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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 721 records · Page 40
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Strong parallel evidence of selection during switchgrass sward establishment in hybrid and lowland ecotypes

Switchgrass sward establishment results in up to 90% seedling mortality. The degree of selection during sward establishment has not been reported using modern genetic methods. Pooled leaf samples were sequenced from replicated swards of 46 half-sib families from two breeding groups (lowland and hybrid) before and through 3 years of stand establishment. Pooled allele frequencies were then assessed using fixation indices (Fst) and an independent data set was used to predict the polygenic impact of establishment selection on two traits (heading date and winter survivorship). Last, the DNA pools were assigned survival rankings to predict the sward survival genomically estimated breeding values within the training data set. Strong and parallel selection occured in both breeding groups. Five genomic regions exceeded the significant threshold of 99.9% in >10 families, indicating consistent selection across families and breeding groups. Polygenic trait predictions determined that establishment selection was partially associated with winter survivorship but resulted in variable heading date alterations. The genomewide variation is consistent with selection for a small number of related parental lines. This study observed strong selection for a small number of hybrid and coastal ecotype individuals which are promising germplasm sources for improved sward survival. This confirms prior reports of sward selection during grassland establishment and highlights the strength of pooled DNA sequencing for survival traits.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

Monitoring the Morphology of M87* in 2009-2017 with the Event Horizon Telescope

The Event Horizon Telescope (EHT) has recently delivered the first resolved images of M87*, the supermassive black hole in the center of the M87 galaxy. These images were produced using 230 GHz observations performed in 2017 April. Additional observations are required to investigate the persistence of the primary image feature—a ring with azimuthal brightness asymmetry—and to quantify the image variability on event horizon scales. To address this need, we analyze M87* data collected with prototype EHT arrays in 2009, 2011, 2012, and 2013. While these observations do not contain enough information to produce images, they are sufficient to constrain simple geometric models. We develop a modeling approach based on the framework utilized for the 2017 EHT data analysis and validate our procedures using synthetic data. Applying the same approach to the observational data sets, we find the M87* morphology in 2009–2017 to be consistent with a persistent asymmetric ring of ∼40 μas diameter. The position angle of the peak intensity varies in time. In particular, we find a significant difference between the position angle measured in 2013 and 2017. These variations are in broad agreement with predictions of a subset of general relativistic magnetohydrodynamic simulations. We show that quantifying the variability across multiple observational epochs has the potential to constrain the physical properties of the source, such as the accretion state or the black hole spin.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Probabilistic machine learning for battery health diagnostics and prognostics—review and perspectives

Abstract Diagnosing lithium-ion battery health and predicting future degradation is essential for driving design improvements in the laboratory and ensuring safe and reliable operation over a product’s expected lifetime. However, accurate battery health diagnostics and prognostics is challenging due to the unavoidable influence of cell-to-cell manufacturing variability and time-varying operating circumstances experienced in the field. Machine learning approaches informed by simulation, experiment, and field data show enormous promise to predict the evolution of battery health with use; however, until recently, the research community has focused on deterministic modeling methods, largely ignoring the cell-to-cell performance and aging variability inherent to all batteries. To truly make informed decisions regarding battery design in the lab or control strategies for the field, it is critical to characterize the uncertainty in a model’s predictions. After providing an overview of lithium-ion battery degradation, this paper reviews the current state-of-the-art probabilistic machine learning models for health diagnostics and prognostics. Details of the various methods, their advantages, and limitations are discussed in detail with a primary focus on probabilistic machine learning and uncertainty quantification. Last, future trends and opportunities for research and development are discussed.

