arXiv subjects

Christian Beck

Publications and source records attributed to Christian Beck.

At least 73 records · Page 4Linked to original sources

arXiv2018

Non-Gaussian power grid frequency fluctuations characterized by Lévy-stable laws and superstatistics

Multiple types of fluctuations impact the collective dynamics of power grids and thus challenge their robust operation. Fluctuations result from processes as different as dynamically changing demands, energy trading, and an increasing share of renewable power feed-in. Here we analyze principles underlying the dynamics and statistics of power grid frequency fluctuations. Considering frequency time series for a range of power grids, including grids in North America, Japan and Europe, we find a substantial deviation from Gaussianity best described as Lévy-stable and q-Gaussian distributions. We present a coarse framework to analytically characterize the impact of arbitrary noise distributions as well as a superstatistical approach which systematically interprets heavy tails and skewed distributions. We identify energy trading as a substantial contribution to today's frequency fluctuations and effective damping of the grid as a controlling factor enabling reduction of fluctuation risks, with enhanced effects for small power grids.

physics.data-an↗

arXiv2018

Optimal control of a commercial building's thermostatic load for off-peak demand response

This paper studies the optimal control of a commercial building's thermostatic load during off-peak hours as an ancillary service to the transmission system operator of a power grid. It provides an algorithmic framework which commercial buildings can implement to cost-effectively increase their electricity demand at night while they are unoccupied, instead of using standard inflexible setpoint control. Consequently, there is minimal or no impact on user comfort, while the building manager gains an additional income stream from providing the ancillary service, and can benefit further by pre-conditioning the building for later periods. The framework helps determine the amount of flexibility that should be offered for the service, and cost optimized profiles for electricity usage when delivering the service. Numerical results show that there can be an economic incentive to participate even if the payment rate for the ancillary service is less than the price of electricity.

math.OC↗

arXiv2018

Stochastic differential equations driven by deterministic chaotic maps: analytic solutions of the Perron-Frobenius equation

We consider discrete-time dynamical systems with a linear relaxation dynamics that are driven by deterministic chaotic forces. By perturbative expansion in a small time scale parameter, we derive from the Perron-Frobenius equation the corrections to ordinary Fokker-Planck equations in leading order of the time scale separation parameter. We present analytic solutions to the equations for the example of driving forces generated by N-th order Chebychev maps. The leading order corrections are universal for N larger or equal to 4 but different for N=2 and N=3. We also study diffusively coupled Chebychev maps as driving forces, where strong correlations may prevent convergence to Gaussian limit behavior.

math.CA↗

arXiv2018

Thermodynamic Properties of the Evershed Flow in the Lower Chromosphere

We used spectropolarimetric observations of a sunspot in active region NOAA 11809 in the Ca ii line at 854.2 nm taken with the SpectroPolarimeter for Optical and Infrared Regions (SPINOR) at the Dunn Solar Telescope to infer thermodynamic parameters along one hundred super-penumbral fibrils that harbor the inverse Evershed flow. The fibrils were identified in line-of-sight (LOS) velocity and line core intensity maps and were located in a segment of the sunspot that showed a regular penumbra in the photosphere. The chromospheric LOS velocity abruptly decreases from 3 to 15 km/s to zero at the inner footpoints of the fibrils that are located from the mid penumbra to about 1.4 spot radii. The spectra often show multiple components, i.e., one at the rest wavelength and one with a strong red shift, which indicates spatially or vertically unresolved structures. The line-core intensity always peaks slightly closer to the umbra than the LOS velocity. An inversion of the spectra with the CAlcium Inversion using a Spectral ARchive (CAISAR) code provided us with temperature stratifications that allowed us to trace individual fibrils through the atmosphere and to determine the angle of the flows relative to the surface without any additional assumptions on the flow topology such as radial symmetry. We find that the fibrils are not horizontal near the downflow points, but make an angle of 30 to 60 degrees to the local vertical. The temperature is enhanced by 200K at log(tau) ~ -2 and up to 2000K at log(tau) ~ -6 over that of the quiet Sun, whereas there is no signature in the low photosphere. Our results are consistent with a critical, i.e., sonic, or super-sonic siphon flow along super-penumbral flux tubes in which accelerating plasma abruptly attains sub-critical velocity through a standing shock in or near the penumbra as predicted by Montesinos & Thomas (1993).

