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At least 379 records · Page 21
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Using genetic algorithms to discover novel ground-state triplet conjugated polymers

Stable ground-state triplet π-conjugated copolymers have many interesting electronic and optoelectronic properties. However, the large number of potential monomer combinations makes it impractical to synthesize or even just use density functional theory (DFT) to calculate their triplet ground-state stability. Furthermore, we present a genetic algorithm implementation that uses the semi-empirical GFN2-xTB to find ground-state triplet polymer candidates. We find more than 1400 polymer candidates with a triplet ground-state stability of up to 4 eV versus the singlet. Additionally, we explore the properties of the monomers of those candidates in order to understand the design rules which promote the formation of a stable ground-state triplet in π-conjugated polymers.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Extreme Compression for Large Scale Data Store

For the last 5 years Accelogic pioneered and perfected a radically new theory of numerical computing codenamed “Compressive Computing”, which has an extremely profound impact on real-world computer science [1]. At the core of this new theory is the discovery of one of its fundamental theorems which states that, under very general conditions, the vast majority (typically between 70% and 80%) of the bits used in modern large-scale numerical computations are absolutely irrelevant for the accuracy of the end result. This theory of Compressive Computing provides mechanisms able to identify (with high intelligence and surgical accuracy) the number of bits (i.e., the precision) that can be used to represent numbers without affecting the substance of the end results, as they are computed and vary in real time. The bottom line outcome would be to provide a state-of-the-art compression algorithm that surpasses those currently available in the ROOT framework, with the purpose of enabling substantial economic and operational gains (including speedup) for High Energy and Nuclear Physics data storage/analysis. In our initial studies, a factor of nearly x4 (3.9) compression was achieved with RHIC/STAR data where ROOT compression managed only x1.4.In this contribution, we will present our concepts of “functionally lossless compression”, have a glance at examples and achievements in other communities, present the results and outcome of our current, ongoing R&D, as well as present a high-level view of our plan to move forward with a ROOT implementation that would deliver a basic solution readily integrated into HENP applications. As a collaboration of experimental scientists, private industry, and the ROOT Team, our aim is to capitalize on the substantial success delivered by the initial effort and produce a robust technology properly packaged as an open-source tool that could be used by virtually every experiment around the world as means for improving data management and accessibility.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Learning effective physical laws for generating cosmological hydrodynamics with Lagrangian deep learning

The goal of generative models is to learn the intricate relations between the data to create new simulated data, but current approaches fail in very high dimensions. When the true data-generating process is based on physical processes, these impose symmetries and constraints, and the generative model can be created by learning an effective description of the underlying physics, which enables scaling of the generative model to very high dimensions. In this work, we propose Lagrangian deep learning (LDL) for this purpose, applying it to learn outputs of cosmological hydrodynamical simulations. The model uses layers of Lagrangian displacements of particles describing the observables to learn the effective physical laws. The displacements are modeled as the gradient of an effective potential, which explicitly satisfies the translational and rotational invariance. The total number of learned parameters is only of order 10, and they can be viewed as effective theory parameters. Further, we combine N-body solver fast particle mesh (FastPM) with LDL and apply it to a wide range of cosmological outputs, from the dark matter to the stellar maps, gas density, and temperature. The computational cost of LDL is nearly four orders of magnitude lower than that of the full hydrodynamical simulations, yet it outperforms them at the same resolution. We achieve this with only of order 10 layers from the initial conditions to the final output, in contrast to typical cosmological simulations with thousands of time steps. This opens up the possibility of analyzing cosmological observations entirely within this framework, without the need for large dark-matter simulations.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Progress toward Accelogic compression in ROOT

