Classical and Quantum Algebraic Screening in a Coulomb Plasma near a Wall: A Solvable Model

The static position correlation in a quantum Coulomb plasma near a wall is studied by means of a model where two quantum charges are embedded in a classical plasma at equilibrium. Three kinds of walls are considered: a wall without electrostatic properties, a dielectric, and an ideal conductor. At large separations y along the wall, the correlation exactly decays as 1/y ³, though no algebraic tail exists for classical charges near an ideal conductor. This tail originates from thermal statistical and purely quantum fluctuations of polarization clouds which are deformed by the geometric constraint due to the wall and by the charges induced by influence inside a wall with electrical properties. The coefficient of the 1/y ³ tail can be calculated explicitly in a weak-coupling and low-delocalization regime. Then classical, diffraction, and purely quantum contributions are disentangled.     

Classical Coulomb systems near a plane wall. II

The equilibrium structure of classical Coulomb systems bounded by a plane hard wall is studied near that wall. A general sum rule is derived for the asymptotic form of the charge-charge correlation function along the wall. The exact results which can be obtained for the two-dimensional one-component plasma provide a test for this new sum rule, as well as for other already known sum rules or their generalizations.This laboratory is associated with the Centre National de la Recherche Scientifique.     

Guest Charge and Potential Fluctuations in Two-Dimensional Classical Coulomb Systems

A known generalization of the Stillinger-Lovett sum rule for a guest charge immersed in a two-dimensional one-component plasma (the second moment of the screening cloud around this guest charge) is more simply retrieved, just by using the BGY hierarchy for a mixture of several species; the zeroth moment of the excess density around a guest charge immersed in a two-component plasma is also obtained. The moments of the electric potential are related to the excess chemical potential of a guest charge; explicit results are obtained in several special cases. Unité Mixte de Recherche No. 8627—CNRS.     

Classical coulomb systems near a plane wall. I

The equilibrium structure of classical Coulomb systems bounded by a plane wall is studied near that wall. Several models are considered: the two-dimensional one-component plasma at a special value of the coupling constant (which makes the model exactly soluble), the two-dimensional and three-dimensional one-component and two-component plasmas in the weak-coupling limit (a Debye-Hückel type of approach is then used). Along a wall, the pair correlation functions decay only as an inverse power of the distancer, namely, asr ^–v for av-dimensional system (v=2,3). The one-body densities are also studied; the first BGY equation is used.     

Large charge fluctuations in classical Coulomb systems

The typical fluctuation of the net electric chargeQ contained in a subregion of an infinitely extended equilibrium Coulomb system is expected to grow only as S, whereS is the surface area of. For some cases it has been previously shown thatQ/S has a Gaussian distribution as ¦¦. Here we study the probability law for larger charge fluctuations (large-deviation problem). We discuss the case when both ¦¦ andQ are large, but now withQ of an order larger than S. For a given value ofQ, the dominant microscopic configurations are assumed to be those associated with the formation of a double electrical layer along the surface of. The probability law forQ is then determined by the free energy of the double electrical layer. In the case of a one-component plasma, this free energy can be computed, for large enoughQ, by macroscopic electrostatics. There are solvable two-dimensional models for which exact microscopic calculations can be done, providing more complete results in these cases. A variety of behaviors of the probability law are exhibited.     

Charge Fluctuations in Finite Coulomb Systems

When described in a grand canonical ensemble, a finite Coulomb system exhibits charge fluctuations. These fluctuations are studied in the case of a classical (i.e., non-quantum) system with no macroscopic average charge. Assuming the validity of macroscopic electrostatics gives, on a three-dimensional finite large conductor of volume V, a mean square charge Q ² which goes as V ^1/3. More generally, in a short-circuited capacitor of capacitance C, made of two conductors, the mean square charge on one conductor is Q ²=TC, where T is the temperature and C the capacitance of the capacitor. The case of only one conductor in a grand canonical ensemble is obtained by removing the other conductor to infinity. The general formula is checked in the weak-coupling (Debye–Hückel) limit for a spherical capacitor. For two-dimensional Coulomb systems (with logarithmic interactions), there are exactly solvable models which reveal that, in some cases, macroscopic electrostatics is not applicable even for large conductors. This is when the charge fluctuations involve only a small number of particles. The mean square charge on one two-dimensional system alone, in the grand canonical ensemble, is expected to be, at most, one squared elementary charge.     

