Abstract Polymer Models with General Pair Interactions

A convergence criterion of cluster expansion is presented in the case of an abstract polymer system with general pair interactions (i.e. not necessarily hard core or repulsive). As a concrete example, the low temperature disordered phase of the BEG model with infinite range interactions, decaying polynomially as 1/r ^d+λ with λ>0, is studied.     

Exploration of Entropy Pair Functional Theory

Evaluation of the entropy from molecular dynamics (MD) simulation remains an outstanding challenge. The standard approach requires thermodynamic integration across a series of simulations. Recent work Nicholson et al. demonstrated the ability to construct a functional that returns excess entropy, based on the pair correlation function (PCF); it was capable of providing, with acceptable accuracy, the absolute excess entropy of iron simulated with a pair potential in both fluid and crystalline states. In this work, the general applicability of the Entropy Pair Functional Theory (EPFT) approach is explored by applying it to three many-body interaction potentials. These potentials are state of the art for large scale models for the three materials in this study: Fe modelled with a modified embedded atom method (MEAM) potential, Cu modelled with an MEAM and Si modelled with a Tersoff potential. We demonstrate the robust nature of EPFT in determining excess entropy for diverse systems with many-body interactions. These are steps toward a universal Entropy Pair Functional, EPF, that can be applied with confidence to determine the entropy associated with sophisticated optimized potentials and first principles simulations of liquids, crystals, engineered structures, and defects.     

A Dynamical Classification of the Range of Pair Interactions

We formalize a classification of pair interactions based on the convergence properties of the forces acting on particles as a function of system size. We do so by considering the behavior of the probability distribution function (PDF) P(F) of the force field F in a particle distribution in the limit that the size of the system is taken to infinity at constant particle density, i.e., in the “usual” thermodynamic limit. For a pair interaction potential V(r) with V(r→∞)∼1/r ^γ defining a bounded pair force, we show that P(F) converges continuously to a well-defined and rapidly decreasing PDF if and only if the pair force is absolutely integrable, i.e., for γ>d−1, where d is the spatial dimension. We refer to this case as dynamically short-range, because the dominant contribution to the force on a typical particle in this limit arises from particles in a finite neighborhood around it. For the dynamically long-range case, i.e., γ≤d−1, on the other hand, the dominant contribution to the force comes from the mean field due to the bulk, which becomes undefined in this limit. We discuss also how, for γ≤d−1 (and notably, for the case of gravity, γ=d−2) P(F) may, in some cases, be defined in a weaker sense. This involves a regularization of the force summation which is generalization of the procedure employed to define gravitational forces in an infinite static homogeneous universe. We explain that the relevant classification in this context is, however, that which divides pair forces with γ>d−2 (or γ<d−2), for which the PDF of the difference in forces is defined (or not defined) in the infinite system limit, without any regularization. In the former case dynamics can, as for the (marginal) case of gravity, be defined consistently in an infinite uniform system.     

The BCS Critical Temperature for Potentials with Negative Scattering Length

We prove that the critical temperature for the BCS gap equation is given by ${T_c = \mu \left( \frac 8\pi {\rm e}^{\gamma -2} + o(1) \right) {\rm e}^{\pi/(2\sqrt \mu a)}}$ in the low density limit μ→ 0, with γ denoting Euler’s constant. The formula holds for a suitable class of interaction potentials with negative scattering length a in the absence of bound states.     

A Remark on High Temperature Polymer Expansion for Lattice Systems with Infinite Range Pair Interactions

We consider the standard high temperature-small activity polymer expansion for lattice spin systems and show how, in many cases using a tree graph equality, various classical results may be improved in some cases or at least proved in a much simpler way.     

