On Current Developments in Partial Differential Equations

The first part of this article is an overview on some recent major developments in the field of analysis and partial different equations.It is a brief presentation given by the author at a round table discussion.The second part is a supplement of various details provided by several outstanding researchers on subjects.     

The Fundamental Equations of Change in Statistical Ensembles and Biological Populations

A recent article in Nature Physics unified key results from thermodynamics, statistics, and information theory. The unification arose from a general equation for the rate of change in the information content of a system. The general equation describes the change in the moments of an observable quantity over a probability distribution. One term in the equation describes the change in the probability distribution. The other term describes the change in the observable values for a given state. We show the equivalence of this general equation for moment dynamics with the widely known Price equation from evolutionary theory, named after George Price. We introduce the Price equation from its biological roots, review a mathematically abstract form of the equation, and discuss the potential for this equation to unify diverse mathematical theories from different disciplines. The new work in Nature Physics and many applications in biology show that this equation also provides the basis for deriving many novel theoretical results within each discipline.     

AgAlSe2晶体高压相变的第一性原理研究

基于密度泛函理论,采用广义梯度近似法,研究了黄铜矿结构AgAlSe₂在高压下的晶体结构、晶格动力学稳定性与电子结构.结果显示:在0 GPa时AgAlSe₂的晶格参数与实验值吻合,在13.9 GPa附近,质量密度、Se-Ag键长、Se-Al键长、晶格常数a突然增大,相对晶胞体积V/V₀、晶格常数c突然减小,声子谱出现虚频,结构变得不稳定,带隙发生突变,数值呈减小趋势.表明AgAlSe₂晶体在13.9 GPa附近发生结构相变.该研究为AgAlSe₂晶体在理论上所能承受的高压提供信息支撑.     

Comparison of Penta and Tetra-pyridyl Cobalt-based Catalysts for Water Reduction: H2 Production Cycle,Solvent Response and Reduction Free Energy

Understanding water reduction towards H₂ generation is crucial to overcome today's renewable energy obstacles. Previous studies have shown the superior H₂ production performances of Cobalt based penta-pyridyl (CoaPPy) and tetra-pyridyl (CoaTPy) complexes in solution. We investigate H₂ production cycles of CoaPPy and CoaTPy complexes immersed in water solution by means of Ab-initio Molecular Dynamics and Density Functional Theory. We monitor dynamic properties of the systems, solvent response and structural changes occurring in the catalysts, by simulating all intermediate steps of the H₂ production cycle. Reduction free energies and reorganization energies are calculated. Our results show that, following the first electron injection, H₂ production proceeds with the singlet spin state. Following the first electron insertion, we observe a significant rearrangement of the hydrogen bonding network in the first solvation shell. The cobalt center turns out to be more accessible for the surrounding water molecules in the case of CoaTPy at all the intermediate steps, which explains its higher catalytic performance over CoaPPy. Following the first reduction reaction, a larger gain in reduction free energy is estimated for CoaTPy with respect to CoaPPy, with a difference of 0.14 eV, in line with the experiments. For the second reduction, instead, CoaPPy shows more negative reduction potential, by 0.41 eV.     

HfSe2 monolayer stability tuning by strain and charge doping

Strain and charge doping are the effective ways to modulate the electronic and phonon properties of materials. The effects of biaxial tensile strains and charge dopings on the stabilities of HfSe₂ monolayer have been systematically investigated using first-principles methods. Its two-dimensional Young's modulus is only 65.4 N/m, and it is easy to be stretched. When the tensile strain is applied on HfSe₂ monolayer, two of its phonon modes soften with one frequency decreasing to zero at critical strain. Our results show that electron and hole dopings could suppress the softening of phonon modes, and significantly enhance the ideal strength by 28% and 36%, respectively. The calculations for electronic structures and phonon dispersions provide the theoretical references for future nano-device designing.     

Diffraction of electromagnetic waves by a resistive half-plane in magneto-ionic plasma

The interaction of magnetic polarized electromagnetic wave with a resistive half-plane in magneto-ionic plasma is investigated. The method of transition boundary is used for the evaluation of the diffracted wave. The excitation of surface waves is examined in the medium and their diffracted fields are derived by using the diffraction theory for grazing incidence. The uniform field expressions are obtained with the aid of the uniform theory for evanescent waves. The behaviours of the derived scattered waves are studied numerically.     

基于八面体理论的岩石循环加-卸载本构模型及修正

探究岩石的受力特点及破坏特性是研究岩石地下工程安全性的关键,诸多学者都期望能在岩石本构模型的研究上取得突破性进展。在此背景下,提出了一种能够描述循环加-卸载条件下岩石的本构模型。首先,假设岩石的微元强度服从八面体剪应力理论并且微元破坏服从Weibull概率公式,将岩石本构中的损伤变量以及岩石微元强度表达式里包含的损伤因子进行本构变换,得到关于应力、应变等其他表现加-卸载下岩石损伤本构模型的参数,表示出岩石微元强度和损伤变量,再将得到的岩石微元强度和损伤变量代入所提出的岩石本构模型中,并进行等式变换得到一个函数表达式。通过将其与实验数据进行拟合对比分析,得出修正后的拟合参数,将其代入函数式中,得到损伤本构模型的修正式。最后将拟合参数进行必要的敏感性分析,得出各拟合参数的实际物理意义。     

Stability of equilibrium solutions of a double power reaction-diffusion equation with a Dirac interaction

In this paper, information about the instability of equilibrium solutions of a nonlinear family of localized reaction-diffusion equations in dimension one is provided. More precisely, explicit formulas to the equilibrium solutions are computed and, via analytic perturbation theory, the exact number of positive eigenvalues of the linear operator associated to the stability problem is analyzed. In addition, sufficient conditions for blow up of the solutions of the equation are also discussed.