Asymmetric mixture of hard particles with Yukawa attraction between unlike ones: a cluster algorithm simulation study

The influence of a Yukawa attraction between unlike species on the structure of an asymmetric mixture of hard particles is investigated. A recently proposed cluster algorithm for the simulation of hard particle fluids is extended in order to treat the Yukawa tail. Starting with nonoverlapping hard core configurations generated by the cluster algorithm, the attractive tail is treated according to a standard Metropolis scheme. Preliminary results on the effect of the Yukawa attraction on the pair distribution of the larger particles and the potential of mean force at infinite dilution confirm important changes in comparison with pure hard sphere mixtures.     

Jamming percolation and glass transitions in lattice models

A new class of lattice gas models with trivial interactions but constrained dynamics is introduced. These models are proven to exhibit a dynamical glass transition: above a critical density rhoc ergodicity is broken due to the appearance of an infinite spanning cluster of jammed particles. The fraction of jammed particles is discontinuous at the transition, while in the unjammed phase dynamical correlation lengths and time scales diverge as exp[C(rhoc-rho)-mu]. Dynamic correlations display two-step relaxation similar to glass formers and jamming systems.     

Distribution of the Coulomb microfield within an ion cluster

The microfield distribution function in clusters was studied by simulation using the molecular dynamics and Monte Carlo methods. The results obtained were compared with microfield distributions in infinite plasma. It was shown that the calculated distributions have the same asymptotics. However, the position of the maximum and the existence of additional extrema depend on the cluster type and size. The dependence of the microfield expectation and variance on the number of cluster particles was also studied.     

Phase Separation Driven by a Fluctuating Two-Dimensional Self-Affine Potential Field

We study phase separation in a system of hard-core particles driven by a fluctuating two-dimensional self-affine potential landscape which evolves through Kardar–Parisi–Zhang (KPZ) dynamics. We find that particles tend to cluster together on a length scale which grows in time. The final phase-separated steady state is characterized by an unusual cusp singularity in the scaled correlation function and a broad distribution for the order parameter. Unlike the one-dimensional case studied earlier, the cluster-size distribution is asymmetric between particles and holes, reflecting the broken reflection symmetry of the KPZ dynamics, and has a contribution from an infinite cluster in addition to a power law part. A study of the surface in terms of coarse-grained depth variables helps understand many of these features.     

Non-coexistence of Infinite Clusters in Two-Dimensional Dependent Site Percolation

This paper presents three results on dependent site percolation on the square lattice. First, there exists no positively associated probability measure on {0,1}^{\mathbb Z2}\{0,1\}^{\mathbb {Z}^{2}} with the following properties: (a) a single infinite 0cluster exists almost surely, (b) at most one infinite 1∗cluster exists almost surely, (c) certain probabilities regarding 1∗clusters are bounded away from zero. Second, the coexistence of an infinite 1∗cluster and an infinite 0cluster has probability zero when the underlying probability measure is ergodic with respect to translations, positively associated, and satisfies the finite energy condition. The third result analyzes the typical structure of infinite clusters of both types in the absence of positive association. Namely, under a slightly sharpened finite energy condition, the existence of infinitely many disjoint infinite self-avoiding 1∗paths follows from the existence of an infinite 1∗cluster. The same holds with respect to 0paths and 0clusters.     

The cluster expansion in statistical mechanics

The Glimm-Jaffe-Spencer cluster expansion from constructive quantum field theory is adapted to treat quantum statistical mechanical systems of particles interacting by finite range potentials. The HamiltonianH ₀+V need be stable in the extended sense thatH ₀+4V+BN0 for someB. In this situation, with a mild technical condition on the potentials, the cluster expansion converges and the infinite volume limit of the correlation functions exists, at low enough density. These infinite volume correlation functions cluster exponentially. We define a class of interacting boson and fermion particle theories with a matter-like potential, 1/r suitably truncated at large distance. This system would collapse in the absence of the exclusion principle—the potential is unstable—but the Hamiltonian is stable. This provides an example of a system for which our method proves existence of the infinite volume limit, that is not covered by the classic work of Ginibre, which requires stable potentials.One key ingredient is a type of Holder inequality for the expectation values of spatially smeared Euclidean densities, a special interpolation theorem. We also obtain a result on the absolute value of the fermion measure, it equals the boson measure.This work was supported in part by NSF Grant MPS 75-10751Michigan Junior Fellow     

Critical behavior of a dynamical percolation model

The critical behavior of the dynamical percolation model, which realizes the molecular-aggregation conception and describes the crossover between the hadronic phase and the partonic phase, is studied in detail. The critical percolation distance for this model is obtained by using the probability P∞ of the appearance of an infinite cluster. Utilizing the finite-size scaling method the critical exponents γ/v and T are extracted from the distribution of the average cluster size and cluster number density. The influences of two model related factors, I.e. The maximum bond number and the definition of the infinite cluster, on the critical behavior are found to be small.     

