F. Cornu

The electrical double layer: A solvable model

In classical equilibrium statistical mechanics, the two‐dimensional two‐component Coulomb gas is exactly solvable at the special value of the reduced inverse temperature Γ=2. This is used for building an exactly solvable model of the electrical double layer. A charged hard wall (primitive electrode), a polarizable interface, an ideal conductor electrode, a semipermeable membrane are studied: the density profiles and correlation functions are computed. The differential capacity and the surface tension are also obtained.

On the two-dimensional Coulomb gas

This is a sequel to a recent work of Gaudin, who studied the classical equilibrium statistical mechanics of the two-dimensional Coulomb gas on a lattice at a special value of the coupling constantГ such that the model is exactly solvable. This model is briefly reviewed, and it is shown that the correlation functions obey the sum rules that characterize a conductive phase. A related model in which the particles are constrained to move on an array of equidistant parallel lines has simpler mathematics, and the asymptotic behavior of its correlation functions is studied in some detail. In the low-density limit, the lattice model is expected to have the same properties as a system of charged, hard disks; the correlation functions, internal energy, and specific heat of the latter are discussed.

Screened Cluster Expansions for Partially Ionized Gases

We consider a partially ionized gas at thermal equilibrium, in the Saha regime. The system is described in terms of a quantum plasma of nuclei and electrons. In this framework, the Coulomb interaction is the source of a large variety of phenomena occurring at different scales: recombination, screening, diffraction, etc. In this paper, we derive a cluster expansion adequate for a coherent treatment of those phenomena. The expansion is obtained by combining the path integral representation of the quantum gas with familiar Mayer diagrammatics. In this formalism, graphs have a clear physical interpretation: vertices are associated with recombined chemical species, while bonds describe their mutual interactions. The diagrammatical rules account exactly for all effects in the medium. Applications to thermodynamics, van der Waals forces and dielectric versus conductive behaviour will be presented in forthcoming papers.

Correlations in the Kosterlitz-Thouless phase of the two-dimensional Coulomb gas

The particle and charge correlations of the two-dimensional Coulomb gas are studied in the dielectric phase. A term-by-term analysis of the low-fugacity expansions suggests that the large-distance behaviors of the particle correlations are governed by multipolar interactions, similar to what happens in a system of permanent dipoles. These behaviors are compatible with the asymptotic structure of the BGY hierarchy equations; on the other hand, a new identity for the dielectric constant ɛ is used to show that the four-particle correlations decay as the dipole-dipole potential 1/r2 when two neutral pairs are separated by a large distancer. Near the zero-density critical point of the Kosterlitz-Thouless transition, we resum the low-fugacity expansions of both 1/ɛ and the charge correlation C(r). We thus retrieve the coupling constant flow equations of the renormalization group as well as the effective interaction energy of the iterated mean-field theory by Kosterlitz and Thouless. The coupling constant at the RG fixed point is then identified with 1/ɛ. The nonanalyticity of 1/ɛ at the transition turns out to coincide with the divergence of the low-fugacity series for this quantity. The leading term in the large-distance behavior of C(r) is found to be the same as for external charges. Moreover, we exhibit the subleading terms which also contribute to 1/ɛ.

van der Waals forces in presence of free charges: An exact derivation from equilibrium quantum correlations

We study interatomic forces in a fluid consisting of a mixture of free charges and neutral atoms in the framework of the quantum many-body problem at nonzero temperature and nonzero density. Of central interest is the interplay between van der Waals forces and screening effects due to free charges. The analysis is carried out in a partially recombined hydrogen plasma in the Saha regime. The effective potentials in the medium between two atoms, or an atom and a charge, or two charges, are determined from the large-distance behavior of equilibrium proton-proton correlations. We show, in a proper low-temperature and low-density scaling limit, that those potentials all decay as r(-6) at large distance r, while the corresponding amplitudes are calculated exactly. In particular, the presence of free charges only causes a partial (nonexponential) screening of the atomic potential, and it does not modify its typical r(-6) decay. That potential reduces to the standard van der Waals form for two atoms in vacuum when the temperature is driven to zero. The analysis is based on first principles: it does not assume preformed atoms and takes into account in a coherent way all effects, quantum mechanical binding, ionization, and collective screening, which originate from the Coulomb potential. Our method relies on the path integral representation of the quantum Coulomb gas.

