Geometric critical exponent inequalities for general random cluster models

A set of new critical exponent inequalities,d(1 –1 /)2 –, dv(1 – 1/), andd> 1, is proved for a general class of random cluster models, which includes (independent or dependent) percolations, lattice animals (with any interactions), and various stochastic cluster growth models. The inequalities imply that the critical phenomena in the models are inevitably not mean-field-like in the dimensions one, two, and three.The present work was reported at the 56th Statistical Mechanics Meeting (Rutgers, December 1986).     

A rigorous control of logarithmic corrections in four-dimensional ϕ4 spin systems

Using Gawedzki and Kupiainen's rigorous block spin transformation method, we study critical phenomena in ⁴ spin systems in four dimensions. In Part I of this work we investigate in detail the renormalization group trajectory of the system not exactly at the critical point.     

Metastable decay rates,asymptotic expansions,and analytic continuation of thermodynamic functions

The grand potentialP(z)/kT of the cluster model at fugacityz, neglecting interactions between clusters, is defined by a power series _n Q _n z ⁿ , whereQ _n , which depends on the temperatureT, is the partition function of a cluster of sizen. At low temperatures this series has a finite radius of convergencez _s . Some theorems are proved showing that ifQ _n , considered as a function ofn, is the Laplace transform of a function with suitable properties, thenP(z) can be analytically continued into the complexz plane cut along the real axis fromz _s to + and that (a) the imaginary part ofP(z) on the cut is (apart from a relatively unimportant prefactor) equal to the rate of nucleation of the corresponding metastable state, as given by Becker-Döring theory, and (b) the real part ofP(z) on the cut is approximately equal to the metastable grand potential as calculated by truncating the divergent power series at its smallest term.     

Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems

We consider quantum unbounded spin systems (lattice boson systems) in -dimensional lattice space Z. Under appropriate conditions on the interactions we prove that in a region of high temperatures the Gibbs state is unique, is translationally invariant, and has clustering properties. The main methods we use are the Wiener integral representation, the cluster expansions for zero boundary conditions and for general Gibbs state, and explicitly -dependent probability estimates. For one-dimensional systems we show the uniqueness of Gibbs states for any value of temperature by using the method of perturbed states. We also consider classical unbounded spin systems. We derive necessary estimates so that all of the results for the quantum systems hold for the classical systems by straightforward applications of the methods used in the quantum case.     

Virial expansion for a polymer with a realistic pair-potential interaction

No modern theory of polymer excluded volume adequately describes the crossover from poor solvent to good solvent conditions; a fundamental difficulty is a singularity in the binary cluster integral. Mayer expansion techniques are applied to a model with a pair interaction between monomers to clarify the distinction between geometric and solvent contributions to excluded volume. Detailed calculations are undertaken for a hard-core potential and a mimic Lennard-Jones potential. The significance of the singularity in the binary cluster integral for calculations in the crossover regime is discussed.     

The covariance matrix of the Potts model: A random cluster analysis

We consider the covariance matrix,G ^mm =q ²<(_x,m);(_y,m)>, of thed-dimensionalq-states Potts model, rewriting it in the random cluster representation of Fortuin and Kasteleyn. In any of theq ordered phases, we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances, thereby attributing algebraic significance to these stochastic geometric quantities. We also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers. For dimensiond=2, we show that this correlation length and the correlation length of the two-point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two. For systems with first-order transitions, this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self-dual point _o. For systems with second order transitions, this relation implies the equality of the correlation length exponents from above and below threshold, as well as an amplitude ratio of two. In the course of proving the above results, we establish several properties of independent interest, including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions, and left continuity of the two-dimensional free boundary condition percolation probability at _o. We also introduce DLR equations for the random cluster model and use them to establish ergodicity of the free measure. In order to prove these results, we introduce a new class of events which we call decoupling events and two inequalities for these events. The first is similar to the FKG inequality, but holds for events which are neither increasing nor decreasing; the second is similar to the van den Berg-Kesten inequality in standard percolation. Both inequalities hold for an arbitrary FKG measure.     

Metastable fluid flow described via a discrete-velocity coagulation-fragmentation model

A discrete-velocity Boltzmann model is introduced. It is based on two principles: (i) clusters of particles move in ³ with seven fixed momenta; (ii) clusters may gain or lose particles according to the rules of Becker-Döring cluster equations. The model provides a kinetic representation of evaporation and condensation. The model is used to obtain macroscopic fluid equations which are valid into the metastable fluid regime, , where is any positive number, is the inelastic Knudsen number, and _s is the saturation density.     