Algebraic connectivity analysis in molecular electronic structure theory II: total exponential formulation of second-quantised correlated methods

The fundamentality of the exponential representation of a second-quantised correlated wave function is emphasised with an accent on the physical sense of cluster amplitudes as cumulants of the correlated ansatz. Three main wave function formalisms, namely, the configuration-interaction theory, the coupled-cluster approach, and the many-body perturbation theory (as well as their extensions, e.g. the equation-of-motion coupled-cluster method, multireference schemes, etc.), are represented in an exponential form, leading to a formulation of the working equations in terms of cluster amplitudes. By expressing the corresponding many-body tensor equations in terms of cluster amplitudes, we could unambiguously check connectivity types and the asymptotic behaviour of all tensors/scalars involved (in the formal limit of an infinite number of correlated particles). In particular, the appearance of disconnected cluster amplitudes corresponds to unphysical correlations. Besides, we demonstrate that the equation-of-motion coupled-cluster approach, as well as certain excited-state configuration-interaction methods, can be recast in a fully connected (exponential) form, thus breaking the common belief that all truncated configuration-interaction methods violate connectivity. Our work is based on the recently developed algebraic framework which can be viewed as a complement to the classical diagrammatic analysis.     

Percolation characteristics of spinodal phase separation in polymer blends

Micrographs of thin films of polymer blends undergoing spinodal phase separation were analyzed by computer. It was found that at the late stages of the phase separation process, the infinite interconnected cluster breaks into droplets. This process can be analyzed in the framework of a percolation phenomenon in two dimensions - the resulting fractual dimensionality D = 1.9 for the infinite cluster is in good agreement with theory.     

Statistical thermodynamics of the formation of an infinite cluster of thermally reversible chemical bonds

A systematic “mean-field” treatment of the thermodynamic equilibrium formation of an infinite cluster of bonds in a system of identical monomers capable of forming from n=0 to n>2 reversible chemical bonds with one another is proposed within the Cayley-tree approximation. For this purpose the difference between the symmetry of the monomers appearing in “point-to-point” and closed bond paths, respectively, is taken into account on the basis of an analysis of the structure of the infinite cluster. Minimization with respect to the distribution of such monomers yields a nontrivial solution corresponding to a lower free energy than the classical solution, which does not allow for the symmetry difference indicated. In addition, it is shown that the classical solution corresponds to the free-energy maximum when the infinite cluster is formed and that the formation of the infinite cluster is a first-order phase transition. The possible form of the phase diagrams of the systems considered is analyzed. Zh. éksp. Teor. Fiz. 115, 979–990 (March 1999)     

Infinite Canonical Super-Brownian Motion and Scaling Limits

We construct a measure valued Markov process which we call infinite canonical super-Brownian motion, and which corresponds to the canonical measure of super-Brownian motion conditioned on non-extinction. Infinite canonical super-Brownian motion is a natural candidate for the scaling limit of various random branching objects on when these objects are critical, mean-field and infinite. We prove that ICSBM is the scaling limit of the spread-out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension. We conjecture that it also arises as the scaling limit in various other models above the upper-critical dimension, such as the incipient infinite lattice tree above 8 dimensions, the incipient infinite cluster for unoriented percolation above 6 dimensions, uniform spanning trees above 4 dimensions, and invasion percolation above 6 dimensions.     

Percolation for low energy clusters and discrete symmetry breaking in classical spin systems

We consider classical lattice systems in two or more dimensions with general state space and with short-range interactions. It is shown that percolation is a general feature of these systems: If the temperature is sufficiently low, then almost surely with respect to some equilibrium state there is an infinite cluster of spins trying to form a ground state. For systems having several stable sets of symmetry-related ground states we show that at low temperatures spontaneous symmetry breaking occurs because in a two-dimensional subsystem there is a unique infinite cluster of this type.     

