arXiv subjects

Christian Beck

Publications and source records attributed to Christian Beck.

At least 19 recordsLinked to original sources

arXiv2026

High-fidelity inference of power grid frequency distributions

Precision measurements of power grid frequency track energy supply-demand imbalances, with persistent fluctuations indicating strain that can lead to outages. Characterizing the statistical properties of grid frequency fluctuations is therefore essential to achieve efficient control and stable operations. Existing models had limited success in reconstructing the complex non-Gaussian features of observed frequency distributions. Here, we introduce a method for statistical inference of an interpretable stochastic process governing frequency fluctuations, modeling frequency through a coarse-grained power imbalance signal combined with nonlinear generator control and Gaussian white noise. We develop an efficient algorithm to infer these latent variables via maximum likelihood estimation, combined with superstatistics. We test our method on a large dataset of new measurements from Great Britain and South Africa. Although these grids have markedly different properties, our predictions for frequency distributions match measurements excellently. Our method uses only frequency data for inference, thus offering an alternative to data-intensive approaches.

physics.soc-ph↗

arXiv2026

Superstatistical Analysis of PDFs and autocorrelation functions for air pollution concentrations in the UK

Conventional statistical models often struggle to fully capture the complex spatio-temporal dynamics, intermittent fluctuations, and heavy-tailed distributions characteristic of real-world air pollution data. Furthermore, existing literature frequently focuses on extreme events, overlooking the persistence of low-pollution states and temporal memory effects. To address these gaps, we apply superstatistical frameworks from non-equilibrium statistical physics to analyse a comprehensive five-year dataset (2020-2025) of hourly air pollutant concentrations across the United Kingdom. Excellent fits of experimentally measured distributions are obtained from our theoretical models. We observe large heterogeneities of the best fitting parameters depending on the locations where the measurements are performed. These parameters form characteristic patterns in the 3-dimensional parameter space and depend on the type of pollutant considered, as well as on the environmental conditions (high traffic, industrial, or rural surroundings). We also investigate autocorrelation functions and provide evidence for differences in day-time and night-time decays of the autocorrelation function. Our investigation mainly focuses onto the dynamics of NO, NO2, PM2.5, PM10, but we also report on some anomalous distributions observed for O3.

physics.ao-ph↗

arXiv2026

Velocity of a Quantum Particle in a Classically Forbidden Region

Recently, Sharoglazova et al. [Nature 643, 67 (2025)] proposed a procedure for determining the speed of a quantum particle in the classically forbidden region of a potential step, and implemented it in a beautiful experiment. The inferred speeds disagree significantly with the Bohmian velocities, which the authors presented as an experimental challenge to Bohmian mechanics. This is puzzling because the speeds are inferred from particle populations in coupled waveguides for which Bohmian mechanics and standard quantum mechanics make identical predictions. We resolve the puzzle by a detailed theoretical analysis of the experimental setup. We show that the speed inference rests on an assumption that fails in the relevant (evanescent) regime according to both Bohmian mechanics and standard quantum mechanics -- namely that the inter-waveguide tunneling time is set by the transverse coupling and is not affected by entanglement with the longitudinal degree of freedom. We also consider a second speed estimate suggested by Sharoglazova et al., which is based on the B\"uttiker dwell time formula for particles in the forbidden region. We show that the authors applied the formula incorrectly, and that a correct application yields exact agreement with the predictions of Bohmian mechanics. Our analysis includes explicit calculations of Bohmian trajectories, dwell times, and longitudinal speeds in the two-dimensional waveguide model of the experiment.

quant-ph↗

arXiv2026

Turbulence: An Entropic Approach

We show that maximizing the generalized entropic functional $S_{q,\delta}$ subject to standard kinetic energy constraints provides generalized canonical distributions that agree perfectly with measured probability densities of velocity differences at distance $r$ in highly-turbulent Taylor-Couette flow. The end point of the turbulent cascade is described by $\delta =\frac{3}{2}$, a parameter value that also plays an important role in black-hole physics. At this point the Kolmogorov length scale $r=\eta$ is reached and all observable eddy structures of the turbulent flow disappear, in certain analogy to what is observed for black holes at the event horizon. Our approach generalizes statistical mechanics to more general nonadditive entropic functionals $S_{q,\delta}$ such that it is applicable to turbulent flows. This approach asymptotically generates stretched $q$-exponentials as generalized canonical distributions relevant for turbulent flow, with a particular dependence of the stretching exponent $\delta^{-1}$ on $q$ that follows from the well-known escort formalism in nonextensive statistical mechanics. Along this particular line in the parameter space, the physics can be described by $S_q$ on its own with suitable escort constraints, leading to the prediction $\delta^{-1} (r) =2-q(r)$, thus allowing for a consistent thermodynamic description since $S_q$ is both trace-form and composable. We show that the above theoretically derived relation is well satisfied by measured high-precision experimental data for Taylor-Couette flow. At the Kolmogorov length scale $r=\eta$, the endpoint of our scenario, one has $\delta =\frac{3}{2}$ and at this point the third moment of velocity differences ceases to exist and all eddies disappear. We point out various analogies with thermodynamic entropic approaches to black hole physics.

