Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “entropy functional”

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.

At least 415 records · Page 23
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Ultrafast conformational dynamics of Rydberg-excited N -methyl piperidine

We have observed the ultrafast conformational dynamics of electronically excited N-methyl piperidine (NMP) using time-resolved Rydberg fingerprint spectroscopy. Optical excitation at various wavelengths ranging from 212 nm to 229 nm leads to the 3s or 3p Rydberg states and induces coherent oscillatory motions with periods of about 700 fs. Furthermore, these coherent motions survive the internal conversion from 3p to 3s but then dephase on a time scale of a few oscillations. Intramolecular vibrational energy redistribution on a picosecond time scale leads to an equilibrium between two conformeric structures that are separated in binding energy by 0.09 eV. Model calculations using the DFT-SIC method are in excellent agreement with the experiments and identify the conformers as the chair and twist structures of NMP. The analysis of the equilibrium parameters at long time delays as a function of excitation wavelength allows for the extraction of thermodynamic parameters for the conformeric transformation. We derive an enthalpy of the chair to twist reaction in the 3s excited state of 62 meV with an entropy of 19.70 J mol –1 K –1 . An activation energy of 276 meV is also obtained with a kinetic model.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2023

A prediction of the thermodynamic, thermophysical, and mechanical properties of CrTaO 4 from first principles

It is reported that the self-forming CrTaO 4 oxide scale can protect refractory high-entropy alloys from oxidation, superior to Cr 2 O 3 . In this paper, the phase stability, mechanical, and thermal properties of three polymorphous phases of CrTaO 4 are systematically investigated from first-principles density functional theory calculations. The mechanical properties predicted using the strain–energy methods indicated that all three phases are mechanically stable. The temperature dependence of elastic constants and polycrystalline moduli of three phases demonstrated the thermal softening as temperature increase. The Helmholtz-free energies as a function of volume and temperature are derived from phonon dispersions within the quasi-harmonic approximation at six strained volumes. The calculated apparent bulk coefficients of thermal expansion of these three phases are evaluated, the highest value approximately 13.4× 10 –6 K –1 within a temperature range of 500–2000 K for the rutile I4 1 md phase. The lattice thermal conductivity calculated by the Debye–Callaway model suggested that the rutile type I4 1 md phase has the lowest value of approximately 2.1 W/m/K at 1800 K. The other two phases, C2/m and P2/c, exhibit higher values due to relatively lower Grüneisen parameters and larger phonon velocities. The melting point of CrTaO 4 is predicted to be between 1975 and 2449 K using ab initio molecular dynamics simulations. Finally, this work provides a comprehensive theoretical understanding of the thermodynamic, mechanical, and thermal properties for the new material CrTaO 4 and serves as an example of a viable computational design strategy for improved oxidation resistance of refractory alloys at high temperatures.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Quantifying Information without Entropy: Identifying Intermittent Disturbances in Dynamical Systems

A system’s response to disturbances in an internal or external driving signal can be characterized as performing an implicit computation, where the dynamics of the system are a manifestation of its new state holding some memory about those disturbances. Identifying small disturbances in the response signal requires detailed information about the dynamics of the inputs, which can be challenging. This paper presents a new method called the Information Impulse Function (IIF) for detecting and time-localizing small disturbances in system response data. The novelty of IIF is its ability to measure relative information content without using Boltzmann’s equation by modeling signal transmission as a series of dissipative steps. Since a detailed expression of the informational structure in the signal is achieved with IIF, it is ideal for detecting disturbances in the response signal, i.e., the system dynamics. Those findings are based on numerical studies of the topological structure of the dynamics of a nonlinear system due to perturbated driving signals. The IIF is compared to both the Permutation entropy and Shannon entropy to demonstrate its entropy-like relationship with system state and its degree of sensitivity to perturbations in a driving signal.

