Cluster numbers by series in directed percolation

Cluster numbersn _s from exact expansions for directed percolation in dimensions 2 to 4 are studied at various probability values. The main considerations as regards the asymptotic behaviour are that: $$(i) n_s \propto \exp ( - const s^Z ) for p< p_c (Z = 1) and p > p_c \left( {Z = 1 - \frac{1}{d}} \right).$$ (ii) at the critical thresholdn _s ～s ^?τ+1, although the data can only be reconciled with existing estimates for the exponent if the scaling ansatz is refined to include a first correction.     

Scattering of electrons due to screw dislocations

The phase dismatching effect on the scattering due to screw dislocations is reformulated to take the discreteness of lattice sites into account. Thet-matrix for an electron scattered from the statep top′ is $$\begin{gathered} t\left( {p,p'} \right) = ip_z T\exp \left\{ {i\left( {p - p'} \right) \cdot m_A } \right\}\exp \left\{ {i\left( {p - p'} \right) \cdot \left( {i + j} \right)/2} \right\} \hfill \\ \cdot \frac{{\left[ {\exp \left( { - ip_y } \right) - \exp \left( {ip'_y } \right)} \right] + \left( {\upsilon _y /\upsilon _x } \right)\left[ {\exp \left( {ip_x } \right) - \exp \left( { - ip'_x } \right)} \right]}}{{1 - \exp \left[ {i\left\{ {\left( {p_x - p'_x } \right) + \left( {\upsilon _y /\upsilon _x } \right)\left( {p_y - p'_y } \right)} \right\}} \right]}} \hfill \\ \end{gathered}$$ for 0≦v _y ≦v _x ≦1 and |p _y |, |p′ _y |?1. Here,v is the group velocity of the incident electron andm _A is the position of the dislocation axis. All vector notations represent vectors in two-dimensional space, the unit vectors of which are represented byi andj. Expressions for |p _y |, |p′ _y |?π and other values ofv are obtained through simple modifications. As an application, the resistivity due to screw dislocations is discussed qualitatively.     

Directed spiral site percolation on the square lattice

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold p _c ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension d _f ≈ 1.733. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as p → p _c with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution P _s(p) show power law behaviour with | p - p _c| in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of P _s(p). The results obtained are in good agreement with other model calculations. Received 10 November 2002 / Received in final form 20 February 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: santra@iitg.ernet.in     

Universal estimate of the gap for the Kac operator in the convex case

The aim of this paper is to prove that ifV is a strictly convex potential with quadratic behavior at ∞, then the quotient μ₂/μ₁ between the largest eigenvalue and the second eigenvalue of the Kac operator defined on L²(? ^m ) by exp ?V(x)/2 · exp Δ_x · exp ?V(x)/2 where Δ_x is the Laplacian on ? ^m satisfies the condition: $${{\mu _2 } \mathord{\left/ {\vphantom {{\mu _2 } {\mu _1 {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} \mathord{\left/ {\vphantom {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} {2,}}} \right. \kern-\nulldelimiterspace} {2,}}}}} \right. \kern-\nulldelimiterspace} {\mu _1 {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} \mathord{\left/ {\vphantom {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} {2,}}} \right. \kern-\nulldelimiterspace} {2,}}}}$$ where σ is such that HessV(x)≥σ>0.     

Symmetries of the pseudo-diffusion equation and related topics

We show in details how to determine and identify the algebra g = {A_i} of the infinitesimal symmetry operators of the following pseudo-diffusion equation (PSDE) LQ ≡ \(\left[ {\frac{\partial }{{\partial t}} - \frac{1}{4}\left( {\frac{{{\partial ^2}}}{{\partial {x^2}}} - \frac{1}{{{t^2}}}\frac{{{\partial ^2}}}{{\partial {p^2}}}} \right)} \right]\) Q(x, p, t) = 0. This equation describes the behavior of the Q functions in the (x, p) phase space as a function of a squeeze parameter y, where t = e ^2y. We illustrate how G _i(λ) ≡ exp[λA _i] can be used to obtain interesting solutions. We show that one of the symmetry generators, A ₄, acts in the (x, p) plane like the Lorentz boost in (x, t) plane. We construct the Anti-de-Sitter algebra so(3, 2) from quadratic products of 4 of the A _i, which makes it the invariance algebra of the PSDE. We also discuss the unusual contraction of so(3, 1) to so(1, 1)? h². We show that the spherical Bessel functions I ₀(z) and K ₀(z) yield solutions of the PSDE, where z is scaling and “twist” invariant.     

