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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Tacho

SAND2022-7470 O Tacho is a performance portable sparse direct solver for use with various sparse matrix problems which typically arise from finite element methods. The code relies on the Kokkos parallel programming model porting to both CPUs and GPUs. It is also part of the Trilinos library. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Kim, Kyungjoo↗
DOE OSTI2022

Resource Recovery and Environmental Protection in Wyoming’s Greater Green River Basin Using Selective Nanostructured Membranes (Final Report)

Produced water (PW) represents a sizable waste stream that is co-generated with oil and natural gas production. In 2021 Wyoming ranked 8th and 9th, respectively in domestic oil and natural gas production. In 2017 Wyoming ranked as the 4th highest generator of PW in the U.S, accounting for 7% of the total volume generated. In the context of being the 3rd most arid state in the U.S., the value of water reuse becomes obvious. PW reuse, and resource recovery, in any form requires some level of treatment to remove particulates, residual (free, dispersed) hydrocarbons, organics, and salts. The level of treatment depends on the requirements of the reuse, or resource recovery, application. PW management systems in Wyoming employ in order of volume of PW managed the following management strategies: reinjection for enhanced oil recovery, surface discharge, deep well injection, evaporation ponds (impoundments), and commercial management/treatment. Complicating treatment efforts are the relatively high concentrations of organics (natural and synthetic), dispersed/free hydrocarbons, benzene-toluene-ethylbenzene, and xylenes (BTEX) compounds, biologicals, salts, and minerals. Hydrocarbons (dispersed/dissolved crude oils) and BTEX compounds, as well as synthetic organics, present economic and environmental concerns. The former represents lost revenue, while the latter results in negative environmental impacts like emissions from surface impoundments. The overall objective of this proposal was to synthesize superhydrophilic/oleophobic and superhydrophobic/oleophilic membranes for selectively concentrating and then separating BTEX compounds and oil and grease (O&G) from PW originating from the Greater Green River Basin (GGRB) in Wyoming. Three specific research aims were pursued to accomplish this overall objective. This final report details the development of the superhydrophobic and superhydrophilic membranes, as well as the design of the membrane module prototypes specifically. The technoeconomic assessment is separately reported in another document. 1. Aim #1 – Material optimization and performance evaluation of superhydrophilic/oleophobic and superhydrophobic/oleophilic membranes made by electrospinning/spraying. 2. Aim #2 – Design and construction of cross-flow membrane modules for selectively concentrating and then separating BTEX/oil from GGRB produced water. 3. Aim #3 – Techno-economic assessment of BTEX/oil recovery, and clean water production, using superhydrophilic/oleophobic and superhydrophobic/oleophilic membrane separation for GGRB PW. Superhydrophobic membranes were synthesized by electrospinning poly(vinylidene fluoride-co-hexafluoropropylene) (PVDF-HFP) nanofibers onto polyester (PET) substrates and electrospraying nano-carbon black/PVDF-HFP onto the nanofibrous layer. These membranes were characterized by high (>8000 liters per square meter per hour per bar (LMH/bar)) permeance values for pure hydrocarbon phases and a high hydrocarbon selectivity (>96%) when treating GGRB PW. All results were obtained when operating the membrane in a crossflow configuration representative of actual field operating conditions. Solvent/oil properties, specifically viscosity and total surface energy/tension, affected permeance across the membrane, which resulted in light mineral oil (394 LMH/bar) and o-xylene (1834 LMH/bar) being characterized by lower permeance values in the pure phase tests. Mixed phase fluxes between 40 to 80 LMH were obtained for the PW when operating the membrane at a feed pressure of 0.3 bar. Flux decreased as the mixed phase concentration in the feed decreased pointing to the importance of maximizing the collision efficiency between the emulsion and the membrane surface and maximizing the emulsion concentration in the feed and the turbulence within the feed channel. These tests demonstrated that the superhydrophobic membranes developed here are a viable hydrocarbon recovery method for GGRB PWs and should be pursued for testing in pilot-scale trials. Superhydrophilic membranes were successfully synthesized via electrospinning/spraying using polyacrylonitrile (PAN) nanofibers as a base nanofibrous matrix. Integration of polyaniline (PANI) into the nanofibrous matrix produced a superior membrane, for water filtration applications, relative to PAN alone and reduced graphene oxide (RGO) when integrated into the nanofibrous matrix. This conclusion was based on the PANI-PAN resistance to flux loss (fouling) when treating model solvent/oil solutions representative of GGRB PWs and field collected PW from the GGRB. The synthesized PAN membranes outperformed a commercially available PAN membrane designed for oil/water separation. This finding indicates that the surface chemical and physical characteristics of the electrospun membranes presents improved properties for filtration of challenging waters like GGRB PWs. The electrospun membranes therefore show promise overall as a substitute for conventionally polymerized membranes in PW treatment applications. The PANI-PAN membrane specifically presents superior performance characteristics for concentration O&G prior to treatment by the hydrocarbon recovery membrane and producing high-quality filtrate for reuse and/or additional treatment (desalination).

