Solution Space Coupling in the Random K-Satisfiability Problem

The random K-satisfiability(K-SAT)problem is very difcult when the clause density is close to the satisfiability threshold.In this paper we study this problem from the perspective of solution space coupling.We divide a given difcult random K-SAT formula into two easy sub-formulas and let the two corresponding solution spaces to interact with each other through a coupling field x.We investigate the statistical mechanical property of this coupled system by mean field theory and computer simulations.The coupled system has an ergodicity-breaking(clustering)transition at certain critical value x d of the coupling field.At this transition point,the mean overlap value between the solutions of the two solution spaces is very close to 1.The mean energy density of the coupled system at its clustering transition point is less than the mean energy density of the original K-SAT problem at the temperature-induced clustering transition point.The implications of this work for designing new heuristic K-SAT solvers are discussed.     

Glassy behavior and jamming of a random walk process for sequentially satisfying a constraint satisfaction formula

Random K-satisfiability (K-SAT) is a model system for studying typical-case complexity of combinatorial optimization. Recent theoretical and simulation work revealed that the solution space of a random K-SAT formula has very rich structures, including the emergence of solution communities within single solution clusters. In this paper we investigate the influence of the solution space landscape to a simple stochastic local search process SEQSAT, which satisfies a K-SAT formula in a sequential manner. Before satisfying each newly added clause, SEQSAT walk randomly by single-spin flips in a solution cluster of the old subformula. This search process is efficient when the constraint density α of the satisfied subformula is less than certain value α_cm; however it slows down considerably as α> α_cm and finally reaches a jammed state at α≈α_j. The glassy dynamical behavior of SEQSAT for α≥α_cm probably is due to the entropic trapping of various communities in the solution cluster of the satisfied subformula. For random 3-SAT, the jamming transition point α_j is larger than the solution space clustering transition point α_d, and its value can be predicted by a long-range frustration mean-field theory. For random K-SAT with K ≥ 4, however, our simulation results indicate that α_j = α_d. The relevance of this work for understanding the dynamic properties of glassy systems is also discussed.     

Asymptotic stability of stationary states in the wave equation coupled to a nonrelativistic particle

We consider the Hamiltonian system consisting of a scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subjected to an external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. It is assumed that the charge density satisfies the Wiener condition, which is a version of the “Fermi Golden Rule.” We prove that in the large time approximation, any finite energy solution, with the initial state close to the some stable stationary solution, is a sum of this stationary solution and a dispersive wave which is a solution of the free wave equation.     

The transition temperature of coupled magnetic planes including fluctuations in the spacer thickness

The introduction of a weak coupling between isolated planes, each of which would order ferromagnetically, always raises the transition temperature. The coupled system will have either ferromagnetic or staggered order depending on the sign of the interplanar coupling. If there are fluctuations in the interplanar coupling caused by variations in the spacer thickness, then each plane will experience a random magnetic field caused by the combination of strong correlations in the neighbouring planes and the random exchange.In this paper, the effect of the random field is treated using mean field theory and the Bethe–Peierls approximation. It is shown that the results may be used to fit the data on some real multilayers in which the transition temperature falls below the result for uncoupled films at some values of the average spacer thickness.     

Phase transitions of indirect excitons in coupled quantum wells: The role of disorder

The theory of what happens to a superfluid in a random field, known as the “dirty boson” problem, directly relates to a real experimental system presently under study by several groups, namely excitons in coupled semiconductor quantum wells. We consider the case of bosons in two dimensions in a random field, when the random field can be large compared to the repulsive exciton–exciton interaction energy, but is small compared to the exciton binding energy. The interaction between excitons is taken into account in the ladder approximation. The coherent potential approximation (CPA) allows us to derive the exciton Green's function for a wide range of the random field strength, and in the weak-scattering limit CPA results in the second-order Born approximation. For quasi-two-dimensional excitonic systems, the density of the superfluid component and the Kosterlitz–Thouless temperature of the superfluid phase transition are obtained, and are found to decrease as the random field increases.     

Casimir Effect in Hemisphere Capped Tubes

In this paper we investigate the vacuum densities for a massive scalar field with general curvature coupling in background of a (2 + 1)-dimensional spacetime corresponding to a cylindrical tube with a hemispherical cap. A complete set of mode functions is constructed and the positive-frequency Wightman function is evaluated for both the cylindrical and hemispherical subspaces. On the base of this, the vacuum expectation values of the field squared and energy-momentum tensor are investigated. The mean field squared and the normal stress are finite on the boundary separating two subspaces, whereas the energy density and the parallel stress diverge as the inverse power of the distance from the boundary. For a conformally coupled field, the vacuum energy density is negative on the cylindrical part of the space. On the hemisphere, it is negative near the top and positive close to the boundary. In the case of minimal coupling the energy density on the cup is negative. On the tube it is positive near the boundary and negative at large distances. Though the geometries of the subspaces are different, the Casimir pressures on the separate sides of the boundary are equal and the net Casimir force vanishes. The results obtained may be applied to capped carbon nanotubes described by an effective field theory in the long-wavelength approximation.     

