Explanation of Relation between Wave Function and Probability Density Based on Quantum Mechanics in Phase Space

The main problem of quantum mechanics is to elucidate why the probability density is the modulus square of wave function. For the purpose of solving this problem, we explored the possibility of deducing the fundamental equation of quantum mechanics by starting with the probability density. To do so, it is necessary to formulate a new theory of quantum mechanics distinguished from the previous ones. Our investigation shows that it is possible to construct quantum mechanics in phase space as an alternative autonomous formulation and such a possibility enables us to study quantum mechanics by starting with the probability density rather than the wave function. This direction of research is contrary to configuration-space formulation of quantum mechanics starting with the wave function. Our work leads to a full understanding of the wave function as the both mathematically and physically sufficient representation of quantum-mechanical state which supplements information on quantum state given solely by the probability density with phase information on quantum state. The final result of our work is that quantum mechanics in phase space satisfactorily elucidates the relation between the wave function and the probability density by using the consistent procedure starting with the probability density, thus corroborating the ontological interpretation of the wave function and withdrawing a main assumption of quantum mechanics.

Steady-state probability density function in wave turbulence under large volume limit

We investigate the possibility for two-mode probability density function （PDF） to have a non-zero flux steady state solution. We take the large volume limit so that the space of modes becomes continuous. It is shown that in this limit all the steady-state twoor higher-mode PDFs are the product of one-mode PDFs. The flux of this steady-state solution turns out to be zero for any finite mode PDF.

Mathematical Wave Functions and 3D Finite Element Modelling of the Electron and Positron

The wave/particle duality of particles in Physics is well known. Particles have properties that uniquely characterize them from one another, such as mass, charge and spin. Charged particles have associated Electric and Magnetic fields. Also, every moving particle has a De Broglie wavelength determined by its mass and velocity. This paper shows that all of these properties of a particle can be derived from a single wave function equation for that particle. Wave functions for the Electron and the Positron are presented and principles are provided that can be used to calculate the wave functions of all the fundamental particles in Physics. Fundamental particles such as electrons and positrons are considered to be point particles in the Standard Model of Physics and are not considered to have a structure. This paper demonstrates that they do indeed have structure and that this structure extends into the space around the particle’s center (in fact, they have infinite extent), but with rapidly diminishing energy density with the distance from that center. The particles are formed from Electromagnetic standing waves, which are stable solutions to the Schrödinger and Classical wave equations. This stable structure therefore accounts for both the wave and particle nature of these particles. In fact, all of their properties such as mass, spin and electric charge, can be accounted for from this structure. These particle properties appear to originate from a single point at the center of the wave function structure, in the same sort of way that the Shell theorem of gravity causes the gravity of a body to appear to all originate from a central point. This paper represents the first two fully characterized fundamental particles, with a complete description of their structure and properties, built up from the underlying Electromagnetic waves that comprise these and all fundamental particles.

Alternative Approach to Time Evaluation of Schrodinger Wave Functions

Time evaluation of wave functions for any quantum mechanical system/particle is essential nevertheless quantum mechanical counterpart of the time dependant classical wave equation does simply not appear. Epistemologically and ontologically considered time dependant momentum operator is initially defined and an Alternative Time Dependant Schrodinger Wave Equation (ATDSWE) is plainly derived. Consequent equation is primarily solved for the free particles, in a closed system, signifying a good agreement with the outcomes of the ordinary TDSWE. Free particle solution interestingly goes further possibly tracing some signs of new pathways to resolve the mysterious quantum world.

Inconsistency of Probability Density in Quantum Mechanics and Its Solution

Probability density and particle conservation in quantum mechanics are discussed. The probability density has inconsistency with particle conservation in any quantum system. The inconsistency can be avoided by maintaining conservation of particle. The conservation coerces, a system should exist in a linear combinations of some eigenstates except ground state. The point is applied to the three exactly solvable quantum systems i.e. a particle in one dimensional well potential, harmonic oscillator and hydrogen atom.

Rogue waves of the sixth-order nonlinear Schrodinger equation on a periodic background

In this paper,we construct the rogue wave solutions of the sixth-order nonlinear Schrodinger equation on a background of Jacobian elliptic functions dn and cn by means of the nonlinearization of a spectral problem and Darboux transformation approach.The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.

Probability Density Analysis of Nonlinear Random Ship Rolling

Ship rolling in random waves is a complicated nonlinear motion that contributes substantially to ship instability and capsizing.The finite element method(FEM)is employed in this paper to solve the Fokker Planck(FP)equations numerically for homoclinic and heteroclinic ship rolling under random waves described as periodic and Gaussian white noise excitations.The transient joint probability density functions(PDFs)and marginal PDFs of the rolling responses are also obtained.The effects of stimulation strength on ship rolling are further investigated from a probabilistic standpoint.The homoclinic ship rolling has two rolling states,the connection between the two peaks of the PDF is observed when the periodic excitation amplitude or the noise intensity is large,and the PDF is remarkably distributed in phase space.These phenomena increase the possibility of a random jump in ship motion states and the uncertainty of ship rolling,and the ship may lose stability due to unforeseeable facts or conditions.Meanwhile,only one rolling state is observed when the ship is in heteroclinic rolling.As the periodic excitation amplitude grows,the PDF concentration increases and drifts away from the beginning location,suggesting that the ship rolling substantially changes in a cycle and its stability is low.The PDF becomes increasingly uniform and covers a large region as the noise intensity increases,reducing the certainty of ship rolling and navigation safety.The current numerical solutions and analyses may be applied to evaluate the stability of a rolling ship in irregular waves and capsize mechanisms.