25 ENERGY STORAGE↗
DOE OSTI2020

A Numerical Study of Hypoxia in Chesapeake Bay Using an Unstructured Grid Model: Validation and Sensitivity to Bathymetry Representation

A three-dimensional unstructured-grid hydrodynamic and water quality model (Semi-implicit Cross-scale Hydroscience Intergrated System Model-Integrated Compartment Model) is applied successfully for Chesapeake Bay. The model is validated with observations of salinity, chlorophyll-a, dissolved oxygen, nutrients, and phytoplankton productions from the year 1991 to 1995 for the mainstem and some major tributaries, based on multiple model skill scores. Model experiments are conducted to test the importance of having (1) an accurate representation of bathymetry to correctly predict hypoxia and other processes and (2) a high-resolution model grid for tributaries to correctly simulate water quality variables. Further, comparison with the model experiment results with bathymetry smoothing indicates that bathymetry smoothing, as commonly used for many systems, changes the stratification and lateral circulation pattern, resulting in more salt intrusion into shallow water regions, and an increase in the freshwater age. Consequently, a model with bathymetry smoothing can lead to an unrealistic prediction of the distribution of hypoxia and phytoplankton production. Local grid refinement shows significant improvement of model simulations on local stratification and water quality variables. Overall, the use of high-resolution unstructured grid model leads to a faithful representation of the complex geometry, and thus a seamless cross-scale capability for simulating water quality processes in the Bay including tributaries and tidal creeks.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Machine learning predictions of irradiation embrittlement in reactor pressure vessel steels

Abstract Irradiation increases the yield stress and embrittles light water reactor (LWR) pressure vessel steels. In this study, we demonstrate some of the potential benefits and risks of using machine learning models to predict irradiation hardening extrapolated to low flux, high fluence, extended life conditions. The machine learning training data included the Irradiation Variable for lower flux irradiations up to an intermediate fluence, plus the Belgian Reactor 2 and Advanced Test Reactor 1 for very high flux irradiations, up to very high fluence. Notably, the machine learning model predictions for the high fluence, intermediate flux Advanced Test Reactor 2 irradiations are superior to extrapolations of existing hardening models. The successful extrapolations showed that machine learning models are capable of capturing key intermediate flux effects at high fluence. Similar approaches, applied to expanded databases, could be used to predict hardening in LWRs under life-extension conditions.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Latent-Space Dynamics for Prediction and Fault Detection in Geothermal Power Plant Operations

This paper presents a latent-space dynamic neural network (LSDNN) model for the multi-step-ahead prediction and fault detection of a geothermal power plant’s operation. The model was trained to learn the dynamics of the power generation process from multivariate time-series data and the effects of exogenous variables, such as control adjustment and ambient temperature. In the LSDNN model, an encoder–decoder architecture was designed to capture cross-correlation among different measured variables. In addition, a latent space dynamic structure was proposed to propagate the dynamics in the latent space to enable prediction. The prediction power of the LSDNN was utilized for monitoring a geothermal power plant and detecting abnormal events. The model was integrated with principal component analysis (PCA)-based process monitoring techniques to develop a fault-detection procedure. The performance of the proposed LSDNN model and fault detection approach was demonstrated using field data collected from a geothermal power plant.

15 GEOTHERMAL ENERGY↗
DOE OSTI2022

A combined first principles study of the structural, magnetic, and phonon properties of monolayer CrI3

The first magnetic 2D material discovered, monolayer (ML) CrI3, is particularly fascinating due to its ground state ferromagnetism. However, because ML materials are difficult to probe experimentally, much remains unresolved about ML CrI3’s structural, electronic, and magnetic properties. Here, we leverage Density Functional Theory (DFT) and high-accuracy Diffusion Monte Carlo (DMC) simulations to predict lattice parameters, magnetic moments, and spin–phonon and spin–lattice coupling of ML CrI3. We exploit a recently developed surrogate Hessian DMC line search technique to determine CrI3’s ML geometry with DMC accuracy, yielding lattice parameters in good agreement with recently published STM measurements—an accomplishment given the ~10% variability in previous DFT-derived estimates depending upon the functional. Strikingly, we find that previous DFT predictions of ML CrI3’s magnetic spin moments are correct on average across a unit cell but miss critical local spatial fluctuations in the spin density revealed by more accurate DMC. DMC predicts that magnetic moments in ML CrI3 are 3.62 μB per chromium and -0.145 μB per iodine, both larger than previous DFT predictions. The large disparate moments together with the large spin–orbit coupling of CrI3’s I-p orbital suggest a ligand superexchange-dominated magnetic anisotropy in ML CrI3, corroborating recent observations of magnons in its 2D limit. We also find that ML CrI3 exhibits a substantial spin–phonon coupling of ~3.32 cm-1. Our work, thus, establishes many of ML CrI3’s key properties, while also continuing to demonstrate the pivotal role that DMC can assume in the study of magnetic and other 2D materials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Multi-Split Variable Refrigerant Flow (VRF) System Building Energy Simulations Using Performance Maps

