Screened activity expansion for the grand potential of a quantum plasma and how to derive approximate equations of state compatible with electroneutrality.

We consider a quantum multicomponent plasma made with S species of point charged particles interacting via the Coulomb potential. We derive the screened activity series for the pressure in the grand-canonical ensemble within the Feynman-Kac path integral representation of the system in terms of a classical gas of loops. This series is useful for computing equations of state for it is nonperturbative with respect to the strength of the interaction and it involves relatively few diagrams at a given order. The known screened activity series for the particle densities can be recovered by differentiation. The particle densities satisfy local charge neutrality because of a Debye-dressing mechanism of the diagrams in these series. We introduce a new general neutralization prescription, based on this mechanism, for deriving approximate equations of state where consistency with electroneutrality is automatically ensured. This prescription is compared to other ones, including a neutralization scheme inspired by the Lieb-Lebowitz theorem and based on the introduction of (S-1) suitable independent combinations of the activities. Eventually, we briefly argue how the activity series for the pressure, combined with the Debye-dressing prescription, can be used for deriving approximate equations of state at moderate densities, which include the contributions of recombined entities made with three or more particles.

Comment on "Direct linear term in the equation of state of plasmas".

In a recent paper [Phys. Rev. E 91, 013108 (2015)], Kraeft et al. criticize known exact results on the equation of state of quantum plasmas, which have been obtained independently by several authors. They argue about a difference in the definition of the direct two-body function Q(x), which appears in virial expansions of thermodynamical quantities, but Q(x) is not a measurable quantity in itself. Differences in definitions of intermediate quantities are irrelevant, and only differences in physical quantities are meaningful. Beyond Kraeft et al.'s broad statement that there is no agreement at order ρ(5/2) in the virial equation for the pressure, we show that their published results for this quantity are in fact in perfect agreement with previous existing expressions.

Quantum partition functions of composite particles in a hydrogen-helium plasma via path integral Monte Carlo.

We compute two- and three-body cluster functions that describe contributions of composite entities, like hydrogen atoms, ions H(-), H2(+), and helium atoms, and also charge-charge and atom-charge interactions, to the equation of state of a hydrogen-helium mixture at low density. A cluster function has the structure of a truncated virial coefficient and behaves, at low temperatures, like a usual partition function for the composite entity. Our path integral Monte Carlo calculations use importance sampling to sample efficiently the cluster partition functions even at low temperatures where bound state contributions dominate. We also employ a new and efficient adaptive discretization scheme that allows one not only to eliminate Coulomb divergencies in discretized path integrals, but also to direct the computational effort where particles are close and thus strongly interacting. The numerical results for the two-body function agree with the analytically known quantum second virial coefficient. The three-body cluster functions are compared at low temperatures with familiar partition functions for composite entities.

Communication: on the origin of the surface term in the Ewald formula.

A transparent derivation of the Ewald formula for the electrostatic energy of a periodic three-dimensional system of point charges is presented. The problem of the conditional convergence of the lattice sum is dealt with by separating off, in a physically natural and mathematically simple way, long-range non-absolutely integrable contributions in the series. The general expression, for any summation order, of the surface (or dipole) term emerges very directly from those long-range contributions.

How to Convert SPME to P3M: Influence Functions and Error Estimates.

We demonstrate explicitly how the two seemingly different particle mesh Ewald methods, the smooth particle mesh Ewald (SPME) and the particle-particle particle mesh (P3M), can be mathematically transformed into each other. This allows us in particular to convert the error estimate of the P3M method in the energy-conserving scheme (also known as "P3M with analytic differentiation") into an error estimate for the SPME method, via a simple change of the lattice Green function. Our error estimate is valid for any values of the SPME parameters (mesh size, spline interpolation order, Ewald splitting parameter, real-space cutoff distance), including odd orders of splines. The problem with the self-forces is avoided thanks to an analytical formula that allows to subtract them directly within the particle mesh calculation. Plots of the accuracy of the SPME forces are provided for a wide range of parameter values. The main use of the error estimate is to allow a simulation program to scan quickly the multidimensional parameter space to find the best set of parameters to achieve a target accuracy at the smallest computational cost. As a byproduct, we show how a SPME code can be transformed into a P3M version by changing a few lines of code. We demonstrate also that the P3M lattice Green function can be approximated by a closed form expression, computable on-the-fly, that provides essentially the same accuracy as the full function.

