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NASA NTRS2023

Observational Constraints Reduce Model Spread but Not Uncertainty in Global Wetland Methane Emission Estimates

The recent rise in atmospheric methane (CH 4 ) concentrations accelerates climate change and offsets mitigation efforts. Although wetlands are the largest natural CH 4 source, estimates of global wetland CH 4 emissions vary widely among approaches taken by bottom-up (BU) process-based biogeochemical models and top-down (TD) atmospheric inversion methods. Here, we integrate in situ measurements, multi-model ensembles, and a machine learning upscaling product into the International Land Model Benchmarking system to examine the relationship between wetland CH 4 emission estimates and model performance. We find that using better-performing models identified by observational constraints reduces the spread of wetland CH 4 emission estimates by 62% and 39% for BU- and TD-based approaches, respectively. However, global BU and TD CH 4 emission estimate discrepancies increased by about 15% (from 31 to 36 TgCH 4 year −1 ) when the top 20% models were used, although we consider this result moderately uncertain given the unevenly distributed global observations. Our analyses demonstrate that model performance ranking is subject to benchmark selection due to large inter-site variability, highlighting the importance of expanding coverage of benchmark sites to diverse environmental conditions. We encourage future development of wetland CH 4 models to move beyond static benchmarking and focus on evaluating site-specific and ecosystem-specific variabilities inferred from observations.

Kuang-Yu Chang↗

NASA NTRS2011

Prediction of Weather Impacted Airport Capacity using Ensemble Learning

Ensemble learning with the Bagging Decision Tree (BDT) model was used to assess the impact of weather on airport capacities at selected high-demand airports in the United States. The ensemble bagging decision tree models were developed and validated using the Federal Aviation Administration (FAA) Aviation System Performance Metrics (ASPM) data and weather forecast at these airports. The study examines the performance of BDT, along with traditional single Support Vector Machines (SVM), for airport runway configuration selection and airport arrival rates (AAR) prediction during weather impacts. Testing of these models was accomplished using observed weather, weather forecast, and airport operation information at the chosen airports. The experimental results show that ensemble methods are more accurate than a single SVM classifier. The airport capacity ensemble method presented here can be used as a decision support model that supports air traffic flow management to meet the weather impacted airport capacity in order to reduce costs and increase safety.

Weather impact↗

NASA NTRS2023

Large-Scale Groundwater Monitoring in Brazil Assisted With Satellite-Based Artificial Intelligence Techniques

Here, we develop and test an artificial intelligence (AI)-based approach to monitor major Brazilian aquifers. The approach combines Gravity Recovery and Climate Experiment (GRACE) data and ground-based hydrogeological measurements from Brazil’s Integrated Groundwater Monitoring Network at hundreds of wells distributed in twelve aquifers across the country. We tested model ensembles based on three AI approaches: Extreme Gradient Boost, Light Gradient Boosting Model and CatBoost, followed by a Linear Regression (LR) step. The approach is further boosted with wavelet and seasonal decomposition processes applied to GRACE data. To determine the AI-based model’s sensitivity to data availability, we propose four experiments combining hydrogeological measurements from different aquifers. Groundwater storage estimates from the Global Land Data Assimilation System (GLDAS) are used as benchmark. A sensitivity analysis shows that the LR-based model ensemble is the best suited and to reproduce groundwater storage change in all studied Brazilian aquifers. Results show that the proposed approach outperforms GLDAS in all experiments, with an RMSE value of 2.68cm for the experiment that covers all monitored wells in Brazil. GLDAS resulted in RMSE=6.76cm. Using our AI model outputs, we quantified the groundwater storage change of two major aquifers, Urucuia and Bauru-Caiuá, over the past two decades: -31km 3 and -6km 3 , respectively. Water loss is driven by a prolonged drought across most of the country and intensification of groundwater pumping for irrigation. This study demonstrates that combining satellite data and AI can be a cost-effective alternative to monitor poorly equipped aquifers at the continental scale, with possible global replicability.