physics.flu-dyn↗

arXiv2026

Stochastic Modeling of Power-Grid Frequency Fluctuations in Low-Inertia Systems via a Gaussian-Core Potential and Superstatistics

Power grid frequency stability is fundamental to the secure operation of modern energy systems, yet the growing penetration of renewables and the associated reduction of system inertia have made frequency fluctuations increasingly non-Gaussian and difficult to model. Existing stochastic models based on standard Ornstein--Uhlenbeck-type restoring terms yield a unimodal frequency distribution and therefore fail to reproduce the bimodal structure, central suppression, and heavy tails widely observed in empirical data. Here, we propose a data-driven stochastic process that combines a Gaussian-core potential with superstatistical modeling, assuming slowly fluctuating coefficients for the grid dynamics. The Gaussian-core potential captures the potential barrier that gives rise to the characteristic double-peak structure of frequency distributions. Fitting the model to frequency data resolved at one-second intervals from the Great Britain grid, we find that the central barrier parameter increases substantially from 2020 to 2025 as the grid inertia progressively decreases. To simulate superstatistics, we use an Euler--Maruyama discretization and sample the drift amplitude from a lognormal distribution, thereby successfully reproducing empirical bimodality and heavy tails, as well as the autocorrelation decay. Our results establish a compact and interpretable model for characterizing the evolving complexity of low-inertia grid frequency dynamics.

physics.soc-ph↗

arXiv2026

Understanding the complexity of frequency and phase angle fluctuations in power grids

Power grids must modernize to meet climate goals while maintaining reliable and stable operating conditions. Yet progress is hindered by a limited understanding of the stochastic processes underlying grid frequency and phase-angle fluctuations, which are induced by the growing penetration of renewable generation, consumer demand fluctuations, and market trading. This issue is particularly acute in Africa, where grids often face weak investment. Here, we present results from a newly collected, large-scale, high-resolution dataset of grid frequency and phase angles for the United Kingdom and South Africa, comprising close to one billion data points. Using superstatistical modeling, we treat market-driven power fluctuations as a slowly varying parameter driving grid dynamics and incorporate nonlinear frequency control. As a result, we derive an analytical model that reproduces multimodal frequency distributions previously obtained from numerical simulations, as well as heavy-tailed fluctuations and double-exponential frequency autocorrelation decays, all in excellent agreement with experimental measurements. Beyond frequency, we also address the so far largely overlooked problem of characterizing spatial phase-angle fluctuations. By comparing our predictions with measurement data, we demonstrate that a low-dimensional effective grid model accurately fits South African data despite the grid's complexity. We also highlight significant differences between the grids of South Africa and the United Kingdom. Our results clarify how energy markets and control policies shape grid dynamics across countries with contrasting infrastructure maturity.

physics.soc-ph↗

arXiv2026

Chromospheric Flashes in a Solar Pore: Insights from Multi-line Spectropolarimetric Diagnostics

Solar pores are strongly magnetized regions lacking a photospheric penumbra and characterized by predominantly vertical magnetic fields. We present a multi-line study of flashes in a solar pore using high-resolution observations from the Swedish 1-m Solar Telescope in Fe~\textsc{i}~6302~\AA, Ca~\textsc{ii}~8542~\AA\ and K, and H-$\beta$, complemented by (E)UV data from \textit{IRIS} and \textit{SDO}/AIA. Bisector analysis and spectral inversions with \textsc{SIR} and \textsc{NICOLE} were used to infer stratifications of temperature, line-of-sight velocity, and magnetic field. Flashes, confined to one half of the pore, exhibit cooler photospheric temperatures ($\Delta T \approx 400$~K), stronger magnetic fields ($\Delta B \approx 250$~G), larger inclinations ($\sim25^{\circ}$ versus $\sim18^{\circ}$), and persistent upflows ($\sim0.5$~km~s$^{-1}$) compared to the quiescent pore. They are co-spatial with enhanced 3- and 5-minute power in the photosphere, while only 3-minute power persists in the chromosphere. Flashes are detected down to $\sim50\%$ line depth in Ca~\textsc{ii}~8542~\AA\ intensity and show central chromospheric upflows ($\sim1$~km~s$^{-1}$) flanked by strong downflows ($\sim8$~km~s$^{-1}$). Temperature enhancements reach $\sim500$~K at $\log\tau \approx -5$ and $\sim2500$~K at $\log\tau \approx -6$, with a bimodal velocity distribution. Flashes correspond one-to-one with radially outward running waves near the pore boundary (5--15~km~s$^{-1}$). Strong Ca~\textsc{ii} core emission, occasional Stokes~$V$ reversals, and H-$\beta$ enhancements indicate that pore flashes are confined to the lower and mid-chromosphere, with little influence on higher atmospheric layers.