Thermodynamics of atomic and ionized hydrogen: analytical results versus equation-of-state tables and Monte Carlo data.

We compute thermodynamical properties of a low-density hydrogen gas within the physical picture, in which the system is described as a quantum electron-proton plasma interacting via the Coulomb potential. Our calculations are done using the exact scaled low-temperature (SLT) expansion, which provides a rigorous extension of the well-known virial expansion-valid in the fully ionized phase-into the Saha regime where the system is partially or fully recombined into hydrogen atoms. After recalling the SLT expansion of the pressure [A. Alastuey et al., J. Stat. Phys. 130, 1119 (2008)], we obtain the SLT expansions of the chemical potential and of the internal energy, up to order exp(-|E_{H}|/kT) included (E_{H}≃-13.6 eV). Those truncated expansions describe the first five nonideal corrections to the ideal Saha law. They account exactly, up to the considered order, for all effects of interactions and thermal excitations, including the formation of bound states (atom H, ions H^{-} and H_{2}^{+}, molecule H_{2},⋯) and atom-charge and atom-atom interactions. Among the five leading corrections, three are easy to evaluate, while the remaining ones involve well-defined internal partition functions for the molecule H_{2} and ions H^{-} and H_{2}^{+}, for which no closed-form analytical formula exist currently. We provide accurate low-temperature approximations for those partition functions by using known values of rotational and vibrational energies. We compare then the predictions of the SLT expansion, for the pressure and the internal energy, with, on the one hand, the equation-of-state tables obtained within the opacity program at Livermore (OPAL) and, on the other hand, data of path integral quantum Monte Carlo (PIMC) simulations. In general, a good agreement is found. At low densities, the simple analytical SLT formulas reproduce the values of the OPAL tables up to the last digit in a large range of temperatures, while at higher densities (ρ∼10^{-2} g/cm^{3}), some discrepancies among the SLT, OPAL, and PIMC results are observed.

Particle-particle particle-mesh method for dipolar interactions: on error estimates and efficiency of schemes with analytical differentiation and mesh interlacing.

The interlaced and non-interlaced versions of the dipolar particle-particle particle-mesh (P(3)M) method implemented using the analytic differentiation scheme (AD-P(3)M) are presented together with their respective error estimates for the calculation of the forces, torques, and energies. Expressions for the optimized lattice Green functions, and for the Madelung self-forces, self-torques and self-energies are given. The applicability of the theoretical error estimates are thoroughly tested and confirmed in several numerical examples. Our results show that the accuracy of the calculations can be improved substantially when the approximate (mesh computed) Madelung self-interactions are subtracted. Furthermore, we show that the interlaced dipolar AD-P(3)M method delivers a significantly higher accuracy (which corresponds approximately to using a twice finer mesh) than the conventional method, allowing thereby to reduce the mesh size with respect to the non-interlaced version for a given accuracy. In addition, we present similar expressions for the dipolar ik-differentiation interlaced scheme, and we perform a comparison with the AD interlaced scheme. Rough tests for the relative speed of the dipolar P(3)M method using ik-differentiation and the interlaced/non-interlaced AD schemes show that when FFT computing time is the bottleneck, usually when working at high precisions, the interlaced AD-scheme can be several times faster than the other two schemes. For calculations with a low accuracy requirement, the interlaced version can perform worse than the ik and the non-interlaced AD schemes.

Behavior of bulky ferrofluids in the diluted low-coupling regime: theory and simulation.