Technique of Probability Density Function Shape Control for Nonlinear Stochastic Systems

The shape control of probability density function(PDF) of the system state is an important topic in stochastic systems. In this paper, we propose a control technique for PDF shape of the state variable in nonlinear stochastic systems. Firstly, we derive and prove the form of the controller by investigating the Fokker-PlanckKolmogorov(FPK) equation arising from the stochastic system. Secondly, an approach for getting approximate solution of the FPK equation is provided. A special function including some parameters is taken as the approximate stationary solution of the FPK equation. We use nonlinear least square method to solve the parameters in the function, and capture the approximate solution of the FPK equation. Substituting the approximate solution into the form of the controller, we can acquire the PDF shape controller. Lastly, some example simulations are conducted to verify the algorithm.

A deep learning method based on prior knowledge with dual training for solving FPK equation

The evolution of the probability density function of a stochastic dynamical system over time can be described by a Fokker–Planck–Kolmogorov(FPK) equation, the solution of which determines the distribution of macroscopic variables in the stochastic dynamic system. Traditional methods for solving these equations often struggle with computational efficiency and scalability, particularly in high-dimensional contexts. To address these challenges, this paper proposes a novel deep learning method based on prior knowledge with dual training to solve the stationary FPK equations. Initially, the neural network is pre-trained through the prior knowledge obtained by Monte Carlo simulation(MCS). Subsequently, the second training phase incorporates the FPK differential operator into the loss function, while a supervisory term consisting of local maximum points is specifically included to mitigate the generation of zero solutions. This dual-training strategy not only expedites convergence but also enhances computational efficiency, making the method well-suited for high-dimensional systems. Numerical examples, including two different two-dimensional(2D), six-dimensional(6D), and eight-dimensional(8D) systems, are conducted to assess the efficacy of the proposed method. The results demonstrate robust performance in terms of both computational speed and accuracy for solving FPK equations in the first three systems. While the method is also applicable to high-dimensional systems, such as 8D, it should be noted that computational efficiency may be marginally compromised due to data volume constraints.

Extended F-Expansion Method and Periodic Wave Solutions for Klein-Gordon-SchrSdinger Equations

我们在场为在数学物理发现非线性的进化方程的周期的波浪答案的一个扩大 F 扩大方法。由使用扩大 F 扩大方法，各种各样的 Jacobi 椭圆形的功能为 theKlein-Gordon-Schroedinger 方程表示的许多周期的波浪答案被获得。在限制盒子中，方程的独居的波浪解决方案和三角法的函数解决方案也被获得。

Symbolic Computation of Extended Jacobian Elliptic Function Algorithm for Nonlinear Differential-Different Equations

The Jacobian elliptic function expansion method for nonlinear differential-different equations and its algorithm are presented by using some relations among ten Jacobian elliptic functions and successfully construct more new exact doubly-periodic solutions of the integrable discrete nonlinear Schrodinger equation. When the modulous m → 1or 0, doubly-periodic solutions degenerate to solitonic solutions including bright soliton, dark soliton, new solitons as well as trigonometric function solutions.

混响室内模拟多径衰落场环境的概率统计模型

已有混响室概率统计模型不能同时模拟多种多径衰落场环境。通过扩展经典混响室平面波叠加模型,建立了更具普适性的混响室内模拟多径衰落场环境的概率统计模型。推导了电场直角分量模值的概率密度函数(PDF),在理论上验证了新模型能同时模拟瑞利分布、莱斯分布及更多多径衰落场环境。通过蒙特卡洛模拟仿真了电场直角分量模值的PDF,并与理论PDF对比,进一步验证了新模型同时模拟多种多径衰落场环境的能力。

Application of Maximum Entropy Principle to Studying the Distribution of Wave Heights in A Random Wave Field

Based on the maximum entropy principle, a probability density function (PDF) is derived for the distribution of wave heights in a random wave field, without any more hypothesis. The present PDF, being a non-Rayleigh form, involves two parameters: the average wave height H— and the state parameter γ. The role of γ in the distribution of wave heights is examined. It is found that γ may be a certain measure of sea state. A least square method for determining γ from measured data is proposed. In virtue of the method, the values of γ are determined for three sea states from the data measured in the East China Sea. The present PDF is compared with the well known Rayleigh PDF of wave height and it is shown that it much better fits the data than the Rayleigh PDF. It is expected that the present PDF would fit some other wave variables, since its derivation is not restricted only to the wave height.