astro-ph.SR↗

arXiv2025

Analyzing Spatio-Temporal Dynamics of Dissolved Oxygen for the River Thames using Superstatistical Methods and Machine Learning

By employing superstatistical methods and machine learning, we analyze time series data of water quality indicators for the River Thames, with a specific focus on the dynamics of dissolved oxygen. After detrending, the probability density functions of dissolved oxygen fluctuations exhibit heavy tails that are effectively modeled using $q$-Gaussian distributions. Our findings indicate that the multiplicative Empirical Mode Decomposition method stands out as the most effective detrending technique, yielding the highest log-likelihood in nearly all fittings. We also observe that the optimally fitted width parameter of the $q$-Gaussian shows a negative correlation with the distance to the sea, highlighting the influence of geographical factors on water quality dynamics. In the context of same-time prediction of dissolved oxygen, regression analysis incorporating various water quality indicators and temporal features identify the Light Gradient Boosting Machine as the best model. SHapley Additive exPlanations reveal that temperature, pH, and time of year play crucial roles in the predictions. Furthermore, we use the Transformer to forecast dissolved oxygen concentrations. For long-term forecasting, the Informer model consistently delivers superior performance, achieving the lowest MAE and SMAPE with the 192 historical time steps that we used. This performance is attributed to the Informer's ProbSparse self-attention mechanism, which allows it to capture long-range dependencies in time-series data more effectively than other machine learning models. It effectively recognizes the half-life cycle of dissolved oxygen, with particular attention to key intervals. Our findings provide valuable insights for policymakers involved in ecological health assessments, aiding in accurate predictions of river water quality and the maintenance of healthy aquatic ecosystems.

cs.LG↗

arXiv2024

Anomalous velocity distributions in slow quantum-tunneling chemical reactions

Recent work [Wild et al., Nature 615, 425 (2023)] has provided an experimental break-through in the realization of a quantum-tunneling reaction involving a proton transfer. The reaction $D^-+H_2 \to H^-+HD$ has an extremely slow reaction rate as it can happen only via quantum tunneling, thus requiring an extremely large density of the reactants in the ion trap. At these high densities strong deviations from Maxwell-Boltzmann statistics are observed. Here we develop a consistent generalized statistical mechanics theory for the above nonequilibrium situation involving quantum effects at high densities. The trapped ions are treated in a superstatistical way and a $q$-Maxwellian velocity distribution with a universal dependence of the entropic index $q$ on the density $n$ of the buffer gas is derived. We show that the velocity distribution of the ions is non-Maxwellian, more precisely $q$-Gaussian, i.e., $p(v) \propto v^2 [1+(q-1)\tilde{\beta} v^2]^{1/(1-q)}$, with entropic index $q>1$ depending on the density $n$ of $H_2$ molecules, in excellent agreement with the experimental observations of Wild et al. Our theory also makes predictions on the statistics of temperature fluctuations in the ion trap which can be tested in future experiments. Through the superstatistical approach, we obtain an analytical expression for $q(n)$ which is consistent with the available experimental data, and which yields $\lim_{n\to 0}q(n)=1$, i.e. recovering the Maxwell-Boltzmann distribution in the ideal gas limit, as well as $\lim_{n\to\infty}q(n)=7/5$.

