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We revisit the classical mechanics problem of a particle moving under the influence of a force that depends on its velocity. Using the properties of the rotation matrix and associated operators, we show that it is possible to find an exact analytical solution to a number of problems where the differential equation of motion depends on the velocity. First, we apply our method to the well-known cases of a particle under the influence of the Lorentz force and Coriolis force, providing the complete analytical solution in each case for the motion of the particle in three dimensions. We also show that the complete solution can be obtained when the centrifugal force is included, showing the applicability to cases where there is simultaneous dependence on the position and velocity. This method, which is not currently discussed in a typical course in elementary mechanics, provides an alternative approach to the traditional methods that are used to solve these types of problems.

We perform ab initio simulations of beryllium (Be) and magnesium oxide (MgO) at megabar pressures and compare their structural and thermodynamic properties. We make a detailed comparison of our two recently derived phase diagrams of Be (Wu et al 2021 Phys. Rev. B 104 014103) and MgO (Soubiran and Militzer 2020 Phys. Rev. Lett. 125 175701) using the thermodynamic integration technique, as they exhibit striking similarities regarding their shape. We explore whether the Lindemann criterion can explain the melting temperatures of these materials through the calculation of the Debye temperature at high pressure. From our free energy calculations, we find that the melting line of both materials is well represented by the Simon–Glazel fit T m(P) = T 0(1 + P/a)1/c , where T 0 = 1564 K, a = 15.8037 GPa and c = 2.4154 for Be, while T 0 = 3010 K, a = 10.5797 GPa and c = 2.8683 for the MgO in the B1. For the B2 phase, we use the values a = 26.1163 GPa and c = 2.2426. Both materials exhibit negative Clapeyron slopes on the boundaries between the two solid phases that are strongly affected by anharmonic effects, which also influence the location of the solid–solid–liquid triple point. We find that the quasi-harmonic approximation underestimates the stability range of the low-pressure phases, namely hcp for Be and B1 for MgO. We also compute the phonon dispersion relations at low and high pressure for each of the phases of these materials, and also explore how the phonon density of states is modified by temperature. Finally, we derive secondary shock Hugoniot curves in addition to the principal Hugoniot curve for both materials, and study their offsets in pressure between solid and liquid branches.

Ramp compression experiments characterize high-pressure states of matter at temperatures well below those present in shock compression. However, because temperature is typically not directly measured during ramp compression, it is uncertain how much heating occurs under these shock-free conditions. Here, we performed a series of ab initio simulations on carbon in order to match the density-stress measurements of Smith et al. [Smith, et al., Nature 511, 330 (2014)]. We considered isotropically as well as uniaxially compressed solid carbon in the diamond and BC8 phases, with and without defects, as well as liquid carbon. Our idealized model ascribes heating during ramp compression to an initially uniaxially compressed cell transforming isochorically into an isotropically (hydrostatic equivalent) compressed state having lower internal energy, hence higher temperature so as to conserve energy. Multiple such heating events can occur during a single ramp experiment, leading to higher temperatures than with isentropic compression. Comparison with experiments shows that heating alone does not explain the equation of state measurements on diamond, instead implying that a significant uniaxial stress component remains present at high compression. The temperature predictions of our ramp compression model remain to be verified with future laboratory measurements., 13 pages, 13 figures, https://link.aps.org/doi/10.1103/PhysRevB.104.134104

We use first principles molecular dynamics simulations coupled to the thermodynamic integration method to study the hcp-bcc transition and melting of beryllium up to a pressure of 1600~GPa. We derive the melting line by equating solid and liquid Gibbs free energies, and represent it by a Simon Glatzel fit $T_m= 1564~\text{K} (1 + P/(15.6032 ~\text{GPa}))^{0.383}$, which is in good agreement with previous two-phase simulations below 6000~K. We also derive the hcp-bcc solid-solid phase boundary and show the quasiharmonic approximation underestimates the stability of the hcp structure, predicting lower transition pressures between hcp and bcc phases. However, our results are consistent with the stability regime predicted by the phonon quasiparticle method. We also predict that hcp-bcc-liquid triple point is located at 164.7~GPa and 4314~K. In addition, we compute the shock Hugoniot curve, and show that it is in good agreement with experiments, intersecting our derived melting curve at $\sim$235~GPa at 4900~K. Finally, we show that an isentropic compression path that intersects the melting curve at both low and high temperature in the liquid regime, can reappear in the solid after a gap as large as 7000~K. Therefore, we predict that a large section of the melting curve could be sampled, in principle, by a ramp compression experiment, where solid and liquid Be would coexist as the sample is compressed., Comment: 15 pages, 13 figures