42 ENGINEERING↗
DOE OSTI2025

Tetrahedral Tilting and Lithium‐Ion Transport in Halide Argyrodites Prepared by Rapid, Microwave‐Assisted Synthesis

Abstract This study demonstrates a rapid, dry, microwave‐assisted (MW) synthesis method that enables preparation of halide argyrodites ( , , ) in less than 20 min. The structures and ion transport properties of the resulting materials are compared with those synthesized by conventional solid‐state synthesis methods. The microwave‐assisted method leads to increased site disorder and elevated Arrhenius prefactors (), which lead to an order of magnitude improvement in the 30 ionic conductivity of MW‐. X‐ray pair distribution function analysis (XPDF) reveals significant rotational disorder of the units, which is impacted by the synthesis method, choice of halide, and presence of / site disorder. These rotational displacements are strongly correlated with ion transport, specifically and entropy of migration (). Overall, this study demonstrates a rapid synthesis route for preparing high‐quality halide argyrodite solid‐state electrolytes in less than 20 min, and further unravels atomistic insights into the interplay of structural disorder, rotational dynamics, and ion transport mechanisms.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Entanglement rearrangement in self-consistent nuclear structure calculations

Background: Entanglement plays a central role in a diverse array of increasingly important research areas, including quantum computation, simulation, measurement, sensing, and communication. Extensive suites of investigations have been performed to better understand entanglement in atomic and molecular quantum many-body systems, while the exploration of entanglement in the structure of nuclei and their reactions is presently in its infancy. Purpose: The goal of this work is to begin investigating the entanglement properties of nuclei from first-principles nuclear many-body calculations. We attempt to identify common features and emergent structures of entanglement that could ultimately lead to new and natural many-body schemes. With an eye toward quantum accelerators in future hybrid-supercomputers, criteria for partitioning nuclear many-body calculations into quantum and classical components may provide advantages in future large-scale computations. Along the way we look for explanations of the relative success of phenomenological models such as the nuclear shell model, and for better ways to match to low-energy nuclear effective field theories and lattice QCD calculations to nuclear many-body techniques that are based upon entanglement. Method: We explore the entanglement between single-particle states in 4 He and 6 He. The patterns of entanglement emerging from different single-particle bases are compared, and possible links with the convergence of observables are explored, in particular, ground-state energies. The nuclear wave functions are obtained by performing active-space no-core configuration-interaction calculations using a two-body nucleon-nucleon interaction derived from chiral effective field theory. Entanglement measures within single-particle bases exhibiting different degrees of complexity are determined, in particular, harmonic oscillator (HO), Hartree-Fock (HF), natural (NAT) and variational natural (VNAT) bases. Specifically, single-orbital entanglement entropy, two-orbital mutual information, and negativity are studied. Results: The entanglement structures in 4 He and 6 He are found to be more localized within NAT and VNAT bases than within a HO basis for the optimal HO parameters we have worked with. In particular a core-valence structure clearly emerges from the full no-core calculation of 6 He. Here, the two-nucleon mutual information shows that the VNAT basis, which typically exhibits good convergence properties, effectively decouples the active and inactive spaces. Conclusions: Measures of one- and two-nucleon entanglement are found to be useful in analyzing the structure of nuclear wave functions, in particular the efficacy of basis states, and may provide useful metrics toward developing more efficient schemes for ab initio computations of the structure and reactions of nuclei, and quantum many-body systems more generally.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Understanding Twinning and Deformation in High Entropy Alloys