Two-Dimensional Coulomb Systems in a Disk with Ideal Dielectric Boundaries

We study two-dimensional Coulomb systems confined in a disk with ideal dielectric boundaries. In particular we consider the two-component plasma in detail. When the coulombic coupling constant =2 the model is exactly solvable. We compute the grand potential, densities and correlations. We show that the grand potential has a universal logarithmic finite-size correction as predicted in previous works. This logarithmic finite-size correction is also found in the free energy of another solvable model: the one-component plasma.     

The charge fluctuations in classical Coulomb systems

We study the asymptotic behavior of the charge fluctuations (Q – (Q )² in infinite classical systems of charged particles, and show, under certain clustering assumptions, that if the charge fluctuations are not extensive, then they are necessarily of the order of the surface ¦¦. Moreover, when the canonical sum rules that are typical for equilibrium states of particles interacting with long-range forces hold true, we prove a central limit theorem for the normalized charge variable ¦¦^–1/2((Q – (Q )) in two and three dimensions. In one dimension, the probability distribution of the charge itself converges. The latter case is illustrated by the example of the one-dimensional Coulomb gas.     

Coulomb Systems with Ideal Dielectric Boundaries: Free Fermion Point and Universality

A two-component Coulomb gas confined by walls made of ideal dielectric material is considered. In two dimensions at the special inverse temperature =2, by using the Pfaffian method, the system is mapped onto a four-component Fermi field theory with specific boundary conditions. The exact solution is presented for a semi-infinite geometry of the dielectric wall (the density profiles, the correlation functions) and for the strip geometry (the surface tension, a finite-size correction of the grand potential). The universal finite-size correction of the grand potential is shown to be a consequence of the good screening properties, and its generalization is derived for the conducting Coulomb gas confined in a slab of arbitrary dimension 2 at any temperature.     

Charge Correlations in a Coulomb System Along a Plane Wall: A Relation Between Asymptotic Behavior and Dipole Moment

Classical Coulomb systems at equilibrium, bounded by a plane dielectric wall, are studied. A general two-point charge correlation function is considered. Valid for any fixed position of one of the points, a new relation is found between the algebraic tail of the correlation function along the wall and the dipole moment of that function. The relation is tested first in the weak-coupling (Debye–Hückel) limit, and afterwards, for the special case of a plain hard wall, on the exactly solvable two-dimensional two-component plasma at coupling =2, and on the two-dimensional one-component plasma at an arbitrary even integer .     

Coulomb systems seen as critical systems: Ideal conductor boundaries

When a classical Coulomb system has macroscopic conducting behavior, its grand potential 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 ad-dimensional (d2) Coulomb system confined in a slab of thicknessW, the grand potential (in units ofk _B T) per unit area has the universal term (d/2)(d)/2^dd/2W^d–1. For a two-dimensional Coulomb system confined, in a disk of radiusR, the grand potential (in units ofk _B T) has the universal term (1/6) lnR. These results, of general validity, are checked on two-dimensional solvable models.Laboratoire Associé au Centre National de la Recherche Scientifique-URA 63.     

