Classical One-component Plasma Research Articles

Variational principles for the hydrodynamics of the classical one-component plasma

Hydrodynamic equations for a one-component plasma are derived as a unification of the Euler equations with long-range Coulomb interaction. By using a variational principle, these equations self-consistently unify thermodynamics, dispersion laws, nonlinear motion, and conservation laws. In the moderate and strong coupling limits, it is argued that these equations work down to the length scale of the interparticle spacing. The use of a variational principle also ensures that closure is achieved self-consistently. Hydrodynamic equations are evaluated in both the Eulerian frame, where the fluid variables depend on the position in the laboratory, and the Lagrangian frame, where they depend on the position in some reference state, such as the initial position. Each frame has its advantages and our final theory combines elements of both. The properties of longitudinal and transverse dispersion laws are calculated for the hydrodynamic equations. A simple step function approximation for the pair distribution function enables simple calculations that reveal the structure of the equations of motion. The obtained dispersion laws are compared to molecular dynamics simulations and the theory of quasilocalized charge approximation. The action, which gives excellent agreement for both longitudinal and transverse dispersion laws for a wide range of coupling strengths, is elucidated. Agreement with numerical experiments shows that such a hydrodynamic approach can be used to accurately describe a one-component plasma at very small length scales comparable to the average interparticle spacing. The validity of this approach suggests considering nonlinear flows and other systems with long-range interactions in the future.

Amplitude modulation and surface wave generation in a complex plasma monolayer.

The response of a two-dimensional plasma crystal to an externally imposed initial perturbation has been explored using molecular dynamics (MD) simulations. A two-dimensional (2D) monolayer of micron-sized charged particles (dust) is formed in the plasma environment under certain conditions. The particles interacting via Yukawa pair potential are confined in the vertical (z[over ̂]) direction by an external parabolic confinement potential, which mimics the combined effect of gravity and the sheath electric field typically present in laboratory dusty plasma experiments. An external perturbation is introduced in the medium by displacing a small central region of particles in the vertical direction. The displaced particles start to oscillate in the vertical direction, and their dynamics get modulated through a parametric decay process generating beats. It has also been shown that the same motion is excited in the dynamics of unperturbed particles. A simple theoretical model is provided to understand the origin of the beat motions of particles. Additionally, in our simulations, concentric circular wavefronts propagating radially outward are observed on the surface of the monolayer. The physical mechanism and parametric dependence of the observed phenomena are discussed in detail. This research sheds light on the medium's ability to exhibit macroscopic softness, a pivotal characteristic of soft matter, while sustaining surface wave modes. Our findings are also relevant to other strongly coupled systems, such as colloids and classical one-component plasmas.

Collective ion dynamics in Coulomb one-component plasmas within the self-consistent relaxation theory.

In this paper, we present the theoretical formalism describing the collective ion dynamics of the nonideal Coulomb classical one-component plasmas on the basis of the self-consistent relaxation theory. The theory is adapted to account for correlations between the frequency relaxation parameters that characterize the three- and four-particle dynamics and the parameters associated with the two-particle dynamics. The dynamic structure factor spectra and dispersion characteristics calculated for a wide range of wave numbers are in agreement with the molecular dynamics simulation data and the results obtained with the theory of the frequency moments. The proposed formalism reproduces all the features inherent to the Coulomb one-component plasmas and requires only knowledge of the coupling parameter and the information about the structure.

Classical bridge functions in classical and quantum plasma liquids

Bridge functions, the missing link in the exact description of strong correlations, are indirectly extracted from specially designed molecular dynamics simulations of classical one-component plasma liquids and accurately parameterized. Their incorporation into an advanced integral equation theory description of Yukawa one-component plasma liquids and a novel dielectric formalism scheme for quantum one-component plasma liquids lead to an unprecedented agreement with available molecular dynamics simulations and new ab initio path integral Monte Carlo simulations, respectively.

Self-consistent relaxation theory of collective ion dynamics in Yukawa one-component plasmas under intermediate screening regimes.

The self-consistent relaxation theory is employed to describe the collective ion dynamics in strongly coupled Yukawa classical one-component plasmas. The theory is applied to equilibrium states corresponding to intermediate screening regimes with appropriate values of the structure and coupling parameters. The information about the structure (the radial distribution function and the static structure factor) and the thermodynamics of the system are sufficient to describe collective dynamics over a wide range of spatial scales, namely, from the extended hydrodynamic to the microscopic dynamics scale. The main experimentally measurable characteristics of the equilibrium collective dynamics of ions-the spectrum of the dynamic structure factor, the dispersion parameters, the speed of sound, and the sound attenuation-are determined within the framework of the theory without using any adjustable parameters. The results demonstrate agreement with molecular dynamics simulations. Thus a direct realization is presented of the key idea of statistical mechanics: for the theoretical description of the collective particle dynamics in equilibrium fluids it is sufficient to know the interparticle interaction potential and the structural characteristics. Comparison with alternative or complementary theoretical approaches is provided.