A combination of high strength and high ductility has been observed in multi-principal element alloys due to twin formation attributed to low stacking fault energy (SFE). In the pursuit of low SFE alloys, a key bottleneck is the lack of understanding of the composition–SFE cor- relations that would guide tailoring SFE via alloy composition. Using density functional theory (DFT), we show that dopant radius, which have been postulated as a key descriptor for SFE in dilute alloys, does not fully explain SFE trends across different host metals. Instead, charge density is a much more central descriptor. It allows us to (1) explain contrasting SFE trends in Ni and Cu host metals due to various dopants in dilute concentrations, (2) explain the large SFE variations observed in the literature even within a given alloy composition due to the nearest neighbor environments in “model” concentrated alloys, and (3) develop a machine learning model that can be used to predict SFEs in multi-elemental alloys. This model opens a possibility to use charge density as a descriptor for predicting SFE in alloys. Furthermore, a descriptor-less machine learning (ML) model based only on charge density images extracted from density functional theory (DFT) is developed to predict stacking fault energies (SFE) in concentrated alloys. The model is based on convolutional neural networks (CNNs) as one of the promising ML techniques for dealing with complex images and data. Identification of correct descriptors is a key bottleneck to develop ML models for predicting materials properties. Often, in most ML models, textbook physical descriptors such as atomic radius, valence charge and electronegativity are used as descriptors which have limitations because these properties change in concentrated alloys when multiple elements are mixed to form a solid solution. We illustrate that, within the scope of DFT, the search for descriptors can be circumvented by electronic charge density, which is the backbone of the Kohn-Sham DFT and describes the system completely. The performance of our model is demonstrated by predicting SFE of concentrated alloys with an RMSE and R2 of 6.18 mJ/m2 and 0.87, respectively, validating the accuracy of the proposed approach.

36 MATERIALS SCIENCE↗
DOE OSTI2025

An energetic link between order and strength in metals: A nanocrystalline strength limit in high-entropy alloys and intermetallic compounds

The metallurgy and materials communities have long understood and exploited fundamental links between chemical and structural ordering in metallic solids to tailor their mechanical properties. We extend these ideas to include prediction of the nanocrystalline strength limit in high-entropy alloys and intermetallic compounds, where a breakdown occurs in the classical Hall-Petch strengthening behavior. The highest reported strength achievable through alloying has rapidly climbed and given rise to new classifications of materials with extraordinary properties, with a notable case being nanocrystalline metals. High-entropy alloys (chemically disordered, concentrated solid solutions) and intermetallic compounds are two boundary cases of how tailored order can be used to manipulate mechanical behavior. Here, we show that the complex electronic-structure mechanisms governing the peak strength of alloys and pure metals can be reduced to a few physically meaningful parameters based on their atomic arrangements and used – with no fitting parameters – to predict the maximum strength of these materials. This includes a generalized energy-based accounting for the degree of structural and chemical ordering that allows for rapid and reasonably accurate prediction of peak strength (validated in the nanocrystalline limit) as a function of temperature. Predictions of maximum strength based on the activation energy (with all materials properties derived from DFT calculations or experiments) for a stress-driven transition to an amorphous state is shown to accurately describe the breakdown in Hall-Petch behavior at the smallest crystallite sizes for pure metals, intermetallic compounds, high-entropy alloys, and metallic glasses. Further, this activation energy is also shown to be directly proportional to interstitial electronic charge density, which is a good predictor of ductility, stiffness (moduli), and phase stability in high-entropy alloys and solid metals generally. The proposed framework suggests the possibility of coupling ordering and intrinsic strength to mechanisms like dislocation nucleation, hydrogen embrittlement, and transport properties, such as through correlations between the activation energies for amorphization with stacking-fault and grain boundary energies. It additionally opens the prospect for greatly accelerated structural materials design and development to address materials challenges limiting more sustainable and efficient use of energy.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Structural complexity of snapshots of two-dimensional Fermi-Hubbard systems

The development of quantum gas microscopy for two-dimensional optical lattices has provided an unparalleled tool to study the Fermi-Hubbard model (FHM) with ultracold atoms. Spin-resolved projective measurements, or snapshots, have played a significant role in quantifying correlation functions, theory verification, and thus the uncovering of underlying physical phenomena such as antiferromagnetism at commensurate filling on bipartite lattices and other charge and spin correlations, as well as dynamical properties at various densities. Here we employ a recent concept, the multiscale structural complexity, and show that when computed for the snapshots (of single spin species, local moments, or total density) it can provide a theory-free property, immediately accessible to experiments. Specifically, after benchmarking results for Ising and $XY$ models, we study the structural complexity of snapshots of the repulsive FHM in the two-dimensional square lattice as a function of doping and temperature. We generate projective measurements using determinant quantum Monte Carlo and compare their complexities against those from the experiment. We demonstrate that these complexities are linked to relevant physical observables such as the entropy and double occupancy. Their behaviors capture the development of correlations and relevant length scales in the system. Furthermore, we provide an open-source code in python which can be implemented into data analysis routines in experimental settings for the square lattice.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2025

