The statistical error of green's function Monte Carlo

The statistical error in the ground state energy as calculated by Green's Function Monte Carlo (GFMC) is analyzed and a simple approximate formula is derived which relates the error to the number of steps of the random walk, the variational energy of the trial function, and the time step of the random walk. Using this formula it is argued that as the thermodynamic limit is approached withN identical molecules, the computer time needed to reach a given error per molecule increases asN ^h where 0.5 <b < 1.5 and as the nuclear chargeZ of a system is increased the computer time necessary to reach a given error grows asZ ^5.5. Thus GFMC simulations will be most useful for calculating the properties of lowZ elements. The implications for choosing the optimal trial function from a series of trial functions is also discussed.     

Mean shape of large semi-flexible tethered vesicles

The mean shapes of decorated icosahedron models for closed semi-flexible tethered networks are determined using molecular dynamics simulation techniques. It is found that the radii of curvature at center of the edges of the icosahedron both scale asN ^1/2 for the system sizes studied, whereN is proportional to the area of the surface. The radius of curvature at center of triangular faces is also found to scale asN ^1/2. Results for the mean vesicle shape are compared to those of minimum energy solutions of networks of similar size and elastic moduli. It is argued that system sizes several orders of magnitude larger than those studied here are necessary in order to observe the asymptotic scaling behavior predicted by Witten and Li.Dedicated to Prof. H. Wagner on the occasion of his 60th birthday     

Dynamical Triangulation Models with Matter:¶High Temperature Region

We consider a canonical ensemble with a fixed number N of triangles for planar dynamical triangulation models with compact spin in the high temperature region. We find the asymptotics of the partition function Z(N) and reveal the analytic properties of the generating function U(x)=∑: Z(N)x ^N . New cluster expansion techniques are developed for this case. For fixed triangulation it would be quite standard but for random triangulations one has to deal with the non-zero entropy of the space between clusters. It is a multiscale expansion, where the role of scale is played by a topological parameter – the maximal length of chains of imbedded not simply connected clusters. Received: 19 June 2001 / Accepted: 12 October 2001     

Overcoming artificial spatial correlations in simulations of superstructure domain growth with parallel Monte Carlo algorithms

We study the applicability of parallelized/vectorized Monte Carlo (MC) algorithms to the simulation of domain growth in two-dimensional lattice gas models undergoing an ordering process after a rapid quench below an order-disorder transition temperature. As examples we consider models with 2×1 andc(2×2) equilibrium superstructures on the square and rectangular lattices, respectively. We also study the case of phase separation (1×1 islands) on the square lattice. A generalized parallel checkerboard algorithm for Kawasaki dynamics is shown to give rise to artificial spatial correlations in all three models. However, only ifsuperstructure domains evolve do these correlations modify the kinetics by influencing the nucleation process and result in a reduced growth exponent compared to the value from the conventional heat bath algorithm with random single-site updates. In order to overcome these artificial modifications, two MC algorithms with a reduced degree of parallelism (hybrid and mask algorithms, respectively) are presented and applied. As the results indicate, these algorithms are suitable for the simulation of superstructure domain growth on parallel/vector computers.     

Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity

The computational complexity of internal diffusion-limited aggregation (DLA) is examined from both a theoretical and a practical point of view. We show that for two or more dimensions, the problem of predicting the cluster from a given set of paths is complete for the complexity class CC, the subset of P characterized by circuits composed of comparator gates. CC-completeness is believed to imply that, in the worst case, growing a cluster of size n requires polynomial time in n even on a parallel computer. A parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths, and then corrects defects in the guessed cluster through a nonlocal annihilation process. The parallel running time of the relaxation algorithm for two-dimensional internal DLA is studied by simulating it on a serial computer. The numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n. Thus the computational resources needed to grow large clusters are significantly less on average than the worst-case analysis would suggest. For a parallel machine with k processors, we show that random clusters in d dimensions can be generated in ((n/k+logk)n ^2/d ) steps. This is a significant speedup over explicit sequential simulation, which takes (n ^1+2/d ) time on average. Finally, we show that in one dimension internal DLA can be predicted in (logn) parallel time, and so is in the complexity class NC.     