Bridge functions of classical one-component plasmas.

In a recent paper, Lucco Castello etal. [arXiv:2107.03537] performed systematic extractions of classical one-component plasma bridge functions from molecular dynamics simulations and provided an accurate parametrization that was incorporated in their isomorph-based empirically modified hypernetted chain approach for Yukawa one-component plasmas. Here the extraction technique and parametrization strategy are described in detail, while the deficiencies of earlier efforts are discussed. The structural and thermodynamic predictions of the updated version of the integral equationtheory approach are compared with extensive available simulation results revealing a truly unprecedented level of accuracy in the entire dense liquid region of the Yukawa phase diagram.

Static properties of a warm dense uniform electron gas

We show how the static dielectric function and other static characteristics of dense warm charged Fermi liquids can be obtained exclusively from the system static structure factor. The non-perturbative self-consistent method of moments is employed to extend onto quantum fluids, a similar reduction stemming from the fluctuation-dissipation theorem and other exact relations for classical one-component plasmas. The results are compared to and complement the numerical data obtained recently by the path-integral Monte Carlo method. Alternative theoretical approaches are discussed and employed as well.

Integral equation theory based dielectric scheme for strongly coupled electron liquids.

In a recent paper, Lucco Castello et al. (arXiv:2107.03537) provided an accurate parameterization of classical one-component plasma bridge functions that was embedded in a novel dielectric scheme for strongly coupled electron liquids. Here, this approach is rigorously formulated, its set of equations is formally derived, and its numerical algorithm is scrutinized. A systematic comparison with available and new path integral Monte Carlo simulations reveals a rather unprecedented agreement especially in terms of the interaction energy and the long wavelength limit of the static local field correction.

Effects of Coulomb coupling on stopping power and a link to macroscopic transport

Molecular dynamics simulations are used to assess the influence of Coulomb coupling on the energy evolution of charged projectiles in the classical one-component plasma. The average projectile kinetic energy is found to decrease linearly with time when να/ωp ≲ 10−2, where να is the Coulomb collision frequency between the projectile and the medium, and ωp is the plasma frequency. Stopping power is obtained from the slope of this curve. In comparison to the weakly coupled limit, strong Coulomb coupling causes the magnitude of the dimensionless stopping power, (a/kBT)dE/dx, to increase, the Bragg peak to shift to several times the plasma thermal speed, and for the stopping power curve to broaden substantially. The rate of change of the total projectile kinetic energy averaged over many independent simulations is shown to consist of two measurable components: a component associated with a one-dimensional friction force and a thermal energy exchange rate. In the limit of a slow and massive projectile, these can be related to the macroscopic transport rates of self-diffusion and temperature relaxation in the plasma. Simulation results are compared with available theoretical models for stopping power, self-diffusion, and temperature relaxation.

Fourth Moment of the Charge Density Induced Around a Guest Charge in Two-Dimensional Jellium

The model under consideration is the classical two-dimensional one-component plasma (jellium) of pointlike particles with charge $e$, interacting pairwisely via the logarithmic Coulomb potential and immersed in a uniform neutralizing background charge density. The system is in thermal equilibrium at the inverse temperature $\beta$, its thermodynamics depends only on the coupling constant $\Gamma=\beta e^2$. We put into an infinite (homogeneous and translationally invariant) plasma a guest particle of charge $Ze$ with $Z>-2/\Gamma$ in order to prevent from the collapse of the jellium charges onto it. The guest particle induces a screening cloud (the excess charge density) in the plasma. The zeroth and second moments of this screening cloud were derived previously for any fluid value of $\Gamma$. In this paper, we propose a formula for the fourth moment of the screening cloud. The derivation is based on the assumption that the fourth moment is, similarly as the second moment, analytic in $Z$ around $Z=0$. An exact treatment of the limit $Z\to\infty$ shows that it is a finite (cube) polynomial in $Z$. The $\Gamma$-dependence of the polynomial coefficients is determined uniquely by considering the limits $Z\to 0$ and $Z\to\infty$, and the compressibility sum rule for $Z=1$. The formula for the fourth moment of screening cloud is checked in the leading and first correction orders of the Debye-H\"uckel limit $\Gamma\to 0$ and at the exactly solvable free-fermion point $\Gamma=2$. Sufficient conditions for sign oscillations of the induced charge density which follow from the second-moment and fourth-moment sum rules are discussed.

