Stochastic Field Research Articles

Predicting the turbulent transport of cosmic rays via neural networks

A fast artificial neural network is developed for the prediction of cosmic ray transport in turbulent astrophysical magnetic fields. The setup is trained and tested on bespoke datasets that are constructed with the aid of test-particle numerical simulations of relativistic cosmic ray dynamics in synthetic stochastic fields. The neural network uses, as input, particle and field properties and estimates transport coefficients 107 faster than standard numerical simulations with an overall error of ∼5%.

Machine learning assisted characterisation and prediction of droplet distributions in a liquid jet in cross-flow

In this paper, an artificial neural network (ANN) is trained with large eddy simulation (LES) data to predict the droplet size distribution (DSD) from the primary atomisation of a liquid jet in gaseous cross-flow (JIC), in terms of the Weber number (We), momentum flux ratio (q) and density ratio. The JIC is simulated considering three We (250, 500, 1000), q (1, 5, 10), and density ratios (10, 100, 1000), respectively. The accuracy of the simulations is enhanced by including the injector geometry as well. The training data are obtained using LES with a stochastic fields transported-probability density function (PDF) method. We initially provide a physical analysis of the droplet distributions observed. We find that for lower density ratios, the resulting spray is mostly dominated by q, influencing the main mechanisms governing the break-up process, which change the DSD shape. This dual mechanism is not present when increasing the density ratio. In the second part of the work, we build an ANN model (based on a multi-layered perceptron) using the DSDs from the LES as a train-and-test dataset, to predict at the end the full DSD for the JIC given as input the three non-dimensional parameters. The DSD from the trained ANN is found to be a good fit for the range investigated, predicting both the stochastic nature and change in shape of the droplet populations upon varying the input parameters. The developed model is intended to enhance future simulations of secondary atomisation in Eulerian-Lagrangian frameworks by providing the initial DSDs.

Methods of Stochastic Field Theory in Non-Equilibrium Systems --- Spontaneous Symmetry Breaking of Ergodicity

Methods of Stochastic Field Theory in Non-Equilibrium Systems --- Spontaneous Symmetry Breaking of Ergodicity

The Algebra and Calculus of Stochastically Perturbed Spacetime with Classical and Quantum Applications

We consider an alternative to dark matter as a potential solution to various remaining problems in physics: the addition of stochastic perturbations to spacetime to effectively enforce a minimum length and establish a fundamental uncertainty at minimum length (ML) scale. To explore the symmetry of spacetime to such perturbations both in classical and quantum theories, we develop some new tools of stochastic calculus. We derive the generators of rotations and boosts, along with the connection, for stochastically perturbed, minimum length spacetime (“ML spacetime”). We find the metric, the directional derivative, and the canonical commutator preserved. ML spacetime follows the Lie algebra of the Poincare group, now expressed in terms of the two-point functions of the stochastic fields (per Ito’s lemma). With the fundamental uncertainty at ML scale a symmetry of spacetime, we require the translational invariance of any classical theory in classical spacetime to also include the stochastic spacetime perturbations. As an application of these ideas, we consider galaxy rotation curves for massive bodies to find that—under the Robertson–Walker minimum length theory—rotational velocity becomes constant as the distance to the center of the galaxy becomes very large. The new tools of stochastic calculus also set the stage to explore new frontiers at the quantum level. We consider a massless scalar field to derive the Ward-like identity for ML currents.

Longitudinal and azimuthal thermo-acoustic instabilities in an industrial gas turbine combustor operating at elevated pressure

This work numerically investigates longitudinal and azimuthal thermo-acoustic instabilities in the swirl-stabilised can-type industrial SGT-100 gas turbine combustor operated at elevated pressures of 3 and 6 bar. Previous experiments have shown that the combustor is susceptible to self-excited flame oscillations sustained by a thermo-acoustic feedback loop at specific operating conditions. In order to gain a better understanding of this feedback loop, a fully compressible large eddy simulation method is applied. The unknown sub-grid scale turbulence-chemistry interactions are modelled via a transported probability density function approach solved by the Eulerian stochastic fields method. First, the reaction zones and global flame topology at both operating pressures are analysed and compared to experimental images providing good qualitative agreement. Radial profiles of time-averaged and root-mean-square quantities furthermore demonstrate good quantitative agreement with the available measurement data. The applied simulation approach is capable of successfully reproducing self-excited thermo-acoustic instabilities in the longitudinal direction. The fundamental frequency of the predicted limit-cycle oscillation matches the experimentally measured frequency with high accuracy. Similar to the experimental observations, the fluctuation amplitudes of the pressure and global heat release rate increase significantly upon increasing the mean operating pressure from 3 to 6 bar. In addition to the dominant longitudinal mode, a high-frequency, low-amplitude azimuthal mode is also identified at both pressures. This azimuthal mode is periodically amplified and attenuated by the superposed longitudinal mode and induces small asymmetric (around the burner circumference) fluctuations of the local fuel and total mixture mass flow rates entering the flame region.