Growth kinetics in multicomponent fluids

The hydrodynamic effects on the late-stage kinetics in spinodal decomposition of multicomponent fluids are examined using a lattice Boltzmann scheme with stochastic fluctuations in the fluid and at the interface. In two dimensions, the three- and four-component immiscible fluid mixture (with a 1024² lattice) behaves like an off-critical binary fluid with an estimated domain growth oft ^0.4±0.03 rather thant ^1/3 as previously estimated, showing the significant influence of hydrodynamics. In three dimensions (with a 256³ lattice), we estimate the growth ast ^0.96±0.05 for both critical and off-critical quenches, in agreement with phenomenological theory.     

Computing the Loewner Driving Process of Random Curves in the Half Plane

We simulate several models of random curves in the half plane and numerically compute the stochastic driving processes that produce the curves through the Loewner equation. Our models include models whose scaling limit is the Schramm-Loewner evolution (SLE) and models for which it is not. We study several tests of whether the driving process is Brownian motion, as it is for SLE. We find that testing only the normality of the process at a fixed time is not effective at determining if the random curves are an SLE. Tests that involve the independence of the increments of Brownian motion are much more effective. We also study the zipper algorithm for numerically computing the driving function of a simple curve. We give an implementation of this algorithm which runs in a time O(N ^1.35) rather than the usual O(N ²), where N is the number of points on the curve.     

通过盐酸刻蚀和生长作用来选择性合成不同尺寸大小的金纳米团簇

Controllable syntheses of different-sized gold nanoclusters are of great significance for their fundamental science and practical applications.In this work,we achieve the controllable and selective syntheses of Au₇ and Au₁₃ clusters through adding HCl to the traditional Au₁₁ synthetic route at different reaction time.Time-dependent mass spectra and UVVis spectra were employed to monitor these two HCl-directed processes,and revealed the distinct roles of HCl as an etchant or a growth promotor,respectively.Furthermore,parallel experiments on independent synthetic routes involving only non-chlorine H⁺(acetic acid) or Cl^-(tetraethy lammonium chloride) instead of HCl were performed,which illustrated the main role of H⁺-etching and Cl^--assisted growth in HCl-directed cluster synthetic routes.We propose the HCl-etching is mainly achieved via the H⁺ action to break the Au (I)-PPh₃ part of clusters,while the HCl-promoted growth is realized via the attachment of Au-Cl species to the pre-formed clusters.     

Are random walks random?

Random walks have been created using the pseudo-random generators in different computer language compilers (BASIC, PASCAL, FORTRAN, C++) using a Pentium processor. All the obtained paths have apparently a random behavior for short walks (2¹⁴ steps). From long random walks (2³³ steps) different periods have been found, the shortest being 2¹⁸ for PASCAL and the longest 2³¹ for FORTRAN and C++, while BASIC had a 2²⁴ steps period. The BASIC, PASCAL and FORTRAN long walks had even (2 or 4) symmetries. The C++ walk systematically roams away from the origin. Using deviations from the mean-distance rule for random walks, d² ∝ N, a more severe criterion is found, e.g. random walks generated by a PASCAL compiler fulfills this criterion to N < 10 000.     

Finite-N fluctuation formulas for random matrices

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic Σ _j ^N =1 (x _j ? 〈x〉) is computed exactly and shown to satisfy a central limit theorem asN → ∞. For the circular random matrix ensemble the p.d.f.’s for the statistics ½Σ _j ^N =1 (θ _j ?π) and ? Σ _j ^N =1 log 2 |sinθ _j/2| are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem asN → ∞.     

Absolute cross sections of electron attachment to molecular clusters. Part II: Formation of (H2O) N? , (N2O) N? , and (N2) N?

