On the equation of state of classical one-component systems with long-range forces

Choquard, Ph.; Favre, Patrick; Gruber, C.

doi:10.1007/bf01011574

Cited by 52 publications

(34 citation statements)

References 8 publications

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“…In the planar limit R → ∞, the contact theorem (for a zero background charge density) [9,35,52] tells us that the contact particle density equals to πΓ σ 2 which is in agreement with our Γ = 2 result (4.7). The term σ/R is the finite-size correction due to curvature of the disc boundary.…”

Section: Free-fermion γ = 2 Couplingsupporting

confidence: 90%

“…In the planar limit R → ∞, the contact particle density is equal to πΓ σ 2 in agreement with the contact theorem [9,35,52]. The 1/R correction due to the wall curvature is exact for an arbitrary coupling Γ .…”

Section: Sum Rules For One-body Densitysupporting

confidence: 76%

“…In two dimensions, also the compressibility [4,54] and higher-moment [26,30] sum rules have been derived. In the semi-infinite case of an electric double layer, the density of particles at the wall was shown to be related to the bulk pressure via the contact theorem [9,35,52]. The Carnie and Chan generalization of the second-moment Stillinger-Lovett condition to inhomogeneous fluids leads to the dipole sum rule [7,8].…”

Section: Introductionmentioning

confidence: 99%

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Counter-Ions Near a Charged Wall: Exact Results for Disc and Planar Geometries

Šamaj

2015

J Stat Phys

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Macromolecules, when immersed in a polar solvent like water, become charged by a fixed surface charge density which is compensated by "counter-ions" moving out of the surface. Such classical particle systems exhibit poor screening properties at any temperature and the trivial bulk regime (far away from the charged surface) with no particles, so the validity of standard Coulomb sum rules is questionable. In the present paper, we concentrate on the two-dimensional version of the model with the logarithmic interaction potential. We go from the finite disc to the semi-infinite planar geometry. The system is exactly solvable for two values of the coupling constant Γ : in the Poisson-Boltzmann mean-field limit Γ → 0 and at the free-fermion point Γ = 2. We show that the finite-size expansion of the free energy does not contain universal term as is usual for Coulomb fluids. For the coupling constant being an arbitrary positive even integer, using an anticommuting representation of the partition function and many-body densities we derive a sequence of sum rules. As a result, the contact density of counter-ions at the wall is available for the disc. The amplitude function, which characterizes the asymptotic inverse-power law behavior of the two-body density along the wall, is found to be related to the particle density profile. The dielectric susceptibility tensor, calculated exactly for an arbitrary coupling and the particle number, exhibits the anticipated disc value in the thermodynamic limit, in spite of zero contribution from the bulk region. Some of the results obtained in the Poisson-Boltzmann limit are generalized to an arbitrary Euclidean dimension.

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Section: Free-fermion γ = 2 Couplingsupporting

confidence: 90%

Section: Sum Rules For One-body Densitysupporting

confidence: 76%

Section: Introductionmentioning

confidence: 99%

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Counter-Ions Near a Charged Wall: Exact Results for Disc and Planar Geometries

Šamaj

2015

J Stat Phys

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“…In this paper we introduce a modified "floating crystal" trial state and use it to prove that three possible definitions of the Jellium ground state energy coincide in the thermodynamic limit. In particular, we resolve a conundrum originating in [3][4][5][6][7] and raised again in [8], where it was observed that the usual floating crystal trial state fails for Coulomb interactions.…”

Section: Introductionmentioning

confidence: 76%

Floating Wigner crystal with no boundary charge fluctuations

2019

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We modify the "floating crystal" trial state for the classical Homogeneous Electron Gas (also known as Jellium), in order to suppress the boundary charge fluctuations that are known to lead to a macroscopic increase of the energy. The argument is to melt a thin layer of the crystal close to the boundary and consequently replace it by an incompressible fluid. With the aid of this trial state we show that three different definitions of the ground state energy of Jellium coincide. In the first point of view the electrons are placed in a neutralizing uniform background. In the second definition there is no background but the electrons are submitted to the constraint that their density is constant, as is appropriate in Density Functional Theory. Finally, in the third system each electron interacts with a periodic image of itself, that is, periodic boundary conditions are imposed on the interaction potential.

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“…However, here Q has a sharp crest along the line (2.3) and looking for the maximum term of (2.4) (or equivalently of (2.1)) gives only one relation : In other words, only the combination of z and t of (2.5) will be relevant for determining the particle numbers (constrained by (2.3)). This maximum-term method gives the familiar relation In E=N* In Therefore, after (2.5) has been used, In E still depends upon the additional variable z, and (/3V) -1 In E is certainly not a bulk physical quantity ; it cannot be the pressure (several different pressures have been defined for jellium models [7,8], here we mean the usual one -~F/~V with the derivative taken for constant values of the particle numbers and of the background total charge in such a way that the system remains neutral). This special feature of the jellium modelst, that (/3V) -1 In E is not the pressure -~F/~V, does not prevent us however from using (2.6) for computing the free energy.…”

Section: Grand-canonical Formalismmentioning

confidence: 98%

Binary ionic mixtures and a solvable model

Jancovici

1984

Molecular Physics

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The theory of solutions of McMillan and Mayer is applied to the jellium model of a binary ionic mixture: two species of charged particles, with charges e and Ze, immersed in a neutralizing background. The density p~ of the particles of charge Ze is considered as small, and is used as an expansion parameter. The free energy, the pair distribution functions, the internal energy, and the pressure of the mixture are expressed as power series in p~ ; the coefficients are integrals of correlation functions defined in the system at ~ =0 (the reference system). Explicit expressions are obtained in the two-dimensional case, at a special temperature, since in that case the reference system (the two-dimensional, one-component plasma) is a solvable model.

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On the equation of state of classical one-component systems with long-range forces

Cited by 52 publications

References 8 publications

Counter-Ions Near a Charged Wall: Exact Results for Disc and Planar Geometries

Counter-Ions Near a Charged Wall: Exact Results for Disc and Planar Geometries

Floating Wigner crystal with no boundary charge fluctuations

Binary ionic mixtures and a solvable model

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