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Large-scale Molecular Dynamics of Dense Plasmas: The Cimarron Project
Abstract
We describe the status of a new time-dependent simulation capability for dense plasmas. The backbone of this multi-institutional effort—the Cimarron Project—is the massively parallel molecular dynamics (MD) code "ddcMD," devel-oped at Lawrence Livermore National Laboratory. The project's focus is material conditions such as exist in inertial confinement fusion experiments, and in many stellar interiors: high temperatures, high densities, significant elec-tromagnetic fields, mixtures of high-and low-Z elements, and non-Maxwellian particle distributions. Of particular importance is our ability to incorporate into this classical MD code key atomic, radiative, and nuclear processes, so that their interacting effects under non-ideal plasma conditions can be investigated. This paper summarizes progress in computational methodology, discusses strengths and weaknesses of quantum statistical potentials as effective in-teractions for MD, explains the model used for quantum events possibly occurring in a collision, describes two new experimental efforts that play a central role in our validation work, highlights some significant results obtained to date, outlines concepts now being explored to deal more efficiently with the very disparate dynamical timescales that arise in fusion plasmas, and provides a careful comparison of quantum effects on electron trajectories predicted by more elaborate dynamical methods.
... Note that the separation in Eq. (32) requires κ < Z N n α + n β n α n β (38) to avoid divergences. This constrain is satisfied except for extreme plasma conditions when the screening length is smaller than the atomic state effective size. ...
... This ratio goes as r −1 for κ → 0, and the interaction approaches the pure Coulomb potential but goes to ∼0.962Z 2 N as κ approaches the maximum value Z N [see Eq. (38)]. The present work (see Sec. V) is consistent to order κ 2 r 2 a [11] retaining up to the quadrupole term in the multipole expansion. ...
... Results excluding the "bound" electrons from the simulations, used in the line shape calculations, are also plotted in Fig. 3 and not surprisingly deviate from Eq. (54) at short distances. Interesting treatments of bound electrons as well as electron recombination and ionization processes using classical MD simulations have been attempted [36][37][38] but are beyond the scope of the present work. ...
Spectral line broadening by plasmas can be computed by solving the equation of motion for the dipole of the radiating system perturbed by a fluctuating potential obtained from computer simulations. Such calculations have relied on the multipole expansion of the radiator-plasma interaction often keeping only the dipole term. With increasing density, however, higher multipoles as well as plasma perturbers overlapping the bound electron wave functions are expected to become important. For hydrogenic systems, the atomic matrix elements of the full Coulomb and screened Coulomb interactions are given by analytical formulas. Using these results, a computer simulation approach that accounts for the full radiator-plasma interaction is developed. One benefit is the removal of inherent strong collision divergences in the multipole expansion approximation. Furthermore, it yields the plasma polarization shift produced by perturbers penetrating the wave function of the radiator bound electrons. The model was applied to hydrogenic argon Ly-α, Ly-β, and Ly-γ spectral lines in a dense argon plasma at free electron densities of 10 24 or 10 25 cm −3 and temperature of 800 eV relevant to plasma diagnostic techniques for inertial confinement fusion implosions.
... -for Multi-component, we will take a dense plasma consisting of hydrogen H þ admixed with fully ionised argon Ar 18þ where the temperature is chosen to be T ¼ 5:18 Â 10 7 K [34]. ...
... In this application, we choose an interesting case in the field of dense plasmas [34]. We consider plasma consisting of hydrogen H þ admixed with fully ionised argon Ar 18þ [34]. ...
... In this application, we choose an interesting case in the field of dense plasmas [34]. We consider plasma consisting of hydrogen H þ admixed with fully ionised argon Ar 18þ [34]. ...
The pair correlation function (PCF) in a non-degenerated plasma, containing multi-species of electric charges (electrons, positive and negative ions and impurities) are calculated in a Multi-Component Plasma (MCP) model. The method stands on the use of the statistical mechanics under the mean-field theory approach where the screening and the quantum effects have been taken into account. The coupled system of the obtained nonlinear integral equations (NIES) governing the pair correlation functions are solved numerically in three-dimensional space with three different methods: Fixed Point Method (FPM), Verlet Method (VM) and Finite Difference Method (FDM). These methods generally give the same results which are in a good agreement with those obtained by the molecular dynamics simulation (MD).
... Because most plasmas are created out of equilibrium, understanding temperature relaxation is critical for modeling the evolution of multi-temperature HEDPs [22][23][24][25][26][27] . Temperature relaxation has been studied extensively for electron-ion systems 16,17,[28][29][30][31][32][33][34][35] . However, most plasma theories are tailored for the case of widely disparate mass (electrons and a single ion species). ...
... These theories have been compared to MD simulations with varying degrees of success 30,35 . However, explicit electron-ion MD simulations often rely on quantum statistical potentials 36,37 which may only be valid in thermodynamic equilibrium 29 . This complicates comparisons of MD simulations with theory because disagreements can be attributed to uncertainties in the interaction potentials instead of theoretical models. ...
New facilities such as the National Ignition Facility and the Linac Coherent Light Source have pushed the frontiers of high energy-density matter. These facilities offer unprecedented opportunities for exploring extreme states of matter, ranging from cryogenic solid-state systems to hot, dense plasmas, with applications to inertial-confinement fusion and astrophysics. However, significant gaps in our understanding of material properties in these rapidly evolving systems still persist. In particular, non-equilibrium transport properties of strongly-coupled Coulomb systems remain an open question. Here, we study ion-ion temperature relaxation in a binary mixture, exploiting a recently-developed dual-species ultracold neutral plasma. We compare measured relaxation rates with atomistic simulations and a range of popular theories. Our work validates the assumptions and capabilities of the simulations and invalidates theoretical models in this regime. This work illustrates an approach for precision determinations of detailed material properties in Coulomb mixtures across a wide range of conditions.
... Temperature relaxation has been studied extensively for electron-ion systems (19,20,(25)(26)(27)(28)(36)(37)(38)(39) and plasma theories are tailored for the case of disparate mass. These theories have been compared to molecular dynamics (MD) simulations with varying degrees of success (37,39). ...
... Explicit electron-ion MD simulations often rely on quantum statistical potentials (12,40) which may only be valid in thermodynamic equilibrium (36). To avoid these problems, fictitious charges (22,38) or alternate techniques (41,42) are sometimes used. ...
New facilities such as the National Ignition Facility and the Linac Coherent Light Source have pushed the frontiers of high energy-density matter. These facilities offer unprecedented opportunities for exploring extreme states of matter, ranging from cryogenic solid-state systems to hot, dense plasmas, with applications to inertial-confinement fusion and astrophysics. However, significant gaps in our understanding of material properties in these rapidly evolving systems still persist. In particular, non-equilibrium transport properties of strongly-coupled Coulomb systems remain an open question. Here, we study ion-ion temperature relaxation in a binary mixture, exploiting a recently-developed dual-species ultracold neutral plasma. We compare measured relaxation rates with atomistic simulations and a range of popular theories. Our work validates the assumptions and capabilities of the simulations and invalidates theoretical models in this regime. This work illustrates an approach for precision determinations of detailed material properties in Coulomb mixtures across a wide range of conditions.
... Strong ion-ion coupling combined with the quantum behavior of the electron fluid make simulation and modeling challenging. One approach that is able to fully capture the strongly coupled behavior of the ions has been molecular dynamics (MD), i.e., a fully atomistic simulation of the ion dynamics that results from numerical integration of the equations of motion [5]. As MD simulations explicitly calculate the ion trajectories they are able to provide transport properties, such as viscosity and thermal diffusivity [6], acoustic properties, such as the sound speed [7,8], and thermodynamic variables, including the equation of state [5,7]. ...
... One approach that is able to fully capture the strongly coupled behavior of the ions has been molecular dynamics (MD), i.e., a fully atomistic simulation of the ion dynamics that results from numerical integration of the equations of motion [5]. As MD simulations explicitly calculate the ion trajectories they are able to provide transport properties, such as viscosity and thermal diffusivity [6], acoustic properties, such as the sound speed [7,8], and thermodynamic variables, including the equation of state [5,7]. The accuracy and predictive power of such simulations is encompassed within the calculation of the interatomic potential, with increased accuracy coming at the expense of computational cost. ...
We perform nonadiabatic simulations of warm dense aluminum based on the electron-force field (EFF) variant of wave-packet molecular dynamics. Comparison of the static ion-ion structure factor with density functional theory (DFT) is used to validate the technique across a range of temperatures and densities spanning the warm dense matter regime. Focusing on a specific temperature and density (3.5 eV, 5.2 g/cm3), we report on differences in the dynamic structure factor and dispersion relation across a variety of adiabatic and nonadiabatic techniques. We find the dispersion relation produced with EFF is in close agreement with the more robust and adiabatic Kohn-Sham DFT.
... Strong ion-ion coupling combined with the quantum behavior of the electron fluid make simulation and modeling challenging. One approach that is able to fully capture the strongly coupled behavior of the ions has been molecular dynamics (MD), i.e., a fully atomistic simulation of the ion dynamics that results from numerical integration of the equations of motion [5]. As MD simulations explicitly calculate the ion trajectories they are able to provide transport properties, such as viscosity and thermal diffusivity [6], acoustic properties, such as the sound speed [7,8], and thermodynamic variables, including the equation of state [5,7]. ...
... One approach that is able to fully capture the strongly coupled behavior of the ions has been molecular dynamics (MD), i.e., a fully atomistic simulation of the ion dynamics that results from numerical integration of the equations of motion [5]. As MD simulations explicitly calculate the ion trajectories they are able to provide transport properties, such as viscosity and thermal diffusivity [6], acoustic properties, such as the sound speed [7,8], and thermodynamic variables, including the equation of state [5,7]. The accuracy and predictive power of such simulations is encompassed within the calculation of the interatomic potential, with increased accuracy coming at the expense of computational cost. ...
We perform non-adiabatic simulations of warm dense aluminum based on the electron-force field (EFF) variant of wave-packet molecular dynamics. Comparison of the static ion-ion structure factor with density functional theory is used to validate the technique across a range of temperatures and densities spanning the warm dense matter regime. Differences in the dynamic structure factor and dispersion relation between adiabatic and non-adiabatic techniques suggest that the explicit inclusion of electrons is necessary to fully capture the low frequency dynamics of the response function.