cond-mat.stat-mech↗

arXiv2024

Generalization of the Gauss Map: A jump into chaos with universal features

The Gauss map (or continued fraction map) is an important dissipative one-dimensional discrete-time dynamical system that exhibits chaotic behaviour and which generates a symbolic dynamics consisting of infinitely many different symbols. Here we introduce a generalization of the Gauss map which is given by $x_{t+1}=\frac{1}{x_t^\alpha} - \Bigl[\frac{1}{x_t^\alpha} \Bigr]$ where $\alpha \geq 0$ is a parameter and $x_t \in [0,1]$ ($t=0,1,2,3,\ldots$). The symbol $[\dots ]$ denotes the integer part. This map reduces to the ordinary Gauss map for $\alpha=1$. The system exhibits a sudden `jump into chaos' at the critical parameter value $\alpha=\alpha_c \equiv 0.241485141808811\dots$ which we analyse in detail in this paper. Several analytical and numerical results are established for this new map as a function of the parameter $\alpha$. In particular, we show that, at the critical point, the invariant density approaches a $q$-Gaussian with $q=2$ (i.e., the Cauchy distribution), which becomes infinitely narrow as $\alpha \to \alpha_c^+$. Moreover, in the chaotic region for large values of the parameter $\alpha$ we analytically derive approximate formulas for the invariant density, by solving the corresponding Perron-Frobenius equation. For $\alpha \to \infty$ the uniform density is approached. We provide arguments that some features of this transition scenario are universal and are relevant for other, more general systems as well.

cond-mat.stat-mech↗

arXiv2024

Spatial analysis of tails of air pollution PDFs in Europe

Outdoor air pollution is estimated to cause a huge number of premature deaths worldwide, it catalyses many diseases on a variety of time scales, and it has a detrimental effect on the environment. In light of these impacts it is necessary to obtain a better understanding of the dynamics and statistics of measured air pollution concentrations, including temporal fluctuations of observed concentrations and spatial heterogeneities. Here we present an extensive analysis for measured data from Europe. The observed probability density functions (PDFs) of air pollution concentrations depend very much on the spatial location and on the pollutant substance. We analyse a large number of time series data from 3544 different European monitoring sites and show that the PDFs of nitric oxide ($NO$), nitrogen dioxide ($NO_{2}$) and particulate matter ($PM_{10}$ and $PM_{2.5}$) concentrations generically exhibit heavy tails. These are asymptotically well approximated by $q$-exponential distributions with a given entropic index $q$ and width parameter $\lambda$. We observe that the power-law parameter $q$ and the width parameter $\lambda$ vary widely for the different spatial locations. We present the results of our data analysis in the form of a map that shows which parameters $q$ and $\lambda$ are most relevant in a given region. A variety of interesting spatial patterns is observed that correlate to properties of the geographical region. We also present results on typical time scales associated with the dynamical behaviour.

physics.ao-ph↗

arXiv2023

Transition to anomalous dynamics in a simple random map

The famous Bernoulli shift (or dyadic transformation) is perhaps the simplest deterministic dynamical system exhibiting chaotic dynamics. It is a piecewise linear time-discrete map on the unit interval with a uniform slope larger than one, hence expanding, with a positive Lyapunov exponent and a uniform invariant density. If the slope is less than one the map becomes contracting, the Lyapunov exponent is negative, and the density trivially collapses onto a fixed point. Sampling from these two different types of maps at each time step by randomly selecting the expanding one with probability $p$, and the contracting one with probability $1-p$, gives a prototype of a random dynamical system. Here we calculate the invariant density of this simple random map, as well as its position autocorrelation function, analytically and numerically under variation of $p$. We find that the map exhibits a non-trivial transition from fully chaotic to completely regular dynamics by generating a long-time anomalous dynamics at a critical sampling probability $p_c$, defined by a zero Lyapunov exponent. This anomalous dynamics is characterised by an infinite invariant density, weak ergodicity breaking and power law correlation decay.

nlin.CD↗

arXiv2023

Space-Time-Matter: Some Notes on the Localization Problem in Relativistic Quantum Theory

This work aims to shed some light on the meaning of the positive energy assumption in relativistic quantum theory and its relation to questions of localization of quantum systems. It is shown that the positive energy property of solutions of relativistic wave equations (such as the Dirac equation) is very fragile with respect to state transformations beyond free time evolution. Paying attention to the connection between negative energy Dirac wave functions and pair creation processes in second quantization, this analysis leads to a better understanding of a class of problems known as the localization problem of relativistic quantum theory (associated for instance with famous results of Newton and Wigner, Reeh and Schlieder, Hegerfeldt or Malament). Finally, this analysis is reflected from the perspective of a Bohmian quantum field theory.

quant-ph↗

arXiv2023

Nonlinear Monte Carlo methods with polynomial runtime for Bellman equations of discrete time high-dimensional stochastic optimal control problems

