ALBERT

Notizen: Abstract Computer displays of colorful and undulating lava-like forms will decorate living rooms of the future. These forms can already be created using mathematical simulations and hardware-assisted, isosurface computer graphics. In particular, the simulations involve “twisted majority rules” computed using a 3-D cellular automation lattice described in this paper.

Notizen: Abstract The Glauber kinetics of Ising spins is considered as a queueing process and simulated “event by event” as first proposed by Bortz, Kalos, and Lebowitz. The advantage of this algorithm compared to the standard single-flip Monte Carlo method is discussed for the situation of slowing down of dynamics. This process is used to generate fluctuations of magnetization and energy in the critical regimeT=Tc of two-dimensional Ising models. The analysis of these fluctuations leads to numerical determination of the critical exponents for dynamics: for the size dependence of correlation time atT c , and Μ for frequency dependence of the power spectrumS(ω)~ω −µ . From the finite-size scaling hypothesis, scaling relations are settled which are confirmed by this numerical experiment.

Notizen: Abstract We study the thermodynamic behavior of a two-component random ferromagnetic Ising system in the presence of given boundary conditions. The system consists of two species A and B occupying the sites of a Bethe lattice which terminates on a surface layer where the spins are kept fixed. We study the interface of the system when the spins on half of the surface of the lattice are fixed opposite to the spins on the other half. More specifically, we study the influence of disorder on the interface width. We find that disorder clearly increases the interface width at temperatures well belowT c, indicating that the interface roughening of disordered Ising systems in 3D real lattices should occur at temperatures significantly below those of the corresponding ordered ones.

Notizen: Abstract A model in statistical physics is presented based on assigning non-Abelian phase factors to the turning points of polygons in three dimensions. This model allows for an exact solution and exhibits an unexpectedly rich phase structure. The model as well as the solution are obtained by a generalization of the methods of Kac and Ward and by mapping the problem to a Markov process as was done by Feynman for the two-dimensional Ising model

Notizen: Abstract The quantum corrections to the law of corresponding states are studied by calculating the critical pressure, temperature, and density to first order in Planck's constanth on an exactly soluble model. The ratio of the critical parameters to the corresponding classical values are found to be (p c/p c 0)1/2=ρc/ρc 0 = Tc/Tc 0 = 1−0.67Λ, withΛ=hρ c 1/3(mkT c)−1/2. The critical ratio is independent ofh to first order. The results are compared with critical data for noble gases and hydrogen isotopes.

Notizen: Abstract A frustration potential is a sum of interactions the terms of which are not simultaneously minimized even in the ground-state spin configurations. Ising models with such potentials can be discussed by the use of contours. The Peierls condition for the phase transition can be properly generalized, taking into account the presence of zero-energy contours. Frustration has some special features in two dimensions, which we study in detail. The connection with models of spin-glasses is discussed.

Notizen: Abstract We study a generalized Ising system consisting of a Bethe lattice on every site of which two spin-1/2 and two isospin-1/2 (or atomic species A) states can be realized, the spin-species interacting with appropriate nearest neighbor couplings. The system is equivalent to that of four states per site and we obtain its exact thermodynamic behavior. The case of a fixed concentration of species is the annealed or liquid magnetic binary alloy. The temperature dependence of the short-range-order (SRO) parameter of such systems is obtained as an application of the theory and discussed in connection with relevant material from the literature. When fixing both the concentration and the nearest neighbor spatial correlation of the species, we obtain Eggarter's formulas for the “frozen-in” species problem, which therefore are only approximately valid even on the Bethe lattice.

Notizen: Abstract We implement a recently proposed exact method for solving distrete statistical models for the 3-dimensional Ising model with open boundary conditions. Our computations were done on the Connection machine because the problem maps very naturally onto massively parallel architectures. We explicitly calculate the number of states of the system at each energy for systems of size 5×4×L z, forL z⩽5. On serial or vector computers, the time for the computation scales with the volumeV likeV2 LλLy . On the Connection Machine, the calculation can be spread across the processors. This decreases the computation requirements by a factor equal to the number of processors. We describe the method, its implementation on the Connection Machine both in PARIS and in FORTRAN, and our results. We also state the requirements for solving larger systems using this method.