... Alongside the quantum methods of KSMD and OFMD, it has been shown that very accurate results can be found by performing classical MD calculations using effective potentials [41][42][43][44][45][46][47]. In these calculations, it is the generation of the effective ion-ion interactions to account for the quantum electron motion that determines the success of the approach. ...
... KSMD-OFMD combines KSMD and OFMD with a new temperature dependent XC potential to generate results over a wide range of temperatures [39,40,109,110]. The MD-PAKS and MD-QSP models perform MD calculations using effective interaction potentials; MD-PAKS uses KS-DFT in a pseudoatom picture [44][45][46][47] while MD-QSP forms potentials based on a calculation of pair density matrices [42,43]. In QMC-CEI calculations, PIMC is used to find both the electronic and ionic contribution to the EOS [51,111]. ...
Material equation-of-state (EOS) models, generally providing the pressure and internal energy for a given density and temperature, are required to close the equations of hydrodynamics. As a result they are an essential piece of physics used to simulate inertial confinement fusion (ICF) implosions. Historically, EOS models based on different physical/chemical pictures of matter have been developed for ICF relevant materials such as the deuterium (D2) or deuterium-tritium (DT) fuel, as well as candidate ablator materials such as polystyrene (CH), glow-discharge polymer (GDP), beryllium (Be), carbon (C), and boron carbide (B4C). The accuracy of these EOS models can directly affect the reliability of ICF target design and understanding, as shock timing and material compressibility are essentially determined by what EOS models are used in ICF simulations. Systematic comparisons of current EOS models, benchmarking with experiments, not only help us to understand what the model differences are and why they occur, but also to identify the state-of-the-art EOS models for ICF target designers to use. For this purpose, the first Equation-of-State Workshop, supported by the US Department of Energy's ICF program, was held at the Laboratory for Laser Energetics (LLE), University of Rochester on 31 May–2nd June, 2017. This paper presents a detailed review on the findings from this workshop: (1) 5–10% model-model variations exist throughout the relevant parameter space, and can be much larger in regions where ionization and dissociation are occurring, (2) the D2 EOS is particularly uncertain, with no single model able to match the available experimental data, and this drives similar uncertainties in the CH EOS, and (3) new experimental capabilities such as Hugoniot measurements around 100 Mbar and high-quality temperature measurements are essential to reducing EOS uncertainty.
... (2). All subsequent studies in the area of plasmas and dusty plasmas have adopted the same scheme [3][4][5], including dusty plasmas in oscillating (AC) external fields [6][7][8][9], lane and band formation in dusty * alessio.zaccone@unimi.it; daniele.gamba1@studenti.unimi.it ...
The Langevin equation is ubiquitously employed to numerically simulate plasmas and dusty plasmas. However, the usual assumption of white noise becomes untenable when the system is subject to an external AC electric field. This is because the charged particles in the plasma, which provide the thermal bath for the particle transport, become themselves responsive to the AC field and the thermal noise is field-dependent and non-Markovian. We theoretically study the particle diffusiv-ity in a Langevin transport model for a tagged charged particle immersed in a dense plasma of charged particles that act as the thermal bath, under an external AC electric field, by properly accounting for the effects of the AC field on the thermal bath statistics. We analytically derive the time-dependent generalized diffusivity D(t) for different initial conditions. The generalized diffusiv-ity exhibits damped oscillatory-like behaviour with initial very large peaks, where the generalized diffusion coefficient is enhanced by orders of magnitude with respect to the infinite-time steady-state value. The latter coincides with the Stokes-Einstein diffusivity in the absence of external field. For initial conditions where the external field is already on at t = 0 and the system is thermalized under DC conditions for t ≤ 0, the short-time behaviour is hyperballistic, M SD ∼ t 4 (where MSD is the mean-squared displacement), leading to giant enhancement of the particle transport. Finally, the theory elucidates the role of medium polarization on the local Lorentz field, and allows for estimates of the effective electric charge due to polarization by the surrounding charges.
... The NVE ensemble presents notable advantages for exploring adiabatic processes, phase transitions, and non-equilibrium dynamics [11,12]. Note that in spite that the symplectic scheme [13][14][15][16][17] can be employed, the unphysical energy drift during long-time simulations has not been fully resolved [18][19][20] due to the error introduced in the force calculation. ...
The computational bottleneck of molecular dynamics is pairwise additive long-range interactions between particles. The random batch Ewald (RBE) method provides a highly efficient and superscalable solver for long-range interactions, but the stochastic nature of this algorithm leads to unphysical self-heating effect during the simulation. We propose an energy stable scheme (ESS) for particle systems by employing a Berendsen-type energy bath. The scheme removes the notorious energy drift, which exists due to the force error even when a symplectic integrator is employed. Combining the RBE with the ESS, the new method provides a perfect solution to the computational bottleneck of molecular dynamics at the microcanonical ensemble. Numerical results for a primitive electrolyte and all-atom pure water systems demonstrate the attractive performance of the algorithm, including its dramatically high accuracy, linear complexity, and overcoming the energy drift for long-time simulations.
... Non-local transport may also require different continuum-level descriptions and not be amenable to off-line property evaluations. The accuracy of MD has been demonstrated as an efficient computational tool to probe the microscopic properties of high-energy density experiments 14 . MD was used to validate various microscopic models that were implemented in an ICF hydrodynamics code: equilibration phenomena, structure, and radiative processes 15 . ...
Throughout computational science, there is a growing need to utilize the continual improvements in raw computational horsepower to achieve greater physical fidelity through scale-bridging over brute-force increases in the number of mesh elements. For instance, quantitative predictions of transport in nanoporous media, critical to hydrocarbon extraction from tight shale formations, are impossible without accounting for molecular-level interactions. Similarly, inertial confinement fusion simulations rely on numerical diffusion to simulate molecular effects such as non-local transport and mixing without truly accounting for molecular interactions. With these two disparate applications in mind, we develop a novel capability which uses an active learning approach to optimize the use of local fine-scale simulations for informing coarse-scale hydrodynamics. Our approach addresses three challenges: forecasting continuum coarse-scale trajectory to speculatively execute new fine-scale molecular dynamics calculations, dynamically updating coarse-scale from fine-scale calculations, and quantifying uncertainty in neural network models.
... Broadly, the stopping power models that are applied in the WDM regime fall into four categories: (1) highly detailed multi-atom first-principles models [54][55][56][57][58], (2) highly efficient average-atom models [42,58,59], (3) models based on variants of the uniform electron gas [60][61][62][63][64][65], and (4) classical or semiclassical models [66][67][68][69][70]. Type- (2), (3), and (4) models can be efficient enough to tabulate results across the wide range of thermodynamic conditions required by radiation-hydrodynamic codes that support ICF development, or even evaluated inline [71]. ...
Stopping power is the rate at which a material absorbs the kinetic energy of a charged particle passing through it -- one of many properties needed over a wide range of thermodynamic conditions in modeling inertial fusion implosions. First-principles stopping calculations are classically challenging because they involve the dynamics of large electronic systems far from equilibrium, with accuracies that are particularly difficult to constrain and assess in the warm-dense conditions preceding ignition. Here, we describe a protocol for using a fault-tolerant quantum computer to calculate stopping power from a first-quantized representation of the electrons and projectile. Our approach builds upon the electronic structure block encodings of Su et al. [PRX Quantum 2, 040332 2021], adapting and optimizing those algorithms to estimate observables of interest from the non-Born-Oppenheimer dynamics of multiple particle species at finite temperature. Ultimately, we report logical qubit requirements and leading-order Toffoli costs for computing the stopping power of various projectile/target combinations relevant to interpreting and designing inertial fusion experiments. We estimate that scientifically interesting and classically intractable stopping power calculations can be quantum simulated with roughly the same number of logical qubits and about one hundred times more Toffoli gates than is required for state-of-the-art quantum simulations of industrially relevant molecules such as FeMoCo or P450.
... The accuracy of MD has been demonstrated as an efficient computational tool to probe the microscopic properties of high-energy density experiments. 13 MD was used to validate various microscopic models that were implemented in an ICF hydrodynamics code: equilibration phenomena, structure, and radiative processes. 14 However, MD cannot currently be performed over time and length scales relevant for ICF experiments as it becomes very expensive at higher temperatures. ...
Throughout computational science, there is a growing need to utilize the continual improvements in raw computational horsepower to achieve greater physical fidelity through scale-bridging over brute-force increases in the number of mesh elements. For instance, quantitative predictions of transport in nanoporous media, critical to hydrocarbon extraction from tight shale formations, are impossible without accounting for molecular-level interactions. Similarly, inertial confinement fusion simulations rely on numerical diffusion to simulate molecular effects such as non-local transport and mixing without truly accounting for molecular interactions. With these two disparate applications in mind, we develop a novel capability which uses an active learning approach to optimize the use of local fine-scale simulations for informing coarse-scale hydrodynamics. Our approach addresses three challenges: forecasting continuum coarse-scale trajectory to speculatively execute new fine-scale molecular dynamics calculations, dynamically updating coarse-scale from fine-scale calculations, and quantifying uncertainty in neural network models.
... In contrast, the field of computational plasma physics has traditionally relied on hydrodynamics [5,6] and kinetic [7,8] codes because the temporal and spatial scales of interest are much too large for MD to be tractable; moreover, the detailed description of discrete particles is less important and instead macroscopic field variables are computed. However, MD plays a central role in a diverse set of plasma subfields concerned with understanding microscopic particles such as astrophysical systems [9,10], dusty plasmas [11,12], ultracold neutral plasmas [13,14], dense plasmas [15,16], low temperature plasmas [17,18], plasma beams [19,20], quark-gluon plasmas [21,22], etc. ...