02 PETROLEUM↗
DOE OSTI2023

Strong Interaction Dynamics and Fermi $β$ Decay in the Nucleon and the Nucleus

Nuclear super-allowed β decay has been used to obtain tight limits on the value of the CKM matrix element V ud that is important for unitarity tests and, therefore, for tests of the standard model. Current requirements on precision are so intense that effects formerly thought too small to matter are now considered relevant. This article is a brief review of personal efforts to include the effects of strong interactions on Fermi β decay. First, I examine the role of isospin violation in the decay of the neutron. The size of the necessary correction depends upon detailed strong-interaction dynamics. The isospin violating parts of the nucleon wave function, important at the low energy of β decay, can be constrained by data taken at much higher energies, via measurements, for example, of ed → e'π ± + X reactions at Jefferson Laboratory. The next point of focus is on the role of nuclear short-ranged correlations, which affect the value of the correction needed to account for isospin violation in extracting the value of V ud . The net result is that effects previously considered as irrelevant are now considered relevant for both neutron and nuclear β decay.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Algorithm to extract direction in 2D discrete distributions and a continuous Frobenius norm

In this study, we present a novel algorithm for determining directionality in 2D distributions of discrete data. We compare a reference dataset with a known direction to a measured dataset with an unknown direction by the Frobenius norm of the difference (FND) to find the unknown direction. To generalize this concept, we develop a continuous Frobenius norm of the difference (CFND) as a continuous analog of the FND and derive its analytical expression. By relating fitted and normalized 2D Gaussian distributions, we show that the CFND approximates the FND, and we validate this relationship with computer simulations. We find that a first-order approximation of the CFND between two similar Gaussian distributions takes the form of an absolute sine function, offering a simple analytical form with potential for specialized applications in segmented inverse beta decay (IBD) neutrino detectors, astronomy, machine learning, and more. Although this method may easily extend to 3D scalar fields, our focus here is on 2D real-valued fields as it directly applies to directionality. Our methodology consists of modeling a 2D Gaussian distribution, binning the data into a histogram, and encoding it as a square matrix. Rotating this matrix around its geometric center and comparing it to a measured dataset using the FND gives us rotational data that we fit with an absolute sine function. The location of the minimum of this fit is the angle closest to the true angle of the direction in the measured dataset. We present the derivation and discuss initial applications of the CFND in our novel algorithm, demonstrating its success in approximating directionality in 2D distributions.

Data Analysis, Statistics and Probability (physics↗
DOE OSTI2020

FY20, 3rd Quarter, Multi-pass Hybrid Laser Arc Welding of Alloy 740H

Although progress has been slowed by telecommuting restrictions at INL and PSU, progress has been made by PSU on modeling and simulation of deep penetration laser welding. Additionally, both INL and PSU has received critical equipment, in the form of high-powered laser wobble heads, needed to verify the modeling and simulation results. The equipment was installed at INL and preliminary tests, based on a test matrix that was developed in this 3rd quarter, will be made early in the 4th quarter. Similarly, the PSU laser wobble head is expected to be installed early in the 4th quarter and preliminary experiments are expected to commence shortly thereafter. Completion dates on a few milestones have been impacted and new dates proposed.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Half-Life and Precision Shape Measurement of the 2⁢𝜈⁢𝛽⁢𝛽 Decay of 130 Te

