Approximate methods for obtaining radial distribution functions of fluids.
Radial distribution functions for fluid particle interactions with pairwise radial forces approximated for three potential models
PARTICLE COLLISION↗
SEARCH · Search NASA
Results for “Pairwise interaction model”
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.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Metabolic model-based ecological modeling for probiotic design
The microbial community composition in the human gut has a profound effect on human health. This observation has lead to extensive use of microbiome therapies, including over-the-counter ‘probiotic’ treatments intended to alter the composition of the microbiome. Despite so much promise and commercial interest, the factors that contribute to the success or failure of microbiome-targeted treatments remain unclear. We investigate the biotic interactions that lead to successful engraftment of a novel bacterial strain introduced to the microbiome as in probiotic treatments. We use pairwise genome-scale metabolic modeling with a generalized resource allocation constraint to build a network of interactions between taxa that appear in an experimental engraftment study. We create induced sub-graphs using the taxa present in individual samples and assess the likelihood of invader engraftment based on network structure. To do so, we use a generalized Lotka-Volterra model, which we show has strong ability to predict if a particular invader or probiotic will successfully engraft into an individual’s microbiome. Furthermore, we show that the mechanistic nature of the model is useful for revealing which microbe-microbe interactions potentially drive engraftment.
59 BASIC BIOLOGICAL SCIENCES↗
Vacancy Relaxation in Cubic Crystals
The configuration of the atoms surrounding a vacancy in four face-centered cubic and three body-centered cubic metals has been computed, using a pairwise, central-force model in which the energy of interaction between two atoms was taken to have the form of a Morse function. Only radial relaxations were considered. The first and second nearest-neighbor relaxations for the face-centered systems were found to be: Pb (1.42,−0.43), Ni (2.14,−0.39), Cu(2.24,−0.40) and Ca (2.73,−0.41, expressed in percentages of normal distances. For the body-centered systems the relaxations out to the fourth nearest neighbors to the vacancy were: Fe (6.07,−2.12, −0.25, −), Ba (7.85, −2.70, 0.70, −0.33) and Na (10.80, −3.14, 3.43, −0.20). The positive signs indicate relaxation toward the vacancy and the negative signs indicate relaxation away from the vacancy. The energies of relaxation (eV) are: Pb (0.162), Ni (0.626), Cu (0.560), Ca (0.400), Fe (1.410), Ba (0.950) and Na (0.172).
Girifalco, L. A.↗
Quantifying uncertainties due to irreducible three-body forces in deuteron-nucleus reactions
Deuteron-induced nuclear reactions are an essential tool for probing the structure of nuclei as well as astrophysical information such as (n, γ) cross sections. The deuteron-nucleus system is typically described within a Faddeev three-body model consisting of a neutron (n), a proton (p), and the target nucleus (A) interacting through pairwise phenomenological potentials. While Faddeev techniques enable the exact description of the three-body dynamics, their predictive power is limited in part by the omission of irreducible neutron-proton-nucleus three-body force (n–p–A 3BF). Here, our goal is to quantify systematic uncertainties stemming from the reduction of deuteron-nucleus (d + A) dynamics to a picture of three pointlike nuclear clusters interacting via pairwise nucleon-nucleus forces, using as testing grounds d + α scattering and the 6 Li ground state. We particularly focus on quantifying uncertainties arising from the full antisymmetrization of the (A + 2)-body system with the target nucleus fixed in its ground state. We adopt the ab initio no-core shell model coupled with the resonating group method (NCSM/RGM) to compute microscopic n–α and p–α interactions, and use them in a three-body description of the d + α system by means of momentum-space Faddeev-type equations. Simultaneously, we also carry out ab initio calculations of d + α scattering and 6 Li ground state by means of six-body NCSM/RGM calculations to serve as a benchmark for the three-body model predictions given by the Faddeev calculations. By comparing the Faddeev and NCSM/RGM results, we show that the irreducible n–p–α 3BF has a non-negligible effect on bound state and scattering observables alike. Specifically, the Faddeev approach yields a 6 Li ground state that is