We present an open-source, performant, pure-python molecular dynamics (MD) suite for non-ideal plasmas. The code, Sarkas, aims to accelerate the research process by providing an MD code but also pre- and post-processing tools. Sarkas offers the ease of use of Python while employing the Numba library to obtain execution speeds comparable to that of compiled languages. The available tools in Sarkas include graphical displays of the equilibration process through a Jupyter interface and the ability to compute quantities such as, radial distribution functions, autocorrelation functions and Green-Kubo relations. Many force laws used to simulate plasmas are included in Sarkas, namely, pure Coulomb, Yukawa, and Moli\`ere pair-potentials. Sarkas also contains quantum statistical potentials and fast Ewald methods are included where necessary. An object-oriented approach allows for easy modification of Sarkas, such as adding new time integrators, boundary conditions and force laws.
... Atomistic models in which the ions, treated through classical molecular dynamics, are coupled with a quantum mechanical treatment of the electrons, have had the most success. The ion trajectories from such simulations can provide transport properties, such as viscosity and thermal diffusivity [9]; acoustic properties, such as the sound speed [10,11]; and thermodynamic variables, including the equation of state [10,12]. The most prevalent of these techniques is density functional theory molecular dynamics (DFT-MD), in which the electrons are treated within the framework of either orbital-free [10] or Kohn-Sham density functional theory [13]. ...
Wave packet molecular dynamics (WPMD) has recently received a lot of attention as a computationally fast tool with which to study dynamical processes in warm dense matter beyond the Born–Oppenheimer approximation. These techniques, typically, employ many approximations to achieve computational efficiency while implementing semi-empirical scaling parameters to retain accuracy. We investigated three of the main approximations ubiquitous to WPMD: a restricted basis set, approximations to exchange, and the lack of correlation. We examined each of these approximations in regard to atomic and molecular hydrogen in addition to a dense hydrogen plasma. We found that the biggest improvement to WPMD comes from combining a two-Gaussian basis with a semi-empirical correction based on the valence-bond wave function. A single parameter scales this correction to match experimental pressures of dense hydrogen. Ultimately, we found that semi-empirical scaling parameters are necessary to correct for the main approximations in WPMD. However, reducing the scaling parameters for more ab-initio terms gives more accurate results and displays the underlying physics more readily.
... Journal Not Specified 2021, xx, 5 2 of 16 state [10,12]. The most prevalent of these techniques is density functional theory molecular dynamics (DFT-MD), in which the electrons are treated within the framework of either orbital-free [10] or Kohn-Sham density functional theory [13]. ...
Wave packet molecular dynamics (WPMD) has recently received a lot of attention as a computationally fast tool to study dynamical processes in warm dense matter beyond the Born-Oppenheimer approximation. These techniques, typically, employ many approximations to achieve computational efficiency while implementing semi-empirical scaling parameters to retain accuracy. We investigate three of the main approximations ubiquitous to WPMD: a restricted basis set, approximations to exchange, and the lack of correlation. We examine each of these approximations in atomic and molecular hydrogen in addition to a dense hydrogen plasma. We find that the biggest improvement to WPMD comes from combining a two Gaussian basis with a semi-empirical correction based on the valence-bond wave function. A single parameter scales this correction to match experimental pressures of dense hydrogen. Ultimately, we find that semi-empirical scaling parameters are necessary to correct for the main approximations in WPMD. However, reducing the scaling parameters for more ab-initio terms gives more accurate results and displays the underlying physics more readily.
... Kohn-Sham calculations at high temperature occur in a range of applications, including the study of warm dense matter and dense plasmas in laser experiments and the interiors of giant planets and stars. [36][37][38][39][40][41] However, these calculations present significant challenges due to the substantially larger number and lesser locality of orbitals that must be computed. Consequently, O(N 3 ) and local-orbital based O(N) methods have very large prefactors, which makes QMD calculations for even small systems intractable. ...
We present an accurate and efficient real-space formulation of the Hellmann–Feynman stress tensor for O(N) Kohn–Sham density functional theory (DFT). While applicable at any temperature, the formulation is most efficient at high temperature where the Fermi–Dirac distribution becomes smoother and the density matrix becomes correspondingly more localized. We first rewrite the orbital-dependent stress tensor for real-space DFT in terms of the density matrix, thereby making it amenable to O(N) methods. We then describe its evaluation within the O(N) infinite-cell Clenshaw–Curtis Spectral Quadrature (SQ) method, a technique that is applicable to metallic and insulating systems, is highly parallelizable, becomes increasingly efficient with increasing temperature, and provides results corresponding to the infinite crystal without the need of Brillouin zone integration. We demonstrate systematic convergence of the resulting formulation with respect to SQ parameters to exact diagonalization results and show convergence with respect to mesh size to the established plane wave results. We employ the new formulation to compute the viscosity of hydrogen at 10⁶ K from Kohn–Sham quantum molecular dynamics, where we find agreement with previous more approximate orbital-free density functional methods.
... Since then, the MD approach has been developed in several advanced codes that model all of the above damage processes. The Cimarron Project [50], developed at Lawrence Livermore National Laboratory, simulates dense plasmas and it has been applied to biological samples [16]. The Xraypac [42] suite of programs was developed specifically for XFEL studies and it includes both a MD part, XMDYN, and an atomic ionization part, XATOM [51], which is based on nonrelativistic quantum electrodynamics and perturbation theory within the Hartree-Fock-Slater model. ...
X-ray free-electron lasers (XFELs) have a unique capability for time-resolved studies of protein dynamics and conformational changes on femto- and pico-second time scales. The extreme intensity of X-ray pulses can potentially cause significant modifications to the sample structure during exposure. Successful time-resolved XFEL crystallography depends on the unambiguous interpretation of the protein dynamics of interest from the effects of radiation damage. Proteins containing relatively heavy elements, such as sulfur or metals, have a higher risk for radiation damage. In metaloenzymes, for example, the dynamics of interest usually occur at the metal centers, which are also hotspots for damage due to the higher atomic number of the elements they contain. An ongoing challenge with such local damage is to understand the residual bonding in these locally ionized systems and bond-breaking dynamics. Here, we present a perspective on radiation damage in XFEL experiments with a particular focus on the impacts for time-resolved protein crystallography. We discuss recent experimental and modelling results of bond-breaking and ion motion at disulfide bonding sites in protein crystals.
... We choose here the expression proposed by Gericke, Murillo, and Schlanges (GMS) [7]. Their formula was successfully validated against molecular dynamics simulations results across a wide range of temperature and density [10,9]. GMS suggested an effective Coulomb logarithm as λ = 0.5 ln 1 + λ 2 D + a 2 ...
One of the difficulties in developing accurate numerical models of radiation flow in a coupled radiation-hydrodynamics setting is accurately modeling the transmission across a boundary layer. The COAX experiment is a platform design to test this transmission including standard radiograph and flux diagnostics as well as a temperature diagnostic measuring the population of excitation levels and ionization states of a dopant embedded within the target material. Using a broad range of simulations, we study the experimental errors in this temperature diagnostic. We conclude with proposed physics experiments that show features that are much stronger than the experimental errors and provide the means to study transport models.
... We choose here the expression proposed by Gericke, Murillo, and Schlanges (GMS) [7]. Their formula was successfully validated against molecular dynamics simulations results across a wide range of temperature and density [9,10]. GMS suggested an effective Coulomb logarithm as where the ion sphere radius is given in terms of the ion density n as = a n 3 4 ...
One of the difficulties in developing accurate numerical models of radiation flow in a coupled radiation-hy-drodynamics setting is accurately modeling the transmission across a boundary layer. The COAX experiment is a platform design to test this transmission including standard radiograph and flux diagnostics as well as a temperature diagnostic measuring the population of excitation levels and ionization states of a dopant embedded within the target material. Using a broad range of simulations, we study the experimental errors in this temperature diagnostic. We conclude with proposed physics experiments that show features that are much stronger than the experimental errors and provide the means to study transport models.
... Accurately describing the electronic structure and physical properties of WDM is crucial to understanding the evolution and internal structure of giant planets and the target dynamics in inertial confinement fusion. [2][3][4][5][6][7][8][9] The direct current (DC) and ultralow-frequency electrical conductivity of WDM are also important to planetary structure and evolution. A recent proposal is to measure terahertz electrical conductivity using terahertz time-domain spectroscopy and to extrapolate DC conductivity with reduced modeldependent uncertainties. ...
There is a growing interest in the electrical conductivity of warm dense matter from terahertz-frequency alternating current to direct current. Herein, using first-principles molecular dynamics simulations, we show that ionic thermal motion in warm dense matter drives thermal fluctuations in the electronic valence band that produce localized states in Lifshitz tails on the top and bottom of the bands. We predict Fermi glass states when these localized states extend and fill the gap between valence and conduction bands. This significantly affects the ultralow-frequency and direct current conductivity because of the very small but nonzero energy gaps between these localized states. An order parameter is proposed to describe the degree of glassiness of an electron energy band using the local density-of-state distribution. To take into account thermal hopping, we introduce electron energy-level broadening as a thermal correction term in the Kubo–Greenwood equation. The calculated terahertz conductivities of warm dense helium and argon show the differences between the Fermi glass and normal metal states.
... Molecular dynamics (MD) simulation is a powerful method for investigating temperature relaxation in strongly coupled plasmas [26][27][28][29][30][31][32][33][34][35][36] because it includes self-consistent collective modes with arbitrary-angle scattering without ad hoc cutoffs. This level of accuracy, however, is the source of the Coulomb catastrophe in CMD in which deeply bound pairs of electrons and ions can form. ...
Theoretical and computational modeling of nonequilibrium processes in warm dense matter represents a significant challenge. The electron-ion relaxation process in warm dense hydrogen is investigated here by nonequilibrium molecular dynamics using the constrained electron force field (CEFF) method. CEFF evolves wave packets that incorporate dynamic quantum diffraction that obviates the Coulomb catastrophe. Predictions from this model reveal temperature relaxation times as much as three times longer than prior molecular dynamics results based on quantum statistical potentials. Through analyses of energy distributions and mean free paths, this result can be traced to delocalization. Finally, an improved GMS [Gericke, Murillo, and Schlanges, Phys. Rev. E 78, 025401 (2008)] model is proposed, in which the Coulomb logarithms are in good agreement with CEFF results.
