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At least 253 records · Page 14

Quantum stabilization of unexpected ordered phases on the honeycomb lattice

In this work, I discuss and showcase the utility of the method called minimally-augmented spin-wave theory (MAGSWT) as a relatively simple semi-analytical approach to the phase diagrams of quantum magnets. It complements numerical methods by providing physical insight into which states are competitive and by yielding approximate phase boundaries that agree well with much more numerically intensive calculations. This approach enabled me to construct the faithful phase diagrams of two paradigmatic honeycomb-lattice models in the quantum S=1/2 limit, the J 1 –J 3 FM-AF and J 1 –J 2 AF models, for the collinear quantum phases that replace or extend the classical ones. The results are in good qualitative and semi-quantitative agreement with state-of-the-art numerical studies for these models, correctly capturing the emergence of unexpected quantum phases and the suppression of classically favored spiral orders by quantum fluctuations. This study provides a much-needed important guidance to the ongoing theoretical and experimental searches of the unconventional quantum states. This work will be of significant and timely interest to both theorists and experimentalists in the field of quantum magnetism, broadly defined.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Insights into distorted lamellar phases with small-angle scattering and machine learning

Lamellar phases are essential in various soft matter systems, with topological defects significantly influencing their mechanical properties. In this report, we present a machine-learning approach for quantitatively analyzing the structure and dynamics of distorted lamellar phases using scattering techniques. By leveraging the mathematical framework of Kolmogorov–Arnold networks, we demonstrate that the conformations of these distorted phases – expressed as superpositions of complex waves – can be reconstructed from small-angle scattering intensities. Through the contour analysis of wave field phase singularities, we obtain the statistics of the spatial distribution of topological defects. Furthermore, we establish that the temporal evolution of these defects can be derived from the time-dependent traveling wave field, informed by the dispersion relation of spectral components. This method opens new avenues for investigating the dynamics of distorted lamellar phases using various dynamic scattering techniques such as neutron spin echo and X-ray photon correlation spectroscopy. These findings enhance our microscopic understanding of how defects influence the physical properties of lamellar materials, with implications for both equilibrium and non-equilibrium states in general lamellar systems.

36 MATERIALS SCIENCE↗

Ensuring Solution Uniqueness in Three-Phase Power System State Estimation

This paper is concerned with the issue of potential non-unique solutions in three-phase state estimation. Theory of observability analysis for positive sequence power system state estimation is based on certain assumptions that avoid possibility of multiple solutions. Also, it is shown that observability of a positive sequence network remains independent of the network parameters or the operating state. When extending single-phase observability analysis directly to the three-phase case, this paper considers the possibility of converging to multiple solutions, i.e. solution non-uniqueness, even for cases where state estimator successfully converges. The study illustrates via numerical examples the likelihood of converging to entirely different solutions for certain network parameters. It also examines how the operating state, particularly under unbalanced loading, leads to solution non-uniqueness. The paper then describes an alternative approach to ensure a unique solution in three-phase state estimation. This method aims to accurately and uniquely estimate the state of any unbalanced three-phase system, irrespective of load imbalance, network configuration, existence of synchronous generators or transformers.

Power System State Estimation, Three-Phase, Distri↗

High-pressure phase transition of olivine-type Mg 2 GeO 4 to a metastable forsterite-III type structure and their equations of state

Germanates are often used as structural analogs of planetary silicates. We have explored the high-pressure phase relations in Mg 2 GeO 4 using diamond-anvil cell experiments combined with synchrotron X-ray diffraction and computations based on density functional theory. Upon room temperature compression, forsterite-type Mg 2 GeO 4 remains stable up to 30 GPa. At higher pressures, a phase transition to a forsterite-III type (Cmc2 1 ) structure was observed, which remained stable to the peak pressure of 105 GPa. Using a third-order Birch Murnaghan fit to the experimental data, we obtained V 0 = 305.1(3) Å3, K 0 = 124.6(14) GPa, and $K'_0$ = 3.86 (fixed) for forsterite-type Mg 2 GeO 4 and V 0 = 263.5(15) Å 3 , K 0 = 175(7) GPa, and $K'_0$ = 4.2 (fixed) for the forsterite-III type phase. The forsterite-III type structure was found to be metastable when compared to the stable assemblage of perovskite/post-perovskite + MgO, as observed during laser-heating experiments. Understanding the phase relations and physical properties of metastable phases is crucial for studying the mineralogy of impact sites, understanding metastable wedges in subducting slabs, and interpreting the results of shock compression experiments.