For the last 7 years, Accelogic pioneered and perfected a radically new theory of numerical computing codenamed “Compressive Computing”, which has an extremely profound impact on real-world computer science [1]. At the core of this new theory is the discovery of one of its fundamental theorems which states that, under very general conditions, the vast majority (typically between 70% and 80%) of the bits used in modern large-scale numerical computations are absolutely irrelevant for the accuracy of the end result. This theory of Compressive Computing provides mechanisms able to identify (with high intelligence and surgical accuracy) the number of bits (i.e., the precision) that can be used to represent numbers without affecting the substance of the end results, as they are computed and vary in real time. The bottom-line outcome will be to provide state-of-the-art compression algorithms --and accompanying software libraries-- able to surpass the performance of the compression engines currently available in the ROOT [7] framework. The resulting technology has the capability to enable substantial economic and operational gains (including speedup) for High Energy and Nuclear Physics data storage/analysis. In our initial studies, a factor of nearly x4 (3.9) compression was achieved with RHIC/STAR data where ROOT compression managed only x1.4 [6].As a collaboration of experimental scientists, private industry, and the ROOT Team, our aim is to capitalize on the substantial success delivered by the initial effort and produce a robust technology properly packaged as an open-source tool that could be used by virtually every experiment around the world as means for improving data management and accessibility.In this contribution, we will present our efforts integrating our concepts of “functionally lossless compression” within the ROOT framework implementation, with the purpose of producing a basic solution readily integrated into HENP applications. We will also present our progress applying this compression through realistic examples of analysis from both the STAR and CMS experiments.

Canal, Ph.↗
DOE OSTI2021

Trailhead for quantum simulation of SU(3) Yang-Mills lattice gauge theory in the local multiplet basis

Maintaining local interactions in the quantum simulation of gauge field theories relegates most states in the Hilbert space to be unphysical—theoretically benign, but experimentally difficult to avoid. Reformulations of the gauge fields can modify the ratio of physical to gauge-variant states often through classically preprocessing the Hilbert space and modifying the representation of the field on qubit degrees of freedom. This paper considers the implications of representing SU(3) Yang-Mills gauge theory on a lattice of irreducible representations in both a global basis of projected global quantum numbers and a local basis in which controlled-plaquette operators support efficient time evolution. Classically integrating over the internal gauge space at each vertex (e.g., color isospin and color hypercharge) significantly reduces both the qubit requirements and the dimensionality of the unphysical Hilbert space. Initiating tuning procedures that may inform future calculations at scale, the time evolution of one and two plaquettes are implemented on one of IBM’s superconducting quantum devices, and early benchmark quantities are identified. The potential advantages of qudit environments, with either constrained two-dimensional hexagonal or one dimensional nearest-neighbor internal state connectivity, are discussed for future large-scale calculations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Leveraging the Higgs to Discover Physics Beyond the Standard Model (Final Technical Report)

The discovery of an apparently Standard Model-like Higgs at the Large Hadron Collider (LHC) heralds the start of a new era in particle physics. While the Higgs marks the completion of the Standard Model framework, it offers far more opportunities in the search for physics beyond the Standard Model. The Higgs boson raises a pressing theoretical problem known as the hierarchy problem: why is an elementary scalar particle so light when quantum corrections tie its mass to the highest energy scales? The Higgs also provides an unprecedented experimental opportunity as a bellwether of new physics: it may be merely the first of several states in the electroweak symmetry breaking sector, while its production and decays may provide unique evidence for additional particles. Research supported by this award leveraged the Higgs boson to explore new physics from both directions, developing novel approaches to solving the hierarchy problem posed by the Higgs boson and directly employing the Higgs as a new tool for discovery. Given that null results at the LHC and other experiments have begun to endanger conventional approaches to the hierarchy problem, research supported by the award identified original solutions to the hierarchy problem wherein the lightest degrees of freedom protecting the Higgs boson carry no Standard Model quantum numbers and thus evade existing searches. The PI's approach combined standard tools of quantum field theory with novel applications of the orbifold reduction of continuous symmetries to define the framework of "neutral naturalness'' and explore its experimental consequences across the energy, intensity, and cosmic frontiers. In employing the Higgs directly as a tool for discovery, research supported by this award articulated a systematic approach to searching for extensions of the Higgs sector at the LHC and pursued four key avenues through which the Higgs can be used to uncover new physics across a range of experiments: (1) as a direct final state probe; (2) as an indirect probe through its couplings; (3) as a portal to states neutral under the Standard Model; and (4) as a source of exotic processes in displaced decays.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Efficient Hamiltonian encoding algorithms for extracting quantum control mechanism as interfering pathway amplitudes in the Dyson series

