Ole J. Heilmann

Theory of monomer-dimer systems

We investigate the general monomer-dimer partition function,P(x), which is a polynomial in the monomer activity,x, with coefficients depending on the dimer activities. Our main result is thatP(x) has its zeros on the imaginary axis when the dimer activities are nonnegative. Therefore, no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomer density is minimal (i.e.x=0). Elaborating on this theme we prove the existence and analyticity of correlation functions (away fromx=0) in the thermodynamic limit. Among other things we obtain bounds on the compressibility and derive a new variable in which to make an expansion of the free energy that converges down to the minimal monomer density. We also relate the monomer-dimer problem to the Heisenberg and Ising models of a magnet and derive Christoffell-Darboux formulas for the monomer-dimer and Ising model partition functions. This casts the Ising model in a new light and provides an alternative proof of the Lee-Yang circle theorem. We also derive joint complex analyticity domains in the monomer and dimer activities. Our considerations are independent of geometry and hence are valid for any dimensionality.

VIOLATION OF THE NONCROSSING RULE: THE HUBBARD HAMILTONIAN FOR BENZENE*

The Hubbard Hamiltonian, while not one of the most successful models for π electrons in benzene, has been extensively investigated in the literature. As part of our general study of that model, we have computed all the energy levels for all values of the repulsion parameter—a task that has not been undertaken before. After extracting all the symmetry of the model we found, to our great surprise, many instances of permanent degeneracy of levels with different symmetry and also crossing of levels of the same symmetry. We can also demonstrate that there is no hidden symmetry to account for these effects. Since these results run counter to one of the oldest folk theorems in quantum chemistry, our otherwise uninspiring graphs may be of general interest.

A finite version of Smoluchowski's coagulation equation

The authors formulate a finite version of Smoluchowskis coagulation equation in which molecules of size N react in the normal way whereas molecules of size N to 2N can be produced but cannot react. They prove that the solution converges to the solution of the infinite system of equations in the limit N to infinity .

Post-gelation solutions to Smoluchowski's coagulation equation

It is proven how the post-gelation behaviour originally suggested by Flory(1993) can be obtained as a result of a limiting process, passing from a finite to an infinite system. In a previous paper by the authors it was shown how the post-gelation behaviour first suggested by Stockmayer(1943) can be obtained by passing to the limit of an infinite system in a different way. It is thus demonstrated that different post-gelation solutions of Smoluchowskis coagulation equation can be obtained by different limiting processes.

Lattice Models for Hydrogen-Bonded Solvents

A class of lattice models for a binary mixture is defined by assuming that one of the components may form bonds to neighboring molecules of the same species. It is assumed that the fugacity of a molecule depends on the number of bonds which connect the molecule to other molecules. If no molecule is allowed to be connected by more than two bonds to other molecules, then no phase transition occurs, while phase transition can occur if more than two bonds are allowed. If only two or no bonds are allowed, then the model can be solved rigorously for certain planar lattices by transforming it to a dimer covering problem; this model shows behavior similar to the Ising model in zero magnetic field.

Phase Transitions in Lattice Gas Models for Water

Water-like lattice gases on the triangular and body-centered cubic lattices are investigated. Molecules may reside on the lattice sites in either of two possible orientations, a hydrogen bond being formed between molecules on neighboring sites if they have the proper orientation with respect to one another. For a range of chemical potential at sufficiently low temperatures, the models are shown to have an ordered phase consisting of an open, hydrogen-bonded, icelike structure. The models are shown to be transitionfree at sufficiently high temperature, indicating the existence of a critical point.

Zeros of the Grand Partition Function for a Lattice Gas

The following three statements about the zeros of the grand partition function of a lattice gas with negative (attractive) interactions are proved: (1) Not all the zeros will be on the unit circle in the high‐temperature limit if forces of higher order than 2‐body are included; (2) in the low‐temperature limit they will, in general, lie on the unit circle; (3) it is possible to have the zeros dense in the complex plane. It is also shown that not all polynomials with positive coefficients and roots on the unit circle are a grand partition function of a lattice gas.

Analytical solutions of Smoluschowski's coagulation equation

It is proven that Smoluchowskis coagulation equation with a kernel, Kij, which satisfies Kij or=0.

Crystalline ordering in lattice models of hard rods with nearest neighbour attraction

Abstract Several lattice models for hard rods with attraction between neighbouring parallel rodes are considered. It is proved that the models for sufficiently low temperature and high fugacity exhibit a phase transition to an ordered crystalline structure. Nothing is proved about the existence of a liquid crystal state.

Some positive definite functions on sets and their application to the Ising model

Consideration of correlation inequalities for Ising ferromagnets with arbitrary spins has led to the discovery of a class of positive definite functions on sets. These functions are linear combinations of the functions which enter into Muirheads Theorem. We prove these functions to be positive definite and also show how they can be applied to the Ising problem to prove Griffiths second inequality for arbitrary spins.