Mechanics of thermally fluctuating membranes

Besides having unique electronic properties, graphene is claimed to be the strongest material in nature. In the press release of the Nobel committee it is claimed that a hammock made of a squared meter of one-atom thick graphene could sustain the wight of a 4 kg cat. More practically important are applications of graphene like scaffolds and sensors which are crucially dependent on the mechanical strength. Meter-sized graphene is even being considered for the lightsails in the starshot project to reach the star alpha centaury. The predicted strength of graphene is based on its very large Young modulus which is, per atomic layer, much larger than that of steel. This reasoning however would apply to conventional thin plates but does not take into account the peculiar properties of graphene as a thermally fluctuating crystalline membrane. It was shown recently both experimentally and theoretically that thermal fluctuations lead to a dramatic reduction of the Young modulus and increase of the bending rigidity for micron-sized graphene samples in comparison with atomic scale values. This makes the use of the standard F\"oppl-von Karman elasticity (FvK) theory for thin plates not directly applicable to graphene and other single atomic layer membranes. This fact is important because the current interpretation of experimental results is based on the FvK theory. In particular, we show that the FvK-derived Schwerin equation, routinely used to derive the Young modulus from indentation experiments has to be essentially modified for graphene at room temperature and for micron sized samples. Based on scaling analysis and atomistic simulation we investigate the mechanics of graphene under transverse load up to breaking. We determine the limits of applicability of the FvK theory and provide quantitative estimates for the different regimes.Comment: to appear in npj 2D Materials and Application

Scaling behavior and strain dependence of in-plane elastic properties of graphene

We show by atomistic simulations that, in the thermodynamic limit, the in-plane elastic moduli of graphene at finite temperature vanish with system size $L$ as a power law $~ L^{-\eta_u}$ with $\eta_u \simeq 0.325$, in agreement with the membrane theory. Our simulations clearly reveal the size and strain dependence of graphene's elastic moduli, allowing comparison to experimental data. Although the recently measured difference of a factor 2 between the asymptotic value of the Young modulus for tensilely strained systems and the value from {\it ab initio} calculations remains unsolved, our results do explain the experimentally observed increase of more than a factor 2 for a tensile strain of only a few permille. We also discuss the scaling of the Poisson ratio, for which our simulations disagree with the predictions of the self-consistent screening approximation.Comment: 5 figure

Long-Term Dependence Characteristics of European Stock Indices

In this paper we show the degrees of persistence of the time series if eight European stock market indices are measured, after their lack of ergodicity and stationarity has been established. The proper identification of the nature of the persistence of financial time series forms a crucial step in deciding whether econometric modeling of such series might provide meaningful results. Testing for ergodicity and stationarity must be the first step in deciding whether the assumptions of numerous time series models are met. Our results indicate that ergodicity and stationarity are very difficult to establish in daily observations of these market indexes and thus various time-series models cannot be successfully identified. However, the measured degrees of persistence point to the existence of certain dependencies, most likely of a nonlinear nature, which, perhaps can be used in the identification of proper empirical econometric models of such dynamic time paths of the European stock market indexes. The paper computes and analyzes the long- term dependence of the equity index data as measured by global Hurst exponents, which are computed from wavelet multi-resolution analysis. For example, the FTSE turns out to be an ultra-efficient market with abnormally fast mean-reversion, faster than theoretically postulated by a Geometric Brownian Motion. Various methodologies appear to produce non-unique empirical measurement results and it is very difficult to obtain definite conclusions regarding the presence or absence of long term dependence phenomena like persistence or anti-persistence based on the global or homogeneous Hurst exponent. More powerful methods, such as the computation of the multifractal spectra of financial time series may be required. However, the visualization of the wavelet resonance coefficients and their power spectrograms in the form of localized scalograms and average scalegrams, forcefully assist with the detection and measurement of several nonlinear types of market price diffusion.Long-Term Dependence, European Stock Indices

