Diameter Distribution of Spherical Primary Grains in the boolean model from small-angle scattering

A method for determining the size-distribution density of small spheres which are arranged, without any interaction between them, at random positions in space and form an isotropic twophase system is presented. In the case of higher volume fractions, frequent overlapping between the spheres is a logical consequence. This effect is an essential feature of the model itself and is well considered and observed. Stereological information, which contains all the necessary data about the unknown size distribution, is obtained from the angular intensity distribution (small-angle scattering experiment) of the particle system concerned. The resulting formula still includes, in a first representation, the volume fraction p as a free parameter. In a second step, a general analytical method for the determination of the volume fraction p from the set covariance of the considered random closed set is derived. Therefore, the unknown diameter distribution can be obtained in every detail, starting from the scattering curve of the sample, by way of two independent working steps.     

Contribution to the theory of mixed ising spin systems

An effective-field theory of mixed spin—1/2 and spin-1 Ising system with random bond and crystal-field interactions is developed. The general expressions for evaluating the thermodynamic quantities are obtained. In the last section, transition temperature and tricritical behavior are examined for the mixed ferromagnetic spin system withz=3, wherez is the coordination number. The obtained results are reasonable in comparison with exact and approximate ones. Owing to the simplicity, the present method can be applied to a wide class of the random systems.     

The effect of thermodynamic fluctuations on the formation of superstructures in random heteropolymers

The behavior of a random AB block copolymer with the same fraction of monomers of each type near the critical point has been studied. In the Brazovskii approximation it is shown that thermodynamic fluctuations give rise to a lamellar structure whose period is greater than the size of a block. The structure appears as a result of a first-order phase transition. The parameters of this transition are calculated and the region of coexistence of the disordered and lamellar phases is found. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 3, 183–187 (10 August 1996)     

Recurrent random walks and the absence of continuous symmetry breaking on graphs

We consider geometrically disordered systems with a continuous symmetry groupG, where the internal degrees of freedom are attached to the vertices of a graph. We show that equilibrium states remainG-invariant at any temperatureT>0 if a random walk on the graph is recurrent. This generalizes a previous result obtained by Cassi.     

Random layered frustration models

We study inhomogeneous two-dimensional Ising models with a random distribution of ferro- and antiferromagnetic couplings,K _ij =±K, or equivalently a random distribution of frustrations. In particular, we considerRandom Layered Frustration models (RLF) where randomness is confined to the vertical direction. These RLF-models are solved exactly, i.e., partition function and free energy are obtained in closed form for an arbitrary random distribution of finite period. The phase transition is of Ising type. A simple formula for the transition temperature is derived which depends only on the mean coupling , but not on other details of the distribution. Both cases,T _c =0 andT _c 0, are possible. Groundstate energy and groundstate degeneracy, or equivalently the rest entropy, are determined. It is found that both the occurence or absence of a phase transition may be accompanied with vanishing or nonvanishing rest entropy. We also show that for the RLF-models a phase transition is excluded when all groundstates are connected with one another by local transformations which presumably holds generally. A remarkable result is that the transition of the ferromagnetic Ising model can be destroyed completely if one replaces an arbitrarily small fraction of ferromagnetic couplings by antiferromagnetic ones in a suitable way.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Convergence of block spins defined by a random field

We study the asymptotic behavior of families of dependent random variables called block spins, which are associated with random fields arising in statistical mechanics. We give sufficient conditions for these families to converge weakly to products of independent Gaussian random variables. We also estimate the error terms involved. In addition we give some conditions which imply that the block spins can converge weakly only to families of normal or degenerate random variables. Central to our proofs is a mixing property which is weaker than strong mixing and which holds for many random fields studied in statistical mechanics. Finally we give a simple method for determining when a stationary random field does not satisfy a strong mixing property. This method implies that the two-dimensional Ising model at the critical temperature is not strong mixing, a result obtained by a different method by M. Cassandro and G. Jona-Lasinio. The method also shows that a stationary, mean-zero, positively correlated Gaussian process indexed by is not strong mixing if its covariance function decreases liket ^– , 0 < < 1.     

