Two-dimensional one-component plasma on a Flamm's paraboloid

We study the classical non-relativistic two-dimensional one-component plasma at Coulomb coupling Gamma=2 on the Riemannian surface known as Flamm's paraboloid which is obtained from the spatial part of the Schwarzschild metric. At this special value of the coupling constant, the statistical mechanics of the system are exactly solvable analytically. The Helmholtz free energy asymptotic expansion for the large system has been found. The density of the plasma, in the thermodynamic limit, has been carefully studied in various situations

Exactly solvable model of the 2D electrical double layer

We consider equilibrium statistical mechanics of a simplified model for the ideal conductor electrode in an interface contact with a classical semi-infinite electrolyte, modeled by the two-dimensional Coulomb gas of pointlike $\pm$ unit charges in the stability-against-collapse regime of reduced inverse temperatures $0\le \beta<2$. If there is a potential difference between the bulk interior of the electrolyte and the grounded interface, the electrolyte region close to the interface (known as the electrical double layer) carries some nonzero surface charge density. The model is mappable onto an integrable semi-infinite sine-Gordon theory with Dirichlet boundary conditions. The exact form-factor and boundary state information gained from the mapping provide asymptotic forms of the charge and number density profiles of electrolyte particles at large distances from the interface. The result for the asymptotic behavior of the induced electric potential, related to the charge density via the Poisson equation, confirms the validity of the concept of renormalized charge and the corresponding saturation hypothesis. It is documented on the non-perturbative result for the asymptotic density profile at a strictly nonzero $\beta$ that the Debye-H\"uckel $\beta\to 0$ limit is a delicate issue.Comment: 14 page

"Screening" of universal van der Waals - Casimir terms by Coulomb gases in a fully-finite two-dimensional geometry

This paper is a continuation of a previous one [Jancovici and Samaj, 2004 J. Stat. Mech. P08006] dealing with classical Casimir phenomena in semi-infinite wall geometries. In that paper, using microscopic Coulomb systems, the long-ranged Casimir force due to thermal fluctuations in conducting walls was shown to be screened by the presence of an electrolyte between the walls into some residual short-ranged force. Here, we aim to extend the study of the screening (cancellation) phenomena to universal Casimir terms appearing in the large-size expansions of the grand potentials for microscopic Coulomb systems confined in fully-finite 2D geometries, in particular the disc geometry. Two cases are solved exactly: the high-temperature (Debye-H\"uckel) limit and the Thirring free-fermion point. Similarities and fundamental differences between fully-finite and semi-infinite geometries are pointed out.Comment: 21 pages, 1 figur

Screening of classical Casimir forces by electrolytes in semi-infinite geometries

We study the electrostatic Casimir effect and related phenomena in equilibrium statistical mechanics of classical (non-quantum) charged fluids. The prototype model consists of two identical dielectric slabs in empty space (the pure Casimir effect) or in the presence of an electrolyte between the slabs. In the latter case, it is generally believed that the long-ranged Casimir force due to thermal fluctuations in the slabs is screened by the electrolyte into some residual short-ranged force. The screening mechanism is based on a "separation hypothesis": thermal fluctuations of the electrostatic field in the slabs can be treated separately from the pure image effects of the "inert" slabs on the electrolyte particles. In this paper, by using a phenomenological approach under certain conditions, the separation hypothesis is shown to be valid. The phenomenology is tested on a microscopic model in which the conducting slabs and the electrolyte are modelled by the symmetric Coulomb gases of point-like charges with different particle fugacities. The model is solved in the high-temperature Debye-H\"uckel limit (in two and three dimensions) and at the free fermion point of the Thirring representation of the two-dimensional Coulomb gas. The Debye-H\"uckel theory of a Coulomb gas between dielectric walls is also solved.Comment: 25 pages, 2 figure

Collective modes and correlations in one-component plasmas

The static and time-dependent potential and surface charge correlations in a plasma with a boundary are computed for different shapes of the boundary. The case of a spheroidal or spherical one-component plasma is studied in detail because experimental results are available for such systems. Also, since there is some knowlegde both experimental and theoretical about the electrostatic collective modes of these plasmas, the time-dependent correlations are computed using a method involving these modes.Comment: 20 pages, plain TeX, submitted to Phys. Rev.

Expanded Vandermonde powers and sum rules for the two-dimensional one-component plasma

The two-dimensional one-component plasma (2dOCP) is a system of $N$ mobile particles of the same charge $q$ on a surface with a neutralising background. The Boltzmann factor of the 2dOCP at temperature $T$ can be expressed as a Vandermonde determinant to the power $\Gamma=q^{2}/(k_B T)$. Recent advances in the theory of symmetric and anti-symmetric Jack polymonials provide an efficient way to expand this power of the Vandermonde in their monomial basis, allowing the computation of several thermodynamic and structural properties of the 2dOCP for $N$ values up to 14 and $\Gamma$ equal to 4, 6 and 8. In this work, we explore two applications of this formalism to study the moments of the pair correlation function of the 2dOCP on a sphere, and the distribution of radial linear statistics of the 2dOCP in the plane

Microscopic origin of universality in Casimir forces

The microscopic mechanisms for universality of Casimir forces between macroscopic conductors are displayed in a model of classical charged fluids. The model consists of two slabs in empty space at distance $d$ containing classical charged particles in thermal equilibrium (plasma, electrolyte). A direct computation of the average force per unit surface yields, at large distance, the usual form of the Casimir force in the classical limit (up to a factor 2 due to the fact that the model does not incorporate the magnetic part of the force). Universality originates from perfect screening sum rules obeyed by the microscopic charge correlations in conductors. If one of the slabs is replaced by a macroscopic dielectric medium, the result of Lifshitz theory for the force is retrieved. The techniques used are Mayer expansions and integral equations for charged fluids.Comment: 31 pages, 0 figures, submitted to Journal of Statistical Physic

The Casimir force at high temperature

The standard expression of the high-temperature Casimir force between perfect conductors is obtained by imposing macroscopic boundary conditions on the electromagnetic field at metallic interfaces. This force is twice larger than that computed in microscopic classical models allowing for charge fluctuations inside the conductors. We present a direct computation of the force between two quantum plasma slabs in the framework of non relativistic quantum electrodynamics including quantum and thermal fluctuations of both matter and field. In the semi-classical regime, the asymptotic force at large slab separation is identical to that found in the above purely classical models, which is therefore the right result. We conclude that when calculating the Casimir force at non-zero temperature, fluctuations inside the conductors can not be ignored.Comment: 7 pages, 0 figure

Renormalized energy concentration in random matrices

We define a "renormalized energy" as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line. The definition is inspired by ideas of [SS1,SS3]. Roughly speaking, it is obtained by subtracting two leading terms from the Coulomb potential on a growing number of charges. The functional is expected to be a good measure of disorder of a configuration of points. We give certain formulas for its expectation for general stationary random point processes. For the random matrix $\beta$-sine processes on the real line (beta=1,2,4), and Ginibre point process and zeros of Gaussian analytic functions process in the plane, we compute the expectation explicitly. Moreover, we prove that for these processes the variance of the renormalized energy vanishes, which shows concentration near the expected value. We also prove that the beta=2 sine process minimizes the renormalized energy in the class of determinantal point processes with translation invariant correlation kernels.Comment: last version, to appear in Communications in Mathematical Physic