Higher-order fluctuation-dissipation relations in plasma physics: Binary Coulomb systems.

A recent approach that led to compact frequency domain formulations of the cubic and quartic fluctuation-dissipation theorems (FDTs) for the classical one-component plasma (OCP) [Golden and Heath, J. Stat. Phys. 162, 199 (2016)10.1007/s10955-015-1395-6] is generalized to accommodate binary ionic mixtures. Paralleling the procedure followed for the OCP, the basic premise underlying the present approach is that a (k,ω) 4-vector rotational symmetry, known to be a pivotal feature in the frequency domain architectures of the linear and quadratic fluctuation-dissipation relations for a variety of Coulomb plasmas [Golden et al., J. Stat. Phys. 6, 87 (1972)10.1007/BF01023681; J. Stat. Phys. 29, 281 (1982)10.1007/BF01020787; Golden, Phys. Rev. E 59, 228 (1999)10.1103/PhysRevE.59.228], is expected to be a pivotal feature of the frequency domain architectures of the higher-order members of the FDT hierarchy. On this premise, each member, in its most tractable form, connects a single (p+1)-point dynamical structure function to a linear combination of (p+1)-order p density response functions; by definition, such a combination must also remain invariant under rotation of their (k_{1},ω_{1}),(k_{2},ω_{2}),...,(k_{p},ω_{p}), (k_{1}+k_{2}+⋯+k_{p},ω_{1}+ω_{2}+⋯+ω_{p}) 4-vector arguments. Assigned to each 4-vector is a species index that corotates in lock step. Consistency is assured by matching the static limits of the resulting frequency domain cubic and quartic FDTs to their exact static counterparts independently derived in the present work via a conventional time-independent perturbation expansion of the Liouville distribution function in its macrocanonical form. The proposed procedure entirely circumvents the daunting issues of entangled Liouville space paths and nested Poisson brackets that one would encounter if one attempted to use the conventional time-dependent perturbation-theoretic Kubo approach to establish the frequency domain FDTs beyond quadratic order.

New developments in the application of the method of moments in Plasma Physics

This review article is based on a number of our research papers complemented by some mathematicaldevelopments which are usually not included to texts in Physics, and which can permit a reader to enterinto the details of the self-consistent method of moments, recently suggested, and understand how it couldbe improved even further. The idea of the method of moments which appeared some 35 years ago is toemploy several sum rules and other exact relations to determine the dynamic properties of strongly coupledclassical or partially degenerate plasmas. Now this approach is complemented by new empirical andmathematical observations which permit to determine dynamic characteristics of strongly coupledcompletely ionized classical one-component plasmas without any data input from simulations or directexperiments and express the dynamic properties of the above systems entirely in terms of their staticcharacteristics like the static structure factors. The obtained results are quite satisfactory and promising.

Enhancement of nuclear reaction rates in asymmetric binary ionic mixtures

Using orbital‐free molecular dynamics simulations we study the structure and dynamics of increasingly asymmetric mixtures such as hydrogen–carbon, hydrogen–aluminium, hydrogen–copper, and hydrogen–silver. We show that, whereas the heavy component structure is close to an effective one‐component plasma (OCP), the light component appears more structured than the corresponding OCP. This effect is related to the crossover towards a Lorentz‐type diffusion triggered by strongly coupled, highly charged heavy ions, and witnessed by the change of temperature scaling laws of diffusion. This over‐correlation translates into an enhancement of nuclear reaction rates much higher than its classical OCP counterpart.

Local Density for Two-Dimensional One-Component Plasma

We study the classical two-dimensional one-component plasma of $N$ positively charged point particles, interacting via the Coulomb potential and confined by an external potential. For the specific inverse temperature $\beta=1$ (in our normalization), the charges are the eigenvalues of random normal matrices, and the model is exactly solvable as a determinantal point process. For any positive temperature, using a multiscale scheme of iterated mean-field bounds, we prove that the equilibrium measure provides the local particle density down to the optimal scale of $N^{o(1)}$ particles. Using this result and the loop equation, we further prove that the particle configurations are rigid, in the sense that the fluctuations of smooth linear statistics on any scale are $N^{o(1)}$.

Direct Determination of Dynamic Properties of Coulomb and Yukawa Classical One-Component Plasmas.

Dynamic characteristics of strongly coupled classical one-component Coulomb and Yukawa plasmas are obtained within the nonperturbative model-free moment approach without any data input from simulations so that the dynamic structure factor (DSF) satisfies the first three nonvanishing sum rules automatically. The DSF, dispersion, decay, sound speed, and other characteristics of the collective modes are determined using exclusively the static structure factor calculated from various theoretical approaches including the hypernetted chain approximation. A good quantitative agreement with molecular dynamics simulation data is achieved.