Multi-split variable refrigerant flow (VRF) systems are highly energy-efficient HVAC (heating, ventilation and air conditioning) technologies that connect a single outdoor unit to multiple independent indoor terminal units using a common refrigerant circuit and a variable-speed compressor. Building energy simulations that incorporate VRF systems help model their unique operational characteristics and predict energy consumption in specific building designs. Traditionally, EnergyPlus models these systems by employing multiple sets of performance curves to characterize both individual terminal units and the outdoor unit. However, producing these curves is labor intensive and error prone, and they often do not capture all the key input and output variables. This paper introduces a novel approach that uses multi-dimensional performance maps to model VRF systems in building environments for space cooling. In this approach, performance maps are developed at the component level—separately for the outdoor unit and for each indoor terminal. The new modeling method is validated within EnergyPlus via a Python plug-in that contains a simple solver loop to coordinate the component-level, indoor, and outdoor unit maps. Furthermore, because performance maps can span more variables than traditional performance curves, they offer the opportunity to implement advanced controls, such as enhanced dehumidification and compressor modulation. A VRF air conditioner’s hardware system was modeled using the DOE/ORNL Heat Pump Design Model, which was automated to produce extensive performance maps for both the indoor and outdoor units.

Shen, Bo [ORNL] (ORCID:0000000336600393)↗
DOE OSTI2024

Machine learning models of intermittent operation of RO wellhead water treatment for salinity reduction and nitrate removal

Machine learning models were developed for intermittent multi-mode operation of a wellhead reverse osmosis water purification and desalination system to predict salt passage, nitrate passage, and permeate flux. The models, based on long short-term memory (LSTM) recurrent neural network (RNN) architecture, included an attention mechanism to increase model performance in proximity of the regulatory limit for nitrate. Training and testing of the models for the Startup, Production, Shutdown and Flushing operational modes were based on operational data (consisting of 22 process variables per data sample) acquired every 2–5 s over a six-month period. The significant sets of model input attributes for the different operational modes were assessed via Spearman ranking correlation, Self-Organizing Map (SOM) analysis and feed forward feature selection (FFFS). Although the variability of nitrate passage, salt passage and permeate flux was significant over the four operational modes, prediction performance for the three outcomes were with R2 and Average Absolute Relative Error (AARE) of 0.78–0.95 and 2.96–6.16 %, respectively. Model updates post membrane elements replacement demonstrated similar levels of prediction accuracy. The study results suggest that there is merit in exploring the utility of multi-mode models for sensor fault detection, data imputation, and for potential use in model-predictive control.

Intermittent RO operation↗
DOE OSTI2023

Technical descriptions of the experimental dynamical downscaling simulations over North America by the CAM–MPAS variable-resolution model