... However, even the simplest OFDFT models are expensive enough to prohibit reaching length scales and timescales relevant for important non- equilibrium processes in heterogeneous experiments. As a result, large-scale MD simulations of HED matter tend to employ simplified potentials [25]. ...
Modeling matter across large length scales and timescales using molecular dynamics simulations poses significant challenges. These challenges are typically addressed through the use of precomputed pair potentials that depend on thermodynamic properties like temperature and density; however, many scenarios of interest involve spatiotemporal variations in these properties, and such variations can violate assumptions made in constructing these potentials, thus precluding their use. In particular, when a system is strongly heterogeneous, most of the usual simplifying assumptions (e.g., spherical potentials) do not apply. Here, we present a multiscale approach to orbital-free density functional theory molecular dynamics (OFDFT-MD) simulations that bridges atomic, interionic, and continuum length scales to allow for variations in hydrodynamic quantities in a consistent way. Our multiscale approach enables simulations on the order of micron length scales and 10’s of picosecond timescales, which exceeds current OFDFT-MD simulations by many orders of magnitude. This new capability is then used to study the heterogeneous, nonequilibrium dynamics of a heated interface characteristic of an inertial-confinement-fusion capsule containing a plastic ablator near a fuel layer composed of deuterium-tritium ice. At these scales, fundamental assumptions of continuum models are explored; features such as the separation of the momentum fields among the species and strong hydrogen jetting from the plastic into the fuel region are observed, which had previously not been seen in hydrodynamic simulations.
... In our simulations, the degeneracy parameter was Θ > 1 (SI Appendix, Fig. S5), indicating that the classic approach is a valid approximation. Alternatively, one can use quantum-based particle dynamics codes, like ddcMD (27), at a higher computational cost. ...
Significance
X-ray Free-Electron Lasers have opened the door to a new era in structural biology, enabling imaging of biomolecules and dynamics that were impossible to access with conventional methods. A vast majority of imaging experiments, including Serial Femtosecond Crystallography, use a liquid jet to deliver the sample into the interaction region. We have observed structural changes in the carrying water during X-ray exposure, showing how it transforms from the liquid phase to a plasma. This ultrafast phase transition observed in water provides evidence that any biological structure exposed to these X-ray pulses is destroyed during the X-ray exposure.
... In hot dense plasma, Coulomb interactions are the fundamental interactions for charged particles, but the Coulomb catastrophe always exists in CMD, and the screened potential is applied widely. 22,40 Here, we extend the electron force field (eFF) method to calculate electrical conductivity in hot dense hydrogen plasma. The eFF 41-46 method can be considered as a development of wave packet molecular dynamics, which has been widely used to investigate the structure and dynamic properties [47][48][49] at relatively low temperatures. ...
Electrical conductivity of hot dense hydrogen is directly calculated by molecular dynamics simulation with a reduced electron force field method, in which the electrons are represented as Gaussian wave packets with fixed sizes. Here, the temperature is higher than electron Fermi temperature (T>300 eV, ρ=40 g/cc). The present method can avoid the Coulomb catastrophe and give the limit of electrical conductivity based on the Coulomb interaction. We investigate the effect of ion-electron coupled movements, which is lost in the static method such as density functional theory based Kubo-Greenwood framework. It is found that the ionic dynamics, which contributes to the dynamical electrical microfield and electron-ion collisions, will reduce the conductivity significantly compared with the fixed ion configuration calculations.
... In the WDM regime, the density covers a wide range-from just below solid density to ten times higher-and the temper ature varies from 0.1 to 100 eV. Structural, thermodynamic and transport properties of WDM are crucial to the fields of astrophysics, planet science, inertial confinement fusion (ICF) and materials science [1][2][3][4][5][6][7][8][9]. In this context, two parameters, viz. the coupling parameter Γ and degenerate parameter θ, define the states of matter-where Γ = Z * 2 /(k B Ta), and θ = T/T F [10]. ...
The structural, thermodynamic and transport properties of warm dense matter (WDM) are crucial to the fields of astrophysical physics, planet science, as well as inertial confinement fusion. WDM refers to the states of matter in a regime of temperature and density between cold condensed matter and hot ideal plasmas, where the density is from near up to 10 times solid density and the temperature is between 0.1 and 100 eV. In the WDM regime, matter exhibits moderately or strongly coupled, partially degenerate. Therefore, the methods which used to deal with condensed matter and isolated atom should be validated for WDM properly. It is therefore a big challenge to understand WDM within a unified theoretical description with reliable accuracy. Here we review the progress in the theoretical study of WDM with state-of-the-art simulations, \ie, quantum Langevin molecular dynamics and first principles path integral molecular dynamics. The related applications for WDM are also included.
... AIMD with the Langevin equation works well from condensed phases up to hot dense plasmas [39], and one can determine the thermal properties of hot dense matter including equation of state (EOS) [43], electronic structures [15] and transport properties [44] in a warm dense regime as the same as what AIMD can obtain. For high temperature states, γ can be obtained using some theoretical model such as the kinetic theory or collision model [39,45]. But for the statistical properties, γ can be chosen as large enough so that the equilibrium states can be reached quickly. ...
The optical properties of atoms in a hot and dense environment are of critical importance and a great challenge. In a warm dense matter, strong coupling effects between ions induce new physics which is difficult to be included in statistical isolated atom models such as the average atom model or the detailed level accounting model. Here we study the optical properties of warm dense Li from ab initio molecular dynamics with Kubo-Greenwood relations. We compare the absorption coefficients with those from detailed level accounting methods, which can show a significant difference. The analyses of ionic structures and charge density distribution show that the strongly coupled ions induce local ordered structures and redistributions of the electrons away from the homogeneous electron gases. Also, we study the electronic structures of warm dense Li within the same theoretical framework, and we discover how the local environment of different atoms affects the density of states of the system, which will directly alter the optical properties of warm dense matter.
... There are a wide range of models on thermal conductivity, from the classical Spitzer formula 2 , commonly used Lee-More model [3][4][5] , to recently developed average-atom models [6][7][8][9] , molecular dynamics simulations 10,11 and density function theory calculations [12][13][14] . Predictions of thermal conductivities using different models vary substantially in the WDM regime 13,[15][16][17] . Experimental measurements are challenging yet especially needed to benchmark and validate these models. ...
Thermal conductivity is one of the most crucial physical properties of matter when it comes to understanding heat transport, hydrodynamic evolution, and energy balance in systems ranging from astrophysical objects to fusion plasmas. In the warm dense matter regime, experimental data are very scarce so that many theoretical models remain untested. Here we present the first thermal conductivity measurements of aluminum at 0.5–2.7 g/cc and 2–10 eV, using a recently developed platform of differential heating. A temperature gradient is induced in a Au/Al dual-layer target by proton heating, and subsequent heat flow from the hotter Au to the Al rear surface is detected by two simultaneous time-resolved diagnostics. A systematic data set allows for constraining both thermal conductivity and equation-of-state models. Simulations using Purgatorio model or Sesame S27314 for Al thermal conductivity and LEOS for Au/Al release equation-of-state show good agreement with data after 15 ps. Discrepancy still exists at early time 0–15 ps, likely due to non-equilibrium conditions.
The uncertainty of Heisenberg and exclusion concepts are applicable when electron wave functions overlap. Since fermions are affected by a spin force with the Brillouin function, Quantum magneto hydrodynamic equations have been developed for studying the spin force effects on the system. The Normal Mode Analysis method is used to solve the QHD equation for nonrelativistic degenerate inhomogeneous quantum plasma. Following linearization, the first order perturbation of the densities and velocities of plasma species are observed, which are used in Poission's equation to obtain dispersion relation. Influence of spin force using Brillouin function for half spin particles along with Bohm Potential will be observed in a quantum plasma model consisting of degenerate electron along with non-degenrate positive ions. The results will have impact on both plasma-aided nanotechnology and the dispersion relation for quantum plasma instability
We report on measurements of the ion-electron energy-transfer cross section utilizing low-velocity ion stopping in high-energy-density plasmas at the OMEGA laser facility. These measurements utilize a technique that leverages the close relationship between low-velocity ion stopping and ion-electron equilibration. Shock-driven implosions of capsules filled with DHe3 gas doped with a trace amount of argon are used to generate densities and temperatures in ranges from 1×1023 to 2×1024 cm−3 and from 1.4 to 2.5 keV, respectively. The energy loss of 1-MeV DD tritons and 3.7-MeV DHe3 alphas that have velocities lower than the average velocity of the thermal electrons is measured. The energy loss of these ions is used to determine the ion-electron energy-transfer cross section, which is found to be in excellent agreement with quantum-mechanical calculations in the first Born approximation. This result provides an experimental constraint on ion-electron energy transfer in high-energy-density plasmas, which impacts the modeling of alpha heating in inertial confinement fusion implosions, magnetic-field advection in stellar atmospheres, and energy balance in supernova shocks.
This chapter is devoted to the generalization of one-particle physical problems for the case of nonequilibrium interacting many-particle systems. This formulation of the problem is based on the fact that real systems consist of interacting many particles with different mass and electric charge in motion. A unique mathematical apparatus for considering many-particle system related with one-particle kinetic equations is the chain of kinetic equations initiated by N. N. Bogoliubov, known in the community as the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) chain.
On the example of nonequilibrium complex systems of particles interacting by means of the generalized Yukawa, Coulomb, and the contact potentials, the definition of the density matrix of these systems is indicated. It is based on the solution of the chain of quantum kinetic equations for density matrices by the method of semigroups. The result is generalized to consider nonequilibrium systems consisting of different types of interacting particles with the generalized Yukawa potential. The possibility of deriving the Gross–Pitaevskii equation from the quantum chain of kinetic BBGKY equations is shown.