Determination of the critical current density and the flux pinning force in single crystals YBa2Cu3O7-δ from the magnetization hysteresis cycles

In this study, the magnetization measurements have been performed on high-temperature superconductor's single crystals YBa₂Cu₃O_7-δ at large ranges of temperature T (15-85 K) and in magnetic fields up to 6T at different values of the angle θ between the applied magnetic field and c-axis. The critical current density Jc deduced from the magnetic hysteresis loops by the Bean formula for H parallel to the c-axis (θ=0°), our results have shown that the critical current density Jc was strongly dependent on the applied magnetic field. The pinning force Fp=Jc×μ₀H was determined from magnetization for H//c, however, a plot of the normalized pinning force density fp= Fp/Fpmax as a function of the reduced magnetic field h= H/Hirr at different temperatures have shown good scaling with the form fp ~h^p(1-h)^q, where p and q are scaling parameters. We also found that the point pinning is more dominant than surface pinning under high temperatures.     

Crossover scaling,correlation function and connectivity in dilute low temperature ferromagnets

For quenched dilute ferromagnets with a fractionp of spins (nearest neighbor exchange energyJ) and a fraction 1 —p of randomly distributed nonmagnetic atoms, a crossover assumption similar to tricritical scaling theory relates the critical exponents of zero temperature percolation theory to the low temperature critical amplitudes and exponents near the critical lineT _c (p)>0. For example, the specific heat amplitude nearT _c (p) is found to vanish, the susceptibility amplitude is found to diverge forT _c (p →p _c ) → 0. (Typically,p _c =20%.) AtT=0 the spin-spin correlation function is argued from a droplet picture to obey scaling homogeneity but (at fixed distance) not to vary like the energy; instead it varies as const + (p _c —p)^2β +? for fixed small distances. A generalization of the correlation function to finite temperatures nearT _c (p) allows to estimate the number of effective percolation channels connecting two sites in the infinite (percolating) network forp>p _c ; this in turn gives, via a dynamical scaling argument, a good approximation for theT=0 percolation exponent 1.6 in the conductivity of random three-dimensional resistor networks. This channel approximation also givesΦ=2 for the crossover exponent; i.e. exp(?2J/kT _c (p)) is an analytic function ofp nearp=p _c . An appendix shows that cluster-cluster correlations atT=0 (excluded volume effects) are responsible for the difference between percolation exponents and the (pure) Ising exponents atT _c (p=1).     

Solid-liquid phase transition in small water clusters: a molecular dynamics simulation study

Water clusters, (H₂O) _n , of varying sizes (n = 8, 12, 16, 20, 24, 28, 32, 36, and 40) have been studied at different temperatures from 0 to 200 K using molecular dynamics simulations. Transitions between solid and liquid phases were investigated to estimate the melting temperature of the clusters. Although the melting temperatures showed non-monotonic behaviour as a function of cluster size, their general tendency follows the classical relationship T _m ∝ n ^?1/3 to the cluster size n. Moreover, it was observed that the liquid-solid surface tension decreased with the cluster size in a similar way to the liquid-vapour surface tension in bulk water. Upon cooling, ice-like crystals were formed from the smaller clusters with n up to 20, while the larger clusters were transformed to glassy structures. The decrease in the glass transition temperature with the cluster size was observed to be much less than the corresponding melting temperature. The mutual order of the melting and glass-transition temperatures were found to be reversed compared with that observed for bulk water.     

Toward the theory of electron and positron states in dielectric clusters

Analytical expressions for the binding energy of electrons and positrons in dielectric clusters, analyzed in this work, neglect the elastic effects. Therefore, we present the density-functional theory for neutral liquid clusters that experience the spontaneous deformation. Using the 1/R-expansion, R being the cluster radius, the exact analytical expressions for the size corrections to the chemical potential, surface tension, and atomic density are derived from the condition of mechanical equilibrium. The problem of calculating these corrections is reduced to calculating the quantities for a liquid with a flat surface. The size compression and tension of density occur in the 1/R and 1/R ² orders respectively. The sizes of charged rigid and elastic critical clusters, for which the electron or positron binding energy is close to zero, are calculated for Xe _N ^? , Kr _N ^? , Ar _N ^? , Ne _N ⁺ , He _N ⁺ . The calculations show significant contribution of self-compression to the binding energy of the excess electron in contrast to the positron.     