GRACE↗

DOE OSTI2025

A dissipation-induced superradiant transition in a strontium cavity-QED system

Driven-dissipative many-body systems are ubiquitous in nature and a fundamental resource for quantum technologies. However, they are also complex and hard to model because they cannot be described by the standard tools in equilibrium statistical mechanics. Probing nonequilibrium critical phenomena in pristine setups can illuminate fresh perspectives on these systems. Here, we use an ensemble of cold 88 Sr atoms coupled to a driven high-finesse cavity to study the cooperative resonance fluorescence (CRF) model, a classic driven-dissipative model describing coherently driven dipoles superradiantly emitting light. We observe its nonequilibrium phase diagram characterized by a second-order phase transition. Below a critical drive strength, the atoms quickly reach the so-called superradiant steady state featuring a macroscopic dipole moment; above the critical point, the atoms undergo persistent Rabi-like oscillations. At longer times, spontaneous emission transforms the second-order transition into a discontinuous first-order transition. Our observations pave the way for harnessing robust entangled states and exploring boundary time crystals in driven-dissipative systems.

Science & Technology - Other Topics↗

NASA NTRS2019

The Salience of Nonlinearities in the Boreal Winter Response to ENSO: Arctic Stratosphere and Europe

The Arctic stratospheric response to El Niño (EN) and La Niña (LN) is evaluated in a 41 member ensemble of the period 1980 to 2009 in the Goddard Earth Observing System Chemistry-Climate Model. We consider whether the responses to EN and LN are equal in magnitude and opposite in sign, whether the responses to moderate and extreme events are proportion-ate, and if the response depends on whether sea surface temperature anomalies (SSTs) peak in the Eastern Pacific (EP) or Central Pacific (CP). There is no indication of any nonlinearities between EN and LN, though in ~ 15% of the ensemble members the stratospheric sudden warming (SSW) frequencies for EN and LN are similar, suggesting that a similar SSW frequency for EN and LN, as has occurred over the past ~ 60 years, can occur by chance. The response to extreme EN events is not proportionate to the amplitude of the underlying SST anomalies in spring. EP EN events preferentially increase zonal wavenumber 1 and decrease zonal wavenumber 2 as compared to CP EN events, however the zonal-mean Arctic stratospheric and subpolar surface response is generally little different between EP EN and CP EN once one accounts for the relative weakness of CP events. These differences between EP and CP events and between moderate and extreme EN events only emerge if at least 25 events are composited, however, due to the small signal-to-noise ratio, and hence these differences may be of little practical benefit.

Sea surface temperature anomalies (SSTs) peak in t↗

NASA NTRS2008

Assimilation of GRACE Terrestrial Water Storage Data into a Land Surface Model

The NASA Gravity Recovery and Climate Experiment (GRACE) system of satellites provides observations of large-scale, monthly terrestrial water storage (TWS) changes. In. this presentation we describe a land data assimilation system that ingests GRACE observations and show that the assimilation improves estimates of water storage and fluxes, as evaluated against independent measurements. The ensemble-based land data assimilation system uses a Kalman smoother approach along with the NASA Catchment Land Surface Model (CLSM). We assimilated GRACE-derived TWS anomalies for each of the four major sub-basins of the Mississippi into the Catchment Land Surface Model (CLSM). Compared with the open-loop (no assimilation) CLSM simulation, assimilation estimates of groundwater variability exhibited enhanced skill with respect to measured groundwater. Assimilation also significantly increased the correlation between simulated TWS and gauged river flow for all four sub-basins and for the Mississippi River basin itself. In addition, model performance was evaluated for watersheds smaller than the scale of GRACE observations, in the majority of cases, GRACE assimilation led to increased correlation between TWS estimates and gauged river flow, indicating that data assimilation has considerable potential to downscale GRACE data for hydrological applications. We will also describe how the output from the GRACE land data assimilation system is now being prepared for use in the North American Drought Monitor.

Reichle, Rolf H.↗

NASA NTRS2014

The Goddard Cumulus Ensemble Model (GCE): Improvements and Applications for Studying Precipitation Processes