Here, we present a new measurement of the 2⁢𝜈⁢𝛽⁢𝛽 half-life of 130 Te (𝑇$^{2⁢𝜈}_{1/2}$) using the first complete model of the CUORE data, based on 1038 kg yr of collected exposure. Thanks to optimized data selection, we achieve a factor of two improvement in precision, obtaining 𝑇$^{2⁢𝜈}_{1/2}$ = (9.32⁢$^{+0.05}_{−0.04}$⁢stat⁢ $^{+0.07}_{−0.07}$⁢syst)×10 20 yr. The signal-to-background ratio is increased by 70% compared to our previous results, enabling the first application of the improved 2⁢𝜈⁢𝛽⁢𝛽 formalism to 130 Te . Within this framework, we determine a credibility interval for the effective axial coupling in the nuclear medium as a function of nuclear matrix elements. We also extract values for the higher-order nuclear matrix element ratios: second-to-first and third-to-first. The second-to-first ratio agrees with nuclear model predictions, while the third-to-first ratio deviates from theoretical expectations. These findings provide essential tests of nuclear models and key inputs for future 0⁢𝜈⁢𝛽⁢𝛽 searches.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Optimizing the Accelerated Recursive Doubling Algorithm for Block Tridiagonal Systems of Equations

The need to solve block tridiagonal systems with hundreds or thousands of right-hand sides for the same block tridiagonal matrix is common in a variety of disciplines. To meet this need, the Accelerated Recursive Doubling Algorithm was developed. After a right-hand side independent phase, the algorithm allows for the quick, online calculation of solutions for different right-hand sides. In this work, we present methods to optimize the Accelerated Recursive Doubling Algorithm in memory usage and computation time in a hybrid parallelization model. The right-hand side independent phase of the naïve implementation takes ≥ 11/3 the amount of memory required to store the tridiagonal matrix, while our implementation reduces the fraction to ≈ 5/3 . The right-hand side dependent phase of the naïve implementation takes ≥ 6 times the amount of memory required to store the right-hand side, while our implementation reduces the fraction to ≈ 3. The computation time for the independent phase is reduced to ≈ 2/3 times that of the naïve implementation, while the computation time for the dependent phase is reduced to ≈ 5/9 . With increasing numbers of shared-memory threads q on every distributed processing element, we have O(q) theoretical speedup.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Differential contagiousness of respiratory disease across the United States

The initial contagiousness of a communicable disease within a given population is quantified by the basic reproduction number, $R_0$. This number depends on both pathogen and population properties. On the basis of compartmental models that reproduce Coronavirus Disease 2019 (COVID-19) surveillance data, we used Bayesian inference and the next-generation matrix approach to estimate region-specific $R_0$ values for 280 of 384 metropolitan statistical areas (MSAs) in the United States (US), which account for 95% of the US population living in urban areas and 82% of the total population. We focused on MSA populations after finding that these populations were more uniformly impacted by COVID-19 than state populations. Our maximum a posteriori (MAP) estimates for $R_0$ range from 1.9 to 7.7 and quantify the relative susceptibilities of regional populations to spread of respiratory diseases.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2020

SPND Sensitivity Calculations Using MCNP and Experimental Data from ACRR

The use of the Monte Carlo N-Particle Transport Code (MCNP) to calculate detector sensitivity for Self-Powered Neutron Detectors (SPNDs) in the Annular Core Research Reactor (ACRR) could be a vital tool in the effort to optimize the design of next-generation SPNDs. Next-generation SPND designs, which consider specific materials and geometry, may provide experimenters with capabilities for advanced mixed field dosimetry. These detectors will need to be optimized for configuration, materials, and geometries and the ability to model and iterate must be available in order to decide on the ideal. SPND design. SPNDs were modeled in MCNP which closely resembled the dimensions and location of actual detectors used in the ACRR. Tallies were used to calculate detector sensitivity. Using metrics from a previous report, oscilloscope data from pulses were manipulated in a Matrix Laboratory computing environment (MATLAB) script to calculate experimental detector sensitivity. This report outlines the process in which experimental data from ACRR pulses verified results from tallies in an MCNP ACRR model. The sensitivity values from experiments and MCNP calculations agreed within one standard deviation. Parametric studies were also performed with MCNP to investigate the effects of materials and dimensions of different SPNDs.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2020