Kohn-Sham density functional theory calculations using conventional diagonalization based methods become increasingly expensive as temperature increases due to the need to compute increasing numbers of partially occupied states. We present a density matrix based method for Kohn-Sham calculations at high temperatures that eliminates the need for diagonalization entirely, thus reducing the cost of such calculations significantly. Specifically, we develop real-space expressions for the electron density, electronic free energy, Hellmann-Feynman forces, and Hellmann-Feynman stress tensor in terms of an orthonormal auxiliary orbital basis and its density kernel transform, the density kernel being the matrix representation of the density operator in the auxiliary basis. Using Chebyshev filtering to generate the auxiliary basis, we next develop an approach akin to Clenshaw-Curtis spectral quadrature to calculate the individual columns of the density kernel based on the Fermi operator expansion in Chebyshev polynomials and employ a similar approach to evaluate band structure and entropic energy components. We implement the proposed formulation in the SPARC electronic structure code, using which we show systematic convergence of the aforementioned quantities to exact diagonalization results, and obtain significant speedups relative to conventional diagonalization based methods. Finally, we employ the new method to compute the self-diffusion coefficient and viscosity of aluminum at 116 045 K from Kohn-Sham quantum molecular dynamics, where we find agreement with previous more approximate orbital-free density functional methods.
We present an open-source, performant, pure-python molecular dynamics (MD) suite for non-ideal plasmas. The code, Sarkas, aims to accelerate the research process by providing an MD code complete with pre- and post-processing tools. Sarkas offers the ease of use of Python while employing the Numba library to obtain execution speeds comparable to that of compiled languages. The available tools in Sarkas include graphical displays of the equilibration process through a Jupyter interface and the ability to compute quantities such as, radial distribution functions, autocorrelation functions and Green-Kubo relations. Many force laws used to simulate plasmas are included in Sarkas, namely, pure Coulomb, Yukawa and Molière pair-potentials. Sarkas also contains quantum statistical potentials and fast Ewald methods are included where necessary. An object-oriented approach allows for easy modification of Sarkas, such as adding new time integrators, boundary conditions and force laws.
Program summary
Program Title: Sarkas
CPC Library link to program files: https://doi.org/10.17632/zwpr5mpwms.1
Developer's respository link: https://github.com/murillo-group/sarkas
Licensing provisions: MIT
Programming language: Python
Nature of problem: Molecular dynamics (MD) is an important tool for non-ideal plasma physics research. The wealth of MD codes available are not designed for plasma physics problems. The available codes are written in low-level languages and do not provide pre- and post-processing libraries. These are instead written by researchers in interpreted languages, forcing researchers to have a high level of computing background.
Solution method: Development of an MD suite for plasma physics, complete with pre- and post-processing tools most commonly used in plasma physics. The suite is entirely written in Python for enhanced user-friendliness. The slow speed of Python is mitigated by using the Numba library which is a just-in-time compiler for Python.
We present the results of the first Charged-Particle Transport Coefficient Code Comparison Workshop, which was held in Albuquerque, NM October 4–6, 2016. In this first workshop, scientists from eight institutions and four countries gathered to compare calculations of transport coefficients including thermal and electrical conduction, electron–ion coupling, inter-ion diffusion, ion viscosity, and charged particle stopping powers. Here, we give general background on Coulomb coupling and computational expense, review where some transport coefficients appear in hydrodynamic equations, and present the submitted data. Large variations are found when either the relevant Coulomb coupling parameter is large or computational expense causes difficulties. Understanding the general accuracy and uncertainty associated with such transport coefficients is important for quantifying errors in hydrodynamic simulations of inertial confinement fusion and high-energy density experiments.
Generating benchmark data on charged-particle stopping power in dense plasma is a challenge, motivating new experimental platforms for facilities such as short-pulse lasers. We report on several experiments that aimed to develop a platform for high-precision stopping power measurements at OMEGA EP, specifically developing target-normal sheath acceleration (TNSA) sources, proton isochoric heating, and radiatively heated foams. Integrated experiments were performed to measure stopping power, but several phenomena have been identified that would need to be mitigated for a stopping-power measurement, including irreproducibility of the TNSA sources, spurious electromagnetic fields around proton-heated targets, and cross-talk between the source and subject targets. This work provides a basis for potential future work towards mitigating these problems and performing stopping-power measurements, and is also valuable data for general understanding of TNSA and proton isochoric heating experiments.
Warm dense carbon is generated at 0.3–2.0 g/cc and 1–7 eV by proton heating. The release equation of state (EOS) after heating and thermal conductivity of warm dense carbon are studied experimentally in this regime using a Au/C dual-layer target to initiate a temperature gradient and two picosecond time-resolved diagnostics to probe the surface expansion and heat flow. Comparison between the data and simulations using various EOSs and thermal conductivity models is quantified with a statistical χ2 analysis. Out of seven EOS tables and five thermal conductivity models, only L9061 with the Lee-More model provides a probability above 50% to match all data.
In this paper, the relaxation properties of a fully ionized, hot, ideal plasma have been studied using the molecular dynamics method. As an example, the classical problem of equalization of the electron and ion temperatures for various mass ratios is considered, the relaxation times for temperatures is determined, and the influence of the number of particles and the type of boundary conditions on the simulation results is studied. The simulation results are compared with the available theoretical results.
We report on the first accurate validation of low-Z ion-stopping formalisms in the regime ranging from low-velocity ion stopping—through the Bragg peak—to high-velocity ion stopping in well-characterized high-energy-density plasmas. These measurements were executed at electron temperatures and number densities in the range of 1.4–2.8 keV and 4×1023–8×1023 cm−3, respectively. For these conditions, it is experimentally demonstrated that the Brown-Preston-Singleton formalism provides a better description of the ion stopping than other formalisms around the Bragg peak, except for the ion stopping at vi∼0.3vth, where the Brown-Preston-Singleton formalism significantly underpredicts the observation. It is postulated that the inclusion of nuclear-elastic scattering, and possibly coupled modes of the plasma ions, in the modeling of the ion-ion interaction may explain the discrepancy of ∼20% at this velocity, which would have an impact on our understanding of the alpha energy deposition and heating of the fuel ions, and thus reduce the ignition threshold in an ignition experiment.
It is now possible to solve protein structures with femtosecond X-ray free-electron laser (XFEL) pulses that were previously inaccessible to continuous synchrotron sources due to radiation damage. The key to this success is that diffraction probes the protein structure on femtosecond timescales, whereas nuclear motion takes tens to hundreds of femtoseconds to have a significant effect on the crystal structure. This is the essential idea behind the diffraction-before-destruction principle that underlies serial femtosecond crystallography (SFX) with XFELs. In practice, the principle works well enough to determine protein structures of comparable resolution to synchrotron protein crystallography, which has led to the many successes of XFEL crystallography to date.
We have developed an experimental platform to generate radiatively heated solid density samples for warm dense matter studies at the OMEGA laser facility. Cylindrical samples of boron and beryllium are isochorically heated by K- and L-shell emission from x-ray converter foils wrapped around the cylinders' radii. X-ray Thomson scattering (XRTS) measures the temperature and the ionization state of the samples as function of time. Temperatures approach 10 eV, and the ionization states are found to be ZB=3 and ZBe=2. Radiation hydrodynamics simulations were performed to confirm a homogeneous plasma state exists in the center of the sample for the duration of the experiment. Results from the study can be extended to improve understanding of radiative heating processes in the warm dense matter regime.
Electronic transport properties of warm dense matter, such as electrical or thermal conductivities and nonadiabatic stopping power, are of particular interest to geophysics, planetary science, astrophysics, and inertial confinement fusion (ICF). One example is the α-particle stopping power of dense deuterium-tritium (DT) plasmas, which must be precisely known for current small-margin ICF target designs to ignite. We have developed a time-dependent orbital-free density functional theory (TD-OF-DFT) method for ab initio investigations of the charged-particle stopping power of warm dense matter. Our current dependent TD-OF-DFT calculations have reproduced the recently well-characterized stopping power experiment in warm dense beryllium. For α-particle stopping in warm and solid-density DT plasmas, the ab initio TD-OF-DFT simulations show a lower stopping power up to ∼25% in comparison with three stopping-power models often used in the high-energy-density physics community.
Classical molecular dynamics simulations of hydrogen plasmas have been performed with an emphasis on the analysis of the equilibration process. The theoretical basis of the simulation model as well as numerically relevant aspects, such as the proper choice and definition of simulation units, are discussed in detail, thus proving a thorough implementation of the computer simulation technique. Because of the lack of experimental data, molecular dynamics simulations are often considered as idealized computational experiments for benchmarking of theoretical models. However, these simulations are certainly challenging and consequently a validation procedure is also demanded. In this work we develop an analytical statistical equilibrium model for computational validity assessment of plasma particle dynamics simulations. Remarkable agreement between model and molecular dynamics results including a classical treatment of the ionization-recombination mechanism is obtained for a wide range of plasma coupling parameter values. Furthermore, the analytical model provides guidance to securely terminate simulation runs once the equilibrium stage has been reached, which in turn gives confidence in the statistics that potentially may be extracted from time histories of simulated physical quantities.
In this paper, we review the following simulation techniques for studying nonideal electron–ion plasmas: classical molecular dynamics, wave packet molecular dynamics and Monte Carlo and path integral Monte Carlo. As example the hydrogen nonideal plasma is considered for a wide range of densities and temperatures. The results for the thermodynamic properties such as the internal energy and pressure are compared. The area of applicability of the classical molecular dynamics and fundamental problems of the wave packet Monte Carlo are discussed.
Over the last decade X-ray free-electron laser (XFEL) sources have been made available to the scientific community. One of the most successful uses of these new machines has been protein crystallography. When samples are exposed to the intense short X-ray pulses provided by the XFELs, the sample quickly becomes highly ionized and the atomic structure is affected. Here we present a webtool dubbed FreeDam based on non-thermal plasma simulations, for estimation of radiation damage in free-electron laser experiments in terms of ionization, temperatures and atomic displacements. The aim is to make this tool easily accessible to scientists who are planning and performing experiments at XFELs.