A theoretical formalism to predict the structure factors observed in dipolar soft-sphere fluids based on a virial expansion of the radial distribution function is presented. The theory is able to account for cases with and without externally applied magnetic fields. A thorough comparison of the theoretical results to molecular-dynamics simulations shows a good agreement between theory and numerical simulations when the fraction of particles involved in clustering is low; i.e., the dipolar coupling parameter is lambda less, similar 2, and the volume fraction is phi less, similar 0.25. When magnetic fields are applied to the system, special attention is paid to the study of the anisotropy of the structure factor. The theory reasonably accounts for the structure factors when the Langevin parameter is smaller than 5.

Simulations of non-neutral slab systems with long-range electrostatic interactions in two-dimensional periodic boundary conditions.

We introduce a regularization procedure to define electrostatic energies and forces in a slab system of thickness h that is periodic in two dimensions and carries a net charge. The regularization corresponds to a neutralization of the system by two charged walls and can be viewed as the extension to the two-dimensional (2D)+h geometry of the neutralization by a homogeneous background in the standard three-dimensional Ewald method. The energies and forces can be computed efficiently by using advanced methods for systems with 2D periodicity, such as MMM2D or P3M/ELC, or by introducing a simple background-charge correction to the Yeh-Berkowitz approach of slab systems. The results are checked against direct lattice sum calculations on simple systems. We show, in particular, that the Madelung energy of a 2D square charge lattice in a uniform compensating background is correctly reproduced to high accuracy. A molecular dynamics simulation of a sodium ion close to an air/water interface is performed to demonstrate that the method does indeed provide consistent long-range electrostatics. The mean force on the ion reduces at large distances to the image-charge interaction predicted by macroscopic electrostatics. This result is used to determine precisely the position of the macroscopic dielectric interface with respect to the true molecular surface.

P3M algorithm for dipolar interactions.

An extension to the P(3)M algorithm for electrostatic interactions is presented that allows to efficiently compute dipolar interactions in periodic boundary conditions. Theoretical estimates for the root-mean-square error of the forces, torques, and the energy are derived. The applicability of the estimates is tested and confirmed in several numerical examples. A comparison of the computational performance of the new algorithm to a standard dipolar-Ewald summation methods shows a performance crossover from the Ewald method to the dipolar P(3)M method for as few as 300 dipolar particles. In larger systems, the new algorithm represents a substantial improvement in performance with respect to the dipolar standard Ewald method. Finally, a test comparing point-dipole-based and charged-pair based models shows that point-dipole-based models exhibit a better performance than charged-pair based models.

The optimal P3M algorithm for computing electrostatic energies in periodic systems.

We optimize Hockney and Eastwood's particle-particle particle-mesh algorithm to achieve maximal accuracy in the electrostatic energies (instead of forces) in three-dimensional periodic charged systems. To this end we construct an optimal influence function that minimizes the root-mean-square (rms) errors of the energies. As a by-product we derive a new real-space cutoff correction term, give a transparent derivation of the systematic errors in terms of Madelung energies, and provide an accurate analytical estimate for the rms error of the energies. This error estimate is a useful indicator of the accuracy of the computed energies and allows an easy and precise determination of the optimal values of the various parameters in the algorithm (Ewald splitting parameter, mesh size, and charge assignment order).

Dielectric permittivity profiles of confined polar fluids.

The dielectric response of a simple model of a polar fluid near neutral interfaces is examined by a combination of linear response theory and extensive molecular dynamics simulations. Fluctuation expressions for a local permittivity tensor epsilon(r) are derived for planar and spherical geometries, based on the assumption of a purely local relationship between polarization and electric field. While the longitudinal component of epsilon exhibits strong oscillations on the molecular scale near interfaces, the transverse component becomes ill defined and unphysical, indicating nonlocality in the dielectric response. Both components go over to the correct bulk permittivity beyond a few molecular diameters. Upon approaching interfaces from the bulk, the permittivity tends to increase, rather than decrease as commonly assumed, and this behavior is confirmed for a simple model of water near a hydrophobic surface. An unexpected finding of the present analysis is the formation of "electrostatic double layers" signaled by a dramatic overscreening of an externally applied field inside the polar fluid close to an interface. The local electric field is of opposite sign to the external field and of significantly larger amplitude within the first layer of polar molecules.