We put together a first-principles equation of state (FPEOS) database for matter at extreme conditions by combining results from path integral Monte Carlo and density functional molecular dynamics simulations of the elements H, He, B, C, N, O, Ne, Na, Mg, Al and Si as well as the compounds LiF, B4C, BN, CH4, CH2, C2H3, CH, C2H, MgO, and MgSiO3. For all these materials, we provide the pressure and internal energy over a density-temperature range from ~0.5 to 50 g/cc and from ~10^4 to 10^9 K, which are based on ~5000 different first-principles simulations. We compute isobars, adiabats and shock Hugoniot curves in the regime of L and K shell ionization. Invoking the linear mixing approximation, we study the properties of mixtures at high density and temperature. We derive the Hugoniot curves for water and alumina as well as for carbon-oxygen, helium-neon, and CH-silicon mixtures. We predict the maximal shock compression ratios of H2O, H2O2, Al2O3, CO, and CO2 to be 4.61, 4.64, 4.64, 4.89, and 4.83, respectively. Finally we use the FPEOS database to determine the points of maximum shock compression for all available binary mixtures. We identify mixtures that reach higher shock compression ratios than their endmembers. We discuss trends common to all mixtures in pressure-temperature and particle-shock velocity spaces. In the supplementary material, we provide all FPEOS tables as well as computer codes for interpolation, Hugoniot calculations, and plots of various thermodynamic functions., 13 figures, will appear in Phys. Rev. E

In order to provide a comprehensive theoretical description of MgSiO$_3$ at extreme conditions, we combine results from path integral Monte Carlo (PIMC) and density functional molecular dynamics simulations (DFT-MD) and generate a consistent equation of state for this material. We consider a wide range of temperature and density conditions from 10$^4$ to 10$^8$ K and from 0.321 to 64.2 g$\,$cm$^{-3}$ (0.1- to 20-fold the ambient density). We study how the L and K shell electrons are ionized with increasing temperature and pressure. We derive the shock Hugoniot curve and compare with experimental results. Our Hugoniot curve is in good agreement with the experiments, and we predict a broad compression maximum that is dominated by the K shell ionization of all three nuclei while the peak compression ratio of 4.70 is obtained when the Si and Mg nuclei are ionized. Finally we analyze the heat capacity and structural properties of the liquid., 14 pages, 14 color figures, Physical Review B (accepted, Dec. 2019)

Using path integral Monte Carlo and density functional molecular dynamics (DFT-MD) simulations, we study the properties of MgSiO$_3$ enstatite in the regime of warm dense matter. We generate a consistent equation of state (EOS) that spans across a wide range of temperatures and densities (10$^4$--10$^7$ K and 6.42--64.16 g cm$^{-3}$). We derive the shock Hugoniot curve, that is in good agreement with the experiments. We identify the boundary between the regimes of thermal ionization and pressure ionization by locating where the internal energy at constant temperature attains a minimum as a function of density or pressure. At low density, the internal energy decreases with increasing density as the weight of free states changes. Thermal ionization dominates. Conversely, at high density, in the regime of pressure ionization, the internal energy increases with density. We determine the boundary between the two regimes and show that the compression maximum along the shock Hugoniot curve occurs because K shell electrons become thermally ionized rather than pressure ionized., Comment: arXiv admin note: text overlap with arXiv:2001.00985

