Environmental Superstatistics

A thermodynamic device placed outdoors, or a local ecosystem, is subject to a variety of different temperatures given by short-tem (daily) and long-term (seasonal) variations. In the long term a superstatistical description makes sense, with a suitable distribution function f(beta) of inverse temperature beta over which ordinary statistical mechanics is averaged. We show that f(beta) is very different at different geographic locations, and typically exhibits a double-peak structure for long-term data. For some of our data sets we also find a systematic drift due to global warming. For a simple superstatistical model system we show that the response to global warming is stronger if temperature fluctuations are taken into account.Comment: 37 figures. Significantly extended version, to appear in Physica A. Added new material in section 6 quantifying the stronger response to global warming if temperature fluctuations are taken into account. Concluding section 7 and several new references adde

Entropies for severely contracted configuration space

We demonstrate that dual entropy expressions of the Tsallis type apply naturally to statistical-mechanical systems that experience an exceptional contraction of their configuration space. The entropic index $\alpha>1$ describes the contraction process, while the dual index $\alpha ^{\prime }=2-\alpha<1$ defines the contraction dimension at which extensivity is restored. We study this circumstance along the three routes to chaos in low-dimensional nonlinear maps where the attractors at the transitions, between regular and chaotic behavior, drive phase-space contraction for ensembles of trajectories. We illustrate this circumstance for properties of systems that find descriptions in terms of nonlinear maps. These are size-rank functions, urbanization and similar processes, and settings where frequency locking takes place

Extreme event statistics of daily rainfall: Dynamical systems approach

We analyse the probability densities of daily rainfall amounts at a variety of locations on the Earth. The observed distributions of the amount of rainfall fit well to a q-exponential distribution with exponent q close to q=1.3. We discuss possible reasons for the emergence of this power law. On the contrary, the waiting time distribution between rainy days is observed to follow a near-exponential distribution. A careful investigation shows that a q-exponential with q=1.05 yields actually the best fit of the data. A Poisson process where the rate fluctuates slightly in a superstatistical way is discussed as a possible model for this. We discuss the extreme value statistics for extreme daily rainfall, which can potentially lead to flooding. This is described by Frechet distributions as the corresponding distributions of the amount of daily rainfall decay with a power law. On the other hand, looking at extreme event statistics of waiting times between rainy days (leading to droughts for very long dry periods) we obtain from the observed near-exponential decay of waiting times an extreme event statistics close to Gumbel distributions. We discuss superstatistical dynamical systems as simple models in this context.Comment: 10 pages, 15 figures. Replaced by final version published in J.Phys.

Incidence of qq-statistics in rank distributions

We show that size-rank distributions with power-law decay (often only over a limited extent) observed in a vast number of instances in a widespread family of systems obey Tsallis statistics. The theoretical framework for these distributions is analogous to that of a nonlinear iterated map near a tangent bifurcation for which the Lyapunov exponent is negligible or vanishes. The relevant statistical-mechanical expressions associated with these distributions are derived from a maximum entropy principle with the use of two different constraints, and the resulting duality of entropy indexes is seen to portray physically relevant information. While the value of the index $\alpha$ fixes the distribution's power-law exponent, that for the dual index $2-\alpha$ ensures the extensivity of the deformed entropy.Comment: Santa Fe Institute working paper: http://www.santafe.edu/media/workingpapers/14-07-024.pdf. see: http://www.pnas.org/content/early/2014/09/03/1412093111.full.pdf+htm

Non-linear, bivariate stochastic modelling of power-grid frequency applied to islands

Mitigating climate change requires a transition away from fossil fuels towards renewable energy. As a result, power generation becomes more volatile and options for microgrids and islanded power-grid operation are being broadly discussed. Therefore, studying the power grids of physical islands, as a model for islanded microgrids, is of particular interest when it comes to enhancing our understanding of power-grid stability. In the present paper, we investigate the statistical properties of the power-grid frequency of three island systems: Iceland, Ireland, and the Balearic Islands. We utilise a Fokker-Planck approach to construct stochastic differential equations that describe market activities, control, and noise acting on power-grid dynamics. Using the obtained parameters we create synthetic time series of the frequency dynamics. Our main contribution is to propose two extensions of stochastic power-grid frequency models and showcase the applicability of these new models to non-Gaussian statistics, as encountered in islands

Non-standard power grid frequency statistics in Asia, Australia, and Europe

The power-grid frequency reflects the balance between electricity supply and demand. Measuring the frequency and its variations allows monitoring of the power balance in the system and, thus, the grid stability. In addition, gaining insight into the characteristics of frequency variations and defining precise evaluation metrics for these variations enables accurate assessment of the performance of forecasts and synthetic models of the power-grid frequency. Previous work was limited to a few geographical regions and did not quantify the observed effects. In this contribution, we analyze and quantify the statistical and stochastic properties of self-recorded power-grid frequency data from various synchronous areas in Asia, Australia, and Europe at a resolution of one second. Revealing non-standard statistics of both empirical and synthetic frequency data, we effectively constrain the space of possible (stochastic) power-grid frequency models and share a range of analysis tools to benchmark any model or characterize empirical data. Furthermore, we emphasize the need to analyze data from a large range of synchronous areas to obtain generally applicable models.Comment: 7 pages; 7 figure

Extreme event statistics of daily rainfall: dynamical systems approach

We analyse the probability densities of daily rainfall amounts at a variety of locations on Earth. The observed distributions of the amount of rainfall fit well to a q-exponential distribution with exponent q close to q approximate to 1.3. We discuss possible reasons for the emergence of this power law. In contrast, the waiting time distribution between rainy days is observed to follow a near-exponential distribution. A careful investigation shows that a q-exponential with q approximate to 1.05 yields the best fit of the data. A Poisson process where the rate fluctuates slightly in a superstatistical way is discussed as a possible model for this. We discuss the extreme value statistics for extreme daily rainfall, which can potentially lead to flooding. This is described by Frechet distributions as the corresponding distributions of the amount of daily rainfall decay with a power law. Looking at extreme event statistics of waiting times between rainy days (leading to droughts for very long dry periods) we obtain from the observed near-exponential decay of waiting times extreme event statistics close to Gumbel distributions. We discuss superstatistical dynamical systems as simple models in this context

q-GAUSSIAN ANALYSIS IN COMPLEX POLYMERS

Recently, we analyzed the temperature dependent q-Gaussian characteristics of weak chaotic transient currents through thin Aluminum-Polymethylmethacrylate-Aluminum films under voltages in the range of +/-10V at 22 degrees C temperature