Ultrafast Photoflash Synthesis of High-Entropy Oxide Nanoparticles

High-entropy metal oxides (HEOs) have recently received growing attention for broad energy conversion and storage applications due to their tunable properties. HEOs typically involve the combination of multiple metal cations in a single oxide lattice, thus bringing distinctive structures, controllable elemental composition, and tunable functional properties. Many synthesis methods for HEOs have been reported, such as solid-state reactions and carbon thermal shock methods. These methods frequently are energy-intensive or require relatively expensive heating equipment. In this work, we report an ultrafast photoflash synthesis method for HEO nanoparticles on diverse substrates. The energy input is provided by a commercial Xe photoflash unit, which triggers exothermic reactions to convert metal salt precursors to HEO nanoparticles within tens of milliseconds. The formation of HEO nanoparticles is attributed to the ultrafast heating (∼106 K/s) and cooling (∼105 K/s) rates of the photoflash and overall high temperature (>1000 K) during the ultrafast synthesis process. When the synthesized CoNiFeCrMn oxide (HEO) is tested as an oxygen evolution reaction electrocatalyst, it shows similar activity to similar materials prepared by other methods. We believe this photoflash synthesis provides a simple method for many others to synthesize diverse HEOs and explore their properties and potential applications.

exothermic reaction synthesis↗
DOE OSTI2024

Effects of electron beam irradiation on CrMnV and CrMnTiV high entropy alloys: Nano-mechanical, structural, and thermodynamic perspectives

Beam exit windows are crucial components of any particle accelerator as they provide an interface between the beamline vacuum and target material at atmospheric media. For high beam power machines, special materials and designs are required to withstand high radiation and mechanical loads, while minimizing energy loss during transition and maximizing window lifetime. This research investigates the impact of electron beam exposure to bulk CrMnV and CrMnTiV high entropy alloys (HEAs) with the primary goal of identifying suitable candidate materials for the design of robust and durable exit window settings. The selection criteria include intrinsic characteristics, power dissipation, and mechanical responses. According to the thermodynamic calculations, both equiatomic CrMnV and the addition of 7% of Ti with equiatomic CrMnV yield solid-solutions phases. The structural and mechanical properties of CrMnV and CrMnTiV samples were tested using field emission scanning electron microscopy, atomic force microscopy, scanning electron microcopy with energy dispersive x-ray spectroscopy, x-ray diffraction, and nanoindentation before and after exposure to a dose of ~66 kGy from a 10 MeV e-beam accelerator. Despite exhibiting beam transmission characteristics comparable to Cr and V, the indentation hardness of HEAs exceeded that of the Cr and V samples by five to six times. The examination of the CrMnTiV irradiated samples revealed organized deformation patterns depicting new features, which we suspect twinning and twin boundaries due to the addition of Ti to CrMnV. Ti, a hexagonal-close-packed crystal structure, is commonly known for deformation twinning behavior.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Theoretical investigation of surface chemical reactivity of compositionally disordered multi-metallic alloys

Alloys are widely used in both academic and industrial research as heterogeneous catalysts for accelerating technologically important reactions ranging from hydrocarbon reforming, selective hydrogenation, to electrochemical reactions involving hydrogen and oxygen that are fundamental and indispensable to a sustainable future energy infrastructure. Most basic catalysis research has been limited to combinations of two elements that are usually ordered in some manner at the nanoscale because doing so allows observable results to be easily rationalized. It however leaves a huge material space consisting of multi metallic, compositionally disordered alloys unexplored. This research demonstrates a viable theoretical approach combining different types of first principles calculations (including Green’s function-based Korringa–Kohn–Rostoker coherent potential approximation (KKR–CPA) and planewave-based density functional theory) to enable the surface chemical reactivity of compositionally complex alloy surfaces to be investigated. We demonstrate the approach by investigating hydrogen adsorption and evolution on two types of compositionally disordered alloys: a quaternary CoCrFeNi high-entropy alloy, and binary Pt-3d base metal alloys.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Molecular-Weight-Dependent Infiltration and Adsorption of Polymers into Nanochannels