58 GEOSCIENCES↗

Advanced Semi-Supervised Learning with Uncertainty Estimation for Phase Identification in Distribution Systems

The integration of advanced metering infrastructure (AMI) into power distribution networks generates valuable data for tasks such as phase identification; however, the limited and unreliable availability of labeled data in the form of customer phase connectivity presents challenges. To address this issue, we propose a semi-supervised learning (SSL) framework that effectively leverages labeled and unlabeled data. Our approach incorporates self-training, label spreading, and Bayesian neural networks (BNNs) to enhance phase identification with AMI data. Our method uses an ensemble of multilayer perceptron classifiers in a self-training setup, iteratively adding high-confidence pseudo-labels to improve robustness. We also apply label spread to propagate labels based on data similarity, which enhances generalization across diverse distributions. In addition, we employ a BNNs with uncertainty estimation, boosting confidence in predictions and reducing phase identification errors. In our case study, we achieved approximately 98% +/- 0.08 accuracy with uncertainty using minimal and unreliable labeled data from a real U.S. utility, Duquesne Light Company. Our SSL approach, combined with uncertainty estimation, provides an efficient solution for phase identification in AMI data, ultimately improving the reliability of smart grid applications.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Advanced Semi-Supervised Learning With Uncertainty Estimation for Phase Identification in Distribution Systems

The integration of advanced metering infrastructure (AMI) into power distribution networks generates valuable data for tasks such as phase identification; however, the limited and unreliable availability of labeled data in the form of customer phase connectivity presents challenges. To address this issue, we propose a semi-supervised learning (SSL) framework that effectively leverages labeled and unlabeled data. Our approach incorporates self-training, label spreading, and Bayesian neural networks (BNNs) to enhance phase identification with AMI data. Our method uses an ensemble of multilayer perceptron classifiers in a self-training setup, iteratively adding high-confidence pseudo-labels to improve robustness. We also apply label spread to propagate labels based on data similarity, which enhances generalization across diverse distributions. In addition, we employ a BNNs with uncertainty estimation, boosting confidence in predictions and reducing phase identification errors. In our case study, we achieved approximately 98% +/- 0.08 accuracy with uncertainty using minimal and unreliable labeled data from a real U.S. utility, Duquesne Light Company. Our SSL approach, combined with uncertainty estimation, provides an efficient solution for phase identification in AMI data, ultimately improving the reliability of smart grid applications.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Uncertainty-Guided Prediction Horizon of Phase-Resolved Ocean Wave Forecasting Under Data Sparsity: Experimental and Numerical Evaluation

Accurate short-term wave forecasting is critical for the safe and efficient operation of marine structures that rely on real-time, phase-resolved ocean wave information for control and monitoring purposes (e.g., digital twins). These systems often depend on environmental sensors (e.g., waverider buoys, wave-sensing LIDAR). Challenges arise when upstream sensor data are missing, sparse, or phase-shifted due to drift. This study investigates the performance of two machine learning models, time-series dense encoder (TiDE) and long short-term memory (LSTM), for forecasting phase-resolved ocean surface elevations under varying degrees of data degradation. We introduce the τ-trimming algorithm, which adapts the prediction horizon based on uncertainty thresholds derived from historical forecasts. Numerical wave tank (NWT) and wave basin experiments are used to benchmark model performance under short- and long-term data masking, spatially coarse sensor grids, and upstream phase shifts. Results show under a 50% probability of upstream data loss, the τ-trimmed TiDE model achieves a 46% reduction in error at the most upstream target, compared to 22% for LSTM. Furthermore, phase misalignment in upstream data introduces a near-linear increase in forecast error. Under moderate model settings, a ±3 s misalignment increases the mean absolute error by approximately 0.5 m, while the same error is accumulated at ±4 s using the more conservative approach. These findings inform the design of resilient, uncertainty-aware wave forecasting systems suited for realistic offshore sensing environments.

