Statistical Analysis of the Random Convective Dispersion Models

The convective dispersion models are derived by the material balance method in an infinitesimal volume. In comparison with the derivation process by the Markov method, they indicate statistical foundation of random particle movement. By numerical method, the mass transport laws are analyzed comparatively for the models.     

Derivation of the Dirac Equation by Conformal Differential Geometry

A rigorous ab initio derivation of the (square of) Dirac’s equation for a particle with spin is presented. The Lagrangian of the classical relativistic spherical top is modified so to render it invariant with respect conformal changes of the metric of the top configuration space. The conformal invariance is achieved by replacing the particle mass in the Lagrangian with the conformal Weyl scalar curvature. The Hamilton-Jacobi equation for the particle is found to be linearized, exactly and in closed form, by an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” of the 4-spinor solution of Dirac’s equation. All quantum features arise from the subtle interplay between the conformal curvature acting on the particle as a potential and the particle motion which affects the geometric “pre-potential” associated to the conformal curvature itself. The theory, carried out here by assuming a Minkowski metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation.     

Derivation of Dirac's Equation from the Evans Wave Equation

The Evans wave equation [1] of general relativity is expressed in spinor form, thus producing the Dirac equation in general relativity. The Dirac equation in special relativity is recovered in the limit of Euclidean or flat spacetime. By deriving the Dirac equation from the Evans equation it is demonstrated that the former originates in a novel metric compatibility condition, a geometrical constraint on the metric vector qused to define the Einstein metric tensor. Contrary to some claims by Ryder, it is shown that the Dirac equation cannot be deduced unequivocally from a Lorentz boost in special relativity. It is shown that the usually accepted method in Clifford algebra and special relativity of equating the outer product of two Pauli spinors to a three-vector in the Pauli basis leads to the paradoxical result X = Y = Z = 0. The method devised in this paper for deriving the Dirac equation from the Evans equation does not use this paradoxical result.     

Irreversibility in the Derivation of the Boltzmann Equation

Uffink and Valente (Found Phys 45:404–438, 2015) claim that there is no time-asymmetric ingredient that, added to the Hamiltonian equations of motion, allows to obtain the Boltzmann equation within the Lanford’s derivation. This paper is a discussion and a reply to that analysis. More specifically, I focus on two mathematical tools used in this derivation, viz. the Boltzmann–Grad (B–G) limit and the incoming configurations. Although none of them are time-asymmetric ingredients, by themselves, I claim that the use of incoming configurations, as taken within the B–G limit, is such a time-asymmetric ingredient. Accordingly, this leads to reconsider a kind of Stoßzahlansatz within Lanford’s derivation.     

Metastates in Mean-Field Models with Random External Fields Generated by Markov Chains

We extend the construction by Külske and Iacobelli of metastates in finite-state mean-field models in independent disorder to situations where the local disorder terms are a sample of an external ergodic Markov chain in equilibrium. We show that for non-degenerate Markov chains, the structure of the theorems is analogous to the case of i.i.d. variables when the limiting weights in the metastate are expressed with the aid of a CLT for the occupation time measure of the chain.     

A Microscopic Derivation of the Time-Dependent Hartree-Fock Equation with Coulomb Two-Body Interaction

We study the dynamics of a Fermi gas with a Coulomb interaction potential, and show that, in a mean-field regime, the dynamics is described by the Hartree-Fock equation. This extends previous work of Bardos et al. [J. Math. Pures Appl. 82(6):665–683, 2003] to the case of unbounded interaction potentials. We also express the mean-field limit as a “superhamiltonian” system, and state our main result in terms of the Heisenberg-picture dynamics of observables. This is a Egorov-type theorem.     

Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases

We prove that the Gross-Pitaevskii equation correctly describes the ground state energy and corresponding one-particle density matrix of rotating, dilute, trapped Bose gases with repulsive two-body interactions. We also show that there is 100% Bose-Einstein condensation. While a proof that the GP equation correctly describes non-rotating or slowly rotating gases was known for some time, the rapidly rotating case was unclear because the Bose (i.e., symmetric) ground state is not the lowest eigenstate of the Hamiltonian in this case. We have been able to overcome this difficulty with the aid of coherent states. Our proof also conceptually simplifies the previous proof for the slowly rotating case. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.     

考虑扩散和吸附作用的聚合物驱替过程渗流数值模拟

采用显式求解饱和度、隐式求解浓度的思路,对考虑扩散和吸附作用的聚合物驱替过程渗流模型进行了数值求解.饱和度方程求解应用了显式全变差递减(TVD)法;浓度方程求解过程中,空间项离散采用Crank Nicolson差分格式,时间项变量进行拟线性处理,保证了计算的稳定性.通过与解析解对比,验证了该方法的有效性.计算实例分析表明,扩散使聚合物在溶液中稀释,导致浓度传播分散;吸附使聚合物损耗,导致浓度传播滞后.同时,计算结果直观反映了聚合物驱重要的"油墙"形成机理.在段塞注入情况下,原油富集区在出口端的突破是介于聚合物浓度前缘突破和聚合物浓度峰值突破之间.     