Solutions and applications of tridiagonal vector recurrence relations

The applications of infinite systems of linear first order differential equations with 2L+1-term recursion formulas are discussed. It is shown that such systems can be written as a system of linear tridiagonal vector equations of dimensionL. A general method is presented by which the initial value problem can be solved by iteration. For special but physically important initial conditions the solution is given by a matrix continued fraction. The eigenvalues of the tridiagonal vector recurrence relations are obtained as the roots of aL×L determinant the elements of which are determined by a matrix continued fraction. The applicability of the method is demonstrated by calculating the eigenvalues of the laser Fokker-Planck operator.     

Shift in Critical Temperature for Random Spatial Permutations with Cycle Weights

We examine a phase transition in a model of random spatial permutations which originates in a study of the interacting Bose gas. Permutations are weighted according to point positions; the low-temperature onset of the appearance of arbitrarily long cycles is connected to the phase transition of Bose-Einstein condensates. In our simplified model, point positions are held fixed on the fully occupied cubic lattice and interactions are expressed as Ewens-type weights on cycle lengths of permutations. The critical temperature of the transition to long cycles depends on an interaction-strength parameter α. For weak interactions, the shift in critical temperature is expected to be linear in α with constant of linearity c. Using Markov chain Monte Carlo methods and finite-size scaling, we find c=0.618±0.086. This finding matches a similar analytical result of Ueltschi and Betz. We also examine the mean longest cycle length as a fraction of the number of sites in long cycles, recovering an earlier result of Shepp and Lloyd for non-spatial permutations.     

SLE on Doubly-Connected Domains and the Winding of Loop-Erased Random Walks

Two-dimensional loop-erased random walks (LERWs) are random planar curves whose scaling limit is known to be a Schramm-Loewner evolution SLE _κ with parameter κ=2. In this note, some properties of an SLE _κ trace on doubly-connected domains are studied and a connection to passive scalar diffusion in a Burgers flow is emphasised. In particular, the endpoint probability distribution and winding probabilities for SLE₂ on a cylinder, starting from one boundary component and stopped when hitting the other, are found. A relation of the result to conditioned one-dimensional Brownian motion is pointed out. Moreover, this result permits to study the statistics of the winding number for SLE₂ with fixed endpoints. A solution for the endpoint distribution of SLE₄ on the cylinder is obtained and a relation to reflected Brownian motion pointed out.     

On the Moments of Traces of Matrices of Classical Groups

We consider random matrices, belonging to the groups U(n), O(n) , SO(n), and Sp(n) and distributed according to the corresponding unit Haar measure. We prove that the moments of traces of powers of the matrices coincide with the moments of certain Gaussian random variables if the order of moments is low enough. Corresponding formulas, proved partly before by various methods, are obtained here in the framework of a unique method, reminiscent of the method of correlation equations of statistical mechanics. The equations are derived by using a version of the integration by parts.Acknowledgement We are grateful to Prof. Z. Rudnick for drawing our attention to the problem and for stimulating discussions.     

Effects of temperature and equivalence ratio on the ignition of n-heptane fuel spray in turbulent flow

Direct numerical simulations were performed to study the autoignition process of n-heptane fuel spray in a turbulent field. For the solution of the carrier gas fluid, the Eulerian method is employed, while for the fuel droplets, the Lagrangian method is used. Droplets are initialized at random locations in a two-dimensional isotropic turbulent field. A chemistry mechanism for n-heptane with 44 species and 112 reactions was adopted to describe the chemical reactions. Three cases with the same initial global equivalence ratio (0.5) and different initial gas phase temperatures (1100, 1200, and 1300 K) were simulated. In addition, two cases with initial global equivalence ratios of 1.0 and 1.5 and initial temperature 1300 K were simulated to examine the effect of equivalence ratio. Evolution of temperature, species mass fraction, reaction rate, and the joint PDF of temperature and equivalence ratio are presented. Effects of the initial gas temperature and equivalence ratio on vaporization and ignition are discussed. A correlation was derived relating ignition delay times to temperature and equivalence ratio. It was confirmed that with the increase of initial temperature, the autoignition occurs earlier. With the increase of the initial equivalence ratio, however, autoignition occurs later due to a larger decrease in gas phase temperature caused by fuel droplet evaporation. The results obtained in this study are expected to be constructive in understanding fuel spray combustion, such as that in homogeneous charge compression ignition systems.     