Hamiltonian encoding is a methodology for revealing the mechanism behind the dynamics governing controlled quantum systems. In this paper, following Mitra and Rabitz \cite{abhra_1}, we define mechanism via pathways of eigenstates that describe the evolution of the system, where each pathway is associated with a complex-valued amplitude corresponding to a term in the Dyson series. The evolution of the system is determined by the constructive and destructive interference of these pathway amplitudes. Pathways with similar attributes can be grouped together into pathway classes. The amplitudes of pathway classes are computed by modulating the Hamiltonian matrix elements and decoding the subsequent evolution of the system rather than by direct computation of the individual terms in the Dyson series. The original implementation of Hamiltonian encoding was computationally intensive and became prohibitively expensive in large quantum systems. This paper presents two new encoding algorithms that calculate the amplitudes of pathway classes by using techniques from graph theory and algebraic topology to exploit patterns in the set of allowed transitions, greatly reducing the number of matrix elements that need to be modulated. These new algorithms provide an exponential decrease in both computation time and memory utilization with respect to the Hilbert space dimension of the system. To demonstrate the use of these techniques, they are applied to two illustrative state-to-state transition problems.

Abrams, Erez [Princeton University, Massachusetts ↗
DOE OSTI2021

Split representation of adaptively compressed polarizability operator

The polarizability operator plays a central role in density functional perturbation theory and other perturbative treatment of first principle electronic structure theories. The cost of computing the polarizability operator generally scales as O(Ne4) where Ne is the number of electrons in the system. The recently developed adaptively compressed polarizability operator (ACP) formulation [L. Lin, Z. Xu and L. Ying, Multiscale Model. Simul. 2017] reduces such complexity to O(Ne 3 ) in the context of phonon calculations with a large basis set for the first time, and demonstrates its effectiveness for model problems. In this paper, we improve the performance of the ACP formulation by splitting the polarizability into a near singular component that is statically compressed, and a smooth component that is adaptively compressed. The new split representation maintains the O(Ne 3 ) complexity, and accelerates nearly all components of the ACP formulation, including Chebyshev interpolation of energy levels, iterative solution of Sternheimer equations, and convergence of the Dyson equations. For simulation of real materials, we discuss how to incorporate nonlocal pseudopotentials and finite temperature effects. In this work, we demonstrate the effectiveness of our method using one-dimensional model problem in insulating and metallic regimes, as well as its accuracy for real molecules and solids.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Large deformations of Tr(Φ 3 ) and the world at infinity

The amplitudes of the non-linear sigma model can be obtained from those of Tr(Φ 3 ) theory by sending the kinematic (Mandelstam) variables to infinity in a certain direction. In this paper we characterize the behavior of Tr(Φ 3 ) amplitudes under a general class of large kinematic shifts called g-vector shifts. The objects that live in this world at infinity retain certain key amplitude-like properties, most notably factorization, and admit descriptions in terms of polytopes, but they are not generally amplitudes of any cognizable theory. We identify particular g-vector shifts that lead at infinity to mixed amplitudes involving two pions and any number of scalars, allowing us to provide polytopal descriptions of these amplitudes.

effective field theories↗
DOE OSTI2024

Balancing Charging Station Utilization and Throughput in Electric Vehicle Charging Stations with Queuing Theory

The rapid increase in electric vehicle (EV) adoption demands enhancements in the efficiency and adaptability of EV supply equipment (EVSE). Traditional EVSE systems often fail to optimize power delivery to meet the variable acceptance rates of EV batteries, resulting in significant energy wastage and reduced operational efficiency. This research addresses these challenges by integrating queuing theory with modular EVSE architectures, offering a dual strategy to optimize the operation of EV charging stations. A simulation model was developed to assess various configurations of charger capacities and outlet numbers. This model aimed to identify the optimal setup that maximizes station utilization while minimizing charging times and maximizing throughput. The model focused on charger capacities ranging from 50 to 250 kW and analyzed the different capacities’ effects on charging times and the number of vehicles served. The results indicate that a charger capacity of 125 kW is optimal, striking a balance between the charging time and the number of EVs served per hour, thus achieving the highest station utilization rate. This capacity allows for servicing a significant number of EVs with moderate increases in charging times. Lower capacities, although capable of serving more vehicles, lead to longer charging times and decreased throughput efficiency. The study underscores the effectiveness of combining queuing theory with flexible, modular charging systems that can dynamically adjust to EV charging demands.