Stochastic multiple fields inflation: Diffusion dominated regime

Stochastic multiple fields inflation: Diffusion dominated regime

Clustering of Floating Tracers in a Random Velocity Field Modulated by an Ellipsoidal Vortex Flow

The influence of a background vortex flow on the clustering of floating tracers is addressed. The vortex flow considered is induced by an ellipsoidal vortex evolving in a deformation. The system exhibits various vortex motion regimes: (1) a steady state, (2) oscillation and (3) rotation of the ellipsoidal vortex core. The latter two induce an unsteady velocity field for the tracer, thus leading to irregular (chaotic) tracer motion. Superimposing a stochastic divergent velocity field onto the deterministic vortex flow allows us to observe significantly different tracer evolution. An ellipsoidal vortex has ellipsoidal symmetry, and the tracer’s trajectories exhibit the same symmetry inside the vortex. Outside the vortex, the external deformation flow symmetry dominates. Diffusion scattering and chaotic advection give tracers the opportunity to leave the region of ellipsoidal symmetry and form a picture of shear flow symmetry. We use the method of characteristics to integrate the floating tracer density evolution equation and the Euler Ito scheme for obtaining the floating tracer trajectories with a random velocity field. The cluster area and cluster mass from the statistical topography are used as the quantitative diagnostics of a floating tracer’s clustering. For the case of a steady ellipsoidal vortex embedded into the deformation flow with a random velocity field component, we found that the clustering characteristics were weakened by the steady vortex. For the cases of an unsteady ellipsoidal vortex, we observed clustering in the floating tracer density field if the contribution of the divergent component was greater than or equal to that of the rotational (nondivergent) component. Even when the initial floating tracer patch was set on the boundary of the oscillating ellipsoidal vortex, we observed the formation of clusters. In the case of a rotating ellipsoidal vortex, we also observed pronounced clustering. Thus, we argue that unsteady ellipsoidal vortex regimes (oscillation and rotation), which induce chaotic motion of the nearby passive tracer’s trajectories, are still conducive to clustering of floating tracers observed in the density field, despite the intense deformation introduced by strain and shear.

Entropy production in the nonreciprocal Cahn-Hilliard model.

We study the nonreciprocal Cahn-Hilliard model with thermal noise as a prototypical example of a generic class of non-Hermitian stochastic field theories, analyzed in two companion papers [Suchanek, Kroy, and Loos, Phys. Rev. Lett. 131, 258302 (2023)10.1103/PhysRevLett.131.258302; Phys. Rev. E 108, 064123 (2023)10.1103/PhysRevE.108.064123]. Due to the nonreciprocal coupling between two field components, the model is inherently out of equilibrium and can be regarded as an active field theory. Beyond the conventional homogeneous and static-demixed phases, it exhibits a traveling-wave phase, which can be entered via either an oscillatory instability or a critical exceptional point. By means of a Fourier decomposition of the entropy production rate, we quantify the associated scale-resolved time-reversal symmetry breaking, in all phases and across the transitions, in the low-noise regime. Our perturbative calculation reveals its dependence on the strength of the nonreciprocal coupling. Surging entropy production near the static-dynamic transitions can be attributed to entropy-generating fluctuations in the longest wavelength Fourier mode and heralds the emerging traveling wave. Its translational dynamics can be mapped on the dissipative ballistic motion of an active (quasi)particle.

Neural network based generation of a 1-dimensional stochastic field with turbulent velocity statistics

We define and study a fully-convolutional neural network stochastic model, NN-Turb, which generates a 1-dimensional field with some turbulent velocity statistics. In particular, the generated process satisfies the Kolmogorov 2/3 law for the second order structure function. It also presents negative skewness across scales (i.e. Kolmogorov 4/5 law) and exhibits intermittency as characterized by skewness and flatness. Furthermore, our model is never in contact with turbulent data and only needs the desired statistical behavior of the structure functions across scales for training.