astro-ph.SR↗

arXiv2018

Generalized statistical mechanics of cosmic rays: Application to positron-electron spectral indices

We apply generalized statistical mechanics developed for complex systems to theoretically predict energy spectra of particle and anti-particle degrees of freedom in cosmic ray fluxes, based on a $q$-generalized Hagedorn theory for transverse momentum spectra and hard QCD scattering processes. QCD at largest center of mass energies predicts the entropic index to be $q=\frac{13}{11}$, whereas the escort duality of the nonextensive thermodynamic formalism predicts an energy split of effective temperature given by $ΔkT =\pm \frac{1}{10} kT_H \approx \pm 18 $ MeV, where $T_H$ is the Hagedorn temperature. We carefully analyse the measured primary cosmic ray data of the AMS-02 collaboration and provide evidence that the predicted temperature split is indeed observed, leading to a different energy dependence of the $e^+$ and $e^-$ spectral indices. Moreover, we observe that at larger energies $E$ the measured $e^+e^-$ flux starts to deviate from our QCD-based statistical mechanics theory, with a crossover scale of $E^*=(50 \pm 10)$ GeV, which could be a hint for WIMP decay or other new physics setting in at this mass scale. Fits using linear combinations of the escort and non-escort $q$-generalized canonical distributions yield excellent agreement with the measured data in the entire energy range.

astro-ph.HE↗

arXiv2017

Possible resonance effect of dark matter axions in SNS Josephson junctions

Dark matter axions can generate peculiar effects in special types of Josephson junctions, so-called SNS junctions. One can show that the axion field equations in a Josephson environment allow for very small oscillating supercurrents, which manifest themselves as a tiny wiggle in the I-V curve, a so-called Shapiro step, which occurs at a frequency given by the axion mass. The effect is very small but perfectly measurable in modern nanotechnological devices. In this paper I will summarize the theory and then present evidence that candidate Shapiro steps of this type have indeed been seen in several independent condensed matter experiments. Assuming the observed tiny Shapiro steps are due to axion flow then these data point to an axion mass of $(106 \pm 6)μ$eV, consistent with what is expected for the QCD axion. In addition to the above small Shapiro resonance effects at frequencies in the GHz region one also expects to see broad-band noise effects at much lower frequencies. Overall this approach provides a novel pathway for the future design of new types of axionic dark matter detectors. The resonant Josephson data summarized in this paper are consistent with a 'vanilla' axion with a coupling constant $f_a=\sqrt{v_{EW}m_{Pl}}=5.48 \cdot 10^{10}$GeV given by the geometric average of the electroweak symmetry breaking scale $v_{EW}$ and the Planck mass $m_{Pl}$.

hep-ph↗

arXiv2017

Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (i) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (ii) a merged formulation of the PDE and the 2BSDE problem, (iii) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (iv) a stochastic gradient descent-type optimization procedure. Numerical results obtained using ${\rm T{\small ENSOR}F{\small LOW}}$ in ${\rm P{\small YTHON}}$ illustrate the efficiency and the accuracy of the method in the cases of a $100$-dimensional Black-Scholes-Barenblatt equation, a $100$-dimensional Hamilton-Jacobi-Bellman equation, and a nonlinear expectation of a $ 100 $-dimensional $ G $-Brownian motion.

math.NA↗

arXiv2017

Transitions between superstatistical regimes: validity, breakdown and applications

Superstatistics is a widely employed tool of non-equilibrium statistical physics which plays an important role in analysis of hierarchical complex dynamical systems. Yet, its "canonical" formulation in terms of a single nuisance parameter is often too restrictive when applied to complex empirical data. Here we show that a multi-scale generalization of the superstatistics paradigm is more versatile, allowing to address such pertinent issues as transmutation of statistics or inter-scale stochastic behavior. To put some flesh on the bare bones, we provide a numerical evidence for a transition between two superstatistics regimes, by analyzing high-frequency (minute-tick) data for share-price returns of seven selected companies. Salient issues, such as breakdown of superstatistics in fractional diffusion processes or connection with Brownian subordination are also briefly discussed.

q-fin.ST↗

arXiv2017

Symbolic dynamics techniques for complex systems: Application to share price dynamics

