The Approximate Invariance of the Average Number of Connections for the Continuum Percolation of Squares at Criticality

We perform Monte Carlo simulations to determine the average excluded area $$of randomly oriented squares, randomly oriented widthless sticks and aligned squares in two dimensions. We find significant differences between our results for randomly oriented squares and previous analytical results for the same. The sources of these differences are explained. Using our results for$$ and Monte Carlo simulation results for the percolation threshold, we estimate the mean number of connections per object $B_c$ at the percolation threshold for squares in 2-D. We study systems of squares that are allowed random orientations within a specified angular interval. Our simulations show that the variation in $B_c$ is within 1.6% when the angular interval is varied from 0 to $\pi/2$

A System with Multiple Liquid-Liquid Critical Points

We study a three-dimensional system of particles interacting via spherically-symmetric pair potentials consisting of several discontinuous steps. We show that at certain values of the parameters desribing the potential, the system has three first-order phase transitions between fluids of different densities ending in three critical points.Comment: 6 pages, 3 figure

Scale Invariance and Nonlinear Patterns of Human Activity

We investigate if known extrinsic and intrinsic factors fully account for the complex features observed in recordings of human activity as measured from forearm motion in subjects undergoing their regular daily routine. We demonstrate that the apparently random forearm motion possesses previously unrecognized dynamic patterns characterized by fractal and nonlinear dynamics. These patterns are unaffected by changes in the average activity level, and persist when the same subjects undergo time-isolation laboratory experiments designed to account for the circadian phase and to control the known extrinsic factors. We attribute these patterns to a novel intrinsic multi-scale dynamic regulation of human activity.Comment: 4 pages, three figure

Multifractal detrended fluctuation analysis of nonstationary time series

We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended fluctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series to those for shuffled series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima (WTMM) method, and show that the results are equivalent.Comment: 14 pages (RevTex) with 10 figures (eps

Lévy flights in random searches

We review the general search problem of how to find randomly located objects that can only be detected in the limited vicinity of a forager, and discuss its quantitative description using the theory of random walks. We illustrate Lévy flight foraging by comparison to Brownian random walks and discuss experimental observations of Lévy flights in biological foraging. We review recent findings suggesting that an inverse square probability density distribution P(ℓ)∼ℓ−2 of step lengths ℓ can lead to optimal searches. Finally, we survey the explanations put forth to account for these unexpected findings