approximately 600 keV shallower than the one obtained with the NCSM/RGM. Additionally, the Faddeev calculations for d + α scattering yield a 3 + resonance that is located approximately 400 keV higher in energy compared to the NCSM/RGM result. The shape of the d + α angular distributions computed using the two approaches also differ, owing to the discrepancy in the predictions of the 3 + resonance energy. The Faddeev three-body model predictions for d + α scattering and 6 Li using microscopic n–α and p–α potentials differ from those computed microscopically with the NCSM/RGM. These discrepancies are due to the n–p–α 3BF, which arises from two-nucleon exchange terms in the microscopic d–α interaction and are not accounted for in the three-body model Faddeev calculations. This study lays the foundation for future parametrizations of the 3BF due to Pauli exclusion principle effects in improved three-body calculations of deuteron-induced reactions.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Machine Learning an Ab-Initio Based Bond-Order Potential for Bismuthene
Bismuthene is a heavy 2D material whose strong spin–orbit coupling and recently observed single-element ferroelectricity have intensified interest in its structural, vibrational, and transport properties. Accurate modeling of these behaviors requires a short-range interatomic potential that can reproduce the underlying bonding physics at a fraction of the computational cost of first-principles methods. However, such a potential is currently unavailable. Here, in this work, we construct a Tersoff bond-order potential for β-bismuthene using a reinforcement-learning framework that integrates a continuous Monte Carlo Tree Search with a simplex-based local optimizer. The optimized parameter sets reproduce first-principles lattice constants, cohesive energy, the equation of state, elastic constants, and phonon dispersion. We validate the models by performing thermal-conductivity calculations and uniaxial fracture simulations our findings confirm the reliability of the resulting models across multiple thermomechanical regimes. Comparison of the three best solutions reveals how differences in pairwise interactions, angular terms, and bond-order behavior govern phonon features and mechanical responses. We demonstrate an interpretable and computationally efficient potential for bismuthene and demonstrate a general reinforcement-learning strategy for developing bond-order models in emerging 2D materials.
deformation↗
Development of a Transferable Density-Functional Tight-Binding Model for Organic Molecules at the Water/Platinum Interface
A computationally efficient and transferable approach for modeling reactions at metal/water interfaces could significantly accelerate our understanding and ultimately the development of new catalytic transformations, particularly in the context of the emerging field of biomass conversion. Here, we present a parametrization of Pt–X (X = H, O, C) density-functional tight-binding (DFTB) for addressing this need. We first constructed Pt–H, Pt–O, and Pt–C repulsive potential splines. These pairwise parameters were then augmented to include many-body interactions using the Chebyshev Interaction Model for Efficient Simulation (ChIMES). We compare the geometrical and energetic performances of both DFTB and DFTB/ChIMES methods with DFT reference data across a variety of organic molecules at the platinum surface from nanoparticles to single-crystal surfaces. DFTB shows limited transferability between extended crystal surfaces and small nanoparticles. This transferability is significantly improved through the introduction of three-body interactions with Pt in DFTB/ChIMES, which provides consistent results across various systems, with reductions in the RMSD from around 30 kcal/mol in DFTB to around 10 kcal/mol. We demonstrate the stability and reliability of the obtained parameters by performing metadynamic simulations for the adsorption of phenol on Pt(111). We observe that DFTB itself is undersolvating the surface, leading to only one or two chemisorbed water molecules in a c(4 × 6) unit cell. In contrast, DFTB/ChIMES leads to a coverage of about 0.5 ML and successfully captures the chemisorbed mode of phenol at both the solid/liquid and the solid/gas interfaces. Furthermore, in agreement with experimental measurements, the adsorption at the solid/liquid interface is significantly weaker than that at the solid/gas interface. As a result, we highlight that even with DFTB, where we can accumulate dynamics for more than 1 ns for a given system, the simulations are not fully converged.