The mesoporous silica shell coating hydrogenolysis nano-catalysts alters the molecular weight distributions of cleaved polymer chains compared to catalysts without a shell. The shell, composed of radially aligned narrow cylindrical nanopores, reduces the formation of low-valued gaseous products and increases the median molecular weight of the product, thus enhancing the value of the products for polymer upcycling. Here, to understand the role of the mesoporous shell, we have studied the spatial distribution of polystyrene chains, used as a model polymer, in the nanochannels in both the melt phase and solution phase. In the melt, we observed from small-angle X-ray scattering experiments that the infiltration rate of the polymer into the nanochannels is inversely proportional to the molecular weight, which is consistent with theory. In theta solution experiments using UV–vis spectroscopy, we found that the shell significantly enhances polymer adsorption compared to nanoparticles without pores. In addition, the degree of polymer adsorption is not a monotonic function of molecular weight but initially increases with the molecular weight before eventually decreasing. The molecular weight for the peak adsorption increases with the pore diameter. This adsorption behavior is rationalized as resulting from a balance between the mixing entropy gain by surface adsorption and the conformational entropy penalty incurred by chains confined in the nanochannels. The spatial distribution of polymer chains in the nanochannels is visualized by energy-dispersive X-ray spectroscopy (EDX), and inverse Abel-transformed data reveals a less uniform polymer distribution along the primary pore axis for longer chains.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Diffusion-mediated chemical concentration variation and void evolution in ion-irradiated NiCoFeCr high-entropy alloy

High-entropy alloys (HEAs) are proposed as potential structural materials for advanced nuclear systems, but little is known about the response of matrix chemistry in HEAs upon irradiation. Here, we reveal a substantial change of matrix chemical concentration as a function of irradiation damage (depth) in equiatomic NiCoFeCr HEA irradiated by 3 MeV Ni ions. After ion irradiation, the matrix contains more Fe/Cr in depth shallower than ~900–1000 nm but more Ni/Co from ~900–1000 nm to the end of the ion-damaged region due to the preferential diffusion of vacancies through Fe/Cr. Preferential diffusion also facilitates migration of vacancies from high radiation damage region to low radiation damage region, leading to no void formation below ~900–1000 nm and void formation around the end of the ion-damaged region at a fluence of 5 × 10 16 cm –2 (~123 dpa, displacements per atom, peak dose under full cascade mode). As voids grow significantly at an increased fluence (8 × 10 16 cm –2 , 196 dpa), the matrix concentration does not change dramatically due to new voids formed below ~900–1000 nm.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Chromium enhances the mechanical performance of 3d transition metal high entropy alloys

A comprehensive examination of the compositional effects on the deformation behavior of high entropy alloys (HEAs) was conducted through nanoindentation, indentation creep, and stress relaxation experiments at ambient temperature. The alloys investigated included NiCoFe, NiCoCr, NiCoCrFe, and NiCoCrFeMn, all of which underwent identical thermomechanical processing. From the experimental results, NiCoCr exhibited the highest maximum shear stress for dislocation nucleation (13 GPa) and nanoindentation hardness (3.8 GPa) among the alloys tested. It also showed the lowest stress sensitivity (0.02) and activation volume (3–4 b 3 ) under creep and stress relaxation conditions, indicative of its superior plastic flow properties. Density functional theory (DFT) calculations further revealed an uneven charge density and larger bonding-length variation in NiCoCr due to the presence of Cr, which increased lattice distortion, impeding dislocation movement. Furthermore, in this study, post-deformation microscopy using transmission electron microscopy (TEM) revealed a high density of stacking faults and twins in NiCoCr that effectively enhanced its creep strength. The results highlight the significance of specific elemental effects, particularly from Cr in this study, over configurational entropy effect (i.e., compositional complexity) in governing the deformation microstructure and mechanical properties of HEAs. Finally, these insights will be instrumental for the design and optimization of advanced alloys for load-bearing applications.

