On the dynamics of statistical fluctuations in heavy ion collisions

We show how statistical fluctuations can be treated within the collective approach to heavy ion reactions. In the classical limit, the equation of motion for the distribution d in the collective variables Q_μ and their conjugate momenta P_μ turns out to be a Fokker-Planck equation. We briefly describe the connection of this equation to one of the Smoluchowski type for a distribution in Q_μ only, often used in heavy ion physics. For anharmonic motion our general Fokker-Planck equation is simplified to be linear in the deviations of the Q_μ mand P_μ from their mean values. The solution of this equation is discussed in terms of a simple Gaussian. The parameters of this Gaussian are determined completely by the first and second moments in Q_μ mand P_μ. The equations for the first moments are identical to the Newton equations including frictional forces. Those for the second moments are linear differential equations of first order and hence easily solvable. The whole derivation is completely analogous to that for the Newton equation reported recently. Here the starting point is the quantum mechanical von Neumann equation rather than the Heisenberg equations. As an intermediate result we obtain and discuss briefly a quantal equation for the reduced density operator d which includes frictional effects.     

Numerical test of approximate theories of the superconducting transition temperature

Various equations for t_c obtained from approximations to the Eliashberg theory are numerically solved and T_cvs λ curves are drawn. Specifically, the weak coupling Kirzhnits, Maksimov and Khomskii (KMK) equation, Eliashberg equation and BCS equation with Bardeen Pines interaction are considered with the α²F function of Nb. The KMK equation gives quite a high value or T_c and the BCS equation overestimates T_c by an even larger factor when compared with the results of the Eliashberg equation.     

具有阻尼项的非线性波动方程的相似约化

利用Clarkson和Kruskal引入的直接约化法,给出了具有阻尼项的非线性波动方程u_tt-2bu_xxt+αu_xxxx=β(uⁿ_x)x(α>0,β≠0,n≥2)三种类型的相似约化.从这些约化方程的Painlevé分析表明该方程在Ablowitz的猜测意义下是不可积的.此外还获得了该方程(n=2)的4种精确类孤波解. 关键词： 波动方程 相似约化 Painlevé分析 精确解     

The zero temperature energy gap of strong coupling superconductors

A simple equation for the zero temperature energy gap of a superconductor is derived which enables one to calculate Δ₀ very quickly with only a modest loss of accuracy. This equation is used to calculate the relative effectiveness of phonons of different frequencies in enhancing Δ₀. It predicts that the optimum phonon frequency for enhancing Δ₀ is close to Δ₀ for any material.     

非线性波动方程的孤波解

用平衡法并结合吴消元法得到了一类较广泛非线性波动方程u_tt-a₁u_xx+a₂u_t+a₃u+a₄u³=0的若干孤波解公式，从而物理学上许多著名的方程，如φ⁴方程、Klein-Gordon方程、Landau-Ginzburg-Higgs方程、非线性电报方程等都可作为该方程的特殊情形得到相应的孤波解 关键词：     

Geometrical relationship between the classical and quantum descriptions of arbitrary force interactions

The paper is a continuation of a series papers by the author in which it is shown that the equations of contemporary quantum theory (the Klein-Gordon equation and the Dirac integral equation) can be interpreted as conditions of harmonicity of linear form in the manifold^sV₄. In this paper, geometrization of classical and quantum interactions of a physical object is performed for the case of an arbitrary external force field in the manifold^kV₄. This manifold is constructed on the same set of surfaces as the manifold^sV₄. The equation representing conditions of harmonicity of linear form in^kV₄ is derived and is analogous to the KleinGordon equation.     

Symmetry Reductions,Group‐Invariant Solutions,and Conservation Laws of a (2+1)‐Dimensional Nonlinear Schrödinger Equation in a Heisenberg Ferromagnetic Spin Chain

Nonlinear spin excitations in ferromagnetic spin chains are studied for spintronic and magnetic devices including magnetic‐field sensors and for high‐density data storage. Here, (2+1)‐dimensional nonlinear Schrödinger equation is investigated, which describes the nonlinear spin dynamics for a Heisenberg ferromagnetic spin chain. Lie point symmetry generators and Lie symmetry groups of that equation are derived. Lie symmetry groups are related to the time, space, scale, rotation transformations, and Galilean boosts of that equation. Certain solutions, which are associated with the known solutions, are constructed. Based on the Lie symmetry generators, the reduced systems of such an equation are obtained. Based on the polynomial expansion and through one of the reduced systems, group‐invariant solutions are constructed. Soliton‐type group‐invariant solutions are graphically investigated and effects of the magnetic coupling coefficients, that is, α₁, α₂, α₃, and α₄, on the soliton's amplitude, width, and velocity are discussed. It is seen that α₁, α₂, α₃, and α₄ have no influence on the soliton's amplitude, but can affect the soliton's velocity and width. Lax pair and conservation laws of such an equation are derived.     