两体相互作用费米系统在自旋轨道耦合和塞曼场中的基态转变

在超冷费米系统中实现人造规范势的突破,吸引了许多新问题的研究,展现了许多新奇的物理现象.本文研究了在环阱中,具有自旋轨道耦合和塞曼作用的两体相互作用费米模型.通过平面波展开的方法,解析求解了两体费米系统的本征能态.系统的总动量为守恒量,可以在不同总动量空间中研究能谱.研究发现:随着塞曼相互作用增大,在不同总动量空间,两体费米系统的本征能量均逐渐降低,系统基态从总动量为零空间转变到有限值空间.从吸引到排斥相互作用,无塞曼相互作用时,基态总动量始终为零,有塞曼相互作用时,基态总动量从零转变为有限值.通过单粒子和基态动量分布研究,本文直观地揭示了由塞曼能级劈裂引起的基态转变.     

Directed Dominating Set Problem Studied by Cavity Method: Warning Propagation and Population Dynamics

The minimal dominating set for a digraph (directed graph) is a prototypical hard combinatorial optimization problem. In a previous paper, we studied this problem using the cavity method. Although we found a solution for a given graph that gives very good estimate of the minimal dominating size, we further developed the one step replica symmetry breaking theory to determine the ground state energy of the undirected minimal dominating set problem. The solution space for the undirected minimal dominating set problem exhibits both condensation transition and cluster transition on regular random graphs. We also developed the zero temperature survey propagation algorithm on undirected Erdös-Rényi graphs to find the ground state energy. In this paper we continue to develope the one step replica symmetry breaking theory to find the ground state energy for the directed minimal dominating set problem. We find the following. (i) The warning propagation equation can not converge when the connectivity is greater than the core percolation threshold value of 3.704. Positive edges have two types warning, but the negative edges have one. (ii) We determine the ground state energy and the transition point of the Erdös-Rényi random graph. (iii) The survey propagation decimation algorithm has good results comparable with the belief propagation decimation algorithm.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

The use of iteration factors in the solution of the NLTE line transfer problem—II. Multilevel atom

The iteration factors method (IFM) developed in Paper I [1] (Atanackovi?-Vukmanovi? and Simonneau, 1994) to solve the NLTE line transfer problem for a two-level atom model, is extended here to deal with a multilevel atom case. At the beginning of each iteration step, for each line transition, angle and frequency averaged depth-dependent iteration factors are computed from the formal solution of radiative transfer (RT) equation and used to close the system of the RT equation moments, non-linearly coupled with the statistical equilibrium (SE) equations. Non-linear coupling of the atomic level populations and the corresponding line radiation field intensities is tackled in two ways. One is based on the linearization of the equations with respect to the relevant variables, and the other on the use of the old (known from the previous iteration) level populations in the line-opacity-like terms of the SE equations. In both cases the use of quasi-invariant iteration factors provided very fast and accurate solution. The properties of the proposed procedures are investigated in detail by applying them to the solution of the prototype multilevel RT problem of Avrett and Loeser [2], and compared with the properties of some other methods.     

原子核Gamow-Teller共振和β衰变寿命的超越平均场描述

原子核的β衰变是决定宇宙中从铁到铀重元素合成的关键核过程之一。原子核β衰变的主导核跃迁是Gamow-Teller(GT)跃迁,因此,研究原子核β衰变寿命的关键是准确描述原子核的GT跃迁。描述原子核GT跃迁和β衰变寿命最常用的理论模型之一为无规相位近似(RPA)模型。然而,由于该模型仅考虑了一粒子一空穴激发组态,因此无法给出GT共振宽度,并容易高估β衰变寿命。为了克服上述困难,基于Skyrme密度泛函,发展了包含粒子振动耦合效应的无规相位近似(RPA+PVC)模型。相比于RPA模型,该模型在组态空间进一步考虑了一粒子一空穴和声子的耦合组态,从而包含了超越平均场的多体关联效应。为了推广至开壳原子核的研究,进一步考虑了对关联效应,发展了包含准粒子振动耦合效应的准粒子无规相位近似(QRPA+QPVC)模型。基于上述模型,研究了幻数原子核和超流原子核的GT跃迁、β衰变和β+/电子俘获。研究发现,采用同一组Skyrme相互作用参数SkM*,上述模型能够重现实验测量的GT共振宽度和跃迁强度分布,部分解释实验观测的GT跃迁强度压低问题,并同时改进对β衰变寿命的描述。该文针对上述最新研究进展进行了综述,并对将来的发展方向给出展望。     