General Dynamical Equations for Dingle’s Space-Times Filled with a Charged Non-perfect Fluid

In this paper we have assumed charged non-perfect fluid as the material content of the space-time. The expression for the “mass function-M(r,y,z,t)” is obtained for the general situation and the contributions from the Ricci tensor in the form of material energy density ρ, pressure anisotropy [\fracp₂+p₃2-p₁][\frac{p_{2}+p_{3}}{2}-p_{1}] , electromagnetic field energy ℰ and the conformal Weyl tensor, viz. energy density of the free gravitational field ε (=\frac-3Y₂4p)(=\frac{-3\Psi_{2}}{4\pi}) are made explicit. This work is an extension of the work obtained earlier by Rao and Hasmani (Math. Today XIIA:71, 1993; New Directions in Relativity and Cosmology, Hadronic Press, Nonantum, 1997) for deriving general dynamical equations for Dingle’s space-times described by this most general orthogonal metric,

Coarsening of mass-selected Au clusters on amorphous carbon at room temperature

Low-energy cluster beam deposition was used to deposit mass-selected Au_n clusters (n = 4, 6, 13 and 20) on amorphous carbon (a-C) substrates. The resulting samples were stored at room temperature under ambient conditions for time periods up to 32 months to analyze the coarsening behaviour of the clusters. Cluster-size distributions were measured in regular time intervals by transmission electron microscopy (TEM). The TEM experiments show a significant increase of the average cluster size with time analogous to classical surface Ostwald ripening (OR). The coarsening of Au clusters can be well described by steady-state diffusion-limited kinetics. The derived surface mass-transport diffusion coefficients at room temperature range between 1.1 and 3.8·10⁻²⁵ m² s⁻¹ for our samples. A detailed analysis of values suggests that, the rate of the surface OR for mass-selected Au_n clusters increases with the cluster size in the sequence: Au₄ ≈ Au₆ < Au₁₃ < Au₂₀ for the investigated range of Au clusters. Given that the initial, on-surface cluster-size distributions are nominally monodisperse, classical OR with cluster coarsening based only on the Gibbs-Thomson effect cannot explain the observed coarsening. The activation of the coarsening process is rationalized by initial variations of the cluster sizes due to the deposition process itself and/or the interaction of the clusters with the substrate. Moreover, the presence of initial deposited Au clusters as different isomers with slightly different chemical potential on the substrate, may also initiate the coarsening by surface OR. Furthermore, we find that the coarsening is most pronounced for the paucidispersed sample with Au_m (10 ? m ? 20) clusters. A possible explanation of this behaviour is the presence of an initial distribution of different cluster sizes directly after deposition.     

Separating Solution of a Quadratic Recurrent Equation

In this paper we consider the recurrent equation

$\Lambda_{p+1}=\frac{1}{p}\sum_{q=1}^pf\bigg(\frac{q}{p+1}\bigg)\Lambda _{q}\Lambda_{p+1-q}$

for p≥1 with f∈C[0,1] and Λ₁=y>0 given. We give conditions on f that guarantee the existence of y ⁽⁰⁾ such that the sequence Λ _p with Λ₁=y ⁽⁰⁾ tends to a finite positive limit as p→∞.

     

The geometric structure of pure SF6 and mixed Ar/SF6 clusters investigated by core level photoelectron spectroscopy

The S 2p core level photoelectron spectra of Sulphurhexafluoride clusters have been investigated together with heterogeneous Ar/SF₆ clusters, created by doping Ar host clusters (with a mean size of 3600 atoms) with the molecule. Surface and bulk features are resolved both in the argon 2p and the sulphur 2p core level photoelectron spectra. For the latter level such features were only observed in the pure cluster case; a single feature characterizes the S 2p core level spectra of SF₆ doped argon clusters. From the chemical shifts, investigated with respect to SF₆ doping pressure. It can be concluded that the host clusters get smaller with increasing doping pressures and that the SF₆ molecules predominantly stay below the cluster surface, whereas the Argon core stays intact. We have neither observed features corresponding to SF₆ on the cluster surface, nor features corresponding to molecules deep inside the bulk in any of the spectra from the pick-up experiments.     

Truncated Connectivities in a Highly Supercritical Anisotropic Percolation Model

We consider an anisotropic bond percolation model on $\mathbb{Z}^{2}$ , with p=(p _h ,p _v )∈[0,1]², p _v >p _h , and declare each horizontal (respectively vertical) edge of $\mathbb{Z}^{2}$ to be open with probability p _h (respectively p _v ), and otherwise closed, independently of all other edges. Let $x=(x_{1},x_{2}) \in\mathbb{Z}^{2}$ with 0<x ₁<x ₂, and $x'=(x_{2},x_{1})\in\mathbb{Z}^{2}$ . It is natural to ask how the two point connectivity function $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})>\mathbb{P}_{\mathbf {p}}(\{0\leftrightarrow x'\})$ . In this note we give an affirmative answer in the highly supercritical regime.     