Convection is the primary transport process in the Earth's atmosphere. About two-thirds of the Earth's rainfall and severe floods derive from convection. In addition, two-thirds of the global rain falls in the tropics, while the associated latent heat release accounts for three-fourths of the total heat energy for the Earth's atmosphere. Cloud-resolving models (CRMs) have been used to improve our understanding of cloud and precipitation processes and phenomena from micro-scale to cloud-scale and mesoscale as well as their interactions with radiation and surface processes. CRMs use sophisticated and realistic representations of cloud microphysical processes and can reasonably well resolve the time evolution, structure, and life cycles of clouds and cloud systems. CRMs also allow for explicit interaction between clouds, outgoing longwave (cooling) and incoming solar (heating) radiation, and ocean and land surface processes. Observations are required to initialize CRMs and to validate their results. The Goddard Cumulus Ensemble model (GCE) has been developed and improved at NASA/Goddard Space Flight Center over the past three decades. It is amulti-dimensional non-hydrostatic CRM that can simulate clouds and cloud systems in different environments. Early improvements and testing were presented in Tao and Simpson (1993) and Tao et al. (2003a). A review on the application of the GCE to the understanding of precipitation processes can be found in Simpson and Tao (1993) and Tao (2003). In this paper, recent model improvements (microphysics, radiation and land surface processes) are described along with their impact and performance on cloud and precipitation events in different geographic locations via comparisons with observations. In addition, recent advanced applications of the GCE are presented that include understanding the physical processes responsible for diurnal variation, examining the impact of aerosols (cloud condensation nuclei or CCN and ice nuclei or IN) on precipitation processes, utilizing a satellite simulator to improve the microphysics, providing better simulations for satellite-derived latent heating retrieval, and coupling with a general circulation model to improve the representation of precipitation processes.

GCE↗

NASA NTRS2018

Impact of Satellite Sea Surface Salinity Observations on ENSO Predictions from the GMAO S2S Forecast System

El Nino/Southern Oscillation (ENSO) has far reaching global climatic impacts and so extending useful ENSO forecasts would be of great benefit for society. However, one key variable that has yet to be fully exploited within coupled forecast systems is accurate estimation of near-surface ocean density. Satellite Sea Surface Salinity (SSS), combined with temperature, help to identify ocean density changes and associated mixing near the ocean surface. We assess the impact of satellite SSS observations for improving near-surface dynamics within ocean analyses and how these impact dynamical ENSO forecasts using the NASA GMAO (Global Modeling and Assimilation Office) Sub-seasonal to Seasonal (S2S_v2.1) coupled forecast system (Molod et al. 2018 - i.e. NASA's contribution to the NMME (National Multi-Model Ensemble) project). For all initialization experiments, all available along-track absolute dynamic topography and in situ observations are assimilated using the LETKF ( Local Ensemble Transform Kalman Filter) scheme similar to Penny et al., 2013. A separate reanalysis additionally assimilates Aquarius V5 (September 2011 to June 2015) and SMAP (Soil Moisture Active Passive) V4 (March 2015 to present) along-track data.We highlight the impact of satellite SSS on ocean reanalyses by comparing validation statistics of experiments that assimilate SSS versus our current prediction system that withholds SSS. We find that near-surface validation versus observed statistics for salinity are slightly degraded when assimilating SSS. This is an expected result due to known biases between SSS (measured by the satellite at approximately 1 centimeter) and in situ measurements (typically measured by Argo floats at 3 meters). On the other hand, a very encouraging result is that both temperature, absolute dynamic topography, and mixed layer statistics are improved with SSS assimilation. Previous work has shown that correcting near-surface density structure via gridded SSS assimilation can improve coupled forecasts. Here we present results of coupled forecasts that are initialized from the GMAO S2S reanalyses that assimilates/withholds along-track (L2) SSS. In particular, we contrast forecasts of the overestimated 2014 El Nino, the big 2015 El Nino, and the minor 2016 La Nina. For each of these ENSO scenarios, assimilation of satellite SSS improves the forecast validation. Improved SSS and density upgrades the mixed layer depth leading to more accurate coupled air/sea interaction.

Hackert, E.↗

NASA NTRS2011

Estimating Evapotranspiration with Land Data Assimilation Systems

Advancements in both land surface models (LSM) and land surface data assimilation, especially over the last decade, have substantially advanced the ability of land data assimilation systems (LDAS) to estimate evapotranspiration (ET). This article provides a historical perspective on international LSM intercomparison efforts and the development of LDAS systems, both of which have improved LSM ET skill. In addition, an assessment of ET estimates for current LDAS systems is provided along with current research that demonstrates improvement in LSM ET estimates due to assimilating satellite-based soil moisture products. Using the Ensemble Kalman Filter in the Land Information System, we assimilate both NASA and Land Parameter Retrieval Model (LPRM) soil moisture products into the Noah LSM Version 3.2 with the North American LDAS phase 2 (NLDAS-2) forcing to mimic the NLDAS-2 configuration. Through comparisons with two global reference ET products, one based on interpolated flux tower data and one from a new satellite ET algorithm, over the NLDAS2 domain, we demonstrate improvement in ET estimates only when assimilating the LPRM soil moisture product.