42 ENGINEERING↗

Prognostic simulations of mixed-phase clouds with model AC-1D v1.0: the impact of aerosol types and freezing parameterizations on ice crystal budgets

Mixed-phase clouds at high latitudes contribute to the uncertainty in predicting cloud feedbacks and climate sensitivity, mainly due to the complexity of microphysical processes that influence the partitioning between the supercooled liquid and ice phases, and hence, cloud radiative effects on regional scales. Particularly in Arctic mixed-phase clouds, the activation of ice-nucleating particles (INPs) from various aerosol populations remains a leading source of uncertainty. We developed an aerosol-cloud one-dimensional (AC-1D) model, which provides a novel framework to prognostically treat INP and ice crystal budgets while explicitly accounting for polydisperse and multicomponent aerosol that activate INPs following different freezing parameterizations. The AC-1D model is informed by large-eddy simulations to probe the impact of INP representation on predicted ice crystal number concentrations (N i ) and ice crystal budgets in mixed-phase Arctic stratus. We apply three immersion freezing (IMF) parameterizations, two time-independent (singular) and one time-dependent (classical nucleation theory), to predict the evolution of the INP reservoir and resulting ice crystal budget from polydisperse mineral dust, organic (humic-like substances), and sea spray aerosol particle size distributions. Our analysis focuses on how variations in aerosol number concentration and cloud system parameters such as cloud cooling rate, cloud-top entrainment rate, and ice crystal fall speed influence the INP reservoir and ice crystal budgets. Furthermore, this study investigates the competitive ice nucleation dynamics in mixed aerosol environments and provides a process-level quantification of the INP budget terms, which directly controls ice crystal budgets. For all studied case scenarios, the aerosol types and associated particle size distributions significantly impact INP and N i , and the choice between a time-dependent and a singular freezing description yields orders-of-magnitude differences in the predicted INP and N i over the 10 h simulation time, reflecting typical cloud lifetimes. Our results show that the influence of cloud cooling, INP entrainment, and sedimentation varies significantly depending on the chosen freezing parameterization. These findings underscore the critical need for robust IMF parameterizations and precise cloud system observations to enhance the accuracy of models in predicting mixed-phase cloud structure and evolution.

Arctic clouds↗

Thermodynamic Cloud Phase Classifications Using Machine Learning at NSA and ANX

Vertically resolved thermodynamic cloud phase classifications are essential for studies of atmospheric cloud and precipitation processes. The Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) THERMOCLDPHASE Value-Added Product (VAP) uses a multi-sensor approach to classify thermodynamic cloud phase by combining lidar backscatter and depolarization, radar reflectivity, Doppler velocity, spectral width, microwave radiometer-derived liquid water path, and radiosonde temperature measurements. The measured voxels are classified as ice, snow, mixed-phase, liquid (cloud water), drizzle, rain, and liq_driz (liquid+drizzle). We use this product as the ground truth to train three machine learning (ML) models to predict the thermodynamic cloud phase from multi-sensor remote sensing measurements taken at the ARM North Slope of Alaska (NSA) observatory: a random forest (RF), a multilayer perceptron (MLP), and a convolutional neural network (CNN) with a U-Net architecture. Evaluations against the outputs of the THERMOCLDPHASE VAP with one year of data show that the CNN outperforms the other two models, achieving the highest test accuracy, F1-score, and mean Intersection over Union (IOU). Analysis of ML confidence scores shows ice, rain, and snow have higher confidence scores, followed by liquid, while mixed, drizzle, and liq_driz have lower scores. Feature importance analysis reveals that the mean Doppler velocity and vertically resolved temperature are the most influential datastreams for ML thermodynamic cloud phase predictions. The ML models’ generalization capacity is further evaluated by applying them at another Arctic ARM site in Norway using data taken during the ARM Cold-Air Outbreaks in the Marine Boundary Layer Experiment (COMBLE) field campaign. Finally, we evaluate the ML models’ response to simulated instrument outages and signal degradation.

