This book brings together two of the most exciting and widely studied subjects in modern physics: namely fractals and surfaces. To the community interested in the study of surfaces and interfaces, it brings the concept of fractals. To the community interested in the exciting field of fractals and their application, it demonstrates how these concepts may be used in the study of surfaces. The authors cover, in simple terms, the various methods and theories developed over the past ten years to study surface growth. They describe how one can use fractal concepts successfully to describe and predict the morphology resulting from various growth processes. Consequently, this book will appeal to physicists working in condensed matter physics and statistical mechanics, with an interest in fractals and their application. The first chapter of this important new text is available on the Cambridge Worldwide Web server: http://www.cup.cam.ac.uk/onlinepubs/Textbooks/textbookstop.html

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We study the statistical properties of volatility, measured by locally averaging over a time window T, the absolute value of price changes over a short time interval deltat. We analyze the S&P 500 stock index for the 13-year period Jan. 1984 to Dec. 1996. We find that the cumulative distribution of the volatility is consistent with a power-law asymptotic behavior, characterized by an exponent mu approximately 3, similar to what is found for the distribution of price changes. The volatility distribution retains the same functional form for a range of values of T. Further, we study the volatility correlations by using the power spectrum analysis. Both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents. Also, both methods show the presence of a crossover at approximately 1.5 days. In addition, we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 U.S. companies over the two-year period Jan. 1994 to Dec. 1995.

The unusual low-temperature behavior of liquid water is interpreted using a simple model based upon connectivity concepts from correlated-site percolation theory. Emphasis is placed on examining the physical implications of the continuous hydrogen-bonded network (or ’’gel’’) formed by water molecules. Each water molecule A is assigned to one of five species based on the number of ’’intact bonds’’ (the number of other molecules whose interaction energy with A is stronger than some cutoff VHB). It is demonstrated that in the present model the spatial positions of the various species are not randomly distributed but rather are correlated. In particular, it is seen that the infinite hydrogen-bonded network contains tiny ’’patches’’ of four-bonded molecules. Well-defined predictions based upon the putative presence of these tiny patches are developed. In particular, we predict the detailed dependence upon (a) temperature, (b) dilution with the isotope D2O, (c) hydrostatic pressure greater than atmospheric, and (d) ’’patch-breaking impurities’’—for four separate response functions, (i) the isothermal compressibility KT, (ii) the constant-pressure specific heat CP, (iii) the constant-volume specific heat CV, and (iv) the thermal expansivity αP, as well as for dynamic properties such as (v) the transport coefficients self-diffusivity Ds and shear viscosity η, (vi) the characteristic rotational relaxation time τch, and (vii) the Angell singularity temperature Ts. The experimentally observed dependence of these seven quantities upon the four parameters (a)–(d) is found in all cases to agree with the predicted behavior. The paradoxical behavior associated with the absence of a glass transition in pure liquid water is also resolved. Finally, we propose certain experiments and computer simulations—some of which are underway—designed to put the proposed percolation model to better tests than presently possible using available information.

The correlation function of a financial index of the New York stock exchange, the S&P 500, is analyzed at 1 min intervals over the 13-year period, Jan 84 -Dec 96. We quantify the correlations of the absolute values of the index increment. We find that these correlations can be described by two different power laws with a crossover time t × ≈ 600 min. Detrended fluctuation analysis gives exponents α 1 = 0.66 and α 2 = 0.93 for t < t × and t > t × respectively. Power spectrum analysis gives corresponding exponents β 1 = 0.31 and β 2 = 0.90 for f > f × and f < f × respectively.A topic of considerable recent interest to both the economics and physics communities is whether there are correlations in economic time series and, if so, how to best quantify these correlations [1,2,3,4]. Here we study the S&P 500 index of the New York stock exchange over a 13-year period (Fig. 1a). We calculate the logarithmic increments g(t) ≡ ln Z(t+1)−ln Z(t) over a fixed time lag of 1 min, where Z(t) denotes the index at time t (t counts the number of minutes during the opening hours of the stock market), and quantify the correlations as follows:(i) We find that the correlation function of g(t) decays exponentially with a characteristic time of the order of 1-10 min, but the absolute value |g(t)| does not. This result is consistent with previous studies on several economic series [2,3,4]. 1(ii) We calculate the power spectrum of |g(t)| (Fig. 2a), and find that the data fit not one but rather two separate power laws: for f > f × the power law exponent is β 1 = 0.31, while for f < f × the exponent β 2 = 0.90 is three times larger; here f × is called the crossover frequency.(iii) We confirm these results using the DFA (detrended fluctuation analysis) method (see Fig.2b), which allows accurate estimates of exponents independent of local trends [5].From the behavior of the power spectrum, we expect that the DFA method will also predict two distinct regions of power law behavior, with exponents α 1 = 0.66 and α 2 = 0.95 for t less than or greater than a characteristic time scale t × ≡ 1/f × , where we have used the general mathematical result [6] that α = (1 + β)/2. The data of Fig. 2b yield α 1 = 0.66, α 2 = 0.93, thereby confirming the consistency of the power spectrum and DFA methods.Also the crossover time is very close to the result obtained from the power spectrum, with t × ≈ 1/f × ≈ 600 min (about 1.5 trading days).We observed the crossover behavior noted above by considering the entire 13-year period studied, so it is natural to enquire whether it will still hold for periods smaller than 13 y.Therefore, we choose a sliding window (with size 1 y) and calculate both exponents α 1 and α 2 within this window as the window is dragged, down the data set. We find (Fig. 1b) that the value of α 1 is very "stable" (independent of the position of the window) fluctuating around the mean value 2/3. Surprisingly, however, the variation of α 2 is much greater, showing sudden jumps when very volatile periods enter or leave the time window.We...

Firms having similar business activities are correlated. We analyze two different cross-correlation matrices C constructed from (i) 30-min price fluctuations of 1000 US stocks for the two-year period 1994-95 and (ii) one-day price fluctuations of 422 US stocks for the 35-year period 1962-96. We find that the eigenvectors of C corresponding to the largest eigenvalues allow us to partition the set of all stocks into distinct subsets. These subsets are similar to business sectors, and are stable for extended periods of time. We find that price fluctuations of these subsets are characterized by power-law decaying time correlations, reminiscent of strongly interacting systems.

We propose and test a model that describes the morphology of cities, the scaling of the urban perimeter of individual cities, and the area distribution of systems of cities. The model is also consistent with observable urban growth dynamics, our results agreeing both qualitatively and quantitatively with urban data. The resulting growth morphology can be understood from interactions among the constituent units forming an urban region, and can be modeled using a correlated percolation model in the presence of a gradient.

We study hydrogen-bond dynamics in liquid water at low temperatures using molecular dynamics simulations. We analyze the dynamics using energetic and geometric definitions of a hydrogen bond, and employ two analysis methods: ͑i͒ a history-dependent correlation function, related to the distribution of bond lifetimes, and ͑ii͒ a history-independent correlation function. For method ͑i͒ we find an approximately Arrhenius temperature dependence of the bond lifetime, and find that the distribution of bond lifetimes is extremely sensitive to the choice of bond definition. For method ͑ii͒ we find-independent of bond definition-that the dynamics are consistent with the predictions of the mode-coupling theory, suggesting that the slow dynamics of hydrogen bonds can be explained in the same framework as standard transport quantities. Our results allow us to clarify the significance of the choice of both bond definition and analysis technique.