Generalized Nonlinear Fluctuation-Dissipation Relation for the One-Component Plasma

We present a novel procedure leading to compact and tractable cubic \((p=3)\) and quartic \((p=4)\) scalar fluctuation-dissipation relations for the classical one-component plasma subjected to a weak external Coulombic perturbation; there is no external magnetic field. The ultimate goal here is to establish a generalized nonlinear fluctuation-dissipation relation in which a single \((p+1)\)-point dynamical structure function is expressed as a linear combination of pth-order external density response functions. When cast in this form, such an architecture, by definition, is invariant with respect to rotation on the (\(p+1\))-sided polygon formed by the wave vector-frequency arguments of the dynamical structure function. The application of this rotation symmetry to suppress unphysical isolated singularities is a basic ingredient of the derivation.

Coupling strength in Coulomb and Yukawa one-component plasmas

In a non-ideal classical Coulomb one-component plasma (OCP), all thermodynamic properties are known to depend only on a single parameter—the coupling parameter Γ. In contrast, if the pair interaction is screened by background charges (Yukawa OCP) the thermodynamic state depends, in addition, on the range of the interaction via the screening parameter κ. How to determine in this case an effective coupling parameter has been a matter of intensive debate. Here we propose a consistent approach for defining and measuring the coupling strength in Coulomb and Yukawa OCPs based on a fundamental structural quantity, the radial pair distribution function (RPDF). The RPDF is often accessible in experiments by direct observation or indirectly through the static structure factor. Alternatively, it is directly computed in theoretical models or simulations. Our approach is based on the observation that the build-up of correlation from a weakly coupled system proceeds in two steps: First, a monotonically increasing volume around each particle becomes devoid of other particles (correlation hole), and second (upon further increase of the coupling), a shell structure emerges around each particle giving rise to growing peaks of the RPDF. Using molecular dynamics simulation, we present a systematic study for the dependence of these features of the RPDF on Γ and κ and derive a simple expression for the effective coupling parameter.

Comparative merits of the memory function and dynamic local-field correction of the classical one-component plasma

The complementarity of the liquid and plasma descriptions of the classical one-component plasma is explored by studying wave number and frequency dependent dynamical quantities: the dynamical structure factor (DSF) and the dynamic local field correction (LFC). Accurate molecular dynamics (MD) simulations are used to validate and test models of the DSF and LFC. Our simulations, which span the entire fluid regime (Γ=0.1-175), show that the DSF is very well represented by a simple and well known memory function model of generalized hydrodynamics. On the other hand, the LFC, which we have computed using MD for the first time, is not well described by existing models.

Effective pair potentials using anab initiovariational approach

We used a variational approach adapted to a quantum molecular-dynamics code to determine the best reference potential for warm dense aluminum. This ab initio variational approach was based on the Gibbs-Bogolyubov inequality. We used many-body reference systems interacting through inverse-power-law potentials, among which the Coulomb potential was a particular case defining the classical one-component plasma model. By comparisons with full quantum molecular-dynamics simulations, we found that the Coulomb potential was not always the best reference potential. We calculated the self-diffusion coefficient and the shear viscosity and discussed the results obtained using the Chisolm-Wallace relation in the warm dense matter regime.

One-component plasma of point charges and of charged rods

An approximate theory is developed to describe the properties of mobile particles with extended charge distributions in the presence of a neutralizing fixed background charge. Long-wavelength fluctuations of the electric potential are handled within a variational perturbation approximation, and the short-wavelength fluctuations are handled within a cumulant (fugacity) expansion. The distinct treatment of these two contributions to the free energy enables the theory to provide quantitative predictions for the properties of these systems from the weak- to the strong-coupling regimes. With this theory, we study three different variations in the classical one-component plasma model: a plasma of point charges, a plasma of particles consisting of 8 linearly bonded point charges (8-mer), and a plasma of line charges. The theory was found to agree well with the available computer simulation data for the electrostatic interaction energy of these systems for all values of the plasma coupling parameter examined ( Gamma=0 to 400). In addition, we find that both the 8-mer rod and the line charge systems form a strongly ordered nematic phase, which is entirely driven by electrostatic interactions. The nematic phase only exists within a finite range of lengths of the charged particles. If the particles are too short or too long, the nematic phase does not appear. Finally, we find that the nematic phase is stable over a broader range of conditions for the line charge system than for the 8-mer rod system; consequently, the phase behavior of the one-component plasma is sensitive to the manner in which the charge is distributed on the particles.