54 ENVIRONMENTAL SCIENCES↗

Comparison of Gas Phase Fragmentation Behaviors of Nuclear Fuel Cycle Ligands in Lanthanide and Americium Metal Ligand Nitrate Clusters

Title (20 word): Comparison of Gas Phase Fragmentation Behaviors of Nuclear Fuel Cycle Ligands in Lanthanide and Americium Metal Ligand Nitrate Clusters Introduction (120 words): Transport of metal ions across the aqueous-organic phase boundary is an essential step in a hydrometallurgical nuclear fuel reprocessing strategy. The study of transport agents for nuclear fuel elements is imperative to guide the design of ligands that boost the separation efficiency of the recovery process from fission products. However, limited studies have been made on the chemistry of these transport agents when complexing with transuranic elements in gas-phase where all surrounding factors are essentially excluded. This work investigates the reagent ligand complexations to transuranic and other metals and their dissociations in the gas phase. Comparisons are made between 4f and 5f elements and between ligands. Methods (120 words): (N,N-diisobutylcarbamoylmethyl)phenyloctylphosphine oxide (CMPO) and N,N,N',N'-tetraoctyldiglycolamide (TODGA) have been selected to complex with metal nitrates. The actinide americium and lanthanides neodymium, samarium, and europium were investigated as part of this work. The lanthanides were selected to act as size and electron configuration analogues of the minor actinides. Metal complexes with two ligands and two nitrates ([M(NO3)2(CMPO)2]+, for example) are studied in Bruker micrOTOF-Q II mass spectrometer equipped with collision-induced dissociation capability. The comparisons of the mass spectra are made in groups of homogenous ligands and mixed TODGA-CMPO ligands clusters. Comparisons are also made based on the complexed metals (Am and lanthanides). Preliminary data (300 words): Collision-induced dissociation mass spectrometry data are collected on two ligands complexed with metal nitrates where the two ligands are homogenous, with (CMPO)2 or (TODGA)2, or heterogeneous, with (TODGA)(CMPO). Several fragmentation patterns are observed among complexes with the CMPO ligand whereas the TODGA ligand commonly dissociates intact from the complex. Most of the metal complexes exhibit similar fragmentation patterns, but there are a few notable deviations in fragmentation patterns between the Am and Ln-bearing complexes. For the [M(CMPO)2(NO3)2]+ complexes, the initial loss of nitrate in the form of nitric acid is observed in all four complexes. However, [Am(CMPO)2(NO3)2]+ exhibits an additional fragmentation not found in the lanthanide complexes. Also, a significantly different ratio of the second nitric acid loss is found in the Am complex. These deviations may indicate the different interaction behaviors between actinides and lanthanides. The [M(TODGA)2(NO3)2]+ complexes exhibit the fragmentation as the loss of one TODGA ligand as an intact form and the loss of nitrate as nitric acid. The Am complex exhibits an additional fragmentation after losing the TOGDA ligand, which is not observed among the Ln complexes. The heterogeneous [M(TODGA)(CMPO)(NO3)2]+ complexes exhibit both similarities and differences between the Am and Ln complexes. For example, the heterogenous Am complex does not exhibit the loss of an intact TODGA ligand while all three Ln complexes do. This indicates that TODGA may bind more strongly to Am than Ln. Additionally, the intensity of the loss of CMPO ligand (as partially or whole) is found to be significantly larger than that of the loss of TODGA (as partially of whole) indicating that TODGA is bound to the metal significantly stronger than CMPO. Planned computational analysis will help understand the deviation in fragmentation behaviors between americium and lanthanide metal centers, or between TODGA and CMPO ligands. Novel aspect (20 words): Gas-phase actinide and lanthanide complex formation and fragmentation provide insight into the coordination environment differences of f-element metals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Per-Phase Control for CHB Converters with Negative-Sequence Current and DC Voltage Balancing Control

This paper presents a per-phase controller for a grid-connected Cascaded H-Bridge (CHB) converter featuring negative sequence current regulation and DC bus voltage balancing capabilities. Unlike conventional three-phase controllers, each phase of the CHB can be separately and controlled to regulate the DC voltage and reactive power flow. per-phase negative sequence current control strategy is incorporated into the proposed controller to ensure compliance with IEEE Standard 2800-2022, which requires the absorption of negative-sequence reactive current during grid voltage disturbances. Furthermore, a feed-forward control-based DC bus voltage balancing method is employed, which eliminates the need for the integrator commonly used in traditional approaches. The performance of the proposed controller is validated under various unbalanced grid scenarios and DC bus voltage imbalances through electromagnetic transient (EMT) simulation using a three-phase, three-level grid-connected CHB–Dual Active Bridge (DAB) converter testbed developed on MATLAB/Simulink. Additionally, the performance of the negative-sequence current regulation is investigated through simulation under different unbalanced grid voltage scenarios.

