Vapour-liquid equilibrium of the charged Yukawa fluid from Gibbs ensemble Monte Carlo simulations and the mean spherical approximation

A Gibbs ensemble Monte Carlo (GEMC) study supplemented by theoretical calculations using the mean spherical approximation (MSA) is reported for the charged Yukawa system. In this system the particles interact via an attractive Yukawa potential and a Coulomb potential. When the Coulomb potential is weak compared to the Yukawa attraction the results obtained for the vapour-liquid equilibrium from the GEMC and the MSA are very similar. On increasing the charge, the system becomes similar to the primitive model of electrolytes, and at high charges similar phenomena, such as ion association, are observed. At these charges, the MSA yields poorer results. Using both methods, an increase of the charge results in an increase of the critical temperature but a decrease of the critical density.     

Universality in some classical Coulomb systems of restricted dimension

Coulomb systems in which the particles interact through thed-dimensional Coulomb potential but are confined in a flat manifold of dimensiond–1 are considered. The actual Coulomb potential acting is defined by particular boundary conditions involving a characteristic macroscopic distanceW in the direction perpendicular to the manifold: either it is periodic of periodW in that direction, or it vanishes on one ideal conductor wall parallel to the manifold at a distanceW from it, or it vanishes on two parallel walls at a distanceW from each other with the manifold equidistant from them. Under the assumptions that classical equilibrium statistical mechanics is applicable and that the system has the macroscopic properties of a conductor, it is shown that the suitably smoothed charge correlation function is universal, and that the free energy and the grand potential have universal dependences onW (universal means independent of the microscopic detail). The casesd=2 are discussed in detail, and the generic results are checked on an exactly solvable model. The cased=3 of a plane parallel to an ideal conductor is also explicitly worked out.Laboratoire associé au Centre National de la Recherche Scientifique-URA D0063.     

Charging energy and stability of charged metallic particles

The stability of a charged metallic particle is investigated within the classical theory. The minimal number of atoms for which a charged spherical particle is stable against the spheroidal deformation, which relaxes the energy cost due to the Coulomb repulsion between extra charges, is derived. It is suggested that, for negatively charged particles, an extra electron virtually bound to particle is emitted before the fragmentation due to the Coulomb repulsion between extra charges will occur.     

Two-dimensional models for an electrode with adsorption sites

Through a new method, the following model is solved exactly in the framework of classical equilibrium statistical mechanics of two-dimensional Coulomb systems, for the special value=2 of the coupling constant: the mobile charges of a one-component plasma are attracted by a line of equidistant sticky adsorption sites embedded in a background, the density of which varies in the direction orthogonal to the line. First the general expressions are given for the densities and correlation functions of nonadsorbed and adsorbed particles. Then these results are used to investigate two models of electrodes with localized adsorption: the externally charged hard wall and the impermeable polarized membrane. In each case the influence of the adsorption upon macroscopic features is studied: the potential drop across the interface, the contact theorem, and the Lippmann equation, which involves the surface free energy.     

General Considerations on the Finite-Size Corrections for Coulomb Systems in the Debye--Hückel Regime

We study the statistical mechanics of classical Coulomb systems in a low coupling regime (Debye--Hückel regime) in a confined geometry with Dirichlet boundary conditions for the electric potential. We use a method recently developed by the authors which relates the grand partition function of a Coulomb system in a confined geometry with a certain regularization of the determinant of the Laplacian on that geometry with Dirichlet boundary conditions. We study several examples of fully confining geometry in two and three dimensions and semi-confined geometries where the system is confined only in one or two directions of the space. We also generalize the method to study systems confined in arbitrary geometries with smooth boundary. We find a relation between the expansion for small argument of the heat kernel of the Laplacian and the large-size expansion of the grand potential of the Coulomb system. This allow us to find the finite-size expansion of the grand potential of the system in general. We recover known results for the bulk grand potential (in two and three dimensions) and the surface tension (for two-dimensional systems). We find the surface tension for three-dimensional systems. For two-dimensional systems our general calculation of the finite-size expansion gives a proof of the existence a universal logarithmic finite-size correction predicted some time ago, at least in the low coupling regime. For three-dimensional systems we obtain a prediction for the curvature correction to the grand potential of a confined system.     

Melting of Coulomb-interacting classical particles in 2D irregular traps

The “melting” of self-formed rigid structures made of a small number of interactingclassical particles confined in an irregular two-dimensional space is investigated usingMonte Carlo simulations. It is shown that the interplay of long-range Coulomb repulsionsbetween these particles and the irregular confinement yields a solid-like phase at lowtemperatures that possesses a bond-orientational order, however, the positional order isdepleted even at the lowest temperatures. Upon including thermal fluctuations, thissolid-like phase smoothly crosses over to a liquid-like phase by destroying thebond-orientational order. The collapse of solidity is shown to be defect mediated, andaided by the proliferation of free disclinations. The behaviour of different physicalobservables across the crossover region is studied. These results will help us quantifyingthe melting found in experiments on systems with confined geometries.     