Peters-Lidard, C. D.↗

NASA NTRS2020

Modelling climate change impacts on maize yields under low nitrogen input conditions in sub‐Saharan Africa

Smallholder farmers in sub‐Saharan Africa (SSA) currently grow rainfed maize with limited inputs including fertilizer. Climate change may exacerbate current production constraints. Crop models can help quantify the potential impact of climate change on maize yields, but a comprehensive multimodel assessment of simulation accuracy and uncertainty in these low‐input systems is currently lacking. We evaluated the impact of varying [CO2], temperature and rainfall conditions on maize yield, for different nitrogen (N) inputs (0, 80, 160 kg N/ha) for five environments in SSA, including cool subhumid Ethiopia, cool semi‐arid Rwanda, hot subhumid Ghana and hot semi‐arid Mali and Benin using an ensemble of 25 maize models. Models were calibrated with measured grain yield, plant biomass, plant N, leaf area index, harvest index and in‐season soil water content from 2‐year experiments in each country to assess their ability to simulate observed yield. Simulated responses to climate change factors were explored and compared between models. Calibrated models reproduced measured grain yield variations well with average relative root mean square error of 26%, although uncertainty in model prediction was substantial (CV = 28%). Model ensembles gave greater accuracy than any model taken at random. Nitrogen fertilization controlled the response to variations in [CO2], temperature and rainfall. Without N fertilizer input, maize (i) benefited less from an increase in atmospheric [CO2], (ii) was less affected by higher temperature or decreasing rainfall and (iii) was more affected by increased rainfall because N leaching was more critical. The model inter-comparison revealed that simulation of daily soil N supply and N leaching plays a crucial role in simulating climate change impacts for low-input systems. Climate change and N input interactions have strong implications for the design of robust adaptation practices across SSA, because the impact of climate change will be modified if farmers intensify maize production with more mineral fertilizer.

Crop simulation model↗

NASA NTRS2006

A Roadmap for using Agile Development in a Traditional System

I. Ensemble Development Group: a) Produces activity planning software for in spacecraft; b) Built on Eclipse Rich Client Platform (open source development and runtime software); c) Funded by multiple sources including the Mars Technology Program; d) Incorporated the use of Agile Development. II. Next Generation Uplink Planning System: a) Researches the Activity Planning and Sequencing Subsystem for Mars Science Laboratory (APSS); b) APSS includes Ensemble, Activity Modeling, Constraint Checking, Command Editing and Sequencing tools plus other uplink generation utilities; c) Funded by the Mars Technology Program; d) Integrates all of the tools for APSS.

uplink planning systems↗

NASA NTRS2006

Simulated convective systems using a cloud resolving model: Impact of large-scale temperature and moisture forcing using observations and GEOS-3 reanalysis

The GCE (Goddard Cumulus Ensemble) model, which has been developed and improved at NASA Goddard Space Flight Center over the past two decades, is considered as one of the finer and state-of-the-art CRMs (Cloud Resolving Models) in the research community. As the chosen CRM for a NASA Interdisciplinary Science (IDS) Project, GCE has recently been successfully upgraded into an MPI (Message Passing Interface) version with which great improvement has been achieved in computational efficiency, scalability, and portability. By basically using the large-scale temperature and moisture advective forcing, as well as the temperature, water vapor and wind fields obtained from TRMM (Tropical Rainfall Measuring Mission) field experiments such as SCSMEX (South China Sea Monsoon Experiment) and KWAJEX (Kwajalein Experiment), our recent 2-D and 3-D GCE simulations were able to capture detailed convective systems typical of the targeted (simulated) regions. The GEOS-3 [Goddard EOS (Earth Observing System) Version-3] reanalysis data have also been proposed and successfully implemented for usage in the proposed/performed GCE long-term simulations (i.e., aiming at producing massive simulated cloud data -- Cloud Library) in compensating the scarcity of real field experimental data in both time and space (location). Preliminary 2-D or 3-D pilot results using GEOS-3 data have generally showed good qualitative agreement (yet some quantitative difference) with the respective numerical results using the SCSMEX observations. The first objective of this paper is to ensure the GEOS-3 data quality by comparing the model results obtained from several pairs of simulations using the real observations and GEOS-3 reanalysis data. The different large-scale advective forcing obtained from these two kinds of resources (i.e., sounding observations and GEOS-3 reanalysis) has been considered as a major critical factor in producing various model results. The second objective of this paper is therefore to investigate and present such an impact of large-scale forcing on various modeled quantities (such as hydrometeors, rainfall, and etc.). A third objective is to validate the overall GCE 3-D model performance by comparing the numerical results with sounding observations, as well as available satellite retrievals.