Adsorption↗
Spatially structured bacterial interactions alter algal carbon flow to bacteria
Phytoplankton account for nearly half of global photosynthetic carbon fixation, and the fate of that carbon is regulated in large part by microbial food web processing. We currently lack a mechanistic understanding of how interactions among heterotrophic bacteria impact the fate of photosynthetically fixed carbon. Here, we used a set of bacterial isolates capable of growing on exudates from the diatom Phaeodactylum tricornutum to investigate how bacteria-bacteria interactions affect the balance between exudate remineralization and incorporation into biomass. With exometabolomics and genome-scale metabolic modeling, we estimated the degree of resource competition between bacterial pairs. In a sequential spent media experiment, we found that pairwise interactions were more beneficial than predicted based on resource competition alone, and 30% exhibited facilitative interactions. To link this to carbon fate, we used single-cell isotope tracing in a custom cultivation system to compare the impact of different "primary" bacterial strains in close proximity to live P. tricornutum on a distal "secondary" strain. We found that a primary strain with a high degree of competition decreased secondary strain carbon drawdown by 51% at the single-cell level, providing a quantitative metric for the "cost" of competition on algal carbon fate. Additionally, a primary strain classified as facilitative based on sequential interactions increased total algal-derived carbon assimilation by 7.6 times, integrated over all members, compared to the competitive primary strain. Our findings suggest that the degree of interaction between bacteria along a spectrum from competitive to facilitative is directly linked to algal carbon drawdown.
genome-scale metabolic model↗
WA-IsoC_MSC1.1.0
The soil microbiome is central to the cycling of carbon and other nutrients and to the promotion of plant growth. Despite its importance, analysis of the soil microbiome is difficult due to its sheer complexity, with thousands of interacting species. Here, we reduced this complexity by developing model soil microbial consortia that are simpler and more amenable to experimental analysis but still represent important microbial functions of the native soil ecosystem. Samples were collected from an arid grassland soil and microbial communities (consisting mainly of bacterial species) were enriched on agar plates containing chitin as the main carbon source. Chitin was chosen because it is an abundant carbon and nitrogen polymer in soil that often requires the coordinated action of several microorganisms for complete metabolic degradation. Several soil consortia were derived that had tractable richness (30-50 OTUs) with diverse phyla representative of the native soil, including Actinobacteria, Bacteroidetes, Firmicutes, Proteobacteria and Verrucomicrobia. The resulting consortia can be stored as glycerol or lyophilized stocks at -80° C and revived while retaining community composition, greatly increasing their use as tools for the research community at large. One of the consortia that was particularly stable was chosen as a model soil consortium (MSC-1) for further analysis. MSC-1 species interactions were studied using both pairwise co-cultivation in liquid media and during growth in soil under several perturbations. Co-abundance analyses highlighted interspecies interactions and helped to define keystone species, including Mycobacterium, Rhodococcus, and Rhizobiales taxa. These experiments demonstrate the success of an approach based on naturally enriching a community of interacting species that can be stored, revived, and shared. The knowledge gained from querying these communities and their interactions will enable better understanding of the soil microbiome and the roles these interactions play in this environment.
54 ENVIRONMENTAL SCIENCES↗
Modeling Sorption of Water and Vanadium Cations by Ion-Exchange Membranes
We develop a macroscopic model for sorption of ions and solvent by ion-exchange membranes and use it to investigate how different membranes behave when exposed to electrolytes for vanadium flow batteries. The model combines a classical expression for the Gibbs free energy of a moderately concentrated electrolyte containing several ions with a term that describes elastic swelling of a polymer membrane. Parameters describing pairwise interactions between mobile ions and fixed membrane ions were fit to published experimental data and discussed in the context of the competition for sites in ion-exchange membranes. The model provides insights into why sorption of vanadium cations by Nafion is lower than predicted by Donnan theory and why switching to an anion-exchange membrane does not dramatically reduce vanadium sorption.
Electrochemistry↗
Cholesterol-induced suppression of Kir2 channels is mediated by decoupling at the inter-subunit interfaces
Cholesterol is a major regulator of multiple types of ion channels. Although there is increasing information about cholesterol binding sites, the molecular mechanisms through which cholesterol binding alters channel function are virtually unknown. In this study, we used a combination of Martini coarse-grained simulations, a network theory-based analysis, and electrophysiology to determine the effect of cholesterol on the dynamic structure of the Kir2.2 channel. We found that increasing membrane cholesterol reduced the likelihood of contact between specific regions of the cytoplasmic and transmembrane domains of the channel, most prominently at the subunit-subunit interfaces of the cytosolic domains. This decrease in contact was mediated by pairwise interactions of specific residues and correlated to the stoichiometry of cholesterol binding events. The predictions of the model were tested by site-directed mutagenesis of two identified residues—V265 and H222—and high throughput electrophysiology.