States of one-dimensional Coulomb systems as simple examples of θ vacua and confinement

The one-dimensional Coulomb system is known to have equilibrium states with nonvanishing electric field. These states are shown here to be analogous, and related, to the vacua which have been discussed for gauge theories in two or more space-time dimensions. The system exhibits confinement of fractional charges, which we dicuss with the purpose of offering a simple example of the-vacua phenomenology. Precise relations and connections between one-dimensional Coulomb gases and two-dimensional Abelian gauge theories, and quantum-mechanical matter systems, are discussed.Supported in part by National Science Foundation Grant PHY-2825390 A01.     

Liouville and Fokker–Planck dynamics for classical plasmas and radiation

We consider a nonequilibrium statistical system formed by many classical non‐relativistic particles of opposite electric charges (plasma) and by the classical dynamical electromagnetic (EM) field. The charges interact with one another directly through instantaneous Coulomb potentials and with the dynamical degrees of freedom of the transverse EM field. The system may also be subject to external influences of: i) either static, but spatially inhomogeneous, electric and magnetic fields (case 1)), or ii) weak distributions of electric charges and currents (case 2)). The particles and the dynamical EM field are described, for any time t > 0, by the classical phase‐space probability distribution functional (CPSPDF) f and, at the initial time (t = 0), by the initial CPSPDF f_in. The CPSPDF f and f_in, multiplied by suitable Hermite polynomials (for particles and field) and integrated over all canonical momenta, yield new moments. The Liouville equation and f_in imply a new nonequilibrium linear infinite hierarchy for the moments. In case 1), f_in describes local equilibrium but global nonequilibrium, and we propose a long‐time approximation in the hierarchy, which introduces irreversibility and relaxation towards global thermal equilibrium. In case 2), the statistical system, having been at global thermal equilibrium, without external influences, for t ≤ 0, is subject to weak external charge‐current distributions: then, new hierarchies for moments and their long‐time behaviours are discussed in outline. As examples, approximate mean‐field (Vlasov) approximations are treated for both cases 1) and 2).     

Electrostatic interaction between two conducting spheres

In the paper we consider the problem of the electrostatic interaction between two charged conducting spheres with arbitrary electrical charges and radiuses. Using the image charges method we determine exact analytical formulas for the force F and for the potential energy W of the interaction between these two spheres as well as for the potential V of the electromagnetic field in an arbitrary point created by them. Our formulas lead to Coulomb’s law for point charges.We theoretically prove the experimentally shown fact that two spheres with the same type (positive or negative) of charges can also attract each other.     

Structure, melting, and potential barriers in mesoscopic clusters of repulsive particles

This paper discusses two-dimensional mesoscopic clusters of particles that repel according to dipole, Coulomb, and logarithmic laws and are confined by an external parabolic potential. These models describe a number of physical systems, in particular, electrons in semiconductor structures or on a liquid-helium surface allowing for image forces, indirect excitons in coupled semi-conductor dots, and a small number of vortices in an island of a second-order superconductor or in superfluid helium. Two competing forms of ordering are detected in the particles in the mesoscopic clusters-the formation of a triangular lattice or of a shell structure. The temperature dependences of the potential energy, the mean-square radial and angular deviations, the radial and angular distributions of the particles, and the distribution of the particles over the local minima are studied. Melting in mesoscopic clusters occurs in two stages: at lower temperatures, there is orientation melting, from the frozen phase into a phase with rotational reorientation of “crystalline” shells with respect to each other; subsequently, a transition occurs in which the radial order disappears. Melting in dipole macroclusters occurs in a single stage. However, in Coulomb and logarithmic macroclusters, orientation melting occurs only for the outer pairs of shells. Orientation melting is also detected in three-dimensional Coulomb clusters. A connection is established between the character of the melting and the ratio of the energy barriers that describe the breakdown of the orientational and radial structure of a cluster. Zh. éksp. Teor. Fiz. 116, 2012–2037 (December 1999)     

Density Profiles in a Classical Coulomb Fluid Near a Dielectric Wall. II. Weak-Coupling Systematic Expansions

In the framework of the grand-canonical ensemble of statistical mechanics, we give an exact diagrammatic representation of the density profiles in a classical multicomponent plasma near a dielectric wall. By a reorganization of Mayer diagrams for the fugacity expansions of the densities, we exhibit how the long-range of both the self-energy and pair interaction are exponentially screened at large distances from the wall. However, the self-energy due to Coulomb interaction with images still diverges in the vicinity of the dielectric wall and the variation of the density is drastically different at short or large distances from the wall. This variation is involved in the inhomogeneous Debye–Hückel equation obeyed by the screened pair potential. Then the main difficulty lies in the determination of the latter potential at every distance. We solve this problem by devising a systematic expansion with respect to the ratio of the fundamental length scales involved in the two coulombic effects at stake. (The application of this method to a plasma confined between two ideally conducting plates and to a quantum plasma will be presented elsewhere). As a result we derive the exact analytical perturbative expressions for the density profiles up to first order in the coupling between charges. The mean-field approach displayed in Paper I is then justified.     