Algorithm for optimal filtering of random Markov signals with pulse-disturbance-amplitude estimation

Optimal filtering problem of random Markov signals with simultaneous estimation of pulse disturbance amplitudes is considered. Linear models of stochastic difference equations in discrete time are used to describe signals, observed processes, and pulse disturbances. Pulse disturbances occur at random times with random amplitudes. A real-time calculation procedure is obtained for the joint a posteriori probability density function of random signals and pulse disturbance amplitudes. A quasioptimal filtering algorithm is derived in the case of scalar signals and scalar observed processes by a partition method. Computer simulation results are presented.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 4, pp. 496–513, April, 1996.     

Survival,Extinction and Approximation of Discrete-time Branching Random Walks

We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly finite) state space is large enough. An analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site. We compare these results with the ones obtained in the continuous-time case and we give some examples and counterexamples.     

New allotropic form of silicon

A new allotropic form of amorphous silicon with sp hybridization of the valence electrons is discovered. The new material consists of linear chains of atoms. A small fraction of the atoms are in the sp ² state. Acting as bridges, these atoms couple the linear chains into a single random network. This conclusion is based on an analysis of experimental data on the effect of annealing and ion implantation on the structure of the short-range order and the properties of amorphous-silicon films obtained by different methods. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 9, 646–649 (10 May 1998)     

Heat conduction in a random medium

The method of time-ordered cumulants is used to investigate the behavior of heat pulses in a one-dimensional medium in which the thermal conductivity is random. A partial differential equation is obtained for the average temperature profile; it is the heat equation modified by the addition of a fourth-order spatial derivative. A solution is obtained by asymptotic series. The first two spatial moments of the average temperature profile are evaluated and are shown to tend to those of a Gaussian whent is large. Finally, an equation is obtained for the covariance function.Alfred P. Sloan Fellow.     

Statistical Analysis of Simulated Random Packings of Spheres

This paper describes two algorithms for the generation of random packings of spheres with arbitrary diameter distribution. The first algorithm is the force‐biased algorithm of Mościński and Bargieł. It produces isotropic packings of very high density. The second algorithm is the Jodrey‐Tory sedimentation algorithm, which simulates successive packing of a container with spheres following gravitation. It yields packings of a lower density and of weak anisotropy. The results obtained with these algorithms for the cases of log‐normal and two‐point sphere diameter distributions are analysed statistically, i. e. standard characteristics of spatial statistics such as porosity (or volume fraction), pair correlation function of the system of sphere centres and spherical contact distribution function of the set‐theoretical union of all spheres are determined. Furthermore, the mean coordination numbers are analysed. These results are compared for both algorithms and with data from the literature based on other numerical simulations or from experiments with real spheres.     

On stochastic complex beam-beam interaction models with Gaussian colored noise

This paper is to continue our study on complex beam-beam interaction models in particle accelerators with random excitations Y. Xu, W. Xu, G.M. Mahmoud, On a complex beam-beam interaction model with random forcing [Physica A 336 (2004) 347-360]. The random noise is taken as the form of exponentially correlated Gaussian colored noise, and the transition probability density function is obtained in terms of a perturbation expansion of the parameter. Then the method of stochastic averaging based on perturbation technique is used to derive a Fokker-Planck equation for the transition probability density function. The solvability condition and the general transforms using the method of characteristics are proposed to obtain the approximate expressions of probability density function to order ε.Also the exact stationary probability density and the first and second moments of the amplitude are obtained, and one can find when the correlation time equals to zero, the result is identical to that derived from the Stratonovich-Khasminskii theorem for the same model under a broad-band excitation in our previous work.     

