arXiv subjects

Andreas Engel

Publications and source records attributed to Andreas Engel.

At least 19 recordsLinked to original sources

arXiv2024

Role of Coupling Asymmetry in the Fully Disordered Kuramoto Model

We investigate the dynamics of phase oscillators in the fully disordered Kuramoto model with couplings of defined asymmetry. The mean-field dynamics is reduced to a self-consistent stochastic single-oscillator problem which we analyze perturbatively and by numerical simulations. We elucidate the influence of the asymmetry on the correlation and response function of the system as well as on the distribution of the order parameter. The so-called volcano transition is shown to be robust with respect to a small degree of coupling asymmetry but to disappear when the antisymmetry in the couplings outweighs the symmetry.

cond-mat.stat-mech↗

arXiv2023

Nature of the Volcano Transition in the Fully Disordered Kuramoto Model

Randomly coupled phase oscillators may synchronize into disordered patterns of collective motion. We analyze this transition in a large, fully connected Kuramoto model with symmetric but otherwise independent random interactions. Using the dynamical cavity method we reduce the dynamics to a stochastic single-oscillator problem with self-consistent correlation and response functions that we study analytically and numerically. We clarify the nature of the volcano transition and elucidate its relation to the existence of an oscillator glass phase.

cond-mat.stat-mech↗

arXiv2023

Regulating ChatGPT and other Large Generative AI Models

Large generative AI models (LGAIMs), such as ChatGPT, GPT-4 or Stable Diffusion, are rapidly transforming the way we communicate, illustrate, and create. However, AI regulation, in the EU and beyond, has primarily focused on conventional AI models, not LGAIMs. This paper will situate these new generative models in the current debate on trustworthy AI regulation, and ask how the law can be tailored to their capabilities. After laying technical foundations, the legal part of the paper proceeds in four steps, covering (1) direct regulation, (2) data protection, (3) content moderation, and (4) policy proposals. It suggests a novel terminology to capture the AI value chain in LGAIM settings by differentiating between LGAIM developers, deployers, professional and non-professional users, as well as recipients of LGAIM output. We tailor regulatory duties to these different actors along the value chain and suggest strategies to ensure that LGAIMs are trustworthy and deployed for the benefit of society at large. Rules in the AI Act and other direct regulation must match the specificities of pre-trained models. The paper argues for three layers of obligations concerning LGAIMs (minimum standards for all LGAIMs; high-risk obligations for high-risk use cases; collaborations along the AI value chain). In general, regulation should focus on concrete high-risk applications, and not the pre-trained model itself, and should include (i) obligations regarding transparency and (ii) risk management. Non-discrimination provisions (iii) may, however, apply to LGAIM developers. Lastly, (iv) the core of the DSA content moderation rules should be expanded to cover LGAIMs. This includes notice and action mechanisms, and trusted flaggers. In all areas, regulators and lawmakers need to act fast to keep track with the dynamics of ChatGPT et al.

cs.CY↗

arXiv2021

Stiffness of random walks with reflecting boundary conditions

We study the distribution of occupation times for a one-dimensional random walk restricted to a finite interval by reflecting boundary conditions. At short times the classical bimodal distribution due to L\'evy is reproduced with walkers staying mostly either left or right to the initial point. With increasing time, however, the boundaries suppress large excursions from the starting point, and the distribution becomes unimodal converging to a $\delta$-distribution in the long time limit. An approximate spectral analysis of the underlying Fokker-Planck equation yields results in excellent agreement with numerical simulations.

cond-mat.stat-mech↗

arXiv2021

Aspects of a phase transition in high-dimensional random geometry

A phase transition in high-dimensional random geometry is analyzed as it arises in a variety of problems. A prominent example is the feasibility of a minimax problem that represents the extremal case of a class of financial risk measures, among them the current regulatory market risk measure Expected Shortfall. Others include portfolio optimization with a ban on short selling, the storage capacity of the perceptron, the solvability of a set of linear equations with random coefficients, and competition for resources in an ecological system. These examples shed light on various aspects of the underlying geometric phase transition, create links between problems belonging to seemingly distant fields and offer the possibility for further ramifications.

cond-mat.stat-mech↗

arXiv2020

Stiffness of Probability Distributions of Work and Jarzynski Relation for Initial Microcanonical and Energy Eigenstates