Abstract. Comprehensive assessment of climate datasets is important for communicating model projections and associated uncertainties to stakeholders. Uncertainties can arise not only from assumptions and biases within the model but also from external factors such as computational constraint and data processing. To understand sources of uncertainties in global variable-resolution (VR) dynamical downscaling, we produced a regional climate dataset using the Model for Prediction Across Scales (MPAS; dynamical core version 4.0) coupled to the Community Atmosphere Model (CAM; version 5.4), which we refer to as CAM–MPAS hereafter. This document provides technical details of the model configuration, simulations, computational requirements, post-processing, and data archive of the experimental CAM–MPAS downscaling data. The CAM–MPAS model is configured with VR meshes featuring higher resolutions over North America as well as quasi-uniform-resolution meshes across the globe. The dataset includes multiple uniform- (240 and 120 km) and variable-resolution (50–200, 25–100, and 12–46 km) simulations for both the present-day (1990–2010) and future (2080–2100) periods, closely following the protocol of the North American Coordinated Regional Climate Downscaling Experiment. A deviation from the protocol is the pseudo-warming experiment for the future period, using the ocean boundary conditions produced by adding the sea surface temperature and sea-ice changes from the low-resolution version of the Max Planck Institute Earth System Model (MPI-ESM-LR) in the Coupled Model Intercomparison Project Phase 5 to the present-day ocean state from a reanalysis product. Some unique aspects of global VR models are evaluated to provide background knowledge to data users and to explore good practices for modelers who use VR models for regional downscaling. In the coarse-resolution domain, strong resolution sensitivity of the hydrological cycles exists over the tropics but does not appear to affect the midlatitude circulations in the Northern Hemisphere, including the downscaling target of North America. The pseudo-warming experiment leads to similar responses of large-scale circulations to the imposed radiative and boundary forcings in the CAM–MPAS and MPI-ESM-LR models, but their climatological states in the historical period differ over various regions, including North America. Such differences are carried to the future period, suggesting the importance of the base state climatology. Within the refined domain, precipitation statistics improve with higher resolutions, and such statistical inference is verified to be negligibly influenced by horizontal remapping during post-processing. Limited (≈50 % slower) throughput of the current code is found on a recent many-core/wide-vector high-performance computing system, which limits the lengths of the 12–46 km simulations and indirectly affects sampling uncertainty. Our experience shows that global and technical aspects of the VR downscaling framework require further investigations to reduce uncertainties for regional climate projection.

58 GEOSCIENCES↗
DOE OSTI2023

An Integrated Approach to Predicting Ash Deposition and Heat Transfer in Coal-Fired Boilers

The overall goal of this project is to develop via measurements and simulations an advanced online technology to predict, monitor and manage fireside ash deposition in a coal-fired boiler allowing for more efficient operations under a range of load conditions and fuel property variability. With this in place fuel sorting and blending can be done upstream and operations can be optimized to compensate for load and fuel properties. In support of this objective, three experimental campaigns were undertaken during the course of the project to measure ash deposition rates within the boiler at different fuel flow rates and its ash composition variability. Simulations of the experimental conditions representing actual geometry, operational scenarios in terms of air flow rates, coal flow rates as well as coal compositions, heating values, and particle size distributions were also carried out. Deposition rates were predicted using a unique particle kinetic energy and viscosity based ash deposition methodology whose validity was ascertained by comparing against deposition rate measurements for widely varying operating conditions and ash compositions in a lab-scale furnace. With a unique end-to-end combustion modeling methodology established and different simulation scenarios carried out, the results from our computational fluid dynamic (CFD) simulations in conjunction with the plant data summarized in this report were used to refine Microbeam Technology Incorporated’s MTI CSPI-CT Tool to predict and monitor fire-side ash deposition under a range of load conditions and fuel property variability in real time.

01 COAL, LIGNITE, AND PEAT↗
DOE OSTI2023

Predicting wind farm operations with machine learning and the P2D‐RANS model: A case study for an AWAKEN site

Abstract The power performance and the wind velocity field of an onshore wind farm are predicted with machine learning models and the pseudo‐2D RANS model, then assessed against SCADA data. The wind farm under investigation is one of the sites involved with the American WAKE experimeNt (AWAKEN). The performed simulations enable predictions of the power capture at the farm and turbine levels while providing insights into the effects on power capture associated with wake interactions that operating upstream turbines induce, as well as the variability caused by atmospheric stability. The machine learning models show improved accuracy compared to the pseudo‐2D RANS model in the predictions of turbine power capture and farm power capture with roughly half the normalized error. The machine learning models also entail lower computational costs upon training. Further, the machine learning models provide predictions of the wind turbulence intensity at the turbine level for different wind and atmospheric conditions with very good accuracy, which is difficult to achieve through RANS modeling. Additionally, farm‐to‐farm interactions are noted, with adverse impacts on power predictions from both models.