Statistical Distribution of Surface Slope in A 3-D Ocean Wave Field

A joint probability density function (PDF) for surface slopes in two arbitrary directions is derived on the basis of Longuet Higgins's linear model for three-dimensionol (3-D) random wave field. and the correlation moments of surface slopes. as parameters in the PDF, are expressed in terms of directional spectrum of ocean waves. So long as the directional spectrum model is given, these parameters are determined. Since the directional spectrum models proposed so far are mostly parameterized by the wind speed and fetch, this allows for substituting these parameters with thc wind speed and fetch. As an example, the wind speed and fetch are taken to be 14 m ' s and 200 km, and the Hasselmann and Donclan directional spectra are, respectively, use to compute these parameters. Some novel results a reobtained. One of the increasing interesting results is that the variances of surface slope in downwind and cross-wind directions determined by the Donclan directional spectra are close to those measured by Cox and Munk (1954). Some discussions are made on these results.

Application of Maximum Entropy Distribution to the Statistical Properties of Wave Groups

The new distributions of the statistics of wave groups based on the maximum entropy principle are presented. The maximum entropy distributions appear to be superior to conventional distributions when applied to a limited amount of information. Its applications to the wave group properties show the effectiveness of the maximum entropy distribution. FFF filtering method is employed to obtain the wave envelope fast and efficiently. Comparisons of both the maximum entropy distribution and the distribution of Longuet-Higgins （1984） with the laboratory wind-wave data show that the former gives a better fit.

Local Galerkin Method for the Approximate Solutions to General FPK Equations

In this paper, the method proposed recently by the author for the solution of probability density function (PDF) of nonlinear stochastic systems is presented in detail and extended for more general problems of stochastic differential equations (SDE), therefore the Fokker Planck Kolmogorov (FPK) equation is expressed in general form with no limitation on the degree of nonlinearity of the SDE, the type of δ correlated excitations, the existence of multiplicative excitations, and the dimension of SDE or FPK equation. Examples are given and numerical results are provided for comparing with known exact solution to show the effectiveness of the method.

Simulation of Random Waves and Associated Laminar Bottom Shear Stresses

This work presents a new approach for simulating the random waves in viscous fluids and the associated bottom shear stresses. By generating the incident random waves in a numerical wave flume and solving the unsteady two-dimensional Navier-Stokes equations and the fully nonlinear free surface boundaiy conditions for the fluid flows in the flume, the viscous flows and laminar bottom shear stresses induced by random waves axe determined. The deterministic spectral amplitude method implemented by use of the fast Fourier transform algorithm was adopted to generate the incident random waves. The accuracy of the numerical scheme is confirmed by comparing the predicted wave spectrum with the target spectrum and by comparing the nanlerical transfer function between the shear stress and the surface elevation with the theoretical transfer function. The maximum bottom shear stress caused by random waves, computed by this wave model, is compared with that obtained by Myrhaug＇ s model （1995）. The transfer function method is also employed to determine the maximum shear stress, and is proved accurate.

Nonlinear dynamics and performance analysis of modified snap-through vibration energy harvester with time-varying potential function

Vibration energy harvesting has emerged as a promising method to harvest energy for small-scale applications.Enhancing the performance of a vibration energy harvester(VEH)incorporating nonlinear techniques,for example,the snap-through VEH with geometric non-linearity,has gained attention in recent years.A conventional snap-through VEH is a bi-stable system with a time-invariant potential function,which was investigated extensively in the past.In this work,a modified snap-through VEH with a time-varying potential function subject to harmonic and random base excitations is investigated.Modified snap-through VEHs,such as the one considered in this study,are used in wave energy harvesters.However,the studies on their dynamics and energy harvesting under harmonic and random excitations are limited.The dynamics of the modified snap-through VEH is represented by a system of differential algebraic equations(DAEs),and the numerical schemes are proposed for its solutions.Under a harmonic excitation,the system exhibits periodic and chaotic motions,and the energy harvesting is superior compared with the conventional counterpart.The dynamics under a random excitation is investigated by the moment differential method and the numerical scheme based on the modified Euler-Maruyama method.The Fokker-Planck equation representing the dynamics is derived,and the marginal and joint probability density functions(PDFs)are obtained by the Monte Carlo simulation.The study shows that the modified snap-through oscillator based VEH performs better under both harmonic and random excitations.The dynamics of the system under stochastic resonance(SR)is investigated,and performance enhancement is observed.The results from this study will help in the development of adaptive VEH techniques in the future.

TPDF-ASOM复合湍流燃烧模型及其检验

先进航空发动机燃烧室设计要求对湍流火焰精确控制,现有模拟方法需提高精度和效率。输运概率密度函数(TPDF)湍流燃烧模型精度高,代数二阶矩(ASOM)湍流燃烧模型计算成本低,类比离散涡模拟思想,基于Da数将湍流燃烧场区分“高精度”和“低成本”2个区域,在输运方程框架下采用随机场TPDF(高精度)和ASOM(低成本)方法重构TPDF-ASOM复合湍流燃烧模型,以提高模拟的整体精度和效率。在大涡模拟(LES)-TPDF程序平台创建ASOM并进一步实现TPDF-ASOM复合湍流燃烧模型,用Flame D实验数据检验所建模型和方法。结果表明:所建模型的预测结果与实验值接近,而且能够兼顾精度和计算效率。