Статистичний метод визначення параметрів поля зсувних напружень в площині ковзання в багатокомпонентному сплаві

A method has been developed in which atomic sizes misfit and elastic modulus misfit at crystal lattice nodes are considered as discrete random variables and the definition of their dispersion allows to obtain analytical expressions for standard deviations and correlation lengths of the short- and long-wave components of stochastic shear stress field created by solute atoms in the glide plane in a multicomponent alloy. This makes it possible to significantly reduce the amount of calculations when determining the shear stress field parameters. The developed method was applied to calculate these parameters for the CrCoNiFeMn alloy. The calculated parameters were well correlated with similar parameters determined from the analysis of shear stress distributions in the glide plane, which were calculated by the method of direct summation of solute atoms contributions. In addition, it was found that there are separate effective crystal lattice distortions for the short- and long-wave components that differ from the average distortion that was proposed earlier. This results from the fact that these components are determined by solute atoms with different distance from the glide plane. Effective distortion is greater, the greater this distance from the glide plane. In addition, there is no single empirical constant for all alloy to determine the yield strength as a function of their shear modulus and average distortion. But the proposed method makes it possible to determine the main parameters of the shear stress field in a specific multicomponent alloy. These parameters can be used to calculate the yield strength of this alloy. Keywords: shear stress, multicomponent alloy, glide plane, solid solution.

AutoKE: An Automatic Knowledge Embedding Framework for Scientific Machine Learning

Imposing physical constraints on neural networks as a method of knowledge embedding has achieved great progress in solving physical problems described by governing equations. However, for many engineering problems, governing equations often have complex forms, including complex partial derivatives or stochastic physical fields, which results in significant inconveniences from the perspective of implementation. In this paper, to effectively automate the process of embedding physical knowledge, a scientific machine learning framework, called AutoKE, is proposed, and a reservoir flow problem with a complex governing equation is taken as an instance. In AutoKE, an emulator comprised of deep neural networks (DNNs) is built for predicting the physical variables of interest. An arbitrarily complex equation can be parsed and automatically converted into a computational graph through the equation parser module, and the fitness of the emulator to the governing equation is evaluated via automatic differentiation. Furthermore, the fixed weights in the loss function are substituted with adaptive weights by incorporating the Lagrangian dual method. Neural architecture search (NAS) is also introduced into the AutoKE to select an optimal network architecture of the emulator according to the specific problem. Finally, we apply transfer learning to enhance the scalability of the emulator. In experiments, the framework is verified by a series of physical problems in which it can automatically embed physical knowledge into an emulator without heavy hand-coding. The results demonstrate that the emulator can not only make accurate predictions, but also be applied to similar problems with high efficiency via transfer learning.

Uncertainty quantification for Fisher-Kolmogorov equation on graphs with application to patient-specific Alzheimer's disease

The Fisher-Kolmogorov equation is a diffusion-reaction PDE that is used to model the accumulation of prionic proteins, which are responsible for many different neurological disorders. Likely, the most important and studied misfolded protein in literature is the Amyloid-$\beta$, responsible for the onset of Alzheimer disease. Starting from medical images we construct a reduced-order model based on a graph brain connectome. The reaction coefficient of the proteins is modelled as a stochastic random field, taking into account all the many different underlying physical processes, which can hardly be measured. Its probability distribution is inferred by means of the Monte Carlo Markov Chain method applied to clinical data. The resulting model is patient-specific and can be employed for predicting the disease's future development. Forward uncertainty quantification techniques (Monte Carlo and sparse grid stochastic collocation) are applied with the aim of quantifying the impact of the variability of the reaction coefficient on the progression of protein accumulation within the next 20 years.

Enhanced diffusion of tracer particles in nonreciprocal mixtures.

We study the diffusivity of a tagged particle in a binary mixture of Brownian particles with nonreciprocal interactions. Numerical simulations reveal that, for a broad class of interaction potentials, nonreciprocity can significantly increase the long-time diffusion coefficient of tracer particles and that this diffusion enhancement is associated with a breakdown of the Einstein relation. These observations are quantified and confirmed via two different and complementary analytical approaches: (i) a linearized stochastic density field theory, which is particularly accurate in the limit of soft interactions, and (ii) a reduced two-body description, which is exact at leading order in the density of particles. The latter reveals that diffusion enhancement can be attributed to the formation of transiently propelled dimers of particles, whose cohesion and speed are controlled by the nonreciprocal interactions.