反铁磁晶体中的非线性TE表面波

将反铁磁晶体中ＴＥ表面波满足的非线性波动方程变形为伯努利微分方程，用反函数形式给出了表面波磁场的精确解．证明非线性ＴＥ表面波存在的必要条件是μ_x＞０．求得了色散方程，分析了非线性ＴＥ波的频率范围．理论结果表明，导波磁场峰值位置具有稳定性，同功率无关；在ε₁＞ε₂条件下，在功率阈值之上，一个传播功率对应两个不同的有效折射率，表面导波具有双稳特性 关键词：     

On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity

The Bardeen-Cooper-Schrieffer integral equation with a positive kernel is studied in full generality. It is shown that, there exists a unique finite transition temperature, T _c so that, if T _c ,the equation possesses a positive solution, representing the onset of the superconducting phase, while if T>T _c ,the only solution of the equation is the trivial one, indicating the occurrence of the normal phase. Moreover, it is demonstrated that such a positive solution may be approximated by a sequence of solutions of the equation restricted on bounded domains. This latter result provides a useful computational scheme for the problem.     

Self consistent code for a 150 GHz gyrotron

A self consistent calculation for gyrotron oscillators is described and applied to a TE₀₃ mode gyromonotron. The RF field profile satisfies a wave equation in which the AC beam current appears as a source. The wave equation is solved simultaneously with the electron equation of motion.     

Multi-component Dirac equation hierarchy and its multi-component integrable couplings system

A general scheme for generating a multi-component integrable equation hierarchy is proposed. A simple 3M-dimensional loop algebra \tilde{X} is produced. By taking advantage of \tilde{X}, a new isospectral problem is established and then by making use of the Tu scheme the multi-component Dirac equation hierarchy is obtained. Finally, an expanding loop algebra \tilde{F}_M of the loop algebra \tilde{X} is presented. Based on the \tilde{F}_M, the multi-component integrable coupling system of the multi-component Dirac equation hierarchy is investigated. The method in this paper can be applied to other nonlinear evolution equation hierarchies.     

Approximate solution methods for linear stochastic difference equations: I. Moments

The cumulant expansion for linear stochastic differential equations is extended to the case of linear stochastic difference equations. We consider a vector difference equation, which contains a deterministic matrix A₀ and a random perturbation matrix A₁(t). The expansion proceeds in powers of ατ_c, where τ_c is the correlation time of the fluctuations in A₁(t) and α a measure for their strength. Compared to the differential case, additional cumulants occur in the expansion. Moreover one has to distinguish between a nonsingular and a singular A₀. We also discuss a limiting situation in which the stochastic difference equation can be replaced by a stochastic differential equation. The derivation is not restricted to the case where in the limit the stochastic parameters in the difference equation are replaced by white noise.     

Equation of state of dense nuclear matter

An equation of state for cold nuclear matter for the region of densities ρ_nm−4ρ_nm, where ρ_nm is empirical nuclear-matter density, is constructed. We begin from the detailed calculation of Day and Wiringa for the two-body interactions; these give a saturation density of ∼2ρ_nm. This density is brought down to ρ_nm by the addition of relativistic corrections. Additional binding is obtained from three-body forces. A reasonable picture is obtained with the Day-Wiringa compression modulus for the two-body calculation, but the picture can be further improved by choosing this to be smaller.Our equation of state is similar to that of Friedman and Pandharipande in the region of nuclear matter denstiy ρ_nm, but, due to higher-order terms in the loop correction, is substantially softer at high density. Basically what happens is that the many-body effects saturate with increasing density, leaving only the two-body interactions.With this equation of state, prompt supernova explosions are very powerful when the compression modulus of neutron-rich matter (twice as many neutrons as protons) is ∼150 MeV, which corresponds to K_nm ∼ 190 MeV for symmetric nuclear matter.Analysis shows that hot nuclear matter formed in heavy ion collisions demands a very stiff equation of state. We understand this as arising from the strong velocity dependence in the real part of the optical model potential which follows chiefly from the Lorentz character of the interactions, the vector mean field growing with increasing density and the scalar one decreasing. This gives a substantial repulsive contribution to the energy per particle and produces a stiff effective equation of state for several hundred MeV heavy-ion collisions. With increasing degree of equilibration the magnitude of the repulsive energy decreases since equilibration decreases the effective momentum. Given the strong velocity dependence in the interaction, the hot equation of state can be reconciled with the cool one.     

