Decay Rates in Probability Metrics Towards Homogeneous Cooling States for the Inelastic Maxwell Model

We quantify the long-time behavior of solutions to the nonlinear Boltzmann equation for spatially uniform freely cooling inelastic Maxwell molecules by means of the contraction property of a suitable metric in the set of probability measures. Existence, uniqueness, and precise estimates of overpopulated high energy tails of the self-similar profile proved in ref. 9 are revisited and derived from this new Liapunov functional. For general initial conditions the solutions of the Boltzmann equation are then proved to converge with computable rate as t → ∞ to the self-similar solution in this distance, which metrizes the weak convergence of measures. Moreover, we can relate this Fourier distance to the Euclidean Wasserstein distance or Tanaka functional proving also its exponential convergence towards the homogeneous cooling states. The findings are relevant in the understanding of the conjecture formulated by Ernst and Brito in refs. 15, 16, and complement and improve recent studies on the same problem of Bobylev and Cercignani⁽⁹⁾ and Bobylev, Cercignani and one of the authors.⁽¹¹⁾     

The one-body and two-body density matrices of finite nuclei with an appropriate treatment of the center-of-mass motion⋆

The one-body and two-body density matrices in coordinate space and their Fourier transforms in momentum space are studied for a nucleus (a nonrelativistic, self-bound finite system). Unlike the usual procedure, suitable for infinite or externally bound systems, they are determined as expectation values of appropriate intrinsic operators, dependent on the relative coordinates and momenta (Jacobi variables) and acting on intrinsic wave functions of nuclear states. Thus, translational invariance (TI) is respected. When handling such intrinsic quantities, we use an algebraic technique based upon the Cartesian representation, in which the coordinate and momentum operators are linear combinations of the creation and annihilation operators and for oscillator quanta. Each of the relevant multiplicative operators can then be reduced to the form: one exponential of the set { } times another exponential of the set { }. In the course of such a normal-ordering procedure we offer a fresh look at the appearance of “Tassie-Barker” factors, and point out other model-independent results. The intrinsic wave function of the nucleus in its ground state is constructed from a nontranslationally-invariant (nTI) one via existing projection techniques. As an illustration, the one-body and two-body momentum distributions (MDs) for the ⁴He nucleus are calculated with the Slater determinant of the harmonic-oscillator model as the trial, nTI wave function. We find that the TI introduces quite important effects in the MDs.     

铁分子Fe2的自旋极化效应

采用密度泛函方法(B3P86)对 Fe_2分子结构进行了优化.计算结果中未观察到自旋污染,基态波函数与高态波函数并未混杂,结果表明,Fe_2中有8个未配对电子,这些电子空间分布不同和自旋平行产生的自旋极化效应,使 Fe_2能量最低.计算结果表明,Fe_2分子的基态是~9∑_g~ ,并非~7Δ_u,进而表明 Fe_2的自旋平行效应比电子自旋配对效应要强.计算得到该分子基态的二阶、三阶和四阶力常数分别为1.4115×10~(-2)aJ/nm~2、-37.1751×10~3aJ/nm~3和 98.7596×10~4aJ/nm~4；光谱数据ω_eχ_e、B_e、α_e分别为0.3522、0.0345、 0.4963×10~(-4)cm~(-1)；离解能为3.5522eV,平衡键长为0.2137nm,振动频率为292.914cm~(-1)；并得到了 Murrel-Sorbie 函数.     

Soluble models for dynamics driven by a super-diffusive noise

We explicitly discuss scalar Langevin type of equations where the deterministic part is linear, but where the integrated noise source is a non-linear diffusion process exhibiting superdiffusive behavior. We calculate transient and stationary probabilities and study the possibility of noise induced transitions from a unimodal to a bimodal probability shape. Illustrations from finance and dynamical systems are given.     

