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Astandard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T → ∞. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T → ∞ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion.Wedemonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent.Wealso compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.
In this paper we report on photoswitchable polymer surfaces with dynamically and reversibly fluctuating topographies. It is well known that when azobenzene containing polymer films are irradiated with optical interference patterns the film topography changes to form a surface relief grating. In the simplest case, the film shape mimics the intensity distribution and deforms into a wave like, sinusoidal manner with amplitude that may be as large as the film thickness. This process takes place in the glassy state without photo-induced softening. Here we report on an intriguing discovery regarding the formation of reliefs under special illumination conditions. We have developed a novel setup combining the optical part for creating interference patterns, an AFM for in situ acquisition of topography changes and diffraction efficiency signal measurements. In this way we demonstrate that these gratings can be “set in motion” like water waves or dunes in the desert. We achieve this by applying repetitive polarization changes to the incoming interference pattern. Such light responsive surfaces represent the prerequisite for providing practical applications ranging from conveyer or transport systems for adsorbed liquid objects and colloidal particles to generation of adaptive and dynamic optical devices.