36 - MATERIALS SCIENCE↗
DOE OSTI2025

Electronic density of states as the descriptor of elastic bond strength, ductility, and local lattice distortion in BCC refractory alloys

Although electronic density of states (DOS) is fundamental to materials properties, its general relationship to mechanical properties of alloys is not well established. In this paper, using density functional theory (DFT) calculations, we show that the electronic occupancy at the Fermi level, N(E f ), obtained from DOS is a key descriptor of alloy strength and ductility. Our comprehensive analysis of numerous body centered cubic (BCC) refractory high entropy alloys (RHEAs) shows an overwhelming correlation that low N(E f ) indicates strong bonds that have high stiffness resulting in high elastic constants. High bond stiffness indicates presence of covalent nature of bonds that are directional in nature resulting in resistance to deformation leading to high bulk (B) and shear (G) moduli. Consequently, N(E f ) provides a direct correlation to the tendency of alloy ductility evidenced in the Pugh ratio (G/B). As stiffer bonds result in lower local lattice distortion (LLD), N(E f ) are LLD are also found to be corelated which opens up a correlation to solid solution strengthening and yield strength. Thus, this work unveils fundamental correlations between N(E f ) and (1) elastic bond strength, (2) ductility, and (3) LLD. These correlations open opportunities for the design of high strength high ductile RHEAs.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Quantitative Simulations of Siloxane Adsorption in Metal–Organic Frameworks

We present a transferable force field (FF) for simulating the bulk properties of linear and cyclic siloxanes and the adsorption of these species in metal–organic frameworks (MOFs). Unlike previous FFs for siloxanes, our FF accurately reproduces the vapor–liquid equilibria of each species in the bulk phase. The quality of our FF combined with the Universal Force Field using standard Lorentz–Berthelot combining rules for MOF atoms was assessed in a wide range of MOFs without open metal sites, showing good agreement with dispersion-corrected density functional theory calculations. Predictions with this FF show good agreement with the limited experimental data for siloxane adsorption in MOFs that is available. As an example of using the FF to predict adsorption properties in MOFs, we present simulations examining entropy effects in binary linear and cyclic siloxane mixture coadsorption in the large-pore MOF with structure code FOTNIN.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Azimuthal dependence of two-particle transverse momentum current correlations

Two-particle transverse momentum correlation functions are a powerful technique for understanding the dynamics of relativistic heavy-ion collisions. Among these, the transverse momentum correlator G 2 (Δη,Δφ) is of particular interest for its potential sensitivity to the shear viscosity per unit of entropy density η/s of the quark-gluon plasma formed in heavy-ion collisions. We use the UrQMD, AMPT, and EPOS models for Au–Au at √ s NN = 200 GeV and Pb–Pb at √ s NN = 2760 GeV to investigate the long range azimuthal dependence of G 2 (Δη,Δφ), and explore its utility to constrain η/s based on charged particle correlations. We find that the three models yield quantitatively distinct transverse momentum Fourier harmonics coefficients $a$ $^{pT}_{n}$. We also observe these coefficients exhibit a significant dependence on η/s in the context of the AMPT model. These observations suggest that exhaustive measurements of the dependence of G 2 (Δφ) with collision energy, system size, collision centrality, in particular, offer the potential to distinguish between different theoretical models and their underlying assumptions. Exhaustive analyses of G 2 (Δφ) obtained in large and small systems should also be instrumental in establishing new constraints for precise extraction of η/s.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Linear Velocity Problems Part 2: Thermodynamic Considerations

A compressible polytropic gas, in one dimensional planar, cylindrical, or spherical coordinate systems, is examined through four thermodynamic properties: mass density, pressure, specific internal energy (SIE), and entropy, under the assumptions that the fluid velocity is linearly proportional to the radial coordinate and that the Euler gas dynamics equations assert a self similar solution class. The behavior of the thermodynamics is almost entirely determined by an arbitrary function that results from the derivation of the density distribution. Through physical and other arguments, the arbitrary function can be plotted spatially with time profiles, providing a deeper understanding of the system and how the arbitrary function affects the behavior of the compressible gas.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