Shie, C.-L.↗

NASA NTRS2018

Sub-Seasonal Forecasting of the Stratospheric Wave Events, Sudden Stratospheric Warmings, and Their Influence on the Troposphere

Stratospheric wave events and major sudden stratospheric warming (SSW) events are well captured in high-resolution global forecasts out to 10 days. Tropospheric influences of SSW include statistically significant shifts in the storm tracks and associated surface temperature and precipitation pattern changes. Since these events can in turn influence the troposphere on time scales of 30-60 days the ability to predict these events and the subsequent long-term response on time scales beyond 10 days is of interest. Here we examine the prediction of stratospheric wave events and their evolution using the NASA GMAO (Global Modeling and Assimilation Office) Sub-Seasonal to Seasonal (S2S) system. This is a recently released subseasonal to seasonal forecast system, GEOS-S2S version 2.1. Compared to GMAO's previous system, the new version runs at higher atmospheric resolution (approximately 1/2 degree globally), contains a substantially improved model of the cryosphere, includes additional interactive aerosol model components, and the ocean data assimilation system has been replaced with a Local Ensemble Transform Kalman Filter. Results are based on a comprehensive series of hindcasts starting from the year 2000. They show that while the S2S system is not as accurate at 10 days in forecasting major SSW events as the NASA GMAO Forward Processing system, it can usefully predict stratospheric anomalies out to 20 days and the subsequent stratospheric/tropospheric evolution beyond 30 days.

Coy, L.↗

NASA NTRS2022

Investigation of the Potential Saturation of Information from Global Navigation Satellite System Radio Occultation Observations with an Observing System Simulation Experiment

The potential impact of large numbers of Global Navigation Satellite System radio occultation (GNSS-RO) observations on numerical weather prediction is investigated using a global observing system simulation experiment (OSSE). The hybrid four-dimensional ensemble variational Gridpoint Statistical Interpolation (GSI) data assimilation system and Global Earth Observing System (GEOS) model are used to ingest up to 100,000 GNSS-RO soundings per day in addition to the current suite of conventional and radiance data. Analysis quality, forecast skill, and forecast sensitivity to observation impact are examined with differing quantities of additional GNSS-RO profiles. It is found that saturation of information from additional RO soundings has not been reached with 100,000 soundings per day. There are some indications of suboptimal performance of the GSI in handling GNSS-RO observations particularly in the middle and lower tropospheric extratropics.

GNSS-RO↗

NASA NTRS2020

Analyzing the Impact of CryoSat-2 Ice Thickness Initialization on Seasonal Arctic Sea Ice Prediction

Twin 5-month seasonal forecast experiments are performed to predict the September 2018 minimum ice extent using the fully coupled Navy Earth System Prediction Capability (ESPC). In the control run, ensemble forecasts are initialized from the operational US Navy Global Ocean Forecasting System (GOFS) 3.1 for the ocean and sea ice but do not assimilate ice thickness data. Another set of forecasts are initialized from the same GOFS 3.1 fields but with sea ice thickness derived from CryoSat-2 (CS2). The Navy ESPC ensemble mean September 2018 minimum sea ice extent initialized with GOFS 3.1 ice thickness was over-predicted by 0.68 M sq.km (5.27 M sq.km) versus the ensemble set of forecasts initialized with CS2 ice thickness that had an error of 0.40 M sq.km (4.99 M sq.km), a 56% reduction in error. The September mean Integrated Ice Edge Error (IIEE) shows a 19% improvement for the entire Arctic with the CS2 data versus the control run. Comparison against Upward Looking Sonar (ULS) ice thickness in the Beaufort Sea reveals a lower bias and RMSE with the CS2 forecasts at all three moorings. Ice concentration at these locations is also improved, but neither set of experiments show ice free conditions as observed at moorings A and D.

time-lagged ensembles↗

NASA NTRS2020

Using a Simple Water Balance Framework to Quantify the Impact of Soil Moisture Initialization on Subseasonal Evapotranspiration and Air Temperature Forecasts