We combine two first-principles computer simulation techniques, path integral Monte-Carlo and density functional theory molecular dynamics, to determine the equation of state of magnesium oxide in the regime of warm dense matter, with densities ranging from 0.35 to 71~g$\,$cm$^{-3}$ and temperatures from 10,000 K to $5\times10^8$~K. These conditions are relevant for the interiors of giant planets and stars as well as for shock wave compression measurements and inertial confinement fusion experiments. We study the electronic structure of MgO and the ionization mechanisms as a function of density and temperature. We show that the L-shell orbitals of magnesium and oxygen hybridize at high density. This results into a gradual ionization of the L-shell with increasing density and temperature. In this regard, MgO behaves differently from pure oxygen, which is reflected in the shape of the MgO principal shock Hugoniot curve. The curve of oxygen shows two compression maxima, while that of MgO shows only one. We predict a maximum compression ratio of 4.66 to occur for a temperature of 6.73 $\times 10^7$ K. Finally we study how multiple shocks and ramp waves can be used to cover a large range of densities and temperatures., 12 pages, 9 figures. arXiv admin note: text overlap with arXiv:2001.00985

The motion of a block sliding on a curve is a well studied problem for flat and circular surfaces, but the necessary conditions for the block to leave the surface deserve a deeper treatment. In this article, we generalize this problem to an arbitrary surface, including the effects of friction, and provide a general expression to determine under what conditions a particle will leave the surface. An explicit integral form for the speed is given, which is analytically integrable for some cases. We demonstrate general criteria to determine the critical speed at which the particle immediately leaves the surface. Three curves, a circle, a cycloid, and a catenary, are analyzed in detail, revealing several interesting features.

A simple model for the stochastic evolution of defects in a material under irradiation is presented. Using the master-equation formalism, we derive an expression for the average number of defects in terms of the power flux and the exposure time. The model reproduces the qualitative behavior of self-healing due to defect recombination, reaching a steady-state concentration of defects that depends on the power flux of the incident radiation and the material temperature, while also suggesting a particular time scale on which the incident energy is most efficient for producing defects, in good agreement with experimental results. Given this model, we discuss the integral damage factor, a descriptor that combines the power flux and the square of the irradiation time. In recent years, the scientific community involved in plasma-facing materials for nuclear fusion reactors has used this parameter to measure the equivalent material damage produced in experiments of various types with different types of radiation and wide ranges of power flux and irradiation time. The integral damage factor is useful in practice but lacks formal theoretical justification. In this simple model, we find that it is directly proportional to the maximum concentration of defects.

We study nonideal mixing effects in the regime of warm dense matter (WDM) by computing the shock Hugoniot curves of BN, MgO, and MgSiO_3. First, we derive these curves from the equations of state (EOS) of the fully interacting systems, which were obtained using a combination of path integral Monte Carlo calculations at high temperature and density functional molecular dynamics simulations at lower temperatures. We then use the ideal mixing approximation at constant pressure and temperature to rederive these Hugoniot curves from the EOS tables of the individual elements. We find that the linear mixing approximation works remarkably well at temperatures above ~2*10^5 K, where the shock compression ratio exceeds ~3.2. The shape of the Hugoniot curve of each compound is well reproduced. Regions of increased shock compression, that emerge because of the ionization of L and K shell electrons, are well represented and the maximum compression ratio on the Hugoniot curves is reproduced with high precision. Some deviations are seen near the onset of the L shell ionization regime, where ionization equilibrium in the fully interacting system cannot be well reproduced by the ideal mixing approximation. This approximation also breaks down at lower temperatures, where chemical bonds play an increasingly import role. However, the results imply that equilibrium properties of binary and ternary mixtures in the regime of WDM can be derived from the EOS tables of the individual elements. This significantly simplifies the characterization of binary and ternary mixtures in the WDM and plasma phases, which otherwise requires large numbers of more computationally expensive first-principles computer simulations., 7 figures, 19 pages, 130 citations