The Soft-Core Coulomb Potential in the Semi-Relativistic Two-Body Basis

We consider the soft-core Coulomb potential within a semi-relativistic two-body framework which arises from the spinless Salpeter equation after some approximations valid for heavy interacting particles. To provide an analytical solution, we propose a novel ansatz solution to the arising Schrödinger-like equation and thereby obtain the ground-state energy. Our results, for the special case of ordinary Coulomb potential, are in complete agreement with the corresponding exact analytical solution. The spectrum is numerically reported for typical values of the potential parameters.     

Positive and negative charged rods alternating along a line: Exact results

The Coulomb system consisting of an equal number of positive and negative charged rods confined to a line with the charges alternating in sign along the line is considered. By replacing the line with a lattice, one can calculate the grand partition function and correlations exactly for one value of the coupling constant. The exact solution exhibits features forbidden in the corresponding continuous system, in which each pair of oppositely charged rods also interact via a short-range repulsive potential, and there is no restriction on the ordering of the charges. The sum rule indicating the phase of the system is identified.     

Exact invariants for a class of three-dimensional time-dependent classical hamiltonians

An exact invariant is derived for three-dimensional Hamiltonian systems of N particles confined within a general velocity-independent potential. The invariant is found to contain a time-dependent function f(2)(t), embodying a solution of a third-order differential equation whose coefficients depend on the explicitly known trajectories of the particle ensemble. Our result is applied to a one-dimensional time-dependent nonlinear oscillator and to a system of Coulomb interacting particles in a time-dependent quadratic external potential.     

Confined Coulomb Systems with Adsorbing Boundaries: The Two-Dimensional Two-Component Plasma

Using a solvable model, the two-dimensional two-component plasma, we study a Coulomb gas confined in a disk and in an annulus with boundaries that can adsorb some of the negative particles of the system. We obtain explicit analytic expressions for the grand potential, the pressure and the density profiles of the system. By studying the behavior of the disjoining pressure we find that without the adsorbing boundaries the system is naturally unstable, while with attractive boundaries the system is stable because of a positive contribution from the surface tension to the disjoining pressure. The results for the density profiles show the formation of a positive layer near the boundary that screens the adsorbed negative particles, a typical behavior in charged systems. We also compute the adsorbed charge on the boundary and show that it satisfies a certain number of relations, in particular an electro-neutrality sum rule.     

N identical particles under quantum confinement: a many-body dimensional perturbation theory approach

Systems that involve N identical interacting particles under quantum confinement appear throughout many areas of physics, including chemical, condensed matter, and atomic physics. In this paper, we present the methods of dimensional perturbation theory, a powerful set of tools that uses symmetry to yield simple results for studying such many-body systems. We present a detailed discussion of the dimensional continuation of the N-particle Schrödinger equation, the spatial dimension D→∞ equilibrium (D⁰) structure, and the normal-mode (D⁻¹) structure. We use the FG matrix method to derive general, analytical expressions for the many-body normal-mode vibrational frequencies, and we give specific analytical results for three confined N-body quantum systems: the N-electron atom, N-electron quantum dot, and N-atom inhomogeneous Bose-Einstein condensate with a repulsive hard-core potential.     

Stark effect on an exciton or impurity in a finite width quantum wire with an electric field confined in a finite region

We study the electronic properties of a quantum system formed by two charged particles moving in a quantum wire (QW) with finite width σ and interacting through a Coulomb potential under an uniform electric field E applied over a spatially confined region of thickness 2a (-a<z<a). The number of electronic states of this finite width system is twice the number of the less realistic system with σ=0.     

Coulomb self-energy of a uniformly charged three-dimensional cylinder

We calculate exactly the Coulomb self-energy of a uniformly charged three-dimensional cylinder. We derive a general analytical formula which, in the limit of zero length, gives also the correct result for the Coulomb self-energy of a uniformly charged two-dimensional disk. The exact analytical expression that we derive can be used in models that incorporate finite thickness effects in studies of two-dimensional electronic systems in the fractional quantum Hall regime as well as models that describe cylindrical beams of charged particles, certain colloidal suspensions of charged rigid rod-like particles and biological systems consisting of macromolecules with cylindrical symmetry.