We consider closed quantum systems (into which baths may be integrated) that are driven, i.e., subject to time-dependent Hamiltonians. As a starting point we assume that, for systems initialized in microcanonical states at some energies, the resulting probability densities of work (work-PDFs) are largely independent of these specific initial energies. We show analytically that this assumption of "stiffness", together with the assumption of an exponentially growing density of energy eigenstates, is sufficient but not necessary for the validity of the Jarzynski relation (JR) for the above microcanonical initial states. This holds, even in the absence of microreversibility. To scrutinize the connection between stiffness and the JR for microcanonical initial states, we perform numerical analysis on systems comprising random matrices which may be tuned from stiff to nonstiff. In these examples we find the JR fulfilled in the presence of stiffness, and violated in its absence, which indicates a very close connection between stiffness and the JR. Remarkably, in the limit of large systems, we find the JR fulfilled, even for pure initial energy eigenstates. As this has no analogue in classical systems, we consider it a genuine quantum phenomenon.

quant-ph↗

arXiv2020

Work statistics in the periodically driven quartic oscillator: classical versus quantum dynamics

In the thermodynamics of nanoscopic systems the relation between classical and quantum mechanical description is of particular importance. To scrutinize this correspondence we study an anharmonic oscillator driven by a periodic external force with slowly varying amplitude both classically and within the framework of quantum mechanics. The energy change of the oscillator induced by the driving is closely related to the probability distribution of work for the system. With the amplitude $\lambda(t)$ of the drive increasing from zero to a maximum $\lambda_{max}$ and then going back to zero again initial and final Hamiltonian coincide. The main quantity of interest is then the probability density $P(E_f|E_i)$ for transitions from initial energy $E_i$ to final energy $E_f$. In the classical case non-diagonal transitions with $E_f\neq E_i$ mainly arise due to the mechanism of separatrix crossing. We show that approximate analytical results within the pendulum approximation are in accordance with numerical simulations. In the quantum case numerically exact results are complemented with analytical arguments employing Floquet theory. For both classical and quantum case we provide an intuitive explanation for the periodic variation of $P(E_f|E_i)$ with the maximal amplitude $\lambda_{max}$ of the driving.

cond-mat.stat-mech↗

arXiv2020

Large systems of random linear equations with non-negative solutions: Characterizing the solvable and unsolvable phase

Large systems of linear equations are ubiquitous in science. Quite often, e.g. when considering population dynamics or chemical networks, the solutions must be non-negative. Recently, it has been shown that large systems of random linear equations exhibit a sharp transition from a phase, where a non-negative solution exists with probability one, to one where typically no such solution may be found. The critical line separating the two phases was determined by combining Farkas' lemma with the replica method. Here, we show that the same methods remain viable to characterize the two phases away from criticality. To this end we analytically determine the residual norm of the system in the unsolvable phase and a suitable measure of robustness of solutions in the solvable one. Our results are in very good agreement with numerical simulations.

cond-mat.dis-nn↗

arXiv2019

On non-negative solutions to large systems of random linear equations

Systems of random linear equations may or may not have solutions with all components being non-negative. The question is, e.g., of relevance when the unknowns are concentrations or population sizes. In the present paper we show that if such systems are large the transition between these two possibilities occurs at a sharp value of the ratio between the number of unknowns and the number of equations. We analytically determine this threshold as a function of the statistical properties of the random parameters and show its agreement with numerical simulations. We also make contact with two special cases that have been studied before: the storage problem of a perceptron and the resource competition model of MacArthur.

cond-mat.dis-nn↗

arXiv2018

Systems of random linear equations and the phase transition in MacArthur's resource-competition model

Complex ecosystems generally consist of a large number of different species utilizing a large number of different resources. Several of their features cannot be captured by models comprising just a few species and resources. Recently, Tikhonov and Monasson have shown that a high-dimensional version of MacArthur's resource competition model exhibits a phase transition from a 'vulnerable' to a 'shielded' phase in which the species collectively protect themselves against an inhomogeneous resource influx from the outside. Here we point out that this transition is more general and may be traced back to the existence of non-negative solutions to large systems of random linear equations. Employing Farkas' Lemma we map this problem to the properties of a fractional volume in high dimensions which we determine using methods from the statistical mechanics of disordered systems.

physics.bio-ph↗

arXiv2017

Stochastic thermodynamics of interacting degrees of freedom: Fluctuation theorems for detached path probabilities

Systems with interacting degrees of freedom play a prominent role in stochastic thermodynamics. Our aim is to use the concept of detached path probabilities and detached entropy production for bipartite Markov processes and elaborate on a series of special cases including measurement-feedback systems, sensors and hidden Markov models. For these special cases we show that fluctuation theorems involving the detached entropy production recover known results which have been obtained separately before. Additionally, we show that the fluctuation relation for the detached entropy production can be used in model selection for data stemming from a hidden Markov model. We discuss the relation to previous approaches including those which use information flow or learning rate to quantify the influence of one subsystem on the other. In conclusion, we present a complete framework with which to find fluctuation relations for coupled systems.

cond-mat.stat-mech↗

arXiv2016

Design of NbN superconducting nanowire single photon detectors with enhanced infrared photon detection efficiency