negative sequence current↗

Development and Validation of Two-Phase Flow Models in MOOSE and Application to Molten Salt Reactors

Two-phase flow in Molten Salt Reactors (MSRs) is important as it impacts reactivity evolution, reactor transient response, and the removal of species dissolved in the molten salt through gas phase transfer. Therefore, accurately predicting the gas distribution and the associated liquid-gas interface area in MSRs is essential for their design and operation. Recently, we integrated a new two-phase model into Idaho National Laboratory (INL)’s Multiphysics Object-Oriented Simulation Environment (MOOSE): a multi-D generalization of a mixture drift-flux model. It provides greater computational efficiency, which is typically preferred for modeling reactor transients. However, the mixture model’s accuracy in capturing void distribution and interfacial area in MSRs still needs to be assessed. This article begins with a description of the mathematical framework for the two-phase model implemented in MOOSE. It then presents validation of these models against relevant experimental data. Finally, the model is applied to the Molten Salt Reactor Experiment case study, analyzing various operational conditions such as different rates of fission product volatilization and diverse cover gas entrainment scenarios at the reactor pump. The article concludes by assessing the suitability of the mixture drift-flux model for capturing the two-phase flow dynamics critical to MSR operations

42 - ENGINEERING↗

Coherent-Precipitation-Stabilized Phase Formation in Over-Stoichiometric Rocksalt-Type Li Superionic Conductors

Rationalizing synthetic pathways is crucial for material design and property optimization, especially for polymorphic and metastable phases. Over-stoichiometric rocksalt (ORX) compounds, characterized by their face-sharing configurations, are a promising group of materials with unique properties; however, their development is significantly hindered by challenges in synthesizability. Here, taking the recently identified Li superionic conductor, over-stoichiometric rocksalt Li–In–Sn–O (o-LISO) material as a prototypical ORX compound, the mechanisms of phase formation are systematically investigated. It is revealed that the spinel-like phase with unconventional stoichiometry forms as coherent precipitate from the high-temperature-stabilized cation-disordered rocksalt phase upon fast cooling. This process prevents direct phase decomposition and kinetically locks the system in a metastable state with the desired face-sharing Li configurations. This insight enables us to enhance the ionic conductivity of o-LISO to be >1 mS cm -1 at room temperature through low-temperature post-annealing. This work offers insights into the synthesis of ORX materials and highlights important opportunities in this new class of materials.

36 MATERIALS SCIENCE↗

Current Advances in i‐MAX Phases and their Two Dimensional Derivative i‐MXenes: Challenges and Opportunities (Adv. Electron. Mater. 21/2025)

The discovery of quaternary (M′ 2/3 M′′ 1/3 ) 2 AX phases has introduced newly ordered i-MAX phases in the MAX phase community. These atomically layered solids display in-plane chemical ordering of M′ and M′′, featuring a frustrated triangular lattice overlaid on an M′ honeycomb arrangement and an A Kagomé lattice. This unique structure gives rise to novel electronic and magnetic properties, paving the way for diverse applications and the creation of new MXenes. Both experimental and theoretical research have confirmed that these i-MAX phases can be chemically exfoliated into single- or multilayered and vacancy-ordered 2D transition metal carbides, known as i-MXenes. These 2D i-MXenes exhibit intriguing optical, electrochemical, piezoelectric, and magnetic properties, which are decidedly reliant on the surface functional groups (-F, -OH, -O). This review encompasses all available theoretical and experimental studies on i-MAX and i-MXenes, with a focus on their fundamental properties, organized in multiple sections. Along with the experimental investigation, significant attention is also directed toward theoretical predictions of potential i-MAX phases and i-MXenes, including their structural, vibrational, electronic, optical, magnetic, mechanical, piezoelectric, and electrochemical properties. This article provides a comprehensive understanding of vital properties of these materials by providing a review of foundational literature with existing challenges, limitations, and future perspectives.

electrochemical, electronic↗

Solution‐Phase Metathesis of Li 3 N and FeCl 3 to Synthesize Fe 2 N/Fe 3 N Nanoparticles