Critical dynamics of the sine-Gordon model ind=2−ε dimensions

The renormalization scheme of Amit, Goldschmidt and Grinstein is extended tod=2?ε dimensions. The exponent ν of the correlation lengthv ^?1=2ε+O(ε²) is in agreement with the result of Kosterlitz for the Coulomb gas. The exponent η of the correlation function of the sine-Gordon field is η=ε+O(ε²). The scaling form of the dynamical structure factorS(q,ω) of the dynamic sine-Gordon ModelA is studied ind=2?ε dimensions. The dynamic exponentz is found to bez=2+(b?1)ε+O(ε²) for ε≧0. The constantb is given by the integral $$b = \int\limits_0^\infty {dss^{ - 2} \exp } \left( { - 2\int\limits_s^\infty {dxx^{ - 1} e^{ - x} } } \right) = 2,371544...$$      

On stochastic spatial patterns and neuronal polarity

To investigate the statistical behavior in the sizes of finite clusters for percolation, cluster size distribution n _s (p) for site and bond percolations at different lattices and dimensions was simulated using a modified algorithm. An equation to approximate the finite cluster size distribution n _s (p) was obtained and expressed as: log?(n _s (p)) = as ? b log?s + c. Based on the analysis of simulation data, we found that the equation is valid for p from 0 to 1 on site and for the bond percolation of two-dimensional (2D) and three-dimensional (3D) lattices. Furthermore, the relationship between the coefficients of the equation and the occupied ratio p was studied using the finite-size scaling method. When \(x = D(p - p_c )L^{y_t }\) , p < p _c , and D was a nonuniversal metric factor. a was found to be related only to p, and the a-x curves of different lattices were nearly overlapped; b was related to the dimensions and p, and the scaled data of the b of all lattices with the same dimension tended to fall on the same curves. Unlike a and b, c apparently had a quadratic relation with x in 2D lattices and linear relation with x in 3D lattices. The results of this paper could significantly reduce the amount of tasks required to obtain numerical data of on the cluster size distribution for p from 0 to p _c .     

Transverse momenta of helium fragments in gold fragmentation at 10.6 GeV/nucleon

The transverse momentum (p_t) distributions of helium fragments from gold fragmentation on different nuclei of nuclear emulsion have been measured and a clear increase of average p _t with target mass is seen. The p _t distributions can be parameterized by a sum of three exponential functions of the form $\sim {\rm exp}(-p_{t}^{2}/B_{i})$ . The differences in p _t distributions in interactions on different targets can be explained by different contributions of the three exponential functions. These contributions depend on the projectile breakup in the collision, and for a given degree of the projectile breakup do not depend on the target mass.     

Cluster generation under pulsed laser ablation of zinc oxide

Neutral and cationic Zn _n O _m clusters of various stoichiometry have been produced by nanosecond laser ablation of ZnO in vacuum and investigated by time-of-flight mass spectrometry. Particular attention was paid to the effect of laser wavelength (in the range from near-IR to UV) on cluster composition. Under 193-nm laser ablation, the charged clusters are essentially substoichiometric with Zn_nO_n-1⁺\mathrm{Zn}_{n}\mathrm{O}_{n-1}^{+} and Zn_nO_n-3⁺\mathrm{Zn}_{n}\mathrm{O}_{n-3}^{+} being the most abundant series. Both sub- and stoichiometric cationic clusters are generated in abundance at 532- and 1064-nm ablation whose composition depends on the cluster size. The reactivity of small stoichiometric Zn_nO_n⁺\mathrm{Zn}_{n}\mathrm{O}_{n}^{+} clusters (n<11) toward hydrogen is found to be high, while oxygen-deficient species are less reactive. The neutral plume particles are mainly stoichiometric with Zn₄O₄ tetramer being a magic cluster. It is suggested that the Zn₄O₄ loss is the dominant fragmentation channel of large zinc oxide clusters upon electron impact. Plume expansion conditions under ZnO ablation with visible and IR laser pulses are shown to be favorable for stoichiometric cluster formation.     