59 BASIC BIOLOGICAL SCIENCES↗
Development of $\mathrm{AMOEBA}$ Polarizable Force Field for Rare-Earth La 3+ Interaction with Bioinspired Ligands
Rare-earth metals (REMs) are crucial for many important industries, such as power generation and storage, in addition to cancer treatment and medical imaging. One promising new REM refinement approach involves mimicking the highly selective and efficient binding of REMs observed in relatively recently discovered proteins. However, realizing any such bioinspired approach requires an understanding of the biological recognition mechanisms. In this report we developed a new classical polarizable force field based on the AMOEBA framework for modeling a lanthanum ion (La 3+ ) interacting with water, acetate, and acetamide, which have been found to coordinate the ion in proteins. The parameters were derived by comparing to high-level ab initio quantum mechanical (QM) calculations that include relativistic effects. The AMOEBA model, with advanced atomic multipoles and electronic polarization, is successful in capturing both the QM distance-dependent La 3+ –ligand interaction energies and experimental hydration free energy. A new scheme for pairwise polarization damping (POLPAIR) was developed to describe the polarization energy in La 3+ interactions with both charged and neutral ligands. Simulations of La3+ in water showed water coordination numbers and ion–water distances consistent with previous experimental and theoretical findings. Water residence time analysis revealed both fast and slow kinetics in water exchange around the ion. This new model will allow investigation of fully solvated lanthanum ion–protein systems using GPU-accelerated dynamics simulations to gain insights on binding selectivity, which may be applied to the design of synthetic analogues.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
A Sheaf Theoretical Approach to Uncertainty Quantification of Heterogeneous Geolocation Information
Integration of multiple, heterogeneous sensors is a challenging problem across a range of applications. Prominent among these are multi-target tracking, where one must combine observations from different sensor types in a meaningful and efficient way to track multiple targets. Because different sensors have differing error models, we seek a theoretically justified quantification of the agreement among ensembles of sensors, both overall for a sensor collection, and also at a fine-grained level specifying pairwise and multi-way interactions among sensors. We demonstrate that the theory of mathematical sheaves provides a unified answer to this need, supporting both quantitative and qualitative data. Furthermore, the theory provides algorithms to globalize data across the network of deployed sensors, and to diagnose issues when the data do not globalize cleanly. We demonstrate and illustrate the utility of sheaf-based tracking models based on experimental data of a wild population of black bears in Asheville, North Carolina. A measurement model involving four sensors deployed among the bears and the team of scientists charged with tracking their location is deployed. This provides a sheaf-based integration model which is small enough to fully interpret, but of sufficient complexity to demonstrate the sheaf’s ability to recover a holistic picture of the locations and behaviors of both individual bears and the bear-human tracking system. A statistical approach was developed in parallel for comparison, a dynamic linear model which was estimated using a Kalman filter. This approach also recovered bear and human locations and sensor accuracies. When the observations are normalized into a common coordinate system, the structure of the dynamic linear observation model recapitulates the structure of the sheaf model, demonstrating the canonicity of the sheaf-based approach. However, when the observations are not so normalized, the sheaf model still remains valid.