Application of high-credible statistical results calculation scheme based on least squares Quasi-Monte Carlo method in multimodal stochastic problems

Solving stochastic problems based on multimodal distributions with high accuracy and efficiency is the focus of current research in the stochastic field. However, since exact statistical results in practical engineering problems are unknown, how to efficiently obtain statistical results with high credibility is of great importance. Therefore, a high-credible statistical results calculation scheme is proposed in this paper. First, the multimodal model is transformed into a weighted sum form of multiple unimodal models by Gaussian mixture model. Then, a high-credible statistical results calculation scheme is designed based on the reusability of low-discrepancy sequences in the quasi-Monte Carlo (QMC) method. The scheme calculates the coefficient of variation for the random response statistical results of the unimodal model across various sample ranges. When the coefficient of variation is less than the tolerance error, which indicates low fluctuation, the statistical results are deemed to have converged to the exact solution. In order to further accelerate the calculation efficiency of the proposed scheme, this paper proposes a least squares quasi-Monte Carlo (LSQMC) method to yield high accuracy statistical moments. Compared with the averageness weights in the QMC method, the non-averageness weights obtained by LSQMC method can more effectively reflect the numerical Characteristics of the random variables and the location distribution of the sampling points in the probability space. Finally, numerical examples have demonstrated that the high-credible statistical results calculation scheme based on the LSQMC method can obtain the high accuracy statistical moments with high efficiency in multimodal stochastic problems.

Tailoring resonant magnetic perturbation to optimize fast-ion confinement during ELM control in KSTAR

3D resonant magnetic perturbation (RMP) is one promising way to control edge localized modes that can cause excessive material erosion of tokamak first walls. However, RMP can lead to undesired degradation of plasma confinement, including fast-particle losses, which can impact the performance and safety of the reactor. This work investigates the optimization of the poloidal spectrum of the 3D field to optimize fast ion confinement during edge localized mode (ELM) suppression. In the initial step, the validity of the modeling framework is tested against experimental data. Simulations successfully replicate an increase in poloidal limiter temperature with different poloidal spectra. Then, the simulation shows improvement of fast ion confinement with a reduction of core resonant response, while edge resonant magnetic fields are maintained above the threshold to sustain the ELM suppression. Reduction of the core resonant fields keeps the Kolmogorov–Arnold–Moser surface and reduces the fast particle losses due to the stochastic magnetic field lines. The results highlight the potential of edge localization of the resonant fields to enhance the performance of fusion reactors, but further investigation is needed to improve the validation of this approach.

On analytical theories for conductivity and self-diffusion in concentrated electrolytes.

Describing analytically the transport properties of electrolytes, such as their conductivity or the self-diffusion of the ions, has been a central challenge of chemical physics for almost a century. In recent years, this question has regained some interest in light of Stochastic Density Field Theory (SDFT) - an analytical framework that allows the approximate determination of density correlations in fluctuating systems. In spite of the success of this theory to describe dilute electrolytes, its extension to concentrated solutions raises a number of technical difficulties, and requires simplified descriptions of the short-range repulsion between the ions. In this article, we discuss recent approximations that were proposed to compute the conductivity of electrolytes, in particular truncations of Coulomb interactions at short distances. We extend them to another observable (the self-diffusion coefficient of the ions) and compare them to earlier analytical approaches, such as the mean spherical approximation and mode-coupling theory. We show how the treatment of hydrodynamic effects in SDFT can be improved, that the choice of the modified Coulomb interactions significantly affects the determination of the properties of the electrolytes, and that comparison with other theories provides a guide to extend SDFT approaches in this context.

Electric-Field Fluctuations as the Cause of Spectral Instabilities in Colloidal Quantum Dots.

Spectral diffusion (SD) represents a substantial obstacle toward implementation of solid-state quantum emitters as a source of indistinguishable photons. By performing high-resolution emission spectroscopy for individual colloidal quantum dots at cryogenic temperatures, we prove the causal link between the quantum-confined Stark effect and SD. Statistically analyzing the wavelength of emitted photons, we show that increasing the sensitivity of the transition energy to an applied electric field results in amplified spectral fluctuations. This relation is quantitatively fit to a straightforward model, indicating the presence of a stochastic electric field on a microscopic scale, whose standard deviation is 9 kV/cm, on average. The current method will enable the study of SD in multiple types of quantum emitters such as solid-state defects or organic lead halide perovskite quantum dots, for which spectral instability is a critical barrier for applications in quantum sensing.