On the free energy of the hopfield model

The general theory of inhomogeneous mean-field systems of Raggio and Werner provides a variational expression for the (almost sure) limiting free energy density of the Hopfield model $$H_{N,p}^{\{ \xi \} } (S) = - \frac{1}{{2N}}\sum\limits_{i,j = 1}^N {\sum\limits_{\mu = 1}^N {\xi _i^\mu \xi _j^\mu S_i S_j } } $$ for Ising spinsS _i andp random patterns ξ^μ=(ξ ₁ ^μ ,ξ ₂ ^μ ,...,ξ _N ^μ ) under the assumption that $$\mathop {\lim }\limits_{N \to \gamma } N^{ - 1} \sum\limits_{i = 1}^N {\delta _{\xi _i } = \lambda ,} \xi _i = (\xi _i^1 ,\xi _i^2 ,...,\xi _i^p )$$ exists (almost surely) in the space of probability measures overp copies of {?1, 1}. Including an “external field” term ?ξ _μ ^p h^μμξ _i=1 ^N ξ _i ^μ S_i, we give a number of general properties of the free-energy density and compute it for (a)p=2 in general and (b)p arbitrary when λ is uniform and at most the two componentsh ^μ1 andh ^μ2 are nonzero, obtaining the (almost sure) formula $$f(\beta ,h) = \tfrac{1}{2}f^{ew} (\beta ,h^{\mu _1 } + h^{\mu _2 } ) + \tfrac{1}{2}f^{ew} (\beta ,h^{\mu _1 } - h^{\mu _2 } )$$ for the free energy, wheref ^cw denotes the limiting free energy density of the Curie-Weiss model with unit interaction constant. In both cases, we obtain explicit formulas for the limiting (almost sure) values of the so-called overlap parameters $$m_N^\mu (\beta ,h) = N^{ - 1} \sum\limits_{i = 1}^N {\xi _i^\mu \left\langle {S_i } \right\rangle } $$ in terms of the Curie-Weiss magnetizations. For the general i.i.d. case with Prob {ξ _i ^μ =±1}=(1/2)±?, we obtain the lower bound 1+4?²(p?1) for the temperatureT _c separating the trivial free regime where the overlap vector is zero from the nontrivial regime where it is nonzero. This lower bound is exact forp=2, or ε=0, or ε=±1/2. Forp=2 we identify an intermediate temperature region between T_*=1?4?² and T_c=1+4?² where the overlap vector is homogeneous (i.e., all its components are equal) and nonzero.T _* marks the transition to the nonhomogeneous regime where the components of the overlap vector are distinct. We conjecture that the homogeneous nonzero regime exists forp≥3 and that T_*=max{1?4?²(p?1),0}.     

A matrix integral solution to two-dimensionalW p-gravity

Thep ^th Gel'fand-Dickey equation and the string equation [L, P]=1 have a common solution τ expressible in terms of an integral overn×n Hermitean matrices (for largen), the integrand being a perturbation of a Gaussian, generalizing Kontsevich's integral beyond the KdV-case; it is equivalent to showing that τ is a vacuum vector for aW _?p ⁺ , generated from the coefficients of the vertex operator. This connection is established via a quadratic identity involving the wave function and the vertex operator, which is a disguised differential version of the Fay identity. The latter is also the key to the spectral theory for the two compatible symplectic structures of KdV in terms of the stress-energy tensor associated with the Virasoro algebra. Given a differential operator $$L = D^p + q_2 (t) D^{p - 2} + \cdots + q_p (t), with D = \frac{\partial }{{dx}},t = (t_1 ,t_2 ,t_3 ,...),x \equiv t_1 ,$$ consider the deformation equations¹ (0.1) $$\begin{gathered} \frac{{\partial L}}{{\partial t_n }} = [(L^{n/p} )_ + ,L] n = 1,2,...,n + - 0(mod p) \hfill \\ (p - reduced KP - equation) \hfill \\ \end{gathered} $$ ofL, for which there exists a differential operatorP (possibly of infinite order) such that (0.2) $$[L,P] = 1 (string equation).$$ In this note, we give a complete solution to this problem. In section 1 we give a brief survey of useful facts about theI-function τ(t), the wave function Ψ(t,z), solution of ?Ψ/?t _n=(L ^n/p) _x Ψ andL ^1/pΨ=zΨ, and the corresponding infinitedimensional planeV ⁰ of formal power series inz (for largez) $$V^0 = span \{ \Psi (t,z) for all t \in \mathbb{C}^\infty \} $$ in Sato's Grassmannian. The three theorems below form the core of the paper; their proof will be given in subseuqent sections, each of which lives on its own right.