97 MATHEMATICS AND COMPUTING↗
Density functionals combined with van der Waals corrections for graphene adsorbed on layered materials
Standard density functionals like Perdew-Burke-Ernzerhof (PBE) or Strongly Constrained and Appropriately Normed (SCAN) need a correction to account for long-range van der Waals (vdW) interaction. The damped Zaremba-Kohn model (dZK) starts from a formula for the vdW interaction of a distant atom with a solid surface, both with known dielectric properties, damps this formula at short range, and then treats an adsorbed molecule or atomic layer as a collection of renormalized atoms. Here, we extend this model to graphene adsorbed on semiconducting layered materials [bulk graphite and hBN, and multilayer transition metal dichalcogenides (TMD)] by including the C 4 asymptotic term and multiple electrostatic image effects due to the two surfaces of the substrate slabs in the vdW calculations. The resulting SCAN-vdW-dZK and PBE-vdW-dZK give approximately the same results for the systems considered here, in agreement with available reference values. The predicted binding energies are roughly 25% lower than those from SCAN+rVV10 (revised Vydrov and Van Voorhis 2010), and ~15% lower than those from PBE+rVV10. Since SCAN+rVV10 usually overbinds, the predicted binding energies by SCAN-vdW-dZK and PBE-vdW-dZK are expected to be closer to the true values. The predicted equilibrium binding distances from SCAN-vdW-dZK and PBE-vdW-dZK are slightly larger (~ 0.1 Å) than those from SCAN+rVV10, and close to those from SCAN. The binding energy depends upon the number of substrate layers more strongly in vdW-dZK than in rVV10. The C 4 -term contributions can be 40% of the total vdW interactions, and the C 5 term contributes about 10%. The effects of images and the back surfaces of slabs can contribute about 4–10%. The vdW interaction energy power laws from the vdW-dZK model for graphene adsorbed on multilayer MoS 2 show slowly varying decay to the pairwise exponent –4 with increasing separation $\textit{D}$, very similar to those obtained from the random-phase approximation and renormalization group approaches by Ambrosetti et al. Both the PBE-vdW-dZK and SCAN-vdW-dZK give a greater increase in interlayer binding energy when the TMD substrate changes from monolayer to four-layer for the graphene/TMD adsorption systems. This is consistent with the relevant electron-energy-loss spectroscopy experimental results showing increased dielectric response from the substrate with increasing substrate layer number.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Learning epistatic polygenic phenotypes with Boolean interactions
Detecting epistatic drivers of human phenotypes is a considerable challenge. Traditional approaches use regression to sequentially test multiplicative interaction terms involving pairs of genetic variants. For higher-order interactions and genome-wide large-scale data, this strategy is computationally intractable. Moreover, multiplicative terms used in regression modeling may not capture the form of biological interactions. Building on the Predictability, Computability, Stability (PCS) framework, we introduce the epiTree pipeline to extract higher-order interactions from genomic data using tree-based models. The epiTree pipeline first selects a set of variants derived from tissue-specific estimates of gene expression. Next, it uses iterative random forests (iRF) to search training data for candidate Boolean interactions (pairwise and higher-order). We derive significance tests for interactions, based on a stabilized likelihood ratio test, by simulating Boolean tree-structured null (no epistasis) and alternative (epistasis) distributions on hold-out test data. Finally, our pipeline computes PCS epistasis p-values that probabilisticly quantify improvement in prediction accuracy via bootstrap sampling on the test set. We validate the epiTree pipeline in two case studies using data from the UK Biobank: predicting red hair and multiple sclerosis (MS). In the case of predicting red hair, epiTree recovers known epistatic interactions surrounding MC1R and novel interactions, representing non-linearities not captured by logistic regression models. In the case of predicting MS, a more complex phenotype than red hair, epiTree rankings prioritize novel interactions surrounding HLA-DRB1 , a variant previously associated with MS in several populations. Taken together, these results highlight the potential for epiTree rankings to help reduce the design space for follow up experiments.
59 BASIC BIOLOGICAL SCIENCES↗
DNA Break Clustering as a Predictor of Cell Death across Various Radiation Qualities: Influence of Cell Size, Cell Asymmetry, and Beam Orientation
Cosmic radiation, composed of high charge and energy (HZE) particles, causes cellular DNA damage that can result in cell death or mutation that can evolve into cancer. In this work, a cell death model is applied to several cell lines exposed to HZE ions spanning a broad range of linear energy transfer (LET) values. We hypothesize that chromatin movement leads to the clustering of multiple double strand breaks (DSB) within one radiation-induced foci (RIF). The survival probability of a cell population is determined by averaging the survival probabilities of individual cells, which is function of the number of pairwise DSB interactions within RIF. The simulation code RITCARD was used to compute DSB. Two clustering approaches were applied to determine the number of RIF per cell. RITCARD outputs were combined with experimental data from four normal human cell lines to derive the model parameters and expand its predictions in response to ions with LET ranging from ∼0.2keV/μmto∼3000keV/μm. Spherical and ellipsoidal nuclear shapes and two ion beam orientations were modeled to assess the impact of geometrical properties on cell death. The calculated average number of RIF per cell reproduces the saturation trend for high doses and high-LET values that is usually experimentally observed. The cell survival model generates the recognizable bell shape of LET dependence for the relative biological effectiveness (RBE). At low LET, smaller nuclei have lower survival due to increased DNA density and DSB clustering. At high LET, nuclei with a smaller irradiation area either because of a smaller size or a change in beam orientation have a higher survival rate due to a change in the distribution of DSB/RIF per cell. If confirmed experimentally, the geometric characteristics of cells would become a significant factor in predicting radiation-induced biological effects.