We have extended a recently developed multispecies, multitemperature Bhatnagar-Gross-Krook model [Haack et al., J. Stat. Phys. 168, 822 (2017)], to include multiphysics capabilities that enable modeling of a wider range of physical conditions. In terms of geometry, we have extended from the spatially homogeneous setting to one spatial dimension. In terms of the physics, we have included an atomic ionization model, accurate collision physics across coupling regimes, self-consistent electric fields, and degeneracy in the electronic screening. We apply the model to a warm dense matter scenario in which the ablator-fuel interface of an inertial confinement fusion target is heated, but for larger length and time scales and for much higher temperatures than can be simulated using molecular dynamics. Relative to molecular dynamics, the kinetic model greatly extends the temperature regime and the spatiotemporal scales over which we are able to model. In our numerical results we observe hydrogen from the ablator material jetting into the fuel during the early stages of the implosion and compare the relative size of various diffusion components (Fickean diffusion, electrodiffusion, and barodiffusion) that drive this process. We also examine kinetic effects, such as anisotropic distributions and velocity separation, in order to determine when this problem can be described with a hydrodynamic model.
The ability to calculate the range of charged fusion products in a target is critical when estimating driver requirements. Additionally, charged particle ranges are a determining factor in the possibility that a burn front will propagate through the surrounding cold fuel layer, igniting the plasma. Performance parameters of the plasma, such as yield, gain, etc therefore rely on accurate knowledge of particle ranges and stopping power over a wide range of densities and temperatures. Further, this knowledge is essential in calculating ignition conditions for a given target design. In this paper, stopping power is calculated for DD and D⁶Li plasmas using a molecular dynamics based model. Emphasis is placed on solid D⁶Li which has been recently considered as a fuel option for fusion propulsion systems.
We present SQDFT: a large-scale parallel implementation of the Spectral Quadrature (SQ) method for $\mathcal{O}(N)$ Kohn-Sham Density Functional Theory (DFT) calculations at high temperature. Specifically, we develop an efficient and scalable finite-difference implementation of the infinite-cell Clenshaw-Curtis SQ approach, in which results for the infinite crystal are obtained by expressing quantities of interest as bilinear forms or sums of bilinear forms, that are then approximated by spatially localized Clenshaw-Curtis quadrature rules. We demonstrate the accuracy of SQDFT by showing systematic convergence of energies and atomic forces with respect to SQ parameters to reference diagonalization results, and convergence with discretization to established planewave results, for both metallic and insulating systems. We further demonstrate that SQDFT achieves excellent strong and weak parallel scaling on computer systems consisting of tens of thousands of processors, with near perfect $\mathcal{O}(N)$ scaling with system size and wall times as low as a few seconds per self-consistent field iteration. Finally, we verify the accuracy of SQDFT in large-scale quantum molecular dynamics simulations of aluminum at high temperature.
Nose has modified Newtonian dynamics so as to reproduce both the canonical and the isothermal-isobaric probability densities in the phase space of an N-body system. He did this by scaling time (with s) and distance (with V¹D/ in D dimensions) through Lagrangian equations of motion. The dynamical equations describe the evolution of these two scaling variables and their two conjugate momenta p/sub s/ and p/sub v/. Here we develop a slightly different set of equations, free of time scaling. We find the dynamical steady-state probability density in an extended phase space with variables x, p/sub x/, V, epsilon-dot, and zeta, where the x are reduced distances and the two variables epsilon-dot and zeta act as thermodynamic friction coefficients. We find that these friction coefficients have Gaussian distributions. From the distributions the extent of small-system non-Newtonian behavior can be estimated. We illustrate the dynamical equations by considering their application to the simplest possible case, a one-dimensional classical harmonic oscillator.
http://www.springer.com/engineering/mechanical+engineering/book/978-3-540-24542-1
In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.
An attempt is made to describe the quantum ideal gases by means of a classical Boltzmann factor using an effective pair potential. We find the n-body distribution function for ideal gases of spin 0 and use the n=2,3 cases, generalized for spin s, with the Born—Green hierarchal relation to obtain an integrodifferential equation for the effective potential. Several numerical solutions for both the Bose—Einstein and Fermi—Dirac ideal gases are presented.
A quantitative comparison between molecular dynamics and particle-in-cell simulations for strongly coupled plasmas is performed. While the particle-in-cell simulation's nonlinear potential solver produces a significant improvement in accuracy over molecular dynamics, the requirement for many grid cells between particles (in the strongly coupled regime) causes the particle-in-cell simulation to be prohibitively expensive.
We present an O(N) multigrid-based method for the efficient calculation of the long-range electrostatic forces needed for biomolecular simulations, that is suitable for implementation on massively parallel architectures. Along general lines, the method consists of: (i) a charge assignment scheme, which both interpolates and smoothly assigns the charges onto a grid; (ii) the solution of Poisson’s equation on the grid via multigrid methods; and (iii) the back interpolation of the forces and energy from the grid to the particle space. Careful approaches for the charge assignment and the force interpolation, and a Hermitian approximation of Poisson’s equation on the grid allow for the generation of the high-accuracy solutions required for high-quality molecular dynamics simulations. Parallel versions of the method scale linearly with the number of particles for a fixed number of processors, and with the number of processors, for a fixed number of particles. © 2001 American Institute of Physics.
We study the problem of electron-ion temperature equilibration in plasmas. We consider pure H at various densities and temperatures and Ar-doped H at temperatures high enough so that the Ar is fully ionized. Two theoretical approaches are used: classical molecular dynamics (MD) with statistical two-body potentials and a generalized Lenard-Balescu (GLB) theory capable of treating multicomponent weakly coupled plasmas. The GLB is used in two modes: (1) with the quantum dielectric response in the random-phase approximation (RPA) together with the pure Coulomb interaction and (2) with the classical (ℏ→0) dielectric response (both with and without local-field corrections) together with the statistical potentials. We find that the MD results are described very well by classical GLB including the statistical potentials and without local-field corrections (RPA only); worse agreement is found when static local-field effects are included, in contradiction to the classical pure-Coulomb case with like charges. The results of the various approaches are all in excellent agreement with pure-Coulomb quantum GLB when the temperature is high enough. In addition, we show that classical calculations with statistical potentials derived from the exact quantum two-body density matrix produce results in far better agreement with pure-Coulomb quantum GLB than classical calculations performed with older existing statistical potentials.
The recently commissioned Linac Coherent Light Source is an X-ray free-electron laser at the SLAC National Accelerator Laboratory. It produces coherent soft and hard X-rays with peak brightness nearly ten orders of magnitude beyond conventional synchrotron sources and a range of pulse durations from 500 to <10 fs (10−15 s). With these beam characteristics this light source is capable of imaging the structure and dynamics of matter at atomic size and timescales. The facility is now operating at X-ray wavelengths from 22 to 1.2 Å and is presently delivering this high-brilliance beam to a growing array of scientific researchers. We describe the operation and performance of this new ‘fourth-generation light source’.
FLASH, the Free-electron LASer in Hamburg, is a worldwide unique source for extremely bright ultra-short laser-like pulses tunable in a wide spectral range in the extreme ultraviolet and soft x-ray region (Ackermann et al 2007 Nat. Photonics 1 336–42). To fully exploit the features of this new generation of light sources, a user facility with efficient radiation transport to the experimental area and novel online photon diagnostics capable of characterizing the unique parameters of the FLASH radiation has been built. It serves a broad user community active in many scientific fields ranging from atomic and molecular physics to plasma and solid state physics as well as chemistry and biology. A special focus is placed on the exploitation of the ultra-short FLASH pulses using pump–probe techniques. Thus, the facility is equipped with optical and THz sources synchronized to FLASH. This paper gives a detailed overview of the FLASH user facility.
Effective two-body potentials for electron–ion plasmas are analysed. Such potentials which have been previously derived by Kelbg and others capture the basic quantum diffraction effects and are exact in the weak coupling limit. Moreover, using path integral Monte Carlo (PIMC) methods, they can be applied to strongly coupled plasmas which include bound states. We investigate the accuracy of the diagonal Kelbg potential (DKP) as well as the off-diagonal Kelbg potential (ODKP) by comparison with accurate numerical solutions of the off-diagonal two-particle Bloch equation. A significant improvement is achieved by correcting the potential value at zero particle separation.
A method for the calculation of the two-particle statistical density matrix for attractive Coulomb forces is described. The path-integral expression for the density matrix is reduced to a modified path integral which involves summation over only one-dimensional paths. This expression is then approximated by an iteration procedure using direct numerical quadratures. The results obtained are related directly to the quantum-mechanical radial-distribution function for a plasma at small ion-electron separations.
Molecular dynamics simulations have been used to compute the thermal
conductivity of the strongly coupled, nearly classical hydrogen plasma.
The relaxation of a suitably defined heat current is significantly
faster than the decay of the microscopic electric current. Electrical
and thermal conductivities are not related by a simple Wiedemann-Franz
law in the dense plasma.
Part I. Introductions: 1. Astrophysics introduction; 2. Theoretical
physics introduction; 3. Computational physics introduction; 4.
Mathematical introduction; Part II. The Continuum Limit: 5. Paradoxical
thermodynamics; 6. Statistical mechanics; 7. Motion in a central
potential; 8. Some famous models; 9. Methods; Part III. Mean Field
Dynamics: 10. Violent relaxation; 11. Internal mass loss; 12. External
influences; Part IV. Microphysics: 13. Exponential orbit instability;
14. Two-body relaxation; 15. From Kepler to Kustaanheimo; Part V.
Gravothermodynamics: 16. Escape and mass segregation; 17. Gravothermal
instability; 18. Core collapse rate for star clusters; Part VI.
Gravitational Scattering: 19. Thought experiments; 20. Mathematical
three-body scattering; 21. Analytical approximations; 22. Laboratory
experiments; 23. Gravitational burning and transmutation; Part VII.
Primordial Binaries: 24. Binaries in star clusters; 25. Triple formation
and evolution; 26. A non-renewable energy source; Part VIII.
Post-Collapse Evolution: 27. Surviving core collapse; 28. Gravothermal
oscillations; 29. Dissolution; Part IX. Star Cluster Ecology: 30.
Stellar and dynamical evolution; 31. Collisions and capture; 32. Binary
star evolution and blue stragglers; 33. Star cluster evolution;
Appendices.