Brownian dynamics of polydisperse colloidal hard spheres: Equilibrium structures and random close packings

Recently we presented a new technique for numerical simulations of colloidal hard-sphere systems and showed its high efficiency. Here, we extend our calculations to the treatment of both 2- and 3-dimensional monodisperse and 3-dimensional polydisperse systems (with sampled finite Gaussian size distribution of particle radii), focusing on equilibrium pair distribution functions and structure factors as well as volume fractions of random close packing (RCP). The latter were determined using in principle the same technique as Woodcock or Stillinger had used. Results for the monodisperse 3-dimensional system show very good agreement compared to both pair distribution and structure factor predicted by the Percus-Yevick approximation for the fluid state (volume fractions up to 0.50). We were not able to find crystalline 3d systems at volume fractions 0.50–0.58 as shown by former simulations of Reeet al. or experiments of Pusey and van Megen, due to the fact that we used random start configurations and no constraints of particle positions as in the cell model of Hoover and Ree, and effects of the overall entropy of the system, responsible for the melting and freezing phase transitions, are neglected in our calculations. Nevertheless, we obtained reasonable results concerning concentration-dependent long-time selfdiffusion coefficients (as shown before) and equilibrium structure of samples in the fluid state, and the determination of the volume fraction of random close packing (RCP, glassy state). As expected, polydispersity increases the respective volume fraction of RCP due to the decrease in free volume by the fraction of the smaller spheres which fill gaps between the larger particles.     

Nonlinear parametric oscillations in certain stochastic systems: A random van der Pol oscillator

Astochastic model for some class ofnonlinear oscillators, which includes a van der Pol-type oscillator with random parameters, is analyzed in thediffusion limit. That is, small random fluctuations and long time are considered, while the nonlinearity is also assumed to be small. We show that there existstationary distributions, independent of the phase of the oscillator, a result proved earlier by R. L. Stratonovich assuming the random perturbations of the frequency to be delta correlated. The time behavior of the moments of the displacement of the oscillator from its rest position is also investigated and the results are compared with the corresponding ones for the linear random oscillator. A numerical study is also performed for the first two moments and plots are given.on leave from the University of Padua, Italy.     

Towards the assignment of the REMPI spectrum of Ph2O using CIS and TD-DFT methods

Tentative assignment of the low-lying vibrational features of the first electronic excited singlet-state S₁ of diphenyl ether (Ph₂O), obtained from resonance-enhanced multiphoton ionisation (REMPI) spectroscopy, is performed using CIS and time-dependent density functional theory (TD-DFT) methods. The potential energy surfaces, regarding the rotation of the phenyl rings relatively to the C–O–C plane, are obtained at the B3LYP/6-31G(d) level of theory, for the ground-state of neutral and cationic Ph₂O and for its first excited singlet state. The torsional barriers of the ground state of diphenyl ether were studied by means of quantum-chemical perturbations of increasing accuracy and an extrapolation to the complete basis set limit and full configuration interaction (FCI) was performed through the use of correlation consistent basis sets and the continued fraction method. The first adiabatic ionisation energy (AIE) of the twist conformer is computed at 8.60 eV in the FCI limit, much higher than the experimental results of Terlouw et al. (8.09±0.03 eV) and Paiva et al. (7.8±0.1 eV). The B3LYP result of 7.82 eV is, however, in reasonable agreement with the result of Paiva et al. The first singlet excitation energy for the twist conformation is found to be 5.5 eV at the CIS/6-311++G(d,p) level of theory. Some features of the experimental REMPI spectrum, previously obtained by one of the authors, are explained and a new insight on the ionisation energy of diphenyl ether is presented.