Dynamic Response of Ising System to a Pulsed Field

The dynamical response to a pulsed magnetic field has been studied here both using Monte Carlo simulation and by solving numerically the meanfield dynamical equation of motion for the Ising model. The ratio R_p of the response magnetisation half-width to the width of the external field pulse has been observed to diverge and pulse susceptibility \chi_p (ratio of the response magnetisation peak height and the pulse height) gives a peak near the order-disorder transition temperature T_c (for the unperturbed system). The Monte Carlo results for Ising system on square lattice show that R_p diverges at T_c, with the exponent $\nu z \cong 2.0$, while \chi_p shows a peak at $T_c^e$, which is a function of the field pulse width $\delta t$. A finite size (in time) scaling analysis shows that $T_c^e = T_c + C (\delta t)^{-1/x}$, with $x = \nu z \cong 2.0$. The meanfield results show that both the divergence of R and the peak in \chi_p occur at the meanfield transition temperature, while the peak height in $\chi_p \sim (\delta t)^y$, $y \cong 1$ for small values of $\delta t$. These results also compare well with an approximate analytical solution of the meanfield equation of motion.Comment: Revtex, Eight encapsulated postscript figures, submitted to Phys. Rev.

Spontaneous magnetization of the Ising model on the Sierpinski carpet fractal, a rigorous result

We give a rigorous proof of the existence of spontaneous magnetization at finite temperature for the Ising spin model defined on the Sierpinski carpet fractal. The theorem is inspired by the classical Peierls argument for the two dimensional lattice. Therefore, this exact result proves the existence of spontaneous magnetization for the Ising model in low dimensional structures, i.e. structures with dimension smaller than 2.Comment: 14 pages, 8 figure

Multifractals of Normalized First Passage Time in Sierpinski Gasket

The multifractal behavior of the normalized first passage time is investigated on the two dimensional Sierpinski gasket with both absorbing and reflecting barriers. The normalized first passage time for Sinai model and the logistic model to arrive at the absorbing barrier after starting from an arbitrary site, especially obtained by the calculation via the Monte Carlo simulation, is discussed numerically. The generalized dimension and the spectrum are also estimated from the distribution of the normalized first passage time, and compared with the results on the finitely square lattice.Comment: 10 pages, Latex, with 3 figures and 1 table. to be published in J. Phys. Soc. Jpn. Vol.67(1998

The Computational Complexity of Generating Random Fractals

In this paper we examine a number of models that generate random fractals. The models are studied using the tools of computational complexity theory from the perspective of parallel computation. Diffusion limited aggregation and several widely used algorithms for equilibrating the Ising model are shown to be highly sequential; it is unlikely they can be simulated efficiently in parallel. This is in contrast to Mandelbrot percolation that can be simulated in constant parallel time. Our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory. In addition, the results may serve as a guide to simulation physics.Comment: 28 pages, LATEX, 8 Postscript figures available from [email protected]

On large deviation properties of Erdos-Renyi random graphs

We show that large deviation properties of Erd\"os-R\'enyi random graphs can be derived from the free energy of the $q$-state Potts model of statistical mechanics. More precisely the Legendre transform of the Potts free energy with respect to $\ln q$ is related to the component generating function of the graph ensemble. This generalizes the well-known mapping between typical properties of random graphs and the $q\to 1$ limit of the Potts free energy. For exponentially rare graphs we explicitly calculate the number of components, the size of the giant component, the degree distributions inside and outside the giant component, and the distribution of small component sizes. We also perform numerical simulations which are in very good agreement with our analytical work. Finally we demonstrate how the same results can be derived by studying the evolution of random graphs under the insertion of new vertices and edges, without recourse to the thermodynamics of the Potts model.Comment: 38 pages, 9 figures, Latex2e, corrected and extended version including numerical simulation result

Fractal chemical kinetics: Reacting random walkers

Computer simulations on binary reactions of random walkers ( A + A → A ) on fractal spaces bear out a recent conjecture: ( ρ −1 − ρ 0 −1 ) ∞ t f , where ρ is the instantaneous walker density and ρ 0 the initial one, and f = d s /2, where d s is the spectral dimension. For the Sierpinski gaskets: d =2, 2 f =1.38 ( d s =1.365); d =3, 2 f =1.56 ( d s =1.547); biased initial random distributions are compared to unbiased ones. For site percolation: d = 2, p =0.60, 2 f = 1.35 ( d s =1.35); d=3, p =0.32, 2 f =1.37 ( d s =1.4); fractal-to-Euclidean crossovers are also observed. For energetically disordered lattices, the effective 2 f (from reacting walkers) and d s (from single walkers) are in good agreement, in both two and three dimensions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45149/1/10955_2005_Article_BF01012924.pd