17 WIND ENERGY↗
DOE OSTI2022

Dynamic logistic regression and variable selection: Forecasting and contextualizing civil unrest

Civil unrest can range from peaceful protest to violent furor, and researchers are working to monitor, forecast, and assess such events to allocate resources better. Twitter has become a real-time data source for forecasting civil unrest because millions of people use the platform as a social outlet. Additionally, daily word counts are used as model features, and predictive terms contextualize the reasons for the protest. To forecast civil unrest and infer the reasons for the protest, we consider the problem of Bayesian variable selection for the dynamic logistic regression model and propose using penalized credible regions to select parameters of the updated state vector. This method avoids the need for shrinkage priors, is scalable to high-dimensional dynamic data, and allows the importance of variables to vary in time as new information becomes available. A substantial improvement in both precision and F1-score using this approach is demonstrated through simulation. Finally, we apply the proposed model fitting and variable selection methodology to the problem of forecasting civil unrest in Latin America. Our dynamic logistic regression approach shows improved accuracy compared to the static approach currently used in event prediction and feature selection.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Enhanced predictability of Eastern North Pacific Tropical cyclone activity using the ENSO Longitude Index

Past studies have indicated that El Nino-Southern Oscillation (ENSO) plays a major role in the interannual variability of Eastern Pacific hurricane activity. The primary mechanism being the eastward displacement of the warm pool during an El Nino, which carries warm water into that basin thereby creating favorable oceanic conditions. Despite this, the question of whether an accurate knowledge of ENSO enhances seasonal predictabiity of Eastern Pacific hurricanes has not been addressed specifically. In this study, we show that unlike traditional indices of ENSO, the ENSO Longitude Index (ELI) is able to predict Eastern Pacific hurricane activity at significant lead times. By capturing changes in the location of deep convection and associated thermocline processes more accurately, ELI explains the most variability in the upper-ocean heat content in the main development region of the Eastern Pacific basin compared to other ENSO indices. These results have substantial implications for operational seasonal forecasts of Eastern Pacific hurricanes.

Balaguru, Karthik↗
DOE OSTI2025

Multiscale and Machine Learning Modeling for Process-informed Microstructure Prediction in Additively Manufactured Materials using MALAMUTE

The Advanced Materials and Manufacturing Technologies (AMMT) program under the Department of Energy Office of Nuclear Energy aims to develop and qualify additively manufactured materials for nuclear applications. One key challenge to this is the microstructural variability observed in the additively manufactured products and their impact on the properties and performance of the material in extreme environments. AMMT is using a combination of high-throughput experimental and modeling techniques to accelerate qualification. Conventionally, in-situ and ex-situ characterizations and testing are performed to correlate different aspects of the additive manufacturing process to the final product and its performance. However, adopting a trial-and-error approach to experimentally evaluate the vast range of process parameters required to capture microstructural variability is cost-prohibitive. Modeling and simulation provide a comparatively inexpensive way to understand and correlate the microstructural evolution to the processing conditions. The modeling and simulation work-packages within the AMMT program aims to use physics-based and machine learning models to develop a digital twin for additive manufacturing that can correlate the process conditions to the final product and establish a process-structure-property-performance (PSPP) correlation. The melting and subsequent solidification that occurs during the additive process is a complex phenomenon that requires multiscale multiphysics analysis. This work package focuses on understanding the role of process variabilities on the unique microstructural characteristics of additively manufactured materials. Microstructural features at the subgrain level, such as compositional micro-heterogeneity and dislocation cells, are of particular interest here since they can influence the creep properties and radiation performance. Idaho National Laboratory’s Multiphysics Object-Oriented Simulation Environment (MOOSE), specifically the MOOSE Application Library for Advanced Manufacturing UTilitiEs (MALAMUTE) software, provides an ideal platform for developing the multiphysics multiscale model to explore the intricacies of the microstructural evolution during the AM processes within a single framework. Furthermore, given that such full-fidelity simulations can be computationally intensive, reduced order models are necessary to explore the PSPP space for additively manufactured materials in an efficient, reliable, and cost-effective way. This work focuses on capturing the microstructural variabilities at the subgrain level that are often missing in the part-scale models. In fiscal year 2025, we significantly advanced upon our work in the last fiscal year, in terms of the predictive capabilities of the physics-based and ML models, by adding the capabilities to capture subgrain-level micro-segregation during solidification using phase-field model and to predict the time-dependent dynamics of the AM process through the MOGPAR model. The alloy solidification model in MOOSE incorporates the thermodynamic properties and free energy relevant to 316 stainless steel. The model demonstrates the Cr and Ni segregation that occurs during solidification, including that the rate of solidification. The microstructural evolution model is connected to the process conditions via the surrogate model developed in this work. This enables predictions of the final microstructure in conjunctions with the manufacturing process. This work supports AMMT's rapid qualification goals by laying the foundation for an efficient and cost-effective model establishing the PSPP correlation for AM. The generated microstructures and predicted micro-segregation can be used by other work packages under AMMT to evaluate the properties and environmental response of the material at the mesoscale. Thus, this work helps to identify the key microstructural features at the subgrain level that are significant in property and performance predictions of additively manufactured components. This work will also provide inputs to the large-scale process variability models to reevaluate and validate assumptions and simplifications made in the part-scale models. Furthermore, through active learning this work can help identify the data need from both modeling and experimental sides for development of a robust digital twin for additive manufacturing and accelerate the AMMT's qualification efforts.