Kumar, Praveen↗
DOE OSTI2020

Hadronic contributions to (g - 2) µ

The Muon g-2 Experiment at Fermilab, which recently started running, plans to reduce the uncertain- ties on the already very precisely measured anomalous magnetic moment of the muon by a factor of four. The goal of this effort is to probe the observed difference of more than three standard deviations between Standard-Model theory and experiment, one of the few persistent hints for physics beyond the Standard Model. The Fermilab experiment collected data from their first run last year with statistics comparable to BNL E821. They expect to release the measurement result in 2019. On the theoretical side, because the muon g - 2 arises from quantum- mechanical loop contributions in the Standard Model, it is sensitive to virtual effects of new particles, and places important constraints on Standard-Model extensions. To leverage the anticipated reduction in experimental errors, and determine unambiguously whether or not new-physics effects contribute to this quantity, the theoretical errors must be made more reliable and reduced to a commensurate precision. The Muon g-2 Theory Initiative was created to facilitate this development. The dominant sources of uncertainty in the Standard-Model prediction of the muon g -2 are from the hadronic contributions. The hadronic vacuum polarization (HVP) provides the leading correction followed by hadronic light-by-light (HLbL) scattering. There are a number of complementary theoretical efforts underway to better understand and quantify these contributions, including dispersive and data driven methods, lattice QCD, and effective field theories. Given the precision goals and the phenomenological importance, it is important to have more than one independent method for each of the two hadronic corrections, each with fully quantified uncertainties. Fostering the development of such methods is a prime goal of the initiative, as this will enable critical cross checks, and, upon combination, may yield gains in precision, to maximize the impact of E989. An important aspect of the Muon g-2 Theory Initiative’s activities are providing platforms that facilitate interactions between the different groups, as well as between the theoretical and experimental g - 2 communities. To this end, several workshops were organized in 2017 and 2018. The first meeting, held at Fermilab (June 3–6, 2017, St. Charles, IL, USA), served to kick-off the Initiative’s activities. Two meetings in early 2018 were focused respectively on the HVP and HLbL corrections. The HVP meeting was held at KEK (February 12–14, 2018, Tsukuba, Japan) and the HLbL meeting at the University of Connecticut (March 12–14, 2018, Storrs, CT, USA). The most recent workshop, which served as the second plenary meeting of the Initiative, was held at the University of Mainz (June 18–22, 2018, Mainz, Germany). An important outcome of these meetings are concrete plans for a first white paper, which is currently being written. We aim to post the white paper just prior to the release of the first E989 measurement, to present a clean theoretical prediction. The first white paper is focused on assessing and improving the reliability of the SM prediction. The purpose of the INT workshop in September 2019 is to start the next stage of the Initiative, focusing on the development of strategies to improve the theory uncertainties beyond the current level towards the E989 precision goal. We aim to accelerate theoretical developments on the hadronic contributions to the muon g - 2 so that the Standard-Model theory error can be brought to the needed precision, again in advance of the next release from E989. Hence the workshop will provide crucial theory support for a US experiment with broad impact. The Muon g-2 Theory Initiative relies on input from representatives of all the different communities that are engaged in this effort. It is therefore important that all these areas are properly represented. The funds from this grant will be used to enable more people, especially early-career scientists, to participate and make essential contributions to the workshop discussions. Since the workshop’s main goal is to kick-off the next stage of the theory initiative’s activities, support for this workshop from the DOE will help the theory community provide crucial theoretical support to the Fermilab Muon g-2 Experiment.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Numerical solution of large scale Hartree–Fock–Bogoliubov equations