cell survival↗
Efficient massively parallel simulation of dynamic channel assignment schemes for wireless cellular communications
Fast, efficient parallel algorithms are presented for discrete event simulations of dynamic channel assignment schemes for wireless cellular communication networks. The driving events are call arrivals and departures, in continuous time, to cells geographically distributed across the service area. A dynamic channel assignment scheme decides which call arrivals to accept, and which channels to allocate to the accepted calls, attempting to minimize call blocking while ensuring co-channel interference is tolerably low. Specifically, the scheme ensures that the same channel is used concurrently at different cells only if the pairwise distances between those cells are sufficiently large. Much of the complexity of the system comes from ensuring this separation. The network is modeled as a system of interacting continuous time automata, each corresponding to a cell. To simulate the model, conservative methods are used; i.e., methods in which no errors occur in the course of the simulation and so no rollback or relaxation is needed. Implemented on a 16K processor MasPar MP-1, an elegant and simple technique provides speedups of about 15 times over an optimized serial simulation running on a high speed workstation. A drawback of this technique, typical of conservative methods, is that processor utilization is rather low. To overcome this, new methods were developed that exploit slackness in event dependencies over short intervals of time, thereby raising the utilization to above 50 percent and the speedup over the optimized serial code to about 120 times.
Greenberg, Albert G.↗
Knowledge graph-aided Bayesian active learning for top- K genetic interaction discovery
In silico methods for predicting the effects of multi-gene perturbations hold great promise for advancing functional genomics, computational drug discovery, and disease modeling. However, the development of these predictive algorithms for mammalian systems has been hampered by limited datasets and high experimental costs. In this study, we present a Bayesian active learning framework designed to discover pairwise host gene knockdowns that effectively inhibit viral proliferation in an in vitro HIV-1 infection model. Our method leverages a biological knowledge graph as side information and employs a computationally efficient batch diversification approach. We evaluated this framework using a dataset of viral load measurements obtained from multi-day dual-gene depletion experiments, encompassing all possible pairwise knockdowns of over 350 host genes associated with HIV infection. We demonstrate that our framework rapidly identifies the most effective gene knockdown pairs for reducing viral load. Furthermore, we show that incorporating side information enhances performance during the early stages of active learning (low data regime), while our batch diversification strategy significantly boosts performance in later stages (high data regime). This framework is general and can be adapted to explore gene interactions in other contexts, such as synthetic lethality prediction and mapping epistatic effects across quantitative trait loci.
Computational biology and bioinformatics↗
Computing the solubility of argon and xenon in molten sodium chloride and potassium chloride salts
Molten salt reactors (MSRs) offer significant advancements in nuclear reactor safety and efficiency by operating at higher temperatures and lower pressures compared to traditional reactors. A critical aspect of MSR operation involves understanding the solubility of fission byproducts, particularly noble gases, in the molten salts used. This study employs molecular dynamics (MD) simulations to compute Henry’s law constants and enthalpies of solvation for argon and xenon in molten sodium chloride (NaCl) and potassium chloride (KCl). We developed a new pairwise potential for the noble gas and salt interactions based on first principles calculations. We then used this potential to calculate Henry’s law constants of the two gases in the molten salts, which were modeled using both a rigid ion model (RIM) and a polarizable ion model (PIM). The solubility calculations, performed using the Widom insertion method, show qualitative agreement with limited experimental data, highlighting the temperature dependence and greater solubility of both gases in KCl compared to NaCl. Additionally, free volume analysis elucidated the role of available space within the molten salts in governing solubility trends. Our findings suggest that PIM trajectories provide more reliable predictions for noble gas solubility than RIM due to their accurate density representation. Furthermore, these results enhance understanding of gas solubility in MSR environments, and the methods can be readily extended to other systems.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