DECT-MULTRA: Dual-Energy CT Image Decomposition With Learned Mixed Material Models and Efficient Clustering

Dual-energy computed tomography (DECT) imaging plays an important role in advanced imaging applications due to its material decomposition capability. Image-domain decomposition operates directly on CT images using linear matrix inversion, but the decomposed material images can be severely degraded by noise and artifacts. This paper proposes a new method dubbed DECT-MULTRA for image-domain DECT material decomposition that combines conventional penalized weighted-least squares (PWLS) estimation with regularization based on a mixed union of learned transforms (MULTRA) model. Our proposed approach pre-learns a union of common-material sparsifying transforms from patches extracted from all the basis materials, and a union of cross-material sparsifying transforms from multi-material patches. The common-material transforms capture the common properties among different material images, while the cross-material transforms capture the cross-dependencies. The proposed PWLS formulation is optimized efficiently by alternating between an image update step and a sparse coding and clustering step, with both of these steps having closed-form solutions. The effectiveness of our method is validated with both XCAT phantom and clinical head data. The results demonstrate that our proposed method provides superior material image quality and decomposition accuracy compared to other competing methods.

47 OTHER INSTRUMENTATION↗
DOE OSTI2025

Competing ionization and dissociation: Extension of the energy-dependent frame transformation to the gerade symmetry of H 2

This article solves two major tasks that frequently arise in the theory of electron collisions with a target molecular cation. First, it extends the energy-dependent frame transformation (EDFT) treatment, which is needed to map fixed-nuclei electron-molecule scattering matrices into an energy-dependent laboratory-frame scattering matrix with vibrational channel indices. The EDFT mapping can now be carried out even when the target molecule possesses multiple low-energy potential curves, significantly transcending previous applications. Second, it implements a method to extract the rest of the full laboratory-frame scattering matrix, i.e., the columns and rows describing input and/or output dissociation channels. The treatment is benchmarked in this article against the essentially exact solution of a refined two-dimensional model of the singlet gerade Σ symmetry of H 2 . Our tests demonstrate that the theory accurately maps fixed-nuclei scattering information, of the type provided by existing electron-molecule computer codes, into a laboratory-frame scattering matrix that includes both ionization and dissociation. Furthermore, this treatment can provide a general framework applicable to a broad class of electron collision processes involving diatomic target ions, suitable for an accurate description of challenging processes such as dissociative recombination.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2025

Operator dynamics in Floquet many-body systems

We study operator dynamics in many-body quantum systems, focusing on generic features of systems that are ergodic, spatially extended, and lack conserved densities. Quantum circuits of various types provide simple models for such systems. We focus on Floquet quantum circuits, comparing their behavior with what has been found previously for circuits that are random in time. Floquet circuits, which have discrete time-translation symmetry, represent an intermediate case between circuits that are random in time and lack any symmetry, and systems with a time-independent Hamiltonian and continuous time-translation invariance. By making this comparison, one of our aims is to identify signatures of time-translation symmetry in Floquet operator dynamics. To characterize behavior we examine a variety of quantities in solvable models and numerically: operator autocorrelation functions; the partial spectral form factor; the out-of-time-order correlator (OTOC); and the paths in operator space that make the dominant contributions to the ensemble-averaged autocorrelation functions. Our most striking result is that ensemble-averaged autocorrelation functions show behavior that is distinctively different in Floquet systems compared to systems in which successive time-steps are independent. Specifically, while average autocorrelation functions decay on a microscopic timescale for circuits that are random in time, in Floquet systems they have a late-time tail with a duration that grows parametrically with the size of the operator support. In the simplest models this tail is separated from the initial decay by a minimum, so that the average autocorrelation function has an intermediate-time peak. The existence of these tails provides a way to understand deviations of the spectral form factor from random matrix behavior at times shorter than the Thouless time. In contrast to this feature in autocorrelation functions, we find no new aspects to the behavior of OTOCs for Floquet models compared to random-in-time circuits. We show that this difference between averaged autocorrelation functions and OTOCs can be understood in terms of the paths in operator space that contribute to the two quantities: paths for the former retain a limited support at late times, while paths for the latter are dominated by operator spreading. Published by the American Physical Society 2025