Reynolds stress models applied to canonical shock-turbulence interaction

Shock waves in high-speed flows can drastically alter the nature of Reynolds stresses in a turbulent flow. We study the canonical interaction of homogeneous isotropic turbulence passing through a normal shock, where the shock wave generates significant anisotropy of Reynolds stresses. Existing Reynolds stress models are applied to this canonical problem to predict the amplification of the stream-wise and transverse normal Reynolds stresses across the shock wave. In particular, the efficacy of the different models for the rapid pressure–strain correlation is evaluated by comparing the results with available direct numerical simulation (DNS) data. The model predictions are found to be grossly inaccurate, especially at high-Mach numbers. We propose physics-based improvement to the Reynolds stress-transport equation in the form of shock-unsteadiness effect and enstrophy amplification for turbulent dissipation rate . The resulting model is found to capture the essential physics of Reynolds stress amplification, and match DNS data for a range of Mach numbers. Numerical error encountered at shock waves are also analysed and the model equations are cast in conservative form to obtain physically consistent results with successive grid refinement. Finally, the proposed model for canonical shock-turbulence interaction is generalised to multi-dimensional flows with shock of arbitrary orientation.     

The two-dimensional Coulomb gas on a sphere: Exact results

At the special value of the reduced inverse temperature=2, the equilibrium statistical mechanics of a two-dimensional Coulomb gas confined to the surface of a sphere is an exactly solvable problem, just as it was for the Coulomb gas in a plane. The thermodynamic quantities and all the correlation functions can be calculated. Use is made of an isomorphism between the classical Coulomb gas and the free Fermi field theory associated with the Dirac operator on the sphere.Laboratory associated with the Centre National de la Recherche Scientifique.     

Classical Coulomb fluids in a confined geometry

It has already been argued that a classical (three-dimensional) Coulomb fluid confined between two parallel walls exhibits ideal gas features when the distance between the walls becomes small; this is confirmed in the present paper. Two-dimensional models of Coulomb fluids (with a logarithmic interaction), confined in a strip, are also studied. These models do not become ideal gases in the narrow strip limit. The correlation functions are also studied. There is a special temperature at which exact results are obtained. At that temperature, the two-dimensional, two-component plasma (two-dimensional Coulomb gas), which is a conductor when unconfined, becomes a dielectric as soon as it is confined in a strip of noninfinite width. This can be understood as a displacement of the Kosterlitz-Thouless transition by the confinement.     

Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries

Chern–Simons (CS) gauge theories in three dimensions and the Poisson sigma model (PSM) in two dimensions are examples of the same theory, if their field equations are interpreted as morphisms of Lie algebroids and their symmetries (on-shell) as homotopies of such morphisms. We point out that the (off-shell) gauge symmetries of the PSM in the literature are not globally well defined for non-parallelizable Poisson manifolds and propose a covariant definition of the off-shell gauge symmetries as left action of some finite-dimensional Lie algebroid.

Our approach allows us to avoid complications arising in the infinite-dimensional super-geometry of the BV- and AKSZ-formalism. This preprint is a starting point in a series of papers meant to introduce Yang–Mills type gauge theories of Lie algebroids, which include the standard YM theory, gerbes, and the PSM.     

二甲基亚锗基锗烯与乙醛生成螺锗杂环化合物反应机理的密度泛函理论研究

用B3LYP/6-31G*方法研究了单重态二甲基亚锗基锗烯(Me₂Ge=Ge:)与乙醛环加成反应的反应机理,根据该反应的势能面可以预言,该反应有一条主反应通道. 该反应所呈现的反应规律为：两反应物通过[2+2]环加成反应首先生成了一锗杂四元环锗烯,由于该锗杂四元环锗烯中Ge:原子的4p空轨道与乙醛的?轨道形成了π→p授受键,从而使锗杂四元环锗烯进一步与乙醛结合生成了一中间体. 由于该中间体中的Ge:原子在过渡态之后发生了sp³杂化,从而使该中间体经过渡态异构化为了一螺锗杂环化合物. 该研究结果从理论上揭示了单重态二甲基亚锗基锗烯(Me₂Ge=Ge:)与乙醛环加成反应的反应机制,奠定了亚锗基锗烯(H₂Ge=Ge:)及其衍生物(X₂Ge=Ge:, X=H, Me, F, Cl, Br, Ph, Ar?)与非对称性π键化合物环加成反应的理论基础.