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

We study the multifractal temporal scaling properties of river discharge and precipitation records. We compare the results for the multifractal detrended fluctuation analysis method with the results for the wavelet transform modulus maxima technique and obtain agreement within the error margins. In contrast to previous studies, we find non-universal behaviour: On long time scales, above a crossover time scale of several months, the runoff records are described by fluctuation exponents varying from river to river in a wide range. Similar variations are observed for the precipitation records which exhibit weaker, but still significant multifractality. For all runoff records the type of multifractality is consistent with a modified version of the binomial multifractal model, while several precipitation records seem to require different models.The analysis of river flows has a long history. Already more than half a century ago the engineer H. E. Hurst found that runoff records from various rivers exhibit 'long-range statistical dependencies' [1]. Later, such long-term correlated fluctuation behaviour has also been reported for many other geophysical records including precipitation data [2,3], see also [4]. These original approaches exclusively focused on the absolute values or the variances of the full distribution of the fluctuations, which can be regarded as the first moment F 1 (s) [1][2][3] and the second moment F 2 (s) [5], respectively. In the last

We study molecular diffusion in nanopores with different types of roughness under the exclusion of mutual molecular collisions, i.e., in the so-called Knudsen regime. We show that the diffusion problem can be mapped onto Levy walks and discuss the roughness dependence of the diffusion coefficients D(s) and D(t) of self- and transport diffusion, respectively. While diffusion is normal in d=3, diffusion is anomalous in d=2 with D(s) approximately ln t and D(t) approximately ln L, where t and L are time and system size, respectively. Both diffusion coefficients decrease significantly when the roughness is enhanced, in remarkable disagreement with earlier findings.

We study molecular diffusion in linear nanopores with different types of roughness in the so-called Knudsen regime. Knudsen diffusion represents the limiting case of molecular diffusion in pores, where mutual encounters of the molecules within the free pore space may be neglected and the time of flight between subsequent collisions with the pore walls significantly exceeds the interaction time between the pore wall and the molecules. We present an extension of a commonly used procedure to calculate transport diffusion coefficients. Our results show that using this extension, the coefficients of self-and transport diffusion in the Knudsen regime are equal for all regarded systems, which improves previous literature data.