Absolute cross sections σ⁻(E, N) of electron attachment to clusters (H₂O) _N , (N₂O) _N , and (N₂) _N for varying electron energy E and cluster size N are measured by using crossed electron and cluster beams in a vacuum. Continua of σ⁻(E) are found that correlate well with the functions of electron impact excitation of molecules’ internal degrees of freedom. The electron is attached through its solvation in a cluster. In the formation of (H₂O) _N ⁻ , (N₂O) _N ⁻ , and (N₂) _N ⁻ , the curves σ⁻(N) have a well-defined threshold because of a rise in the electron thermalization and solvation probability with N. For (H₂O)₉₀₀, (N₂O)₃₅₀, and (N₂)₂₆₀ clusters at E = 0.2 eV, the energy losses by the slow electron in the cluster are estimated as 3.0 × 10⁷, 2.7 × 10⁷, and 6.0 × 10⁵ eV/m, respectively. It is found that the growth of σ⁻ with N is the fastest for (H₂O) _N and (N₂) _N clusters at E → 0 as a result of polarization capture of the s-electron. Specifically, at E = 0.1 eV and N = 260, σ⁻ = 3.0 × 10⁻¹³ cm² for H₂O clusters, 8.0 × 10⁻¹⁴ cm² for N₂O clusters, and 1.4 × 10⁻¹⁵ cm² for N₂ clusters; at E = 11 eV, σ⁻ = 9.0 × 10⁻¹⁶ cm² for (H₂O)₂₀₀ clusters, 2.4 × 10⁻¹⁴ cm² for (N₂O)₃₅₀ clusters, and 5.0 × 10⁻¹⁷ cm² for (N₂)₂₆₀ clusters; finally, at E = 30 eV, σ⁻ = 3.6 × 10⁻¹⁷ cm² for (N₂O)₁₀ clusters and 3.0 × 10⁻¹⁷ cm² for (N₂)₁₂₅ clusters. Original Russian Text ? A.A. Vostrikov, D.Yu. Dubov, 2006, published in Zhurnal Tekhnicheskoĭ Fiziki, 2006, Vol. 76, No. 12, pp. 1–15.     

57Fe Mössbauer effect study of well-crystallized goethite (α-FeOOH)

Well-crystallized, natural goethite has been investigated as a function of temperature. The saturation magnetic hyperfine field was found to be 507 kOe and the Néel temperatureT _N=(400±2) K. The antiferromagnetic structure is stable in external fields of up to 6O kOe and the spin direction is at right angles with respect to the EFG's principal axis. AtT/T _N<0.5, the reduced hyperfine field varies asT ² and in the range 0.80<T/T _N<0.97 asT ^1/3. From the temperature dependence of the isomer shift, the Debije temperature was found to be 440 K from which a relative Mössbauer fraction of 0.96 against hematite was evaluated.     

Loop-erased self-avoiding random walk in two and three dimensions

If(n) is the position of the self-avoiding random walk in ^d obtained by erasing loops from simple random walk, then it is proved that the mean square displacementE(¦(n)¦²) grows at least as fast as the Flory predictions for the usual SAW, i.e., at least as fast asn ^3/2 ford=2 andn ^6/5 ford=3. In particular, if the mean square displacement of the usual SAW grows liken ^1.18... ind=3, as expected, then the loop-erased process is in a different universality class.     

The distribution of Lyapunov exponents: Exact results for random matrices

Simple exact expressions are derived for all the Lyapunov exponents of certainN-dimensional stochastic linear dynamical systems. In the case of the product of independent random matrices, each of which has independent Gaussian entries with mean zero and variance 1/N, the exponents have an exponential distribution asN. In the case of the time-ordered product integral of exp[N ^–1/2 dW], where the entries of theN×N matrixW(t) are independent standard Wiener processes, the exponents are equally spaced for fixedN and thus have a uniform distribution as N.John S. Guggenheim Memorial Fellow. Research supported in part by NSF Grant MCS 80-19384     

Simple finite-difference algorithm for calculating waveguide modes

Abstact A simple algorithm for solving the finite-difference (FD) equations for the mode eigenvalues and field distributions of a linear waveguide is presented. By applying the discretized Helmholtz operator column-by-column to an index distribution defined on an N x N grid, a matrix whose size is only a few times N x N is obtained. This yields a reduction in computation time and space compared with the other classical FD approaches which involve an N ² x N ² matrix. Our method is tested against problems for which the exact solutions are known, and we find a high degree of accuracy. Despite the existence of fast algorithms for the treatment of the classical N ² x N ² matrix, our new algorithm presents some advantages over existing FD methods of comparable speed, including: the ability to find all the modes and associated field profiles, very high numerical stability, and no numerical approximations in the procedure. In addition, some general optimum expressions for the domain size and density of grid points which are consistent with the desired precision are provided, and apply to any FD method including ours.     