Using two first-principles computer simulation techniques, path integral Monte Carlo and density functional theory molecular dynamics, we derive the equation of state of magnesium in the regime of warm dense matter, with densities ranging from 0.43 to 86.11 g cm − 3 and temperatures from 20 000 K to 5 × 10 8 K. These conditions are relevant for the interiors of giant planets and stars as well as for shock compression measurements and inertial confinement fusion experiments. We study ionization mechanisms and the electronic structure of magnesium as a function of density and temperature. We show that the L shell electrons, 2s and 2p energy bands, merge at high densities. This results in gradual ionization of the L-shell with increasing density and temperature. In this regard, Mg differs from MgO, which is also reflected in the shape of its principal shock Hugoniot curve. For Mg, we predict a single broad pressure-temperature region, where the shock compression ratio is approximately 4.9. Mg thus differs from Si and Al plasmas that exhibit two well-separated compression maxima on the Hugoniot curve for L and K shell ionizations. Finally, we study multiple shocks and effects of preheat and precompression.

Ab initio molecular dynamics simulations carried out for tetragonal and orthorhombic distortions of iron closely follow the results of static-lattice electronic-structure calculations in revealing that the body-centered cubic (bcc) phase of Fe is mechanically unstable at pressures of 0.3–1.5 TPa and temperatures up to 7000 K. Crystal-structural instabilities originate in the static lattice for the bcc configuration, and are consistent with recent results from both static and dynamic high-pressure experiments. Both theory and experiment thus show that the close-packed (hexagonal, hcp and face-centered cubic, fcc) crystal structures of iron are those relevant to the cores of Earth-like planets.

Ultrahigh-pressure phase boundary between solid and liquid SiO2 is still quite unclear. Here we present predictions of silica melting curve for the multimegabar pressure regime, as obtained from first principles molecular dynamics simulations. We calculate the melting temperatures from three high pressure phases of silica (pyrite-, cotunnite-, and Fe2P-type SiO2) at different pressures using the Z method. The computed melting curve is found to rise abruptly around 330 GPa, an increase not previously reported by any melting simulations. This is in close agreement with recent experiments reporting the α-PbO2–pyrite transition around this pressure. The predicted phase diagram indicates that silica could be one of the dominant components of the rocky cores of gas giants, as it remains solid at the core of our Solar System’s gas giants. These results are also relevant to model the interior structure and evolution of massive super-Earths.

Using classical molecular dynamics, we study the time evolution of out-of-plane standing waves on a square and circular single-layer graphene membrane. We explore the first six normal modes, obtaining the oscillation frequencies from the atomic motion. The modes show long-time stability in the harmonic regime, with no decoupling on multiple frequencies. Using the frequencies of oscillation, we calculated the transverse speed of sound in graphene and the tension on the membrane.

The Z method is a novel technique that allows to calculate the melting temperature of materials at different pressures from the microcanonical ensemble. In this work, we apply this method to study the melting behavior of silica at high pressures, determining melting temperatures and dynamical properties.

Temperature is a key physical quantity that is used to describe equilibrium between two bodies in thermal contact. In computer simulations, the temperature is usually estimated by means of the equipartition theorem, as an average over the kinetic energy. However, recent studies have shown that the temperature can be estimated using only the particles positions, which has been called configurational temperature. Through classical molecular dynamics simulations of 108-argon-atoms system, we compare the performance of four different temperature estimators: the usual kinetic temperature and three configurational temperatures, Our results show that the different estimators converge to the same value, but their fluctuations are different.

By combining density functional molecular dynamics simulations with a thermodynamic integration technique, we determine the free energy of metallic hydrogen and silica, SiO2, at megabar pressures and thousands of degrees Kelvin. Our ab initio solubility calculations show that silica dissolves into fluid hydrogen above 5000 K for pressures from 10 and 40 Mbars, which has implications for the evolution of rocky cores in giant gas planets like Jupiter, Saturn, and a substantial fraction of known extrasolar planets. Our findings underline the necessity of considering the erosion and redistribution of core materials in giant planet evolution models, but they also demonstrate that hot metallic hydrogen is a good solvent at megabar pressures, which has implications for high-pressure experiments.