Dimension Reduction Method-Based Stochastic Wind Field Simulations for Dynamic Reliability Analysis of Communication Towers

The communication tower is a lifeline engineering that ensures the normal operation of wireless communication systems. Extreme wind disasters are inevitable while it is in service. Two dimension-reduction (DR) probabilistic representations based on proper orthogonal decomposition (POD) and wavenumber spectral representation (WSR), say DR-POD and DR-WSR, were thus proposed in this study. In order to determine the least representative sample size that satisfies the engineering accuracy requirements, the simulation error and simulation duration of 10 simulation points distributed along the height direction of the communication tower under different representative sample numbers were compared. Furthermore, for the fluctuating wind field with different numbers of simulation points distributed along the height of the communication tower, the simulation accuracy as well as efficiency of the DR-POD and the DR-WSR were compared. Finally, a high-rise communication tower structure’s wind-induced dynamic response study and wind-resistance reliability analysis were performed utilizing an alliance of the probability density evolution method (PDEM) and two DR probabilistic models, taking 10 load points into account. The structural dynamic analysis illustrates that the reliability of the communication tower structure and the wind-induced dynamic response allying the two DR probabilistic models with the PDEM have outstanding consistency.

Simulation study of interaction between energetic particles and magnetohydrodynamic modes in the JT-60SA inductive scenario with a flat q≈1 profile

Interactions between energetic particles (EPs), an internal kink mode, and other magnetohydrodynamic (MHD) instabilities in the inductive scenario of JT-60SA (scenario # 2) are simulated with MEGA, a global EP–MHD hybrid code. For this scenario, it was predicted by TOPICS, an integrated transport code that the internal kink mode can be unstable and the sawtooth relaxation results in a flat safety factor (q) profile with q ≈ 1 for r/a⩽0.6 . In this equilibrium, it is found in the simulation results that the stability of the internal kink mode depends strongly on the bulk plasma pressure gradient ( ∇Pb ). In the n = 1 simulations where the toroidal mode number is restricted to n = 0 and 1, the pressure-driven internal kink mode is dominant. In the presence of co-passing EPs generated by the negative-ion-based neutral beam (NB), these EPs transfer energy to the internal kink mode; however, the EP driving rate ( γh ) is much lower than the driving rate from the bulk plasma pressure gradient ( γP ). The mode’s frequency is less than 1 kHz because the toroidal orbit frequency (ω φ ) and poloidal orbit frequency (ω θ ) of the co-passing EPs are approximately equal within the q = 1 surfaces. This mode affects the EP and bulk plasma pressure redistributions. The feasibility of stabilizing the internal kink mode using trapped EPs is also investigated. It is found that the trapped EPs with energy 85 keV generated by the positive-ion-based NBs cannot stabilize the internal kink mode. Stabilization is observed when the injection energy is greater than 500 keV. In the multi-n simulations, where n⩽8 modes are retained, the most unstable modes are high n interchange modes with poloidal number m = n whose linear growth rates exceed that of the pressure-driven internal kink mode observed in the n = 1 simulations. The overlapping of these modes creates a stochastic magnetic field, leading to stronger EP and bulk plasma pressure redistributions than those observed in the n = 1 simulations. During the nonlinear phase, the transition from the high m = n modes to the low m = n modes is observed where the dominant mode is the m/n=1/1 mode with an internal kink-like structure. These low m = n modes are generated by the nonlinear coupling of the high m = n modes. The EP kinetic effect has a minor contribution to the dynamics of these nonlinearly generated m = n modes.

A quasi-linear model of particle, momentum and heat transport in a stochastic magnetic field

We present a quasi-linear treatment of the drift-kinetic equation in the presence of a stochastic magnetic field, which provides a self-contained description of particle, parallel momentum and heat transport. Explicit analytical expressions, which satisfy the Onsager reciprocal relations, are obtained by approximating the distribution function by a local shifted Maxwellian. This theory completes previous formulations (Harvey et al., Phys. Rev. Lett., vol. 47, 1981, p. 102) by including the momentum transport and by generalizing the derivation from the cylindrical tokamak configuration to an arbitrary cylindrical pinch. Application to the reversed field pinch provides satisfactory results.