Notizen: Abstract The Earth is continuously expelling gases and liquids from great depths—juvenile volatiles from the mantle and recycled metamorphic products. Some of these fluids ascend through liquid rock in volcanic processes, but others utilize fractures and faults as conduits through the solid lithosphere. The latter process may have a major influence on earthquakes, since fluids at near lithostatic pressures appear to be required to activate deep faults that would otherwise remain locked. Fluids can be driven upward through solid rock by buoyancy, but only if present in sufficient concentration to form large-scale domains occupying interconnected fracture porosity. A growing fluid domain becomes so mobilized only when it attains the critical vertical dimension required for hydrostatic instability. This dimension, depending on the ultimate compressive yield strength of the rock, may be as much as several kilometers. Any column of fluid ascending through fractures in the solid lithosphere from a prolific deep source must become organized into a vertical sequence of discrete domains, separated by fluid-pressure discontinuities. This is required because a continuous hydrostatic-fluid-pressure profile extending from an arbitrarily deep source to the surface cannot be permitted by the finite strength of rock. A vertically stacked sequence of domains allows the internal fluid-pressure profile to approximate the external rock-stress profile in a stepwise fashion. The pressure discontinuity below the base of the uppermost hydrostatic domain may be responsible for some occurrences of so-called anomalous geopressures. An ascending stream of fluid that percolates upward from a deep source through a column of domains must encounter a sequence of abrupt pressure decreases at the transitions between successive domains. If supercritical gases act as solvents, the dissolved substances may drop out of solution at such pressure discontinuities, resulting in a local concentration of minerals and other substances. At great depths, brittle fracture would normally be prevented by high pressure and temperature, with all excessive stress discharged by ductile flow. Rock strata invaded by an ascending fluid domain are weakened, however, because cracks generated or reactivated by the high-pressure fluid can support the overburden, greatly reducing internal friction. This reduction of strength may cause a previously stressed rock to fail, resulting in hydraulic shear fracture. Thus, earthquakes may be triggered by the buoyant migration of deep-source fluids. The actual timing of the failure that leads to such an earthquake may be determined by the relatively rapid inflation of a fluid domain and not by any significant increase in the probably much slower rate of regional tectonic strain. Many earthquake precursory phenomena may be secondary symptoms of an increase in pore-fluid pressure, and certain coseismic phenomena may result from the venting of high-pressure fluids when faults break the surface. Instabilities in the migration of such fluid domains may also contribute to or cause the eruption of mud volcanoes, magma volcanoes, and kimberlite pipes.

Notizen: Abstract We consider a classical system, in a ν-dimensional cube Ω, with pair potential of the formq(r) + γ v φ(γr). Dividing Ω into a network of cells ω1, ω2,..., we regard the system as in a metastable state if the mean density of particles in each cell lies in a suitable neighborhood of the overall mean densityρ, withρ and the temperature satisfying $$f_0 (\rho ) + \tfrac{1}{2}\alpha \rho ^2 〉 f(\rho ,0 + )$$ and $$f''_0 (\rho ) + 2\alpha 〉 0$$ wheref(ρ, 0+) is the Helmholz free energy density (HFED) in the limit γ→ 0; α = ∫ φ(r)d v r andf 0 (ρ) is the HFED for the caseφ = 0. It is shown rigorously that, for periodic boundary conditions, the conditional probability for a system in the grand canonical ensemble to violate the constraints at timet 〉 0, given that it satisfied them at time 0, is at mostλt, whereλ is a quantity going to 0 in the limit $$|\Omega | \gg \gamma ^{ - v} \gg |\omega | \gg r_0 \ln |\Omega |$$ Here,r 0 is a length characterizing the potentialq, andx ≫ y meansx/y → +∞. For rigid walls, the same result is proved under somewhat more restrictive conditions. It is argued that a system started in the metastable state will behave (over times ≪λ −1) like a uniform thermodynamic phase with HFED f0(ρ) + 1/2αρ2, but that having once left this metastable state, the system is unlikely to return.