On potential and field fluctuations in classical charged systems

Using electrostatic identities the potential and microfield in a plasma, important for determining line shapes, are expressed as limits of local quantities. These are shown to be well defined for typical configurations of macroscopic, i.e., infinite systems (under some mild clustering assumptions). Their covariance contains a slowly decaying part (¦x¦^–1, for the potential) whose coefficient is universal whenever the Stillinger-Lovett second moment condition holds. We show further that the contributions from distant regions (which are equal to suitable averages over local regions) have a Gaussian distribution.Supported in part by AFOSR Grant No. 82-0016.Supported in part by the Swiss National Foundation for Scientific Research.     

Density Correlations in the Two-Dimensional Coulomb Gas

We consider a two-dimensional Coulomb gas of positive and negative pointlike unit charges interacting via a logarithmic potential. The density (rather than the charge) correlation functions are studied. In the bulk, the form-factor theory of an equivalent sine-Gordon model is used to determine the density correlation length. At the surface of a rectilinear plain wall, the universality of the asymptotic behavior of the density correlations is suggested. A scaling analysis implies a local form of the compressibility sum rule near a hard wall. A symmetry of the Coulomb system with respect to the Möbius conformal transformation, which induces a gravitational source acting on the particle density, is established. Among the consequences, a universal term of the finite-size expansion of the grand potential is derived exactly for a disk geometry of the confining domain.     

On potential and field fluctuations in two-dimensional classical charged systems

We supplement a previous paper on three-dimensional systems by studying the electric potential and field fluctuations in two-dimensional Coulomb systems. The novelty in two dimensions is that the fluctuations of the potential at a point are infinite in the thermodynamic limit. However, the potential difference between two points has finite fluctuations, which resemble the ones which occur in the three-dimensional case. The field fluctuations are also rather similar in both cases. The correlations do not have a fast decay. Explicit results are obtained for a solvable model; the fluctuations of the potential are Gaussian with an infinite variance.This laboratory is associated with the Centre National de la Recherche Scientifique.     

Decay of correlations in classical fluids with long-range forces

We show with simple arguments that, as a consequence of the Poisson equation, the correlations of a charged system at equilibrium decay faster than any inverse power, if they are integrable and monotonous at infinity. For all other longrange systems (with potential(x)b¦x¦^–s , ¦x¦ , 0v,s} 2), the decay is bounded below by an inverse power.Partially supported by the Swiss National Foundation for Scientific Research.     

Equilibrium equations and symmetries of classical Coulomb systems

In this paper we establish the validity of the BBGKY equilibrium equations for Coulomb states which have been obtained as thermodynamic limit of finite volume states. We also give a new derivation of thel-sum rules for phases constructed by the cluster expansion. These sum rules are interpreted as Ward identities associated to a symmetry of the screening phase.Supported by the Swiss National Foundation for Scientific Research.     

Surface correlations in a quantum mechanical one-component plasma

The time-dependent pair correlation function of a quantum mechanical onecomponent plasma bounded by a plane hard wall is studied near that wall. Along the wall, this function has an algebraic asymptotic form: it decays only as the inverse cube (square) of the distance for a three (two)-dimensional system (the case of fermions at zero temperature is excluded from the present study). The amplitude of the asymptotic form obeys a universal sum rule. Similar results hold at the plane interface between two different one-component plasmas.This laboratory is associated with the Centre National de la Recherche Scientifique     

Coulomb systems seen as critical systems: Finite-size effects in two dimensions

It is known that the free energy at criticality of a finite two-dimensional system of characteristic sizeL has in general a term which behaves like logL asL; the coefficient of this term is universal. There are solvable models of two-dimensional classical Coulomb systems which exhibit the same finite-size correction (except for its sign) although the particle correlations are short-ranged, i.e., noncritical. Actually, the electrical potential and electrical field correlationsare critical at all temperatures (as long as the Coulomb system is a conductor), as a consequence of the perfect screening property of Coulomb systems. This is why Coulomb systems have to exhibit critical finite-size effects.Laboratoire Associé au Centre National de la Recherche Scientifique     

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 constant 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.this laboratory is associated with the Centre National de la Recherche Scientifique