Synthesis and structural studies of polynuclear ruthenium clusters derived from reactions of 1,2,3,4-tetraphenyl-1,2,3,4-tetraphospholane with [Ru3(CO)12]

Reactions of 1,2,3,4-tetraphenyl-1,2,3,4-tetraphospholane (I) with triruthenium dodecacarbonyl at different temperatures result in the cleavage of P-P bonds and even P-C bond(s) in I to afford a series of new ruthenium cluster derivatives containing phosphido and phosphinidene ligands: a penta-ruthenium wing-tip bridged butterfly cluster [Ru₅(CO)₁₁(μ₄-PPh)(μ₃-PPh){(μ₄-η²-(PPh)₂CH₂}] (1), a hepta-ruthenium polyhedral (consisting of two fused square pyramids with a co-apex) cluster [Ru₇(CO)₁₅(μ₄-PPh)₂{(μ₂-PPh)₂CH₂}](2), a linked penta-ruthenium cluster [Ru₄(CO)₁₀(μ₄-PPh)(μ₃-PPh)₂(μ₃-η²-PPhCH₂)Ru(CO)₃] (3), and a hepta-nuclear polyhedral (consisting of two fused square pyramids with different apexes) cluster [Ru₇(CO)₁₅(μ₄-PPh)₂{(μ₂-PPh)₂CH₂}](4). Clusters 2 and 4 are isomeric and differ only in the connection of the two square pyramids in the Ru₇ polyhedron. All the newly obtained clusters have been fully characterized by spectroscopic (IR, FABMS, ¹H- and ³¹P-NMR spectroscopy) and analytical techniques, and their molecular structures are established by single crystal X-ray diffraction analysis.     

Reaction of Tetracopper(I)-Phosphonitocavitand with PhSeSiMe3: Synthesis and Structure of a Polyanionic Cluster [pyH]6[(CuCl)9(μ3-SePh)5(μ4-SePh)]

Treatment of tetracopper(I)-phosphonitocavitand [1·Cu₄(μ-Cl)₄(μ₄-Cl)] (2) (1 = tetraphosphonitocavitand [rccc-2,8,14,20-tetrakis-(iso-butyl)-phosphonitocavitand (C₄₄H₄₈O₈P₄Ph₄)]) with PhSeSiMe₃ in THF at low temperature afforded a novel polyanionic cluster [pyH]₆[(CuCl)₉(μ₃-SePh)₅(μ₄-SePh)] (4) as a major product along with a new tetracopper(I)-phosphonitocavitand (3) with a centered μ₃-Cl. Molecular structure of anionic cluster in 4 consists of six PhSe⁻ bridging ligands containing five μ₃-SePh and one exceptional μ₄-SePh bridging nine copper atoms, of which eight copper atoms have trigonal coordination geometry and the other has distorted tetrahedral geometry. Dedicated to Professor Han-Qin Liu on the occasion of his 70th birthday.     

STUDIES ON cis-trans-ISOMERIZATION AND REACTION KINETICS OF A μ, η~2-VINYL LIGAND IN A TRIOSMIUM CLUSTER

The dihydride cluster H_2O_(s_3)(CO)_(10) reacts with ethyne to give a cis-adduct [O_(s_3)H(μ,η~2-CH=CH_2)(CO)_(10)]. The high deuterated cluster D_2O_(s_3)(CO)_(10), the reaction of D_2O_(s_3)(CO)_(10)with ethyne C_2H_2 to form the vinyl product cluster [O_(s_3)D(μ, η~2-CH=CHD)(CO)_(10)], and thereaction of H_2O_(s_3)(CO)_(10) with the deuterated ethyne C_2D_2 to form [O_(s_3)H(μ,η~2-CD=CDH)(CO)_(10)] have been reported primarily in this paper. The two cis-adducts undergo cis-trans-isomerization of μ,η~2-vinyl ligands under a small amount of the nucleophile pyridine. Usingthe combination of deuteration and dynamic ~1H,~2H NMR spectra, we report the experimen-tal evidence determined by the dynamic ~1H, ~2H NMR spectra, mechanism, and kinetic dataof the isomerization.