Yoshimura, Takato (ORCID:0000000309159846)↗
DOE OSTI2026

Enhancing photoionization rate calculations in low-temperature plasmas using spectral methods

Photoionization plays a central role in the development of streamer discharges and other non-equilibrium plasma phenomena. It creates seed electrons, which are essential for positive streamer propagation, allowing the ionization front to move forward. Because of this, accurate modeling of photoionization is very important for predicting streamer behavior and plasma evolution. The photoionization process in air (N 2 – O 2 mixture) is often described by the Zheleznyak model (1982). This model is usually solved through Helmholtz-type equations that approximate the Zheleznyak photoionization model (Zheleznyak et al. 1982) as Partial Differential Equations (PDEs). Conventional numerical methods, such as the Finite Difference Method (FDM) or Finite Volume Method (FVM), are widely used to solve these equations. Although they are prevalent, the computational cost of these methods due to their need for matrix operations and iterative solver is demanding. To address this challenge, this work develops a spectral solver based on the Fast Fourier Transform (FFT) combined with Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST) to calculate the photoionization rate efficiently in an axisymmetric cylindrical domain. This method naturally satisfies the boundary conditions used in the model and converts the PDE into algebraic ones in spectral space. Thus, avoids the need for iterative matrix solvers. When compared with FDM results, it is demonstrated that the new solver not only maintains accuracy, but also reduces the computational cost, showing a performance increase of approximately 100 compared to FDM over a wide range of problem sizes. The method is parallelized using Message Passing Interface (MPI) and has been integrated into a fluid plasma model for streamer simulation. Here, this FFT-based approach provides a fast and reliable alternative for calculating photoionization in fluid models, helping large-scale plasma simulations run faster and efficiently, and allows higher-resolution simulation without extra computational cost.

Axisymmetric system↗
DOE OSTI2022

Easy-plane anisotropic-exchange magnets on a honeycomb lattice: Quantum effects and dealing with them

We provide analytical and numerical insights into the phase diagram and other properties of the extended Kitaev-Heisenberg model on the honeycomb lattice in the easy-plane limit, in which interactions are only between spin components that belong to the plane of magnetic ions. This parameter subspace allows for a much-needed systematic quantitative investigation of spin excitations in the ordered phases and of their generic features. Specifically, we demonstrate that in this limit one can consistently take into account magnon interactions in both zero-field zigzag and field-polarized phases. For the nominally polarized phase, we propose a regularization of the unphysical divergences that occur at the critical field and are plaguing the 1/S approximation in this class of models. For the explored parameter subspace, all symmetry-allowed terms of the standard parametrization of the extended Kitaev-Heisenberg model, such as K, J, and Γ, are significant, making the offered consideration relevant to a much wider parameter space. Furthermore, the dynamical structure factor near the paramagnetic critical point illustrates this relevance by showing features that are reminiscent of the ones observed in α–RuCl 3 , underscoring that they are not unique and should be common to a wide range of parameters of the model and, by extension, to other materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Application of the immobilized low-activity waste glass corrosion model to the static dissolution of 24 statistically-designed alkali-borosilicate waste glasses