The cluster expansion for potentials with exponential fall-off

Continuing the work of a previous paper, the Glimm-Jaffe-Spencer cluster expansion from constructive quantum field theory is adapted to treat quantum statistical mechanical systems of particles interacting by potentials that fall off exponentially at large distance. The HamiltonianH ₀+V need be stable in the extended sense thatH ₀+4V+BN0 for someB. In this situation, with a mild technical condition on the potentials, the cluster expansion converges and the infinite volume limit of the correlation functions exists, at low enough density. These infinite volume correlation functions cluster exponentially. A natural system included in the present treatment is that of matter with ther ^–1 potential replaced bye ^–ar/r. The Hamiltonian is stable, but the system would collapse in the absence of the exclusion principle—the potential is unstable. Therefore this system cannot be handled by the classic work of Ginibre, which requires stable potentials.This work was supported in part by NSF Grant MPS 75-10751Michigan Junior Fellow     

Transport for Stochastic System with Infinite Locally Coupled Oscillators

We consider the transport of particles for spatially periodic system with infinite locally coupled oscillators dirven by additive and multiplicative noises.A formula of the probability current derived by us shows that the coupling among the infinite oscillators is an ingredinent for transport.This coupling of the oscillators can induce transport of particles in the absence of the correlation of the additive and multiplicative noises,even without the multiplicative noise.     

Spontaneous breaking of molecular identity and phase diagrams of thermally reversible association systems with alternating molecules

The global phase behavior of a mixture of molecules A _f and B _f, each containing f functional groups of, respectively, types A and B capable of forming thermally reversible chemical bonds, is considered. Contrary to the traditional approach based on the consideration of an infinite cluster of labile bonds that appears in such systems on the Bethe lattice (i.e., in the Cayley tree approximation) under certain conditions, we additionally take into account the contribution to the thermodynamics from the cluster fragments forming mesoscopic cycles. It is shown, within the framework of the suggested mesoscopic cyclization approximation, which is based on the concept of spontaneous breaking of molecular identity upon the formation of an infinite cluster, that this contribution is finite. Phase diagrams are constructed for the systems considered. The presence of a point of equal concentrations, where two liquid phases coexist and one of them contains an infinite cluster of thermally reversible bonds, is the specific feature of the phase diagrams in the approximation suggested.     

On the cluster approach for calculating electronic energies of solid surfaces

A scheme for calculating electronic energy states of infinite solid surface systems by a cluster approach under the framework of the method of linear combinations of atomic orbitals is presented. The basis functions consist of atomic-like orbitals confined within a cluster whereas the Hamiltonian is that of the infinite solid. The latter circumvents the difficulty arising from the auxiliary boundary of the cluster which is not the true surface of the solid. All the multicenter integrals appearing in the Hamiltonian matrix can be evaluated exactly by means of the technique of Gaussian orbitals. This cluster-basis method is applied to the chlorine-adsorbed silicon (111) surface using several different clusters. The results are compared with those of the same Hamiltonian with basis functions extending over the entire solid in the Bloch-sum form. Criteria for optimal selection of clusters are suggested.     

Scalar particles in a narrow-band periodic potential

A system of an infinite number of spinless particles in a narrow-band periodic potential is treated. The dimension of the space is arbitrary, the tight-binding approximation is used, and it is assumed that the filling fraction is nearly one electron per atom. After a preliminary discussion of the Hartree approximation, the full Schrödinger equation is considered and a rigorous spectral perturbation theory in the kinetic energy term is set up. To get rid of the lack of relative boundedness of this perturbation, a vacuum state is constructed and its energy renormalized to zero, and passage is made to an excitonic representation in which the quasiparticles appear naturally as local perturbations of the vacuum. In this representation, relative boundedness is recovered and Rayleigh-Schrödinger expansions can be used to find cluster expansions for all local observables.     

On the Coulomb operator for embedded cluster calculations in periodic systems

In performing ab-initio or DFT embedded cluster calculations in infinite periodic systems one unsolved problem is how to embed the quantum cluster in the general case. We show that the Ewald real space sum can be implemented as an operator in the Hamiltonian when the reciprocal space sum is made small.     

Convergence of spherical harmonic expansions for the evaluation of hard-sphere cluster integrals

ForN particles (N>2), by means of a spherical harmonic expansion of Silverstone and Moats, a 3N-dimensional cluster may be reduced to 2N+1 trivial integrals andN–1 interesting integrals. For hard spheres, theN–1 interesting integrals are products of polynomials integrated between binomial bounds. With simple clusters, closed forms are obtained; for more complex clusters, infinite series inl (ofY _lm ) appear. It is here shown for representative cases that these series converge exponentially rapidly, the leading pair of terms accounting for all but a few tenths of a percent of the total cluster integral.     

High-Dimensional Incipient Infinite Clusters Revisited

The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models in three different ways, extending previous work by the second-named author and Járai. We show that each construction yields the same measure, indicating that the IIC is a robust object. Furthermore, our constructions apply to spread-out versions of both finite-range and long-range percolation models. We also get estimates on structural properties of the IIC, such as the volume of the intersection between the IIC and Euclidean balls.