Density profiles in a classical Coulomb fluid near a dielectric wall. I. Mean-field scheme

The equilibrium density profiles in a classical multicomponent plasma near a hard wall made with a dielectric material characterized by a relative dielectric constant ∈w are studied from the first Born–Green–Yvon (BGY) equation combined with Poisson equation in a regime where Coulomb coupling is weak inside the fluid. In order to prevent the collapse between charges with opposite signs or between each charge and its dielectric image inside the wall when ∈w>1, hard-core repulsions are added to the Coulomb pair interaction. The charge-image interaction cannot be treated perturbatively and the density profiles vary very fast in the vicinity of the wall when ∈w≠1. The formal solution of the associated inhomogeneous Debye–Hückel equations will be given in Paper II, together with a systematic fugacity expansion which allows to retrieve the results obtained from the truncated BGY hierarchy. In the present paper the exact density profiles are calculated analytically up to first order in the coupling parameter. The expressions show the interplay between three effects: the geometric repulsion from the impenetrable wall; the electrostatic effective attraction (∈w>1) or repulsion (∈w<1) due to its dielectric response; and the Coulomb interaction between each charge and the potential drop created by the electric layer which appears as soon as the system is not symmetric. We exhibit how the charge density profile evolves between a structure with two oppositely-charged layers and a three-layer organization when ∈w varies. (The case of two ideally conducting walls will be displayed elsewhere).

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 two-dimensional one-component plasma in a doubly periodic background: Exact results

We revisit the equilibrium classical statistical mechanics of the two-dimensional one-component plasma, for the special value Γ=2 of the coupling constant. Using a new method, we find that the model is solvable (then-body densities can be explicitly computed) for a larger class of inhomogeneous backgrounds. In particular, we can deal with a doubly periodic background; this is a classical model for a crystal made of fixed ions and mobile electrons. At Γ=2, this system is conducting: the correlations have a fast decay, and the Stillinger-Lovett screening sum rule is obeyed.

Local detailed balance: a microscopic derivation

Thermal contact is the archetype of non-equilibrium processes driven by constant non-equilibrium constraints when the latter are enforced by reservoirs exchanging conserved microscopic quantities. At a mesoscopic scale only the energies of the macroscopic bodies are accessible together with the configurations of the contact system. We consider a class of models where the contact system, as well as macroscopic bodies, have a finite number of possible configurations. The global system with only discrete degrees of freedom has no microscopic Hamiltonian dynamics, but it is shown that, if the microscopic dynamics is assumed to be deterministic and ergodic and to conserve energy according to some specific pattern, and if the mesoscopic evolution of the global system is approximated by a Markov process as closely as possible, then the mesoscopic transition rates obey three constraints. In the limit where macroscopic bodies can be considered as reservoirs at thermodynamic equilibrium (but with different intensive parameters) the mesoscopic transition rates turn into transition rates for the contact system and the third constraint becomes local detailed balance ; the latter is generically expressed in terms of the microscopic exchange entropy variation, namely the opposite of the variation of the thermodynamic entropy of the reservoir involved in a given microscopic jump of the contact system configuration. For a finite-time evolution after contact has been switched on we derive a fluctuation relation for the joint probability of the heat amounts received from the various reservoirs. The generalization to systems exchanging energy, volume and matter with several reservoirs, with a possible conservative external force acting on the contact system, is given explicitly.

Three-dimensional Lorentz model in a magnetic field: Exact and Chapman–Enskog solutions

We derive the exact solution of the Boltzmann kinetic equation for the three-dimensional Lorentz model in the presence of a constant and uniform magnetic field. The velocity distribution of the electrons reduces exponentially fast to its spherically symmetric component. In the long time hydrodynamic limit there remains only the diffusion process governed by an anisotropic diffusion tensor. The systematic way of building the Chapman–Enskog solutions is described.