A unified theory of equilibrium and nonequilibrium quantum plasmas is presented, using an approach originally developed by Bogoliubov. The results, which are valid in both high temperature-low density and (for Fermi gases) low temperature-high density regions, include: (1) an expression for the equilibrium correlation function (and, hence, the thermodynamic functions) which agrees with results previously obtained by other methods, (2) the kinetic equation for homogeneous "steady state" systems recently derived by Balescu using a different approach, and (3) a new kinetic equation which is valid for inhomogeneous and/or rapidly varying systems.
Liquid and solid hydrogen is considered at high densities (Wigner-Seitz radius rs = 1.78) and temperatures of T = 300 K and T = 3000 K. We perform ab initio calculations based on the time dependent variational principle where the Pauli principle is included at the level of two-particle exchange. The properties of the system are analyzed in terms of the pair-correlation function and the velocity autocorrelation function.
We consider the relationship between three variational principles which generate approximate time evolution in a parametrized manifold of wavefunctions. These are: the McLachlan variational principle, the Dirac-Frenkel variational principle and the time-dependent variational principle. We show that if the manifold can be parametrized by pairs of “complementary” parameters, the abovementioned principles are equivalent. The condition of complementarity, which is a sufficient one, is demonstrated to be satisfied in a number of important applications. However, a case of non-equivalence in the recent literature warns against liberal assumptions about the equivalence of the time-dependent variational principles.
The evolution of wave packets under the influence of a He´non–Heiles potential has been investigated by direct numerical solution of the time-dependent Schro¨dinger equation. Coherent state Gaussians with a variety of mean positions and momenta were selected as initial wave functions. Three types of diagnostics were used to identify chaotic behavior, namely, phase space trajectories computed from the expected values of coordinates and momenta, the correlation function P(t)=‖〈&psgr;(0)‖&psgr;(t)〉‖2, and the uncertainty product or phase space volume V(t)=ΔxΔyΔpxΔpy. The three approaches lead to a consistent interpretation of the system’s behavior, which tends to become more chaotic as the energy expectation value of the wave packet increases. The behavior of the corresponding classical system, however, is not a reliable guide to regular or chaotic behavior in the quantum mechanical system.
We introduce a new identity satisfied by solutions of the Vlasov–Poisson system. It has the property that all quantities which appear have a definite sign, and this allows us to prove new results on the time decay of the solutions in the plasma physical case.
This paper deals with the ground state of an interacting electron gas in an external potential v(r). It is proved that there exists a universal functional of the density, Fn(r), independent of v(r), such that the expression Ev(r)n(r)dr+Fn(r) has as its minimum value the correct ground-state energy associated with v(r). The functional Fn(r) is then discussed for two situations: (1) n(r)=n0+n(r), n/n01, and (2) n(r)= (r/r0) with arbitrary and r0. In both cases F can be expressed entirely in terms of the correlation energy and linear and higher order electronic polarizabilities of a uniform electron gas. This approach also sheds some light on generalized Thomas-Fermi methods and their limitations. Some new extensions of these methods are presented.
The quantity -rV(r) for ions, formed by removing a 1s electron from the neutral atom, is computed by the approach of Herman and Skillman. A straight-line approximation of -rV(r) is made, leading to an exactly solvable one-electron Schrödinger equation. The discrete and continuum orbitals are used to compute Auger KLL and KLM transition rates, radiative rates, and fluorescence yields for the elements Be-Ar. Comparison with experimental K-shell fluorescence yields indicates the calculations are 25% too high for Mg and Al and within 5% for Ar. Comparison of the individual Auger transition intensities for F, Ne, Na, and Mg indicates differences of 50%. This 50% difference between calculated and measured individual Auger transition intensities persists up to Ar, where the sum of the individual intensities is in better than 7% agreement with that derived from the fluorescence yield and K-state width.
With regard to the question if from isotherm measurements one can obtain an experimental test for the existence of Bose statistics in real gases, as is required by theory, we prove the following general theorem. The "Zustandsumme" of a non-ideal Bose or Fermi gas is given by the classical integral provided one replaces the Boltzmann exp(-phiijkT) factor by: e-phiijkT(1+/-exp [-4pi2mkTrij2h2] for each pair of molecules (ij). For the second virial coefficient B, this has, i.e., in a Bose gas, as a consequence that: B=Bnon-ideal class+Bideal Bose+B' where: B'=2piN∞drr2(1-e- phi(r)kT) exp [-4pi2mkTr2h2]. Only at very low temperatures do the last two terms in (2) become appreciable. They are then of the same order of magnitude, but have opposite signs. Due to this fact, due to the lack of precise knowledge of the molecular forces, and due to the absence of accurate measurements of B at very low temperatures, one can as yet not decide from isotherm measurements alone whether or not real gases obey the Bose statistics.
The plasma conditions in isochorically heated beryllium are measured by collective x-ray Thomson scattering. The collectively scattered Cl Ly-α x-ray line at 2.96keV shows up- and down-shifted plasmon signals. From the detailed balance relation, i.e., the ratio of the up-shifted to the down-shifted plasmon intensities, the plasma temperature can be determined independent of model assumptions. Results are shown for an experiment in which a temperature of 18eV was measured. Using detailed balance for temperature measurement will be important to validate models that calculate the static ion–ion structure factor Sii(k).
Quantum-mechanical phase-space distributions, introduced by Wigner in 1932, provide an intuitive alternative to the usual wave-function approach to problems in scattering and reaction theory. The aim of the present work is to collect and extend previous efforts in a unified way, emphasizing the parallels among problems in ordinary quantum theory, nuclear physics, chemical physics, and quantum field theory. The method is especially useful in providing easy reductions to classical physics and kinetic regimes under suitable conditions. Section II, dealing in detail with potential scattering of a spinless nonrelativistic particle, provides the background for more complex problems. Following a brief description of the two-body problem, the authors address the N-body problem with special attention to hierarchy closures, Boltzmann-Vlasov equations, and hydrodynamic aspects. The final section sketches past and possibly future applications to a wide variety of problems.
Physical principles domlnating the electron-ion threebody recombination ; process and forming the basis of a general method for calculating the electron ; temperature and the net rate of three-body recombination are discussed. ; Calculations and data are presented in graphical and tabular forms. (L.N.N.);
A Kramerslike classical pseudopotential which includes Diffraction and Symmetry effects is worked out for kBT>=1 Ry.
A regularization of Kepler motion in the three-dimensional space R3 is developed using a simple mapping of a four-dimensional space R4 onto R3. In R4 the equations of any undisturbed Kepler motion are linear differential equations with constant coefficients thus remaining completely regular at the center of attraction. This agreeable property of the equations makes them well suited for computation of perturbations. Many classical theories of perturbations in rectangular coordinates begin with a linearization of the equations of motion. In our theory such a preconditioning of the problem is avoided since the equations in R4 are already linear. From a practical point of view it may be a disadvantage that the regularized methods require more integrations than the classical procedures. The paper was written during P. Kustaanheimo's stay at the Zürich in summer 1964. No knowledge of spinors is needed for understanding the paper. The authors are indebted to IBM for Sponsoring this work.
Prompted by the need to simulate large molecular or gravitational systems and the availability of multiprocessor computers, alternatives to the standard Ewald calculation of Coulombic interactions have been developed. The two most popular alternatives, the fast multipole method (FMM) and the particle-particle particle-mesh (P3M) method are compared here to the Ewald method for a single processor machine. Parallel processor implementations of the P3M and Ewald methods are compared. The P3M method is found to be both faster than the FMM and easier to implement efficiently as it relies on commonly available software (FFT subroutines). Both the Ewald and P3M method are easily implemented on parallel architectures with the P3M method the clear choice for large systems.
The growing effort in inertial confinement fusion (ICF) research, with the upcoming new MJ class laser facilities, NIF in USA and LMJ in France, and the upgraded MJ z-pinch ZR facility in the USA, makes the appearance of this book by Atzeni and Meyer-ter-Vehn very timely. This book is an excellent introduction for graduate or masters level students and for researchers just entering the field. It is written in a very pedagogical way with great attention to the basic understanding of the physical processes involved. The book should also be very useful to researchers already working in the field as a reference containing many key formulas from different relevant branches of physics; experimentalists will especially appreciate the presence of `ready-to-use' numerical formulas written in convenient practical units. The book starts with a discussion of thermonuclear reactions and conditions required to achieve high gain in ICF targets, emphasizing the importance of high compression of the D-T fuel, and compares the magnetic confinement fusion and inertial confinement fusion approaches. The next few chapters discuss in detail the basic concepts of ICF: the hydrodynamics of a spherically imploding capsule, ignition and energy gain. This is followed by a thorough discussion of the physics of thermal waves, ablative drive and hydrodynamic instabilities, with primary focus on the Rayleigh--Taylor instability. The book also contains very useful chapters discussing the properties of hot dense matter (ionization balance, equation of state and opacity) and the interaction of laser and energetic ion beams with plasma. The book is based on and reflects the research interests of the authors and, more generally, the European activity in this area. This could explain why, in my opinion, some topics are covered in less detail than they deserve, e.g. the chapter on hohlraum physics is too brief. On the other hand, the appearance in the book of an interesting chapter on the concept of fast ignition is also a reflection of the research interests of the authors. Altogether, the book is very well written, contains a wealth of useful information and the reviewer highly recommends it to the interested reader.
Quantum-mechanical kinetic equations are derived for a homogeneous, isotropic system of charged particles and photons. A hierarchy of equations is introduced by the use of the Wigner distribution operators and quasiphoton creation and annihilation operators. The method of Bogoliubov is used to truncate the hierarchy and to obtain kinetic equations. The resulting kinetic equations contain effects due to the processes of particle-particle scattering, particle-photon scattering, and single and double emission-absorption of photons by particles. In addition, there are terms representing shielding of the particles and photons, as well as other many-body effects. Photon self-energy corrections are also discussed.
Emission of Visible Light by Hot Dense Metals
We analyse the dissipative features exhibited by numerical simulations of the Vlasov equation with self-consistent mean-fields. We show both from a formal and from a numerical point of view that the system relaxes towards a classical Boltzmann distribution. In the case of fermions this implies loss of the quantal or at least semi-classical Pauli exclusion principle, contrarily to what would be naively expected from the equation itself. This effect is at least to be traced back to a numerical dissipation through the test particle method. An application to a typical case encountered in nuclear physics is presented. This relaxation effect should play a crucial role in the description of heavy-ion collisions performed with means of Boltzmann-like kinetic equation. The Vlasov dissipation in that case compete with the physical Boltzmann relaxation. We show that associated time scales indeed overlap for typical values of the parameters of the calculation.