The Hartree–Fock–Bogoliubov (HFB) theory is the starting point for treating superconducting systems. However, the computational cost for solving large scale HFB equations can be much larger than that of the Hartree–Fock equations, particularly when the Hamiltonian matrix is sparse, and the number of electrons N is relatively small compared to the matrix size N b . We first provide a concise and relatively self-contained review of the HFB theory for general finite sized quantum systems, with special focus on the treatment of spin symmetries from a linear algebra perspective. We then demonstrate that the pole expansion and selected inversion (PEXSI) method can be particularly well suited for solving large scale HFB equations. For a Hubbard-type Hamiltonian, the cost of PEXSI is at most $\mathcal{O}$(N b 2 ) for both gapped and gapless systems, which can be significantly faster than the standard cubic scaling diagonalization methods. We show that PEXSI can solve a two-dimensional Hubbard-Hofstadter model with N b up to 2.88 × 10 6 , and the wall clock time is less than 100 s using 17 280 CPU cores. Finally, this enables the simulation of physical systems under experimentally realizable magnetic fields, which cannot be otherwise simulated with smaller systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Landau-level composition of bound exciton states in magnetic field

Here, we present a theory that studies the state composition of a bound exciton in magnetic field. Using a basis set made of products of free electron and hole wave functions in Landau gauge, we derive a secular equation which shows the relation between Landau levels (LLs) of the electron and hole when a bound exciton is formed. Focusing on excitons in the light cone, we establish a scattering selection rule for the interaction of an electron in LL 𝑛 e with a hole in LL 𝑛 h . We solve the resulting secular equation and identify a simple pairing law, 𝑛 e =𝑛 h +𝑙, which informs us on the construction of a bound exciton state with magnetic quantum number 𝑙, and on the interaction of the exciton magnetic moment with magnetic field. We obtain good agreement between theory results and recent measurements of the diamagnetic shifts of exciton states in WSe 2 monolayers.

2-dimensional systems↗
DOE OSTI2024

Self-similar Reynolds-averaged mechanical–scalar turbulence models for Rayleigh–Taylor mixing induced by power-law accelerations in the small Atwood number limit

Analytical self-similar solutions to two-, three-, and four-equation Reynolds-averaged mechanical–scalar turbulence models describing turbulent Rayleigh–Taylor mixing driven by a temporal power-law acceleration are derived in the small Atwood number (Boussinesq) limit. The solutions generalize those previously derived for constant acceleration Rayleigh–Taylor mixing for models based on the turbulent kinetic energy K and its dissipation rate ε, together with the scalar variance S and its dissipation rate χ [O. Schilling, “Self-similar Reynolds-averaged mechanical–scalar turbulence models for Rayleigh–Taylor, Richtmyer–Meshkov, and Kelvin–Helmholtz instability-induced mixing in the small Atwood number limit,” Phys. Fluids 33, 085129 (2021)]. The turbulent fields are expressed in terms of the model coefficients and power-law exponent, with their temporal power-law scalings obtained by requiring that the self-similar equations are explicitly time-independent. Mixing layer growth parameters and other physical observables are obtained explicitly as functions of the model coefficients and parameterized by the exponent of the power-law acceleration. Values for physical observables in the constant acceleration case are used to calibrate the two-, three-, and four-equation models, such that the self-similar solutions are consistent with experimental and numerical simulation data corresponding to a canonical (i.e., constant acceleration) Rayleigh–Taylor turbulent flow. The calibrated four-equation model is then used to numerically reconstruct the mean and turbulent fields, and turbulent equation budgets across the mixing layer for several values of the power-law exponent. Finally, the reference solutions derived here can be used to understand the model predictions for strongly accelerated or decelerated Rayleigh–Taylor mixing in the large Reynolds number limit.

42 ENGINEERING↗
DOE OSTI2025

Upper bounds for 21st-century surface air temperatures in the Western United States

