Some solvable models of nonuniform classical fluids

A model classical fluid is constructed by assuming that the direct correlation functionc(r – r) is independent of any applied external field. Thermodynamic consistency requires thatc(r – r) 0, and permits explicit representation of the model by a many-body interaction potential. In the canonical ensemble, the model shows a phase transition to an infinite density condensed phase, but in the grand canonical ensemble only an anomalous transition to zero density vapor is found to stably exist.     

Optimization with extremal dynamics

We explore a new general-purpose heuristic for finding high-quality solutions to hard discrete optimization problems. The method, called extremal optimization, is inspired by self-organized criticality, a concept introduced to describe emergent complexity in physical systems. Extremal optimization successively updates extremely undesirable variables of a single suboptimal solution, assigning them new, random values. Large fluctuations ensue, efficiently exploring many local optima. We use extremal optimization to elucidate the phase transition in the 3-coloring problem, and we provide independent confirmation of previously reported extrapolations for the ground-state energy of +/-J spin glasses in d = 3 and 4.     

Dimensional reduction of integrals of orthogonal invariants

The integration of a function of scalar products ofp vectors in R^D, where D > p, is reduced to that of a corresponding function of vectors in R^p.     

Collective modes in a one-dimensional nonuniform fluid model

A one-dimensional fluid with short-range repulsive interaction and one period of cosinusoidal attraction in a periodic container is transformed to a two-mode format. The system has both high-temperature single-phase regions and lowtemperature two-phase regions with a very broad spatial interface that can be stabilized by a weak external field. The case of vanishing external field brings out properties of the mode amplitude dependence which one expects to extend to more complex systems.     

Finite sections of toeplitz-like matrices

Determinants of large finite sections of Toeplitz and Toeplitz-like matrices are evaluated by an expansion in which the deviation from the identity is parametrically increased. Classical results are reproduced and the inverse matrix expanded as well. The expansion is asymptotically valid only for low-order terms, and so a reordered expansion is introduced. It has the desired asymptotic character and suggests that sharp bounds are also available.