Eigenstate expansions in a soluble model of identical particle scattering

Solutions to the Schrödinger equation and the inhomogeneous equation for the case of two identical particles interacting with a center of force are studied. Eigenstate expansions for solving each equation are explicitly introduced and their properties discussed. The case when the interparticle interaction v₁₂ is zero is then examined; this is a completely soluble problem. The eigenstate expansion solutions for the Schrödinger and inhomogeneous equations are used to explore the means by which the correct solution is obtained. Finally, approximate solutions, obtained by truncating the eigenfunction expansions, are introduced. It is seen that both methods lead to the correct amplitude when τ₁₂ = 0, even though the approximate solution to the inhomogeneous equation does not lead, in the end, to an antisymmetric solution.     

Zero Tension Kardar-Parisi-Zhang Equation in (d + 1)–Dimensions

The joint probability distribution function (PDF) of the height and its gradients is derived for a zero tension d + 1-dimensional Kardar-Parisi-Zhang (KPZ) equation. It is proved that the height's PDF of zero tension KPZ equation shows lack of positivity after a finite time t _c . The properties of zero tension KPZ equation and its differences with the case that it possess an infinitesimal surface tension is discussed. Also potential relation between the time scale t _c and the singularity time scale t _c.v→0 of the KPZ equation with an infinitesimal surface tension is investigated.     

Modification of Tao–Mason equation of state to ionic liquids

In our previous paper, we extended the Tao and Mason equation of state (TM EOS) to refrigerant fluids, using the speed of sound data. Here, we predict the equation of state for ionic liquids (ILs). The considered ILs are [Bmim][PF₆], [C₂mim][NtF₂], [C₃mim][NtF₂], [C₆mim][NtF₂], [C₇mim][NtF₂], [C₂mim][EtOSO₃], [Bmim][MeSO₄], [Bmim][OcSO₄], and [C₄mim][dca]. The equation of state consists of three temperature-dependent parameters: the second virial coefficient, a constant for scaling the softness of repulsive force, and an effective hard-sphere diameter equivalent to the van der Waals co-volume. The second virial coefficients of ILs are scare and there is no accurate potential energy function to allow their theoretical calculation. In this work, the second virial coefficient have been calculated using corresponding states correlation based on temperature and density at normal boiling point. The other two parameters of the equation of state can be calculated using a scaling rule. Analysis of our predicted results shows that the Tao?CMason equation of state is capable of accurately predicting the density of ILs at any temperature and pressure. The overall average absolute deviation densities for 1,633 data points are 2.05%. Also, the density of ILs obtained from the TM EOS has been compared with those calculated from vdW?CCS?C?? and Peng?CRobinson (PR) equation of state. Our results are in favor of the preference of the TM EOS over the two other equations of state. The overall average absolute deviation for 1,633 data points calculated by vdW?CCS?C?? and PR equation of state are 6.63% and 12.19%, respectively.     

The elimination of fast variables in complex chemical reactions. III. Mesoscopic level (irreducible case)

The master equation for a complex chemical reaction cannot always be reduced to a simpler master equation, even if there are fast and slow individual reaction steps. Nevertheless the elimination of intermediates can be carried out with the help of the-expansion. This is illustrated with a well-known complex reaction: the dissociation of N₂O₅. It is shown that the intrinsic fluctuations in the N₂O₅ decay are larger than those implied by the master equation suggested by the macroscopic rate law.     

金属玻璃Ni30Zr70的动态粘度和稳态粘度

本文测量了金属玻璃Ni₃₀Zr₇₀的动态粘度和稳态粘度,结果表明,由于结构弛豫的影响,动态粘度随升温速率的减小而线性增加,在玻璃转变温度T_g以下,动态粘度与温度的关系可以用Arrhenius方程进行描述,稳态粘度与温度的依赖关系满足Fulcher-Vogel方程。从4种不同升温速率的高温(结晶开始温度T_x)动态粘度值,给出金属玻璃Ni₃₀Zr₇₀的平衡粘度。在630—670K     

氮化硅α，β和γ相的解析状态方程和热物理性质

对固体通用状态方程考虑热效应的Einstein模型进行修正，提出一种利用参考温度下的实验数据计算绝对零度下物质参数的方法，并用于Baonza方程以考虑热效应.结果表明，对于Baonza方程，零温下的参数可以解析地求解出来，常用的热物理量也可以推出解析表达式.将得出的公式用于研究氮化硅α，β和γ相的热物理性质.通过对文献的分析确定了这三种相的方程参数，计算的体积随压强变化和热膨胀系数随温度变化的结果与实验数据符合很好.     

A simple and accurate equation for the superconducting transition temperature

Reasonable and consistent approximations are used to reduce the simplified form of the integral equations for the superconducting transition temperature used by Allender, Bray and Bardeen to a very simple, and remarkably accurate, equation for T_c. This equation is compared with the McMillan equation and found to give just as good agreement with experiment for intermediate-coupling materials and considerably better agreement for strong-coupling materials.