Past studies have shown that accurate soil moisture initialization can contribute significant skill to near-surface air temperature (T2M) forecasts at subseasonal leads. The mechanisms by which soil moisture contributes such skill are examined here with a simple water balance-based model that captures the essence of soil moisture behavior in a state-of-the-art subseasonal-to-seasonal (S2S) forecasting system. The simple model successfully transforms initial soil moisture contents into average “forecasted” ET values at 16-30 day lead that agree well, during summer, with the values forecasted by the full NASA GEOS S2S system, indicating that soil moisture initialization dominates over forecasted meteorology in determining ET fluxes at subseasonal leads. When the simple model’s ET anomalies are interpreted in terms of T2M anomalies, a similar conclusion is reached for T2M: soil moisture initialization explains much (about 50% in the eastern half of the continental US) of the T2M anomalies produced by the full GEOS S2S system at 16-30 day lead, and the T2M forecasts produced by the simple model capture about half of the skill attained by the full system. The simple model’s framework is particularly conducive to an analysis of uncertainty in forecasts. Drier soils are generally found to induce larger uncertainty in ET (and thus T2M) forecasts, a result linked to the functional form relating ET to soil moisture in the simple model and verified by an analysis of the ensemble spreads within the forecasts produced by the full GEOS S2S system

Randal D Koster↗

NASA NTRS2013

Assessing the Impact of Pre-gpm Microwave Precipitation Observations in the Goddard WRF Ensemble Data Assimilation System

The forthcoming Global Precipitation Measurement (GPM) Mission will provide next generation precipitation observations from a constellation of satellites. Since precipitation by nature has large variability and low predictability at cloud-resolving scales, the impact of precipitation data on the skills of mesoscale numerical weather prediction (NWP) is largely affected by the characterization of background and observation errors and the representation of nonlinear cloud/precipitation physics in an NWP data assimilation system. We present a data impact study on the assimilation of precipitation-affected microwave (MW) radiances from a pre-GPM satellite constellation using the Goddard WRF Ensemble Data Assimilation System (Goddard WRF-EDAS). A series of assimilation experiments are carried out in a Weather Research Forecast (WRF) model domain of 9 km resolution in western Europe. Sensitivities to observation error specifications, background error covariance estimated from ensemble forecasts with different ensemble sizes, and MW channel selections are examined through single-observation assimilation experiments. An empirical bias correction for precipitation-affected MW radiances is developed based on the statistics of radiance innovations in rainy areas. The data impact is assessed by full data assimilation cycling experiments for a storm event that occurred in France in September 2010. Results show that the assimilation of MW precipitation observations from a satellite constellation mimicking GPM has a positive impact on the accumulated rain forecasts verified with surface radar rain estimates. The case-study on a convective storm also reveals that the accuracy of ensemble-based background error covariance is limited by sampling errors and model errors such as precipitation displacement and unresolved convective scale instability.

microwave observations↗

DOE OSTI2025

Lectures on statistical mechanics

Presented here is a transcription of the lecture notes from Professor Allan N. Kaufman’s graduate statistical mechanics course Physics 212A and 212B at the University of California Berkeley from the 1972–1973 academic year. 212A addressed equilibrium statistical mechanics with topics: fundamentals (micro-canonical and sub-canonical ensembles, adiabatic law and action conservation, fluctuations, pressure, and virial theorem), classical fluids and other systems (equation of state, deviations from ideality, virial coefficients and van der Waals potential, canonical ensemble and partition function, quasistatic evolution, grand-canonical ensemble and partition function, chemical potential, simple model of a phase transition, quantum virial expansion, numerical simulation of equations of state, and phase transition), chemical equilibrium (systems with multiple species and chemical reactions, law of mass action, Saha equation, chemical equilibrium including ionization and excited states), and long-range interactions (including Coulomb, dipole, and gravitational interactions, Debye–Hückel theory, and shielding). 212B addressed nonequilibrium statistical mechanics with topics: fundamentals (definitions: realizations, moments, characteristic function, and discrete variables), Brownian motion (Langevin equation, fluctuation–dissipation theorem, spatial diffusion, Boltzmann’s H-theorem), Liouville and Klimontovich equations, Landau equation (derivation, elaboration, and H-theorem, and irreversibility), Markov processes and Fokker–Planck equation (derivations of the Fokker–Planck equation and a master equation), linear response and transport theory (linear Boltzmann equation, linear response theory of Kubo and Mori, relation of entropy production to electrical conductivity, transport relations and coefficients, normal mode solutions of the transport equations, sketch of a generalized Langevin equation method for transport theory), and an introduction to nonequilibrium quantum statistical mechanics.

plasma dynamics↗