Soft magnetic materials play key roles in the flow of energy in electrically driven machines and power conversion electronics, and there is a great need for improvements in their magnetic properties to provide the right combination of high saturation magnetization, low coercivity, and high permeability. Most phases of iron nitride (Fe x N) are soft magnetic materials with these characteristics, but they exist as numerous phases which are not all stoichiometric compounds. While the production and magnetic properties of the different phases of bulk iron nitride are well known, accessing phase‐pure nanoscale iron nitride consistently remains a challenge. Most methods for the synthesis of iron nitride nanoparticles require complicated apparatus to achieve high‐temperature nitriding of nanoparticle precursors with gaseous nitrogen sources such as ammonia. The first solution‐phase metathesis reaction between FeCl 3 and Li 3 N in oleylamine is developed to directly synthesize Fe 2 N/Fe 3 N nanoparticles, requiring only a fume hood, glove box, standard chemistry laboratory glassware, and equipment. Finally, the ≈10–15 nm spheres display nearly soft magnetic behavior with a saturation magnetization ≈50–60 A m 2 kg −1 , coercivities between 40–50 kA m −1 , and susceptibility values from 0.0001–0.0006 m 3 kg −1 , well within the ranges reported with other published Fe x N nanoparticle synthesis methods.

iron nitride↗

The phase space distance between collider events

How can one fully harness the power of physics encoded in relativistic N-body phase space? Topologically, phase space is isomorphic to the product space of a simplex and a hypersphere and can be equipped with explicit coordinates and a Riemannian metric. This natural structure that scaffolds the space on which all collider physics events live opens up new directions for machine learning applications and implementation. Here we present a detailed construction of the phase space manifold and its differential line element, identifying particle ordering prescriptions that ensure that the metric satisfies necessary properties. We apply the phase space metric to several binary classification tasks, including discrimination of high-multiplicity resonance decays or boosted hadronic decays of electroweak bosons from QCD processes, and demonstrate powerful performance on simulated data. Our work demonstrates the many benefits of promoting phase space from merely a background on which calculations take place to being geometrically entwined with a theory’s dynamics.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Reduced-order condensed-phase kinetic models for polyethylene, polypropylene and polystyrene thermochemical recycling

Thermochemical recycling of plastic waste (PW) into chemicals and energy vectors requires coupling particle and reactor-scale simulations to accurate condensed phase pyrolysis mechanisms for each constituent. This work proposes a methodology to derive reduced-order condensed-phase kinetic models from validated semi-detailed kinetic mechanisms. Two types of kinetic models are obtained for polyethylene (PE), polypropylene (PP) and polystyrene (PS): reduced semi-detailed models and multi-step fully lumped ones. These families offer different compromises between accuracy and computational cost. The former employ 50–100 gas + liquid species and describe both the radical degradation and the detailed carbon distribution of the products. Conversely, the latter involves 5–10 species per polymer tracking only the main petroleum cuts. The kinetic mechanisms are complemented by the definition of thermochemical properties of gas, liquid, and solid-phase species, accounting for phase-transitions through pseudo-chemical reactions. Model validations are performed by comparison with experimental data and the original semi-detailed mechanisms in terms of mass loss, heat fluxes and product distribution profiles. The resulting CHEMKIN-like condensed-phase models are attached as Supplementary Material and as a GitHub repository. Extending the proposed approach to other polymers and coupling it with existing subsets in the CRECK kinetic framework (e.g., biomass, PVC, PET) offers a powerful tool to model thermochemical recycling of PW and biomass/PW mixtures.

kinetics↗

A collision operator for describing dissipation in noncanonical phase space

The phase space of a noncanonical Hamiltonian system is partially inaccessible due to dynamical constraints (Casimir invariants) arising from the kernel of the Poisson tensor. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in a thermodynamic equilibrium described by a Maxwell–Boltzmann distribution. However, the time scale required to reach Maxwell–Boltzmann statistics is often much longer than the time scale over which a given system achieves a state of thermal equilibrium. Examples include diffusion in rigid mechanical systems, as well as collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we focus on self-organizing phenomena over spacetime scales such that particle interactions respect the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for general noncanonical Hamiltonian systems, applicable to fast, localized interactions. This collision operator depends on the interaction exchanged by colliding particles and on the Poisson tensor encoding the noncanonical phase space structure, is consistent with entropy growth and conservation of particle number and energy, preserves the interior Casimir invariants, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell–Boltzmann statistics due to the noncanonical phase space structure, and how self-organization and collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator.

Boltzmann equation↗