We propose an optimized design for nanowire superconducting single photon detectors, using the recently discovered position dependent detection efficiency in these devices. This knowledge allows an optimized the design of meandering wire NbN detectors by altering the field distribution across the wire. In order to calculate the response of the detectors with different geometries, we use a monotonic local detection efficiency from a nanowire and optical absorption distribution via finite-different-time-domain simulations. The calculations predict a trade-off between average absorption and the edge effect leading to a predicted optimal wire width close to 100 nm for 1550 nm wavelength, which drops to 50 nm wire width for 600 nm wavelength. The absorption at the edges can be enhanced by depositing a silicon nanowire on top of the superconducting nanowire, which improves both the total absorption efficiency as well as the internal detection efficiency of meandering wire structures.

physics.ins-det↗

arXiv2016

Large deviations in Taylor dispersion

We establish a link between the phenomenon of Taylor dispersion and the theory of empirical distributions. Using this connection, we derive, upon applying the theory of large deviations, an alternative and much more precise description of the long-time regime for Taylor dispersion.

cond-mat.stat-mech↗

arXiv2016

Level 2 large deviation functionals for systems with and without detailed balance

Large deviation functions are an essential tool in the statistics of rare events. Often they can be obtained by contraction from a so-called level 2 large deviation {\em functional} characterizing the empirical density of the underlying stochastic process. For Langevin systems obeying detailed balance, the explicit form of this functional has been known ever since the mathematical work of Donsker and Varadhan. We rederive the Donsker-Varadhan result by using stochastic path-integrals and then generalize it to situations without detailed balance including non-equilibrium steady states. The proper incorporation of the empirical probability flux turns out to be crucial. We elucidate the relation between the large deviation functional and different notions of entropy production in stochastic thermodynamics and discuss some aspects of the ensuing contractions. Finally, we illustrate our findings with examples.

cond-mat.stat-mech↗

arXiv2015

Position-Dependent Local Detection Efficiency in a Nanowire Superconducting Single-Photon Detector

We probe the local detection efficiency in a nanowire superconducting single-photon detector along the cross-section of the wire with a spatial resolution of 10 nm. We experimentally find a strong variation in the local detection efficiency of the device. We demonstrate that this effect explains previously observed variations in NbN detector efficiency as function of device geometry.

cond-mat.supr-con↗

arXiv2014

Stochastic thermodynamics of rapidly driven systems

We present the stochastic thermodynamics analysis of an open quantum system weakly coupled to multiple reservoirs and driven by a rapidly oscillating external field. The analysis is built on a modified stochastic master equation in the Floquet basis. Transition rates are shown to satisfy the local detailed balance involving the entropy flowing out of the reservoirs. The first and second law of thermodynamics are also identified at the trajectory level. Mechanical work is identified by means of initial and final projections on energy eigenstates of the system. We explicitly show that this two step measurement becomes unnecessary in the long time limit. A steady-state fluctuation theorem for the currents and rate of mechanical work is also established. This relation does not require the introduction of a time reversed external driving which is usually needed when considering systems subjected to time asymmetric external fields. This is understood as a consequence of the secular approximation applied in consistency with the large time scale separation between the fast driving oscillations and the slower relaxation dynamics induced by the environment. Our results are finally illustrated on a model describing a thermodynamic engine.

cond-mat.stat-mech↗

arXiv2014

Detection Mechanism in SNSPD: Numerical Results of a Conceptually Simple, Yet Powerful Detection Model

In a recent publication we have proposed a numerical model that describes the detection process of optical photons in superconducting nanowire single-photon detectors (SNSPD). Here, we review this model and present a significant improvement that allows us to calculate more accurate current distributions for the inhomogeneous quasi-particle densities occurring after photon absorption. With this new algorithm we explore the detector response in standard NbN SNSPD for photons absorbed off-center and for 2-photon processes. We also discuss the outstanding performance of SNSPD based on WSi. Our numerical results indicate a different detection mechanism in WSi than in NbN or similar materials.

cond-mat.supr-con↗

arXiv2014

Prior-predictive value from fast-growth simulations: Error analysis and bias estimation

Variants of fluctuation theorems recently discovered in the statistical mechanics of non-equilibrium processes may be used for the efficient determination of high-dimensional integrals as typically occurring in Bayesian data analysis. In particular for multimodal distributions, Monte-Carlo procedures not relying on perfect equilibration are advantageous. We provide a comprehensive statistical error analysis for the determination of the prior-predictive value in a Bayes problem building on a variant of the Jarzynski equation. Special care is devoted to the characterization of the bias intrinsic to the method. We also discuss the determination of averages over multimodal posterior distributions with the help of a variant of the Crooks theorem. All our findings are verified by extensive numerical simulations of two model systems with bimodal likelihoods.

physics.data-an↗