BCS theory for trapped ultracold fermions

We develop an extension of the well-known BCS-theory to systems with trapped fermionic atoms. The theory fully includes the quantized energy levels in the trap. The key ingredient is to model the attractive interaction between two atoms by a pseudo-potential which leads to a well defined scattering problem and consequently to a BCS-theory free of divergences. We present numerical results for the BCS critical temperature and the temperature dependence of the gap. They are used as a test of existing semi-classical approximations. Received 18 December 1998     

The large-deviation principle and the BCS model

The BCS model is investigated by the functional integrals method and Euclidean quantum field theory technique. It permits us to apply some version of the Large Deviation Principle and get the exact solution which was obtained earlier by the approximation Hamiltonian method.     

Quadratic Interaction Functional for General Systems of Conservation Laws

For the Glimm scheme approximation \({u_\varepsilon}\) to the solution of the system of conservation laws in one space dimension

$$u_t + f(u)_x = 0, \qquad u(0, x) = u_0(x) \in \mathbb{R}^n,$$

with initial data u ₀ with small total variation, we prove a quadratic (w.r.t. Tot. Var. (u ₀)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux f are made (apart from smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems.

More precisely, we obtain the following results:

• a new analysis of the interaction estimates of simple waves;
• a Lagrangian representation of the derivative of the solution, i.e., a map \({\mathtt{x}(t, w)}\) which follows the trajectory of each wave w from its creation to its cancellation;
• the introduction of the characteristic interval and partition for couples of waves, representing the common history of the two waves;
• a new functional \({\mathfrak{Q}}\) controlling the variation in speed of the waves w.r.t. time.

This last functional is the natural extension of the Glimm functional for genuinely nonlinear systems.

The main result is that the distribution \({D_{t} \hat \sigma_k(t,w)}\) is a measure with total mass \({\leq \mathcal{O}(1) {\rm Tot. Var.} (u_0)^2}\) , where \({\hat{\sigma}_k(t, w)}\) is the speed given to the wave w by the Riemann problem at the grid point \({(i\varepsilon, \mathtt{x}(i\varepsilon, w)), t \in [i\varepsilon, (i + 1)\varepsilon)}\).     

超导BCS理论的建立

介绍了巴丁、库珀和施里弗的生平,回顾了BCS理论的建立过程,综述了BCS理论建立的影响和意义,以及由此带给人们的有益启示.     

适用于配对近似计算的集体对凝聚组态变分方法

集体对凝聚组态(collective-pair condensate)的能量变分可用于判断在原子核低激发态中哪些集体对是重要的,并且可以同时给出这些重要集体对的具体结构。这在一定程度上缓解了此前原子核配对近似(Nucleon Pair Approximation, NPA)计算一直深受困扰的集体对不确定性问题。针对过渡区原子核132Ba的试探性计算体现了这种方法的优越性。它可以给出132Ba的三轴形变参数,它也定量地说明了这一原子核$I\!=\!10$基带回弯效应的主导集体对种类。更为重要的是,中子负宇称对凝聚状态的变分计算解释了为什么负宇称集体对这种回弯效应可以呈现出巨大影响。     

The External Field Dependence of the BCS Critical Temperature

We consider the Bardeen–Cooper–Schrieffer free energy functional for particles interacting via a two-body potential on a microscopic scale and in the presence of weak external fields varying on a macroscopic scale. We study the influence of the external fields on the critical temperature. We show that in the limit where the ratio between the microscopic and macroscopic scale tends to zero, the next to leading order of the critical temperature is determined by the lowest eigenvalue of the linearization of the Ginzburg–Landau equation.     

The Lax Pair and Boundary K-Matrices for XXC Model

The Lax pair and corresponding boundary K-matrices for the XXC model are derived. The time-independent transfer matrix with the spectral parameter is constructed, which means the system to be Lax integrable. Our K-matrices include the result used by Arnaudon and Martssarani (D. Arnaudon and Z. Maassarani, lapth-695/98, laval-phy-22/98) as a special case.     