轴向磁场下感应耦合放电模式转换的实验研究

为探究轴向磁场对纯Ar感应耦合等离子体放电模式转换的影响,设计并搭建一整套等离子体产生装置展开实验研究,引入阻抗分析法对放电模式转换进行判断,并得到了朗缪尔探针法的验证。实验发现,当气压为10 Pa时,轴向磁场强度的增加使得E-H和H-E模式转换的阈值功率增大;同时,随着轴向磁场的增强,放电中心区域的电子密度不断降低。初步分析认为,这是由于带电粒子在洛伦兹力作用下做回旋运动,导致高能电子在垂直磁场方向上的碰撞减少,降低了电子密度以及感应耦合效率。进一步分析电子能量概率函数(EEPF)发现,在E模式下,轴向磁场对电子运动的约束作用更加明显,高能电子(>27 eV)所占比例增多,电子能量分布更加均匀。     

Typical universal entanglers

A universal entangler is a very powerful fault-tolerant entangling device for generating quantum entanglements from any joint states. Our paper aims to address the construction of universal entanglers. We prove that universal entanglers may be obtained from random unitary gates according to the Harr measure. The success probability is close to 1 for large system spaces. This result represents the typical density of entanglement subspaces in large state spaces. It also partially solves an open problem of universal bipartite entanglers and is explained by some experiment simulations.     

QCD phase transitions from relativistic hadron models

The models of translationally invariant infinite nuclear matter in the relativistic mean field models are very interesting and simple, since the nucleon can connect only to a constant vector and scalar meson field. Can one connect these to the complicated phase transitions of QCD? For an affirmative answer to this question, one must consider models where the coupling contstants to the scalar and vector fields depend on density in a nonlinear way, since as such the models are not explicitly chirally invariant. Once this is ensured, indeed one can derive a quark condensate indirectly from the energy density of nuclear matter which goes to zero at large density and temperature. The change to zero condensate indicates a smooth phase transition.     

QCD phase transitions from relativistic hadron models

The models of translationally invariant infinite nuclear matter in the relativistic mean field models are very interesting and simple, since the nucleon can connect only to a constant vector and scalar meson field. Can one connect these to the complicated phase transitions of QCD? For an affirmative answer to this question, one must consider models where the coupling contstants to the scalar and vector fields depend on density in a nonlinear way, since as such the models are not explicitly chirally invariant. Once this is ensured, indeed one can derive a quark condensate indirectly from the energy density of nuclear matter which goes to zero at large density and temperature. The change to zero condensate indicates a smooth phase transition.     

Random excitation of a vibratory system with autoparametric interaction

The paper is concerned with the broad band random excitation of a two degree of freedom vibratory system with non-linear coupling of autoparametric type. A general equation for the evolution of the moments of any order of the response co-ordinates is derived by using stochastic calculus and found to represent an infinite hierarchy set. Consideration is given to the determination of the mean square stability boundary for unimodal response with no transverse motion of the coupled system. Two approximate solutions are obtained. These are first of all a solution based on a Gaussian closure technique applied to the system moment equations which allows the stability condition to be determined from the eigenvalues of a four by four matrix, and secondly a perturbation solution which leads to a simple analytical expression for the stability boundary. The two methods give results in close agreement for low values of system damping, but which differ appreciably at high damping levels. Finally, results are obtained from an investigation of the response regions of a laboratory model excited from a random noise generator. The experimental results are found to give excellent correlation with the predicted instability boundaries in the close neighbourhood of internal resonance but show a distinct indication of a wider instability region than predicted by both analytical methods.     

Hall conductivity for two-dimensional magnetic systems

A Kubo inspired formalism is proposed to compute the longitudinal and transverse dynamical conductivities of an electron in a plane (or a gas of electrons at zero temperature) coupled to the potential vector of an external local magnetic field, with the additional coupling of the spin degree of freedom of the electron to the local magnetic field (Pauli Hamiltonian). As an example, the homogeneous magnetic field Ha]] conductivity is rederived. The case of the vortex at the origin is worked out in detail. This system happens to display a transverse Ha]] conductivity (P breaking effect) which is subleading in volume compared to the homogeneous field case, but diverging at small frequencies like 1/ω². A perturbative analysis is proposed for the conductivity in the random magnetic impurity problem (Poissonian vortices in the plane). At first order in perturbation theory, the Ha]] conductivity displays oscillations close to the classical straight line conductivity of the mean magnetic field.