Multi-Fractal Spectral Analysis of the 1987 Stock Market Crash

The multifractal model of asset returns captures the volatility persistence of many financial time series. Its multifractal spectrum computed from wavelet modulus maxima lines provides the spectrum of irregularities in the distribution of market returns over time and thereby of the kind of uncertainty or randomness in a particular market. Changes in this multifractal spectrum display distinctive patterns around substantial market crashes or drawdowns. In other words, the kinds of singularities and the kinds of irregularity change in a distinct fashion in the periods immediately preceding and following major market drawdowns. This paper focuses on these identifiable multifractal spectral patterns surrounding the stock market crash of 1987. Although we are not able to find a uniquely identifiable irregularity pattern within the same market preceding different crashes at different times, we do find the same uniquely identifiable pattern in various stock markets experiencing the same crash at the same time. Moreover, our results suggest that all such crashes are preceded by a gradual increase in the weighted average of the values of the Lipschitz regularity exponents, under low dispersion of the multifractal spectrum. At a crash, this weighted average irregularity value drops to a much lower value, while the dispersion of the spectrum of Lipschitz exponents jumps up to a much higher level after the crash. Our most striking result, therefore, is that the multifractal spectra of stock market returns are not stationary. Also, while the stock market returns show a global Hurst exponent of slight persistence 0.5Financial Markets, Persistence, Multi-Fractal Spectral Analysis, Wavelets

Diffusion-limited deposition with dipolar interactions: fractal dimension and multifractal structure

Computer simulations are used to generate two-dimensional diffusion-limited deposits of dipoles. The structure of these deposits is analyzed by measuring some global quantities: the density of the deposit and the lateral correlation function at a given height, the mean height of the upper surface for a given number of deposited particles and the interfacial width at a given height. Evidences are given that the fractal dimension of the deposits remains constant as the deposition proceeds, independently of the dipolar strength. These same deposits are used to obtain the growth probability measure through Monte Carlo techniques. It is found that the distribution of growth probabilities obeys multifractal scaling, i.e. it can be analyzed in terms of its $f(\alpha)$ multifractal spectrum. For low dipolar strengths, the $f(\alpha)$ spectrum is similar to that of diffusion-limited aggregation. Our results suggest that for increasing dipolar strength both the minimal local growth exponent $\alpha_{min}$ and the information dimension $D_1$ decrease, while the fractal dimension remains the same.Comment: 10 pages, 7 figure

Closing the Window on Strongly Interacting Dark Matter with IceCube

We use the recent results on dark matter searches of the 22-string IceCube detector to probe the remaining allowed window for strongly interacting dark matter in the mass range 10^4<m_X<10^15 GeV. We calculate the expected signal in the 22-string IceCube detector from the annihilation ofsuch particles captured in the Sun and compare it to the detected background. As a result, the remaining allowed region in the mass versus cross sectionparameter space is ruled out. We also show the expected sensitivity of the complete IceCube detector with 86 strings.Comment: 5 pages, 7 figures. Uppdated figures 2 and 3 (y-axis normalization and label) . Version accepted for publication in PR

Matrix-Valued Little q-Jacobi Polynomials

Matrix-valued analogues of the little q-Jacobi polynomials are introduced and studied. For the 2x2-matrix-valued little q-Jacobi polynomials explicit expressions for the orthogonality relations, Rodrigues formula, three-term recurrence relation and their relation to matrix-valued q-hypergeometric series and the scalar-valued little q-Jacobi polynomials are presented. The study is based on a matrix-valued q-difference operator, which is a q-analogue of Tirao's matrix-valued hypergeometric differential operator.Comment: 16 pages, various corrections and minor additions, incorporating referee's comment

Diffusion-limited deposition of dipolar particles

Deposits of dipolar particles are investigated by means of extensive Monte Carlo simulations. We found that the effect of the interactions is described by an initial, non-universal, scaling regime characterized by orientationally ordered deposits. In the dipolar regime, the order and geometry of the clusters depend on the strength of the interactions and the magnetic properties are tunable by controlling the growth conditions. At later stages, the growth is dominated by thermal effects and the diffusion-limited universal regime obtains, at finite temperatures. At low temperatures the crossover size increases exponentially as T decreases and at T=0 only the dipolar regime is observed.Comment: 5 pages, 4 figure