The symbolic dynamics technique is well-known for low-dimensional dynamical systems and chaotic maps, and lies at the roots of the thermodynamic formalism of dynamical systems. Here we show that this technique can also be successfully applied to time series generated by complex systems of much higher dimensionality. Our main example is the investigation of share price returns in a coarse-grained way. A nontrivial spectrum of Renyi entropies is found. We study how the spectrum depends on the time scale of returns, the sector of stocks considered, as well as the number of symbols used for the symbolic description. Overall our analysis confirms that in the symbol space transition probabilities of observed share price returns depend on the entire history of previous symbols, thus emphasizing the need for a modelling based on non-Markovian stochastic processes. Our method allows for quantitative comparisons of entirely different complex systems, for example the statistics of symbol sequences generated by share price returns using 4 symbols can be compared with that of genomic sequences.

nlin.CD↗

arXiv2016

Cosmological flux noise and measured noise power spectra in SQUIDs

The understanding of the origin of $1/f$ magnetic flux noise commonly observed in superconducting devices such as SQUIDS and qubits is still a major unsolved puzzle. Here we discuss the possibility that a significant part of the observed low-frequency flux noise measured in these devices is ultimately seeded by cosmological fluctuations. We consider a theory where a primordial flux noise field left over in unchanged form from an early inflationary or quantum gravity epoch of the universe intrinsically influences the phase difference in SQUIDs and qubits. The perturbation seeds generated by this field can explain in a quantitatively correct way the form and amplitude of measured low-frequency flux noise spectra in SQUID devices if one takes as a source of fluctuations the primordial power spectrum of curvature fluctuations as measured by the Planck collaboration. Our model predicts flux noise with a spectrum given by an $1/f^{2-n_s}$ spectrum, where $n_s=0.96$ is the spectral index of the near-scale invariant primordial density fluctuations. For the typical amplitude of this cosmologically generated universal flux noise we theoretically calculate the average value $δΦ/Φ_0 =3.41 \cdot 10^{-6}$ at 1Hz. These theoretical predictions are in excellent agreement with recent low-frequency flux noise measurements of various experimental groups. Magnetic flux noise, so far mainly considered as a nuisance for electronic devices, may thus contain valuable information about fluctuation spectra in the very early universe.

physics.gen-ph↗

arXiv2016

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.

cond-mat.stat-mech↗

arXiv2016

Transition from lognormal to chi-square superstatistics for financial time series

Share price returns on different time scales can be well modelled by a superstatistical dynamics. Here we provide an investigation which type of superstatistics is most suitable to properly describe share price dynamics on various time scales. It is shown that while chi-square superstatistics works well on a time scale of days, on a much smaller time scale of minutes the price changes are better described by lognormal superstatistics. The system dynamics thus exhibits a transition from lognormal to chi-square superstatistics as a function of time scale. We discuss a more general model interpolating between both statistics which fits the observed data very well. We also present results on correlation functions of the extracted superstatistical volatility parameter, which exhibits exponential decay for returns on large time scales, whereas for returns on small time scales there are long-range correlations and power-law decay.

q-fin.ST↗

arXiv2016

Superstatistics of Blaschke products

We consider a dynamics generated by families of maps whose invariant density depends on a parameter a and where a itself obeys a stochastic or periodic dynamics. For slowly varying a the long-term behavior of iterates is described by a suitable superposition of local invariant densities. We provide rigorous error estimates how good this approximation is. Our method generalizes the concept of superstatistics, a useful technique in nonequilibrium statistical mechanics, to maps. Our main example are Blaschke products, for which we provide rigorous bounds on the difference between Birkhoff density and the superstatistical composition.

nlin.CD↗

arXiv2015

Cross-Calibrating Sunspot Magnetic Field Strength Measurements from the McMath-Pierce Solar Telescope and the Dunn Solar Telescope

In this article we describe a recent effort to cross-calibrate data from an infrared detector at the McMath-Pierce Solar Telescope and the Facility InfraRed Spectropolarimeter (FIRS) at the Dunn Solar Telescope. A synoptic observation program at the McMath-Pierce has measured umbral magnetic field strengths since 1998, and this data set has recently been compared with umbral magnetic field observations from SOHO MDI and SDO HMI. To further improve on the data from McMath-Pierce, we compared the data with measurements taken at the Dunn Solar Telescope with far greater spectral resolution than has been possible with space instrumentation. To minimise potential disruption to the study, concurrent umbral measurements were made so that the relationship between the two datasets can be most accurately characterised. We find that there is a strong agreement between the umbral magnetic field strengths recorded by each instrument, and we reduced the FIRS data in two different ways to successfully test this correlation further.

