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Percolation clusters as partial justification of classical nucleation theory
J. Aerosol Sci., 1977,Vol.8, p. 369.PergamonPress.Printedin Great Britain.
CURRENT WORK ON NUCLEATION
Compiled by D. STAUFFER Institut fiJr Theoretische Physik, Universit~it, 5000 K/51n 41, GFR
Brief abstracts of work which has not yet been published may be submitted to the compiler for inclusion in this section.
Shape and size of clusters in the Ising model. By C. DOMB and E. STOLL;IBM Ziirich Research Laboratory, 8803 Riischlokon, Switzerland. The cyclomatic number of a cluster is introduced as a measure of its degree of compactness or ramification. Using Monte Carlo data for a two-dimensional Ising model, estimates are given of the average number of spins and the average number of cycles per cluster as a function of temperature. The results are related to the Whitney polynomial studied recently by Temperley and Lieb. An exact calculation by these authors at the critical temperature enables the pattern of behavior in the critical region to be conjectured.
One-dimensional kinetic lsing model in a field, By A. BAUMG~RTNERand K. BINDER; Fachrichtung Theoretische Physik, Universit~it, 6600 Saarbriicken 11, West Germany. Kinetic equations are derived for the concentration n [ ( t ) of clusters containing 1 up (or down) spins for a generalized Glauber-Ising chain in nonzero magnetic field H. While the initial relaxation time is found exactly, the relaxation function is obtained numerically. Applications to biopolymers are discussed, and approximations of Schwarz are shown to be inaccurate.
Essential singularity in the percolation model. By H. KUNZ (Laboratoire de Physique Theorique, Ecole Polytechnique Federale, Lausanne, Switzerland) and B. SOUILL~a~D. It is proven that the analogue of the free energy for two- and three-dimensional percolation problems has a singularity at zero external field, as soon as percolation appears. At large concentrations at least this function is infinitely differentiable, whereas it is analytic at small concentrations.
Percolation clusters as partial justification of classical nucleation theory. By D. ST~,UFVER;Institut for Theoretische Physik, Universit~it, 5000 K~ln 41, W. Germany. The capillarity approximation of classical nucleation theory assumes that the concept of surface area and surface tension can be used even for the very small droplets (clusters) of critical size; and one calculates the equilibrium number of clusters from this surface free energy. By using the cluster numbers of percolation theory (exact results for small sizes; Monte Carlo simulations for large sizes) we show that one can quite accurately extrapolate the large-cluster behavior down to smaller sizes, even up to a single molecule.
369
CURRENT WORK ON NUCLEATION
Compiled by D. STAUFFER Institut fiJr Theoretische Physik, Universit~it, 5000 K/51n 41, GFR
Brief abstracts of work which has not yet been published may be submitted to the compiler for inclusion in this section.
Shape and size of clusters in the Ising model. By C. DOMB and E. STOLL;IBM Ziirich Research Laboratory, 8803 Riischlokon, Switzerland. The cyclomatic number of a cluster is introduced as a measure of its degree of compactness or ramification. Using Monte Carlo data for a two-dimensional Ising model, estimates are given of the average number of spins and the average number of cycles per cluster as a function of temperature. The results are related to the Whitney polynomial studied recently by Temperley and Lieb. An exact calculation by these authors at the critical temperature enables the pattern of behavior in the critical region to be conjectured.
One-dimensional kinetic lsing model in a field, By A. BAUMG~RTNERand K. BINDER; Fachrichtung Theoretische Physik, Universit~it, 6600 Saarbriicken 11, West Germany. Kinetic equations are derived for the concentration n [ ( t ) of clusters containing 1 up (or down) spins for a generalized Glauber-Ising chain in nonzero magnetic field H. While the initial relaxation time is found exactly, the relaxation function is obtained numerically. Applications to biopolymers are discussed, and approximations of Schwarz are shown to be inaccurate.
Essential singularity in the percolation model. By H. KUNZ (Laboratoire de Physique Theorique, Ecole Polytechnique Federale, Lausanne, Switzerland) and B. SOUILL~a~D. It is proven that the analogue of the free energy for two- and three-dimensional percolation problems has a singularity at zero external field, as soon as percolation appears. At large concentrations at least this function is infinitely differentiable, whereas it is analytic at small concentrations.
Percolation clusters as partial justification of classical nucleation theory. By D. ST~,UFVER;Institut for Theoretische Physik, Universit~it, 5000 K~ln 41, W. Germany. The capillarity approximation of classical nucleation theory assumes that the concept of surface area and surface tension can be used even for the very small droplets (clusters) of critical size; and one calculates the equilibrium number of clusters from this surface free energy. By using the cluster numbers of percolation theory (exact results for small sizes; Monte Carlo simulations for large sizes) we show that one can quite accurately extrapolate the large-cluster behavior down to smaller sizes, even up to a single molecule.
369