Discrete time stochastic optimal control problems and Markov decision processes (MDPs), respectively, serve as fundamental models for problems that involve sequential decision making under uncertainty and as such constitute the theoretical foundation of reinforcement learning. In this article we study the numerical approximation of MDPs with infinite time horizon, finite control set, and general state spaces. Our set-up in particular covers infinite-horizon optimal stopping problems of discrete time Markov processes. A key tool to solve MDPs are Bellman equations which characterize the value functions of the MDPs and determine the optimal control strategies. By combining ideas from the full-history recursive multilevel Picard approximation method, which was recently introduced to solve certain nonlinear partial differential equations, and ideas from $Q$-learning we introduce a class of suitable nonlinear Monte Carlo methods and prove that the proposed methods do overcome the curse of dimensionality in the numerical approximation of the solutions of Bellman equations and the associated discrete time stochastic optimal control problems.

math.OC↗

arXiv2023

Sustained heating of the chromosphere and transition region over a sunspot light bridge

Sunspot light bridges (LBs) exhibit a wide range of short-lived phenomena in the chromosphere and transition region. In contrast, we use here data from the Multi-Application Solar Telescope (MAST), the Interface Region Imaging Spectrograph (IRIS), Hinode, the Atmospheric Imaging Assembly (AIA), and the Helioseismic and Magnetic Imager (HMI) to analyze the sustained heating over days in an LB in a regular sunspot. Chromospheric temperatures were retrieved from the the MAST Ca II and IRIS Mg II lines by nonlocal thermodynamic equilibrium inversions. Line widths, Doppler shifts, and intensities were derived from the IRIS lines using Gaussian fits. Coronal temperatures were estimated through the differential emission measure, while the coronal magnetic field was obtained from an extrapolation of the HMI vector field. At the photosphere, the LB exhibits a granular morphology with field strengths of about 400 G and no significant electric currents. The sunspot does not fragment, and the LB remains stable for several days. The chromospheric temperature, IRIS line intensities and widths, and AIA 171 \AA and 211 \AA intensities are all enhanced in the LB with temperatures from 8000 K to 2.5 MK. Photospheric plasma motions remain small, while the chromosphere and transition region indicate predominantly red-shifts of 5-20 km/s with occasional supersonic downflows exceeding 100 km/s. The excess thermal energy over the LB is about 3.2x10^26 erg and matches the radiative losses. It could be supplied by magnetic flux loss of the sunspot (7.5x10^27 erg), kinetic energy from the increase in the LB width (4x10^28 erg), or freefall of mass along the coronal loops (6.3x10^26 ,erg).

astro-ph.SR↗

arXiv2022

Information Shift Dynamics Described by Tsallis $q=3$ Entropy on a Compact Phase Space

Recent mathematical investigations have shown that under very general conditions exponential mixing implies the Bernoulli property. As a concrete example of a statistical mechanics which is exponentially mixing we consider a Bernoulli shift dynamics by Chebyshev maps of arbitrary order $N\geq 2$, which maximizes Tsallis $q=3$ entropy rather than the ordinary $q=1$ Boltzmann-Gibbs entropy. Such an information shift dynamics may be relevant in a pre-universe before ordinary space-time is created. We discuss symmetry properties of the coupled Chebyshev systems, which are different for even and odd $N$. We show that the value of the fine structure constant $\alpha_{el}=1/137$ is distinguished as a coupling constant in this context, leading to uncorrelated behaviour in the spatial direction of the corresponding coupled map lattice for $N=3$.

cond-mat.stat-mech↗

arXiv2022

An efficient Monte Carlo scheme for Zakai equations

In this paper we develop a numerical method for efficiently approximating solutions of certain Zakai equations in high dimensions. The key idea is to transform a given Zakai SPDE into a PDE with random coefficients. We show that under suitable regularity assumptions on the coefficients of the Zakai equation, the corresponding random PDE admits a solution random field which, for almost all realizations of the random coefficients, can be written as a classical solution of a linear parabolic PDE. This makes it possible to apply the Feynman--Kac formula to obtain an efficient Monte Carlo scheme for computing approximate solutions of Zakai equations. The approach achieves good results in up to 25 dimensions with fast run times.

math.NA↗

arXiv2022

Initial analysis of the impact of the Ukrainian power grid synchronization with Continental Europe

When Russia invaded Ukraine on the 24\textsuperscript{th} of February 2022, this led to many acts of solidarity with Ukraine, including support for its electricity system. Just 20 days after the invasion started, the Ukrainian and Moldovan power grids were synchronized to the Continental European power grid to provide stability to these grids. Here, we present an initial analysis of how this synchronization affected the statistics of the power grid frequency and cross-border flows of electric power within Continental Europe. We observe faster inter-area oscillations, an increase in fluctuations and changes in the cross-border flows in and out of Ukraine and surrounding countries as an effect of the synchronization with Continental Europe. Overall these changes are small such that the now connected system can be considered as stable as before the synchronization.

physics.soc-ph↗