Magnetization of Ising model in nonzero magnetic field

Journal ArticleKnowing only the zero-field magnetization (e.g., Yang's result) of the Ising model in any number of dimensions, one can construct a lower bound on m(h), the magnetization in finite field. Knowledge of u, the internal energy per bond, enables a more efficient lower bound to be constructed. Both are applications of the Griffiths inequality, as recently generalized by Kelly and Sherman, and should prove useful in the lattice gas problem where it is essential to know m(h)

Two-dimensional Ising model in a finite magnetic field

Journal ArticleWe present the results of numerical calculations giving accurate estimates of the magnetization of the two-dimensional Ising model on a square lattice. Moreover, we argue that these results are strict lower bounds to the correct magnetization M(H, T). The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of M(H, T) converge to the correct answer for the infinite lattice everywhere except for a small region in the T-H plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for H=0, which reproduces exactly the critical-point behavior of the full Ising model

Critical curves and thermodynamic phases of lattice fluids and antiferromagnets with structured interactions

Journal ArticleWe study the classical lattice gas with hard cores, nearest-neighbor repulsive forces, and next-nearest-neighbor attractive forces. We find several thermodynamic phases (vapor, liquid, and solid) and unusually shaped critical curves, yet no critical point or triple point. We find and discuss discontinuities in m(h) in the analogous magnetic systems. The principal calculational tool is molecular field theory but accurate numerical solutions of the transfer matrix are also obtained as a check on some of the results. It-is conjectured that the critical line separating vapor and liquid phases (bounded by the triple point on one end and the critical point on the other) exists in the exact solution of a lattice gas with structured, relatively long-ranged interactions, but cannot be found in the molecular field approximation

Short-range versus long-range order in a model binary alloy

Journal ArticleWe calculate the free energy of a linear binary metallic alloy using an exact transfer-matrix formalism. We obtain for the first time the electronic band structure when there is nonvanishing short-range order and calculate the temperature dependence of the short-range order parameter. Following Foo and Amar, we also introduce long-range order via two sublattices, but we find this necessarily raises the free energy. We conclude that the first-order phase transition of Foo and Amar is spurious, being an artifact of their assumption of long-range order and neglect of short-range order

Second-order phase transition in a model random alloy

Journal ArticleWe investigate the existence and stability of long-range order in a simple model of a binary alloy. We find that for electron concentrations in the vicinity of one per atom, in bipartite lattices, long-range order of the atoms exists at low temperatures, and we find that the system undergoes a second-order phase transition. We use a generalized form of the coherent-potential approximation (CPA) of Soven and find that this CPA reproduces exactly at least eight moments of the density of states. We derive an exact expression for the critical temperature when the alternating potential is much larger than the bandwidth

Viscoelasticity near the gel-point: a molecular dynamics study

We report on extensive molecular dynamics simulations on systems of soft spheres of functionality f, i.e. particles that are capable of bonding irreversibly with a maximum of f other particles. These bonds are randomly distributed throughout the system and imposed with probability p. At a critical concentration of bonds, p_c approximately equal to 0.2488 for f=6, a gel is formed and the shear viscosity \eta diverges according to \eta ~ (p_c-p)^{-s}. We find s is approximately 0.7 in agreement with some experiments and with a recent theoretical prediction based on Rouse dynamics of phantom chains. The diffusion constant decreases as the gel point is approached but does not display a well-defined power law.Comment: 4 pages, 4 figure

A model for gelation with explicit solvent effects: Structure and dynamics

We study a two-component model for gelation consisting of $f$-functional monomers (the gel) and inert particles (the solvent). After equilibration as a simple liquid, the gel particles are gradually crosslinked to each other until the desired number of crosslinks has been attained. At a critical crosslink density the largest gel cluster percolates and an amorphous solid forms. This percolation process is different from ordinary lattice or continuum percolation of a single species in the sense that the critical exponents are new. As the crosslink density $p$ approaches its critical value $p_c$, the shear viscosity diverges: $\eta(p)\sim (p_c-p)^{-s}$ with $s$ a nonuniversal concentration-dependent exponent.Comment: 6 pages, 9 figure