astro-ph.SR↗

arXiv2014

Extreme Value Laws for Superstatistics

We study the extreme value distribution of stochastic processes modeled by superstatistics. Classical extreme value theory asserts that (under mild asymptotic independence assumptions) only three possible limit distributions are possible, namely: Gumbel, Fréchet and Weibull distribution. On the other hand, superstatistics contains three important universality classes, namely $χ^2$-superstatistics, inverse $χ^2$-superstatistics, and lognormal superstatistics, all maximizing different effective entropy measures. We investigate how the three classes of extreme value theory are related to the three classes of superstatistics. We show that for any superstatistical process whose local equilibrium distribution does not live on a finite support, the Weibull distribution cannot occur. Under the above mild asymptotic independence assumptions, we also show that $χ^2$-superstatistics generally leads an extreme value statistics described by a Fréchet distribution, whereas inverse $χ^2$-superstatistics, as well as lognormal superstatistics, lead to an extreme value statistics associated with the Gumbel distribution.

cond-mat.stat-mech↗

arXiv2014

Anomalous flows in a sunspot penumbra

High-resolution spectropolarimetric observations of active region NOAA 11271 were obtained with the spectro-polarimeter on board Hinode to analyze the properties of an anomalous flow in the photosphere in a sunspot penumbra. We detect a blue-shifted feature that appeared on the limb-side penumbra of a sunspot and that was present intermittently during the next two hours. It exhibited a maximum blue-shift of 1.6 km/s, an area of 5.2 arcsec^2, and an uninterrupted lifetime of 1 hr. The blue-shifted feature, when present, lies parallel to red-shifts. Both blue and red shifts flank a highly inclined/horizontal magnetic structure that is radially oriented in the penumbra. The low-cadence SP maps reveal changes in size, radial position in the penumbra and line-of-sight velocity of the blue-shifted feature, from one scan to the other. There was an increase of nearly 500 G in the field strength and a marginal reduction in the field inclination of about 10 deg with the onset of the blue-shifts. In the chromosphere, intense, arc-shaped brightenings were observed close to the location of the blue-shifts, that extend from the edge of the umbral core to the penumbra-quiet Sun boundary. The strongest and largest brightenings were observed about 30 min after the strongest blue-shifts were detected at the photosphere. The close spatial proximity of the two phenomenon strongly suggests a causal relationship. The blue-shifted feature represents plasma motion that could be related to a magnetic structure that rises in the solar atmosphere and subsequently reconnects with the ambient chromospheric magnetic field of the sunspot or an inverse Evershed flow, which would be unique in the photosphere. This transient phenomena is presumably related to the dynamic stability of the sunspot because the corresponding umbral core separated two days later at the location of the blue-shifts and fragmented subsequently.

astro-ph.SR↗

arXiv2014

Lagrangian quantum turbulence model based on alternating superfluid/ normal fluid stochastic dynamics

Inpired by recent measurements of the velocity and acceleration statistics of Lagrangian tracer particles embedded in a turbulent quantum liquid we propose a new superstatistical model for the dynamics of tracer particles in quantum turbulence. Our model consists of random sequences S/N/S/... where the particle spends some time in the superfluid (S) and some time in the normal fluid (N). This model leads to a superposition of power law distributions generated in the superfluid and Gaussian distributions in the normal liquid, in excellent agreement with experimental measurements. We include memory effects into our analysis and present analytic predictions for probability densities and correlation functions.

cond-mat.stat-mech↗

arXiv2014

Superstatistical analysis of sealevel fluctuations

We perform a statistical analysis of measured time series of sea levels at various coastal locations in the UK, measured at time differences of 15 minutes over the past 20 years. When the astronomical tide and other deterministic components are subtracted, a stochastic signal remains which is well-described by a superstatistical model. We do various tests on the measured time series, and compare the data of 5 different UK locations. Overall it appears that $χ^2$-superstatistics is best suitable to describe the data, in particular when one looks at the dynamics of sealevel {\em differences} on short time scales.

cond-mat.stat-mech↗