However,for cases where v>0.01,the frictional force demonstrates nonlinear dependence on sliding velocity.
文摘Applicability of Feynman path integral approach to numerical simulations of quantum dynamics of an electron in real time domain is examined.Coherent quantum dynamics is demonstrated with one dimensional test cases(quantum dot models)and performance of the Trotter kernel as compared with the exact kernels is tested.Also,a novel approach for finding the ground state and other stationary sates is presented.This is based on the incoherent propagation in real time.For both approaches the Monte Carlo grid and sampling are tested and compared with regular grids and sampling.We asses the numerical prerequisites for all of the above.
基金The authors gratefully acknowledge useful discussions with Axel Pelster and Vladimir Slavni´c.This work was supported in part by the Ministry of Education and Science of the Republic of Serbia,under project No.ON171017,and bilateral project NAD-BEC funded jointly with the German Academic Exchange Service(DAAD),and by the European Commission under EU FP7 projects PRACE-1IP,HP-SEE and EGI-InSPIRE.
文摘We present Path Integral Monte Carlo C code for calculation of quantum mechanical transition amplitudes for 1Dmodels.The SPEEDUP C code is based on the use of higher-order short-time effective actions and implemented to themaximal order p=18 in the time of propagation(Monte Carlo time step),which substantially improves the convergence of discretized amplitudes to their exact continuum values.Symbolic derivation of higher-order effective actions is implemented in SPEEDUP Mathematica codes,using the recursive Schrodinger equation approach.In addition to the general 1D quantum theory,developed Mathematica codes are capable of calculating effective actions for specific models,for general 2D and 3D potentials,as well as for a general many-body theory in arbitrary number of spatial dimensions.
文摘Multidimensional tunneling appears in many problems at nano scale.The high dimensionality of the potential energy surface(e.g.many degrees of freedom)poses a great challenge in both theoretical and numerical description of tunneling.Numerical simulation based on Schrodinger equation is often prohibitively expensive.We propose an accurate,efficient,robust and easy-to-implement numerical method to calculate the ground state tunneling splitting based on imaginary-time path integral(‘instanton’formulation).The method is genuinely multi-dimensional and free from any additional ad hoc assumptions on potential energy surface.It enables us to calculate the effects of all coupling modes on the tunneling degree of freedom without loss.We also review in this paper some theoretical background and survey some recent work from other groups in calculating multidimensional quantum tunneling effects in chemical reactions.
基金supported by HK RGC(ECS-209813)NSF of China(NSFC-21303151)+2 种基金HKBU FRG(FRG2/12-13/037)startup funds(38-40-088 and 40-49-495)to K.-Y.WongThe computing resources for our work summarized in this Review were supported in part by Minnesota Supercomputing Institute,and High Performance Cluster Computing Centre and Office of Information Technology at HKBU(sciblade&jiraiya).
文摘Feynman’s path integral reformulates the quantum Schrödinger differential equation to be an integral equation.It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems.In this Review,the molecular Schrödinger equation will first be introduced,together with the BornOppenheimer approximation that decouples electronic and internuclear motions.Some effective semiclassical potentials,e.g.,centroid potential,which are all formulated in terms of Feynman’s path integral,will be discussed and compared.These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrödinger’s energy eigenvalues.As a result,path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques.To complement these techniques,we will examine how Kleinert’s variational perturbation(KP)theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential.To enable the powerful KP theory to be practical for many-body molecular systems,we have proposed a new path-integral method:automated integrationfree path-integral(AIF-PI)method.Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method,we have used it to perform ab initio path-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions.The computational procedure of using our AIF-PI method,along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy(AMAZE),are also highlighted in this review.