ETC相互作用对NLC实验中顶夸克对产生过程CP破坏的贡献

在Topcolor辅助的多标度人工色(TOPMTC)模型框架下计算了扩充人工色(ETC)相互作用对顶夸克对产生过程中CP破坏参数δ的贡献(δ=σ[e–e→t(–)–t(–)]–σ[e–e→t(+)–t(+)]/σ(e–e→t–t)).计算结果表明,对于合理的参数取值,ETC相互作用可对参数δ产生较大的修正(8.65×10^－3≤δ≤1.09×10^－2).期望在将来的NLC实验中能观测到此修正效应.     

The Energy-Momentum Tensor for Electromagnetic Interactions

We compute the energy tensor and the energy-momentum tensor for electrodynamics coupled to the current of a charged scalar field and for electrodynamics coupled tothe current of a Dirac spinor field, without using the equations of motion.     

Overcomplete Free Energy Functional for D = 1 Particle Systems with Next Neighbor Interactions

We deduce an overcomplete free energy functional for D=1 particle systems with next neighbor interactions, where the set of redundant variables are the local block densities _i of i interacting particles. The idea is to analyze the decomposition of a given pure system into blocks of i interacting particles by means of a mapping onto a hard rod mixture. This mapping uses the local activity of component i of the mixture to control the local association of i particles of the pure system. Thus it identifies the local particle density of component i of the mixture with the local block density _i of the given system. Consequently, our overcomplete free energy functional takes on the hard rod mixture form with the set of block densities _i representing the sequence of partition functions of the local aggregates of particle numbers i. The system of equations for the local particle density of the original system is closed via a subsidiary condition for the block densities in terms of . Analoguous to the uniform isothermal-isobaric technique, all our results are expressible in terms of effective pressures. We illustrate the theory with two standard examples, the adhesive interaction and the square-well potential. For the uniform case, our proof of such an overcomplete format is based on the exponential boundedness of the number of partitions of a positive integer (Hardy-Ramanujan formula) and on Varadhan's theorem on the asymptotics of a class of integrals.     

The Two-Nucleon Mechanism for the Neutrinoless Double Beta Decay and the BCS Nuclear Wave Function

A method of using the Bardeen-Cooper-Schrieffer (BCS) nuclear wave function to treat the two-nucleon mechanism for neutiinoless double beta decay process 0⁺ → 0⁺ is proposed.The neutrinoless decay mode and the neutrinoless decay accompanied by a Majoron emission mode of ⁸²Se are studikd. Our cdculated results show that to reproduce the experimental value of γ^(ov) > 1.8 × 10²² yr for neutrinoless double beta decai of ⁸²Se the Majorana neutrino mass m_v < 6.2 eV and the mixing parameter of right-handed current η < 7.0 × 10^-6 In the emission with a Majoron mode the effective Majoron coupling to neutrino is deduced from the experimental value of γ^(ov,H) > 4.4 × 10²⁰ yr for ⁸²Se with the result H0> < 6.2 × 10^-4.     

Self-consistent selection of a superconducting representation for the BCS model

Taking the BCS Hamiltonian written in second-quantized form, a modified form of Umezawa's self-consistent field theory method is applied, and a unitarily nonequivalent representation is selected in which the Hamiltonian obviously describes a superconducting system. This result is not at all obvious, since the original Hamiltonian is completely symmetric, and there is no reason a priori for expecting it to describe an asymmetric superconducting configuration. All higher order terms are accounted for, and in doing so, one finds the existence of the energy-gap condition for Cooper pairs. The representation is picked out without using the adiabatic theorem or pair approximation, as is typically done in the self-consistent method for superconductivity.     

A new method for the thermomodynamics of the BCS model

Using large deviations in combination with the Berezin-Lieb inequalities, we analyse the phase-transition in the BCS model with non-constant energies and interactions.     

Compact solution for a multizone BCS and TTN crack model

This paper evaluates a compact formula incorporating both the Bilby-Cottrell-Swinden (BCS) and the Taira-Tanaka-Nakai (TTN) by considering a rectilinear crack traversing an arbitrary number of zones with different friction stresses. Symmetry is not required. The expressions for the dislocation density and the plastic displacement as well as the stress distribution in the plane of the crack are obtained.