36 - MATERIALS SCIENCE↗
DOE OSTI2024

The influence of burn severity on dissolved organic carbon concentrations across a stream network differs based on seasonal wetness conditions

Large, high-severity wildfires in many regions across the globe have increased concerns about their impacts on carbon cycling in watersheds. Altered sources of carbon and changes in catchment hydrology after wildfire can lead to shifts in dissolved organic carbon (DOC) concentrations in streams, which can have negative impacts on aquatic ecosystem health and downstream drinking-water treatment. Despite its importance, post-fire DOC responses remain relatively unconstrained in the literature, and we lack critical knowledge of how burn severity, landscape elements, and climate interact to affect DOC concentrations. To improve our understanding of the impact of burn severity on DOC concentrations, we measured DOC at 129 sites across a stream network extending upstream, within, and downstream of a large, high-severity wildfire in Oregon, USA. We collected samples across the study sub-basin during four distinct seasonal wetness conditions. We used our high-spatial-resolution data to develop spatial stream network (SSN) models to predict DOC across the stream network and to improve our understanding of the controls on DOC concentrations. Spatially, we found no obvious wildfire signal – instead, we observed a pattern of increasing DOC concentrations from the high-elevation headwaters to the sub-basin outlet, while the mainstem maintained consistently low DOC concentrations. This suggests that effects from large wildfires may be “averaged” out at higher stream orders and larger spatial scales. When we grouped DOC concentrations by burn severity group, we observed a significant decrease in the variability of DOC concentrations in the moderate and high burn severity sub-catchments. However, our SSN models were able to predict decreases in DOC concentrations with increases in burn severity across the stream network. Decreases in DOC concentrations were also highly variable across seasonal wetness conditions, with the greatest (–1.40 to –1.64 mg L –1 ) decrease occurring in the high-severity group during the wetting season. Additionally, our models indicated that in all seasons, baseflow index was more influential in predicting DOC concentrations than burn severity was, indicating that groundwater discharge can obscure the impacts of wildfire in a stream network. Overall, our results suggested that landscape characteristics can regulate the DOC response to wildfire. Moreover, our results also indicated that the seasonal timing of sampling can influence the observed response of DOC concentrations to wildfire.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Formation of Amorphous Carbon Multi‐Walled Nanotubes from Random Initial Configurations

Amorphous carbon nanotubes (a‐CNT) with up to four walls and sizes ranging from 200 to 3200 atoms have been simulated, starting from initial random configurations and using the Gaussian Approximation Potential. The important variables (like density, height, and diameter) required to successfully simulate a‐CNTs were predicted with the machine learning random forest technique. The width of the a‐CNT models ranged between 0.55–2 nm with an average inter‐wall spacing of 0.31 nm. The topological defects in a‐CNTs were analyzed and new defect configurations were observed. The electronic density of states and localization in these phases were discussed and delocalized electrons in the π subspace were identified as an important factor for inter‐layer cohesion. Spatial projection of the electronic conductivity favors axial transport along connecting hexagons, while non‐hexagonal parts of the network either hinder or bifurcate the electronic transport. A vibrational density of states was calculated and is potentially an experimentally comparable fingerprint of the material. The appearance of a low‐frequency radial breathing mode was discussed and the thermal conductivity at 300 K was estimated using the Green‐Kubo formula.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