Soliton Propagation in Optical Fibers with Random Polarization Dispersion

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Derivation of the Lorentz Boost from the Evans Wave Equation

The Lorentz boost is derived from the Evans wave equation of generally covariant unified field theory by constructing the Dirac spinor from the tetrad in the SU(2) representation space of non-Euclidean spacetime. The Dirac equation in its wave formulation is then deduced as a well-defined limit of the Evans wave equation. By factorizing the dAlembertian operator into Dirac matrices, the Dirac equation in its original first differential form is obtained from the Evans wave equation. Finally, the Lorentz boost is deduced from the Dirac equation using geometrical arguments. A self-consistency check of the Evans wave equation is therefore forged by deducing therefrom the Lorentz boost in the appropriate limit. This procedure demonstrates that the Evans wave equation governs the properties of matter and anti-matter in general relativity and unified field theory and leads both to Fermi-Dirac and Bose-Einstein statistics in general relativity.     

Front Speed in the Burgers Equation with a Random Flux

We study the large-time asymptotic shock-front speed in an inviscid Burgers equation with a spatially random flux function. This equation is a prototype for a class of scalar conservation laws with spatial random coefficients such as the well-known Buckley–Leverett equation for two-phase flows, and the contaminant transport equation in groundwater flows. The initial condition is a shock located at the origin (the indicator function of the negative real line). We first regularize the equation by a special random viscous term so that the resulting equation can be solved explicitly by a Cole–Hopf formula. Using the invariance principle of the underlying random processes and the Laplace method, we prove that for large times the solutions behave like fronts moving at averaged constant speeds in the sense of distribution. However, the front locations are random, and we show explicitly the probability of observing the head or tail of the fronts. Finally, we pass to the inviscid limit, and establish the same results for the inviscid shock fronts.     

各向异性颗粒PDF输运方程及其颗粒湍流扩散

根据Minier的模型,获得颗粒所见流体脉动速度的朗之万模型,并推导颗粒所见流体湍流脉动速度自关联函数,该函数具有各向异性的特征,进一步由流体脉动速度朗之万模型和颗粒运动方程获得相应的颗粒PDF输运方程.为描述在网格生成湍流中的颗粒扩散运动,将颗粒PDF输运方程作相应的简化,获得一个具有颗粒湍流各向异性的解析解,将方程的解与Wellsand Stock的颗粒湍流扩散实验结果进行比较,获得了很好的结果.揭示了具有各向异性特征的颗粒PDF输运方程的优越性.     

Derivation of the Time Dependent Two Dimensional Focusing NLS Equation

We present a microscopic derivation of the two-dimensional focusing cubic nonlinear Schrödinger equation starting from an interacting N-particle system of Bosons. The interaction potential we consider is given by \(W_\beta (x)=N^{-1+2 \beta }W(N^\beta x)\) for some spherically symmetric and compactly supported potential \(W \in L^\infty ({\mathbb {R}}^2, {\mathbb {R}})\). The class of initial wave functions is chosen such that the variance in energy is small. Furthermore, we assume that the Hamiltonian \( H_{W_\beta , t}=-\sum _{j=1}^N \Delta _j+\sum _{1\le j< k\le N} W_\beta (x_j-x_k) +\sum _{j=1}^N A_t(x_j)\) fulfills stability of second kind, that is \( H_{W_\beta , t} \ge -\,CN\). We then prove the convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in either Sobolev trace norm, if \(\Vert A_t\Vert _p < \infty \) for some \(p>2\), or in trace norm, for more general external potentials. For trapping potentials of the form \(A(x)=C |x|^s\; , C>0\), the condition \( H_{W_\beta , t} \ge -\,CN\) can be fulfilled for a certain class of interactions \(W_\beta \), for all \(0< \beta < \frac{s+1}{s+2}\), see Lewin et al. (Proc Am Math Soc 145:2441–2454, 2017).     

On the Derivation of the Time-Dependent Equation of Schrödinger

Few have done more than Martin Gutzwiller to clarify the connection between classical time-dependent motion and the time-independent states of quantum systems. Hence it seems appropriate to include the following discussion of the origins of the time-dependent Schrödinger equation in this volume dedicated to him.     

Some Considerations on the Derivation of the Nonlinear Quantum Boltzmann Equation

In this paper we analyze a system of Nidentical quantum particles in a weak-coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula. For short times, we rigorously prove that a subseries of the latter, converges to the solution of the Boltzmann equation which is physically relevant in the context. In particular, we recover the transition rate as it is predicted by Fermi's Golden Rule. However, we are not able to prove that the quantity neglected while retaining a subseries of the complete original perturbative expansion, indeed vanishes in the limit: we only give plausibility arguments in this direction. The present study holds in any space dimension d≥2.     

Derivation of an Improved Hodgkin-Huxley Model for Potassium Channel by Means of the Fokker-Planck Equation

This paper demonstrates the derivation of Hodgkin-Huxley-like equations from the Fokker-Planck equation. The primary result is that instead of the familiar equation expressing the potassium conductance as a function of the variablen which obeys a first order differential equation, the expression , whereL = 2.7, is to be used. This form is obtained by solving analytically an approximate solution to a Fokker-Planck partial difference equation. Instead of the Hodgkin-Huxley interpretation as the probability of occupying the conducting state, the parameter n(t) is now interpreted as the position of the “peak” of the population distribution function P(N, t), which changes in time described by the Fokker-Planck equation. This new approach enables close fitting of the experimental voltage clamp data for potassium conductance. In addition, the Cole-Moore shift paradox can be quantitatively explained in terms of the shift of the distribution function P(N,t) by the initial clamped transmembrane potentialV _i before the final clamped transmembrane potentialV _f is applied, thus increasing the time necessary for the establishment of equilibrium.