The last decade has seen a large number of severe heatwaves that were unprecedented in the observational record, highlighting challenges associated with observationally-based statistical quantification of the likelihood and magnitude of future extreme temperatures. An alternative to such probabilistic assessments is identification of upper bounds that quantify the hottest surface air temperatures that can possibly be achieved by the end of the 21st century. Theory, simulations, and observational analyses support the existence of a finite upper bound for surface air temperature; however, estimates for future upper-bound values that are realistic and usable for planning remain unavailable. Here, we combine atmospheric theory with large ensembles of dynamically downscaled projections to estimate historical and end-of-century upper bounds for surface air temperatures. A number of physical mechanisms can influence upper bounds, and at the end of the 21st century, estimates based on mechanisms that yield more moderate upper-bounds produce values around 60∘C for much of the Western United States and in excess of 80∘C for the hottest parts of the domain. Even cooler high-altitude locations have end-of-century upper bounds over 50∘C. Although these upper-bound estimates might seem implausibly large, increases in the upper bounds over the 21st century are similar to increases in dynamically downscaled peak surface temperatures after adjusting those downscaled temperatures to eliminate the possibly biased model trends in surface specific humidity. While upper bound estimates are high relative to historical observations, they nonetheless suggest that heatwave intensity risk is bounded, with uncertainty dominated by projections of surface and upper-level humidity.

Risser, Mark D↗
DOE OSTI2020

Distillation of maximally correlated bosonic matter from many-body quantum coherence

We construct quantum coherence resource theories in symmetrized Fock space (QCRTF), thereby providing an information-theoretic framework that connects analyses of quantum coherence in discrete-variable (DV) and continuous variable (CV) bosonic systems. Unlike traditional quantum coherence resource theories, QCRTF can be made independent of the single-particle basis and allow to quantify coherence within and between particle number sectors. For example, QCRTF can be formulated in such a way that neither Bose-Einstein condensates nor Heisenberg-Weyl coherent states are considered as quantum many-body coherence resources, whereas spin-squeezed and quadrature squeezed states are. The QCRTF framework is utilized to calculate the optimal asymptotic distillation rate of maximally correlated bosonic states both for particle number conserving resource states and resource states of indefinite particle number. In particular, we show how to generate a uniform superposition of maximally correlated bosonic states from a state of maximal bosonic coherence with asymptotically unit efficiency using only free operations in the QCRTF.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Tidal Love numbers and Green’s functions in black hole spacetimes

Tidal interactions play a crucial role in deciphering gravitational wave signals emitted by the coalescence of binary systems. They are usually quantified by a set of complex coefficients which include tidal Love numbers, describing the conservative response to an external perturbation. In the static case, these are found to vanish exactly for asymptotically flat black holes in general relativity in four space-time dimensions, and recently they have been generalized to dynamical interactions. In the context of response theory, the retarded Green’s function provides the complete description of the behavior of dynamical systems. Here, in this work we investigate the relation between Love numbers and Green’s functions, and highlight the relevance of radiation reaction effects to their connection. As a special case, we discuss Banados-Teitelboim-Zanelli black holes, where the absence of radiative modes allows us to make a direct link between them.

classical black holes↗
DOE OSTI2025

Degrees of Rate Control in Interconnected Reaction Networks

Overall reactions in interconnected networks exhibit net, forward, and reverse rates that are governed by both constitutive elementary steps in the pathway of interest and branching elementary steps that lead to alternative products. Accordingly, steps in branching pathways exhibit negative net, forward, and reverse degrees of rate control, as they reduce reaction flux to the desired product. We here contextualize the forward and reverse degrees of rate control in terms of kinetic resistances (inverse of rates) and leverage the additive nature of kinetic resistance to decouple kinetic driving forces contributed by constitutive elementary steps and branching points (nodal species) in interconnected networks. Regardless of the network connectivity, forward and reverse degrees of rate control are shown to converge at equilibrium. Away from equilibrium, we identify two critical features of interconnected networks: stoichiometric regularity─condition where all stoichiometric numbers are unity─and pathway symmetry around nodal species─condition where branching pathways share the same rate constants, stoichiometry, and species concentrations/activities─that result in (i) equal forward, reverse, and consequently net degrees of rate control and (ii) forward and reverse degrees of rate control that exhibit constant offsets, respectively, across all extents of reaction. Furthermore, our discourse further provides a mathematical description for the influence of stoichiometric irregularity and pathway asymmetry on forward and reverse degrees of rate control. Altogether, the presented work details the effects of network (inter)connectivity and stoichiometry on reaction kinetics and, in doing so, establishes general protocols for capturing these effects as additive terms in the formulation of forward and reverse degrees of rate control.

10 SYNTHETIC FUELS↗