Results of "molecular dynamics" simulations are reported for a model of a fully ionized strongly coupled hydrogen plasma. Quantum effects are taken into account through the use of effective pair potentials; at short distances, these differ significantly from the bare Coulomb potential. Static properties of the plasma are shown to be well described by hypernetted chain theory. The ion- and electron-velocity autocorrelation functions have been computed and the electrical conductivity turns out to be roughly twice that expected on the basis of the electron self-diffusion coefficient. The predictions of Vlasov theory for the damping and dispersion of the plasmon mode are found to be in generally poor agreement with the results of the computer "experiments", but the collective dynamical properties are successfully described by a memory-function scheme in which explicit account is taken of ion-electron correlations. Prospects for future work are briefly reviewed.
In order to facilitate the comparison of the time-dependent Hartree-Fock approximation with other classical theories and to help guide our intuition in understanding the underlying physics, we study the time-dependent Hartree-Fock approximation from a classical viewpoint. We show that the time-dependent Hartree-Fock approximation is approximately equivalent to a purely classical pseudoparticle simulation. In this simulation, a collection of pseudoparticles is introduced to discretize the phase space of spatial and momentum coordinates. The dynamics is completely determined by following the pseudoparticle trajectories which are the same as the trajectories of real particles moving in the self-consistent field. As an application of the concept of the pseudoparticle simulation, we study the origin of the nonfusion events in nearly-head-on heavy-ion collisions as obtained in the time-dependent Hartree-Fock approximation. It is argued that for these nearly-head-on collisions, the emergence of the most energetic pseudonucleons of one nucleus outside the far surface of the other nucleus initiates a coherent flow-through motion because of self-consistency and leads to the breakup of the composite system. Based on this picture, we obtain quantitative estimates of the threshold energies and the low-l fusion window which agree quite well with the time-dependent Hartree-Fock results.NUCLEAR REACTIONS Time-dependent Hartree-Fock approximation. Vlasov equation. Pseudoparticle simulation. Low-l fusion window in heavy-ion collisions.
By solving the hypernetted chain (HNC) equation for the pair distribution functions, and by Monte Carlo simulations, we have calculated the equation of state and the pair structure of dense binary mixtures of classical point ions in a rigid, uniform background of opposite charge. All of our results indicate that the excess internal and free energies of mixing are negligible compared to the energies of the mixture or the pure phases, for ionic-charge ratios Z2/Z1=2 and 3, in the strong coupling (high density or low temperature) regime. This feature allows us to write down a simple equation of state for such mixtures, which is used to determine the phase diagram of pressure-ionized H-He and H-Li mixtures in a rigid background of degenerate electrons. We then treat the polarization of the electron gas by the ionic-charge distribution by perturbation theory and include quantum corrections to the free energy of the ions. Both effects do not drastically modify the phase diagrams. The applicability of our results in astrophysical situations is discussed.
The gradient correction to statistical kinetic free energy, well known at zero temperature, is generalized to finite temperatures, following the prescriptions of the density-functional formalism. The coefficient of the gradient term, a function of electron density, is explicitly determined, and an accurate approximation, suitable for numerical computation, is given. The corrected kinetic free energy, with a phenomenological extrapolation to all temperatures of the exchange and correlation contribution, is applied to equation-of-state calculations. Results are presented in the case of Be, Al, and Cu, for temperatures up to 50 eV and compressions 0.1, 1, and 10, and the influence of the gradient correction is discussed.
Based on the time‐dependent variational principle for a Hartree wave function, a semiclassical approximation for a strongly correlated plasma is derived. This procedure maps the quantum dynamical problem to a classical one with one additional degree of freedom per particle and is considered the natural extension of classical molecular dynamics. As a test case, a single electron in a Coulomb potential is studied. For the full many‐body problem the pair distribution function, the velocity autocorrelation function, the conductivity, and the diffusion constants are calculated for a plasma. We also consider liquid molecular hydrogen and lithium as a further test case which is sensitive to the treatment of the Pauli exclusion principle. © 1994 American Institute of Physics.
A method for the calculation of the two‐particle statistical density matrix for attractive Coulomb forces is described. The path‐integral expression for the density matrix is reduced to a modified path integral which involves summation over only one‐dimensional paths. This expression is then approximated by an iteration procedure using direct numerical quadratures. The results obtained are related directly to the quantum‐mechanical radial‐distribution function for a plasma at small ion‐electron separations.
Explicitly time dependent methods for semiclassical dynamics are explored using variational principles. The Dirac–Frenkel–McLachlan variational principle for the time dependent Schrödinger equation and a variational correction procedure for wavefunctions and transition amplitudes are reviewed. These variational methods are shown to be promising tools for the solution of semiclassical problems where the correspondence principle, classical intuition, or experience suggest reasonable trial forms for the time dependent wavefunction. Specific trial functions are discussed for several applications, including the curve crossing problem. The useful semiclassical content of the time dependent Hartree approximation is discussed. Procedures for the variational propagation of density matrices are also derived.
The spectral method utilizes numerical solutions to the time‐dependent Schrödinger equation to generate the energy eigenvalues and eigenfunctions of the time‐independent Schrödinger equation. Accurate time‐dependent wave functions ψ(r, t) are generated by the split operator FFT method, and the correlation function 〈ψ(r, 0) ‖ ψ(r, t)〉 is computed by numerical integration. Fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system. Analysis of the location of these peaks reveals the eigenvalues with high accuracy. Additional Fourier transforms of ψ(r, t) with respect to time generate the eigenfunctions. Previous applications of the method were to two‐dimensional potentials. In this paper energy eigenvalues and wave functions obtained with the spectral method are presented for vibrational states of three‐dimensional Born–Oppenheimer potentials applicable to SO2, O3, and H2O. The energy eigenvalues are compared with results obtained with the variational method. It is concluded that the spectral method is an accurate tool for treating a variety of practical three‐dimensional potentials.
A method is presented for deriving a set of kinetic equations for a system of electrons and phonons in a simplified model of a metal. By employing the second quantization representation for the creation and annihilation operators of the electrons and the phonons, an hierarchy of equations for the distribution functions and correlation functions is introduced. This hierarchy is studied, using an approach originally developed by Bogoliubov, where both truncation of the hierarchy and irreversibility are achieved under general assumptions. A set of kinetic equations is obtained for an homogeneous system, where the electrons dynamically shield both each other and the phonons.
Contents: I. Introduction. II. Basic formulation: Difference scheme.
Individual time-step algorithm. III. Ahmad-Cohen scheme: Principles and
procedures. Standard algorithm. Parameters and timing tests. IV.
Comoving coordinates: Basic principles. Implementation of the AC method.
Mergers and inelastic effects. V. Planetary perturbations and
collisions: Grid method formulation. Collision algorithm and model
parameters. VI. Two-body regularization: KS transformations and
equations of motion. Standard N-body treatment. Algorithm and
parameters. Regularized AC procedures. VII. Three-body regularization:
Isolated triple systems. General N-body algorithm. VIII. Star-cluster
simulations: Models of open clusters. Postcollapse evolution.
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A procedure for the evaluation of the path integrals for the two-particle statistical density matrix at low temperatures is described. It is applied to both helium and neon, the first being the paradigm of a quantum mechanical gas and the second exhibiting weaker, but still significant, quantum effects at temperatures above the triple point. The density independent part of the pair-correlation functions and the second virial coefficients are obtained from these quantities. Three- and many-particle quantum properties are examined by approximating the Slater sum by a product of Boltzmann factors, each factor containing an effective pair-potential defined from the results of the two-particle case. The radial distribution function, obtained from a Percus-Yevick equation using this effective potential, demonstrates the validity of using the effective potential approximation as a tool for studying quantum properties by giving satisfactory results for helium and good agreement with experiment for neon.
A simple form of multi-ion interaction has been constructed for the purpose of atomistic simulation of transition metals. The model energy consists of a bonding term, which is the square-root of a site density ρi, summed over atoms i, and a repulsive pairwise term of the form The site density ρi is defined as sum over neighbouring sites j of a cohesive potential (R ij). Both V and are assumed to be short-ranged and are parameterized to fit the lattice constant, cohesive energy and elastic moduli of the seven body-centred-cubic (b.c.c.) transition metals. The result is a simple model which, unlike a pair-potential model, can account for experimental vacancy-formation energies and does not require an externally applied pressure to balance the “Cauchy pressure”.
An accurate and efficient algorithm for dynamics simulations of particles with attractive 1/r singular potentials is introduced. The method is applied to semiclassical dynamics simulations of electron–proton scattering processes in the Wigner-transform time-dependent picture, showing excellent agreement with full quantum dynamics calculations. Rather than avoiding the singularity problem by using a pseudopotential, the algorithm predicts the outcome of close-encounter two-body collisions for the true 1/r potential by solving the Kepler problem analytically and corrects the trajectory for multiscattering with other particles in the system by using standard numerical techniques (e.g., velocity Verlet, or Gear Predictor corrector algorithms). The resulting integration is time-reversal symmetric and can be applied to the general multibody dynamics problem featuring close encounters as occur in electron–ion scattering events, in particle–antiparticle dynamics, as well as in classical simulations of charged interstellar gas dynamics and gravitational celestial mechanics.
The time-dependent variational principle associates to a Hamiltonian quantum system a set of trajectories running on a classical phase space with canonical equations of motion. When the quantum observables generate a Lie group G and the states are taken as functions on an appropriate homogeneous quotient space space := G/G0 under under G, they can be equipped with a classical Poisson bracket which reproduces the commutators of the Lie group. The quantum system maps into a classical system whose equations of motion are governed by the expectation value of the quantum Hamiltonian H. We consider examples of this construction and show that the analysis of generalized classical systems provides insight into quantum many-body dynamics like chemical reactions.