arXiv subjects
Publications and source records attributed to B. Jancovici.
The pressure in a classical Coulomb fluid at equilibrium is obtained from the Maxwell tensor at some point inside the fluid, by a suitable statistical average. For fluids in an Euclidean space, this is a fresh look on known results. But, for fluids in a curved space, a case which is of some interest, the unambiguous results from the Maxwell tensor approach have not been obtained by other methods.
The structure function $S(k;β)$ for the one-dimensional one-component log-gas is the Fourier transform of the charge-charge, or equivalently the density-density, correlation function. We show that for $|k| < {\rm min} (2πρ, 2 πρβ)$, $S(k;β)$ is simply related to an analytic function $f(k;β)$ and this function satisfies the functional equation $f(k;β) = f(-2k/β;4/β)$. It is conjectured that the coefficient of $k^j$ in the power series expansion of $f(k;β)$ about $k=0$ is of the form of a polynomial in $β/2$ of degree $j$ divided by $(β/2)^j$. The bulk of the paper is concerned with calculating these polynomials explicitly up to and including those of degree 9. It is remarked that the small $k$ expansion of $S(k;β)$ for the two-dimensional one-component plasma shares some properties in common with those of the one-dimensional one-component log-gas, but these break down at order $k^8$.
In a two-dimensional two-component plasma, the second moment of the number density correlation function has the simple value $\{12 π[1-(Γ/4)]^2\}^{-1}$, where $Γ$ is the dimensionless coupling constant. This result is derived directly by using diagrammatic methods.
In a two-dimensional two-component plasma, the second moment of the density correlation function has the simple value {12 pi [1-(gamma/4)]^2}^{-1}, where gamma is the dimensionless coupling constant. This result is derived by using analogies with critical systems.
We study the equilibrium statistical mechanics of classical two-dimensional Coulomb systems living on a pseudosphere (an infinite surface of constant negative curvature). The Coulomb potential created by one point charge exists and goes to zero at infinity. The pressure can be expanded as a series in integer powers of the density (the virial expansion). The correlation functions have a thermodynamic limit, and remarkably that limit is the same one for the Coulomb interaction and some other interaction law. However, special care is needed for defining a thermodynamic limit of the free energy density. There are sum rules expressing the property of perfect screening. These generic properties can be checked on the Debye-Hückel approximation, and on two exactly solvable models~: the one-component plasma and the two-component plasma, at some special temperature.
Exact and asymptotic formulas relating to dynamical correlations for overdamped Brownian motion are obtained. These formulas include a generalization of the $f$-sum rule from the theory of quantum fluids, a formula relating the static current-current correlation to the static density-density correlation, and an asymptotic formula for the small-$k$ behaviour of the dynamical structure factor. Known exact evaluations of the dynamical density-density correlation in some special models are used to illustrate and test the formulas.
An exact solution is given for a two-dimensional model of a Coulomb gas, more general than the previously solved ones. The system is made of a uniformly charged background, positive particles, and negative particles, on the surface of a sphere. At the special value $Γ= 2$ of the reduced inverse temperature, the classical equilibrium statistical mechanics is worked out~: the correlations and the grand potential are calculated. The thermodynamic limit is taken, and as it is approached the grand potential exhibits a finite-size correction of the expected universal form.
This paper compares two methods of statistical mechanics used to study a classical Coulomb system S near an ideal conductor C. The first method consists in neglecting the thermal fluctuations in the conductor C and constrains the electric potential to be constant on it. In the second method the conductor C is considered as a conducting Coulomb system the charge correlation length of which goes to zero. It has been noticed in the past, in particular cases, that the two methods yield the same results for the particle densities and correlations in S. It is shown that this is true in general for the quantities which depend only on the degrees of freedom of S, but that some other quantities, especially the electric potential correlations and the stress tensor, are different in the two approaches. In spite of this the two methods give the same electric forces exerted on S.
Coulomb systems in which the particles interact through the $d$-dimensional Coulomb potential but are confined in a flat manifold of dimension $d - 1$ are considered. The Coulomb potential is defined with some boundary condition involving a characteristic macroscopic distance $W$ in the direction perpendicular to the manifold~: either it is periodic of period $W$ in that direction, or it vanishes on one ideal conductor wall parallel to the manifold at a distance $W$ from it, or it vanishes on two parallel walls at a distance $W$ 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 on $W$ (universal means independent of the microscopic detail). The cases $d = 2$ are discussed in detail, and the generic results are checked on an exactly solvable model. The case $d = 3$ of a plane parallel to an ideal conductor is also explicitly worked out.
The classical two-dimensional one-component plasma is an exactly solvable model, at some special temperature, even when the one-body potential acting on the particles has a quadrupolar term. As a supplement to a recent work of Di Francesco, Gaudin, Itzykson, and Lesage [{\it Int. J. Mod. Phys.} {\bf A9}, 4257 (1994)] about an $N$-particle system ($N$ large but finite), a macroscopic argument is given for confirming that the particles form an elliptical blob, the analogy between the classical plasma and a quantum $N$-fermion system in a magnetic field is used for the microscopic approach, and a microscopic calculation of the surface charge-surface charge correlation function is performed; an expected universal form is shown to be realized by this correlation function.
From the laws of macroscopic electrostatics of conductors (in particular the existence of screening) taken for granted, one can deduce universal properties for the thermal fluctuations in a classical Coulomb system at equilibrium. The universality is especially apparent in the long-range correlations of the electrical potentials and fields. The charge fluctuations are derived from the field fluctuations. This is a convenient way for studying the surface charge fluctuations on a conductor with boundaries. Explicit results are given for simple geometries. The potentials and the fields have Gaussian fluctuations, except for a short-distance cutoff.
The grand potential of a classical Coulomb system has universal finite-size corrections similar to the ones which occur in the free energy of a simple critical system : the massless Gaussian field. Here, the Coulomb system is assumed to be confined by walls made of an ideal conductor material; this choice corresponds to simple (Dirichlet) boundary conditions for the Gaussian field. For a $d$-dimensional ($d>or=2$) Coulomb system confined in a slab of thickness $W$, the grand potential (in units of $kT$) per unit area has the universal term $Gamma(d/2) zeta(d)/2^d pi^{d/2}W^{d-1}$. For a two-dimensional Coulomb system confined in a disk of radius $R$, the grand potential (in units of $kT$) has the universal term $(1/6) ln R$. These results, of general validity, are checked on two-dimensional solvable models.