Towardsab Initio simulations of concentrated suspensions

Determination of many-body interactions between particles of arbitrary shape in a viscous fluid is a key problem in the simulation of concentrated suspensions. Three-dimensional flows involving such complex fluid-solid boundaries are beyond the scope of spatial methods, even on supercomputers. Boundary integral methods convert the three-dimensional PDE to a two-dimensional integral equation. Unfortunately, conventional boundary methods yield Fredholm integral equations of the first kind, and dense linear systems which are too large for accurate solution. We have pursued a different boundary integral formulation, which yields Fredholm integral equations of the second kind; these arc amenable to iterative solution. The velocity representation involves a compact operator, so a discrete spectrum results. Wielandt deflations give dramatic reductions in the spectral radius and accurate solutions are obtained after only a few iterations (typically less than 10). An analytic construction of the spectrum for sphere sphere interactions confirms these numerical results. The mathematics is similar to that encountered in the mixing ofd-atomic orbitals to form bonding/antibonding molecular orbitals in transition metals. The memory-saving version of our code can be implemented directly on a dedicated MicroVAX to solve problems involving clusters of less than a dozen particles. For a fixed number of processors, the algorithm grows essentially asN ², whereN is the system size, so computational times are readily estimated on more powerful super-minicomputers and supercomputers using standard dot-product benchmarks. The algorithm is especially ideal for gigaflop and teraflop parallel array processors under construction in a number of computer companies; an analysis of the spectrum reveals that asynchronous iterative methods will converge, leading the way to a rigorous formulation of screening concepts for suspended particles of arbitrary shape.     

Factorisations for partition functions of random Hermitian matrix models

The partition functionZ _N , for Hermitian-complex matrix models can be expressed as an explicit integral over ^N , whereN is a positive integer. Such an integral also occurs in connexion with random surfaces and models of two dimensional quantum gravity. We show thatZ _N can be expressed as the product of two partition functions, evaluated at translated arguments, for another model, giving an explicit connexion between the two models. We also give an alternative computation of the partition function for the ⁴-model. The approach is an algebraic one and holds for the functions regarded as formal power series in the appropriate ring.     

A boson representation for SU (N) lattice gauge theories

SU(N) lattice gauge theories are reformulated in terms of fields varying over non-compact spaces ^N , transforming asN dimensional representations of SU(N) and integrated with Gaussian measure. This reformulation is equivalent to a boson operator representation. Strong coupling expansions based on this formalism do not involve SU(N) vector coupling coefficients.     

A path-functional field theory of lattice gauge models and the large-N limit

We transform lattice gauge models to a theory of functional fields defined on a set of closed paths. Some relevant properties of the formalism are discussed in detail, with emphasis on symmetry and topological structure. We then investigate the large-N limit of the U(N) lattice gauge model in arbitrary dimensions using this formalism. Assuming the existence of the limit, we show, to arbitrary order of the strong coupling expansion parameter (g²N)^?, which is kept fixed, that for the leading contribution in the limit: (i) the flow of indices in color space can be represented by planar diagrams; (ii) when the diagrams are immersed in space-time they are random surfaces without handles; (iii) there are interactions of the surfaces which can be depicted as the formation of multisheet bubblesw in the surfaces. This formalism also makes it possible to set up a gauge-invariant mean-field approximation.     

Multiple trapping of random walkers on periodic lattices

Formulas are obtained for the mean absorption time of a set ofk independent random walkers on periodic space lattices containingq traps. We consider both discrete (here we assume simultaneous stepping) and continuous-time random walks, and find that the mean lifetime of the set of walkers can be obtained, via a convolution-type recursion formula, from the generating function for one walker on the perfect lattice. An analytical solution is given for symmetric walks with nearest neighbor transitions onN-site rings containing one trap (orq equally spaced traps), for both discrete and exponential distribution of stepping times. It is shown that, asN , the lifetime of the walkers is of the form Ta_kN², whereT is the average time between steps. Values ofa _k, 2 Sk 6, are provided.