Glass corrosion models that capture the complex mechanisms of the glass-water reaction enable the prediction of nuclear waste glass durability in disposal scenarios. Parameterization of such models is challenging because of the need to capture changes in corrosion behavior with time, reaction conditions, and glass composition. In this report we describe and employ the immobilized low activity waste (ILAW) glass corrosion model (IGCM) in geochemical simulations of static dissolution tests, at two temperatures (40 °C and 90 °C), for a matrix of 24 enhanced low activity waste (eLAW) glasses statistically designed to cover a processable composition space defined by 8 major glass components (Al 2 O 3 , B 2 O 3 , CaO, Na 2 O, SiO 2 , SnO 2 , ZrO 2 , and Others as defined in the text). The IGCM includes a first-order chemical affinity term and an ion-exchange term that represents the net exchange of Na+ ions in the pristine glass with protons in aqueous solution. Constant, time-dependent, and time- and pH-dependent functional forms of the ion-exchange term are evaluated to reproduce the change in corrosion behavior with time in saturated, static dissolution tests. The agreement with measured aqueous concentrations of the main glass components (B, Na, and Si) improved significantly upon addition of a time-dependent term, which therefore constitutes a simple representation of the glass-water reaction progress. Due to the limited changes in pH in the static dissolution tests, past a short initial period of rapid increase, addition of a pH dependent term did not appreciably improve the fits, indicating that comprehensive model parameterization requires more than one type of glass corrosion test to capture a wide range of solution chemistries. IGCM parameters were found to be dependent on glass composition, and the parameter sets generated in this work will enable the development of composition–parameter correlation models that offer the promise of predicting IGCM parameters, and thus glass corrosion behavior, solely based on glass composition.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2020

A description of structured waves in shock compressed particulate composites

Dynamic compression of composite materials is of scientific interest because the mechanical mismatch between internal phases challenges continuum theories. Typical assumptions about steady wave propagation and quasi-instantaneous state changes require reexamination along with the need for time-dependent models. To that end, data and models are presented here for the shock compression of an idealized particulate composite. To serve as a generic representative of this material class, a polymer matrix was filled with tungsten particles, ranging from 1 to 50 vol. %. This creates a simple microstructure containing randomly scattered particles with an extreme impedance mismatch to the binding matrix. These materials were parallel plate impact loaded by Al flyers traveling at 1.8–5.0 km/s. Velocimetry provided records of the equilibrium state and the compression wave structure for each case with trends quantified by an empirical fit. The same quantities were also studied as a function of the wave's propagation distance. A homogenized viscoelastic model then made it possible to progress from cataloging trends to making predictions. Starting from a Mie–Gruneisen equation of state, additional time varying terms were added to capture the transient response. After calibration, accurate predictions of the steady wave structure were possible.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Robustness of Vacancy-Bound Non-Abelian Anyons in the Kitaev Model in a Magnetic Field

Non-Abelian anyons in quantum spin liquids (QSLs) provide a promising route to fault-tolerant topological quantum computation. In the exactly solvable Kitaev honeycomb model, such anyons of the QSL state can be bound to nonmagnetic spin vacancies and endowed with non-Abelian statistics by an infinitesimal magnetic field. Here, we investigate how this approach for stabilizing non-Abelian anyons extends to a finite magnetic field represented by a proper Zeeman term. Through large-scale density-matrix renormalization group simulations, we compute the vacancy-anyon binding energy as a function of magnetic field for both the ferromagnetic and antiferromagnetic Kitaev models. Here, we find that anyon binding remains robust within the entire QSL phase for the ferromagnetic Kitaev model but breaks down already inside this phase for the antiferromagnetic Kitaev model. To compute a binding energy several orders of magnitude below the magnetic energy scale, we introduce both a refined definition and an extrapolation scheme based on carefully tailored perturbations.

Xiao, Bo [Oak Ridge National Laboratory (ORNL), Oa↗
DOE OSTI2023

Adjoint DSMC for nonlinear spatially-homogeneous Boltzmann equation with a general collision model

We derive an adjoint method for the Direct Simulation Monte Carlo (DSMC) method for the spatially homogeneous Boltzmann equation with a general collision law. This generalizes our previous results in Caflisch et al., which was restricted to the case of Maxwell molecules, for which the collision rate is constant. The main difficulty in generalizing the previous results is that a rejection sampling step is required in the DSMC algorithm in order to handle the variable collision rate. We find a new term corresponding to the so-called score function in the adjoint equation and a new adjoint Jacobian matrix capturing the dependence of the collision parameter on the velocities. The new formula works for a much more general class of collision models.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