Exact Energy Computation of the One Component Plasma on a Sphere for Even Values of the Coupling Parameter

Salazar, Robert; Téllez, Gabriel

doi:10.1007/s10955-016-1562-4

Cited by 11 publications

(11 citation statements)

References 28 publications

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“…The first one is based on the expansion of the power of van der monde determinants in the basis of Jack polynomials [32,42,44]. The second one is based on the mapping onto a 1D lattice fermion system; this formalism has been introduced in Ref.…”

Section: Mapping Onto 1d Fermionsmentioning

confidence: 99%

Finite-Size Effects in Non-neutral Two-Dimensional Coulomb Fluids

Šamaj¹

2017

J Stat Phys

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Thermodynamic potential of a neutral two-dimensional (2D) Coulomb fluid, confined to a large domain with a smooth boundary, exhibits at any (inverse) temperature β a logarithmic finite-size correction term whose universal prefactor depends only on the Euler number of the domain and the conformal anomaly number c = −1. A minimal free boson conformal field theory, which is equivalent to the 2D symmetric two-component plasma of elementary ±e charges at coupling constant Γ = βe 2 , was studied in the past. It was shown that creating a non-neutrality by spreading out a charge Qe at infinity modifies the anomaly number to c(Q, Γ ) = −1 + 3Γ Q 2 . Here, we study the effect of non-neutrality on the finite-size expansion of the free energy for another Coulomb fluid, namely the 2D one-component plasma (jellium) composed of identical pointlike e-charges in a homogeneous background surface charge density. For the disk geometry of the confining domain we find that the non-neutrality induces the same change of the anomaly number in the finite-size expansion. We derive this result first at the free-fermion coupling Γ ≡ βe 2 = 2 and then, by using a mapping of the 2D one-component plasma onto an anticommuting field theory formulated on a chain, for an arbitrary coupling constant.

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Section: Mapping Onto 1d Fermionsmentioning

confidence: 99%

Finite-Size Effects in Non-neutral Two-Dimensional Coulomb Fluids

Šamaj¹

2017

J Stat Phys

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“…A special case of the confining domain for the Coulomb system is the surface of a sphere [6,10,30]. For such geometry, by combining a stereographic projection of the sphere onto an infinite plane with linear response theory (TCP, Ref.…”

Section: Introductionmentioning

confidence: 99%

Logarithmic Finite-Size Correction in Non-neutral Two-Component Plasma on Sphere

Šamaj

2018

Preprint

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We consider a general two-component plasma of classical pointlike charges +e (e is say the elementary charge) and −Ze (valency Z = 1, 2, . . .), living on the surface of a sphere of radius R. The system is in thermal equilibrium at the inverse temperature β, in the stability region against collapse of oppositely charged particle pairs βe 2 < 2/Z. We study the effect of the system excess charge Qe on the finite-size expansion of the (dimensionless) grand potential βΩ. By combining the stereographic projection of the sphere onto an infinite plane, the linear response theory and the planar results for the second moments of the species density correlation functions we show that for any βe 2 < 2/Z the large-R expansion of the grand potential is of the form βΩ ∼ A V R 2 + [χ/6 − β(Qe) 2 /2] ln R, where A V is the non-universal coefficient of the volume (bulk) part and the Euler number of the sphere χ = 2. The same formula, containing also a non-universal surface term proportional to R, was obtained previously for the disc domain (χ = 1), in the case of the symmetric (Z = 1) two-component plasma at the collapse point βe 2 = 2 and the jellium model (Z → 0) of identical e-charges in a fixed neutralizing background charge density at any coupling βe 2 being an even integer. Our result thus indicates that the prefactor to the logarithmic finite-size expansion does not depend on the composition of the Coulomb fluid and its non-universal part −β(Qe) 2 /2 is independent of the geometry of the confining domain.

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“…Although, analytic solutions for any value of Γ are limited, there are several studies of the 2dOCP in diverse geometries specially for positives integers values of Γ [13,14,15,16,17]. Previously, authors of [17] described a way to compute the excess energy of the 2dOCP on the sphere based on the expansion the Vandermonde determinant to the power Γ. The main purpose of this work is to obtain the excess energy of the 2dOCP on a hard and soft disk for even values of Γ.…”

Section: Introductionmentioning

confidence: 99%

“…The main purpose of this work is to obtain the excess energy of the 2dOCP on a hard and soft disk for even values of Γ. For this aim, we shall show that the approach of [17] applied on the flat geometry may be used to obtain some analytical results of the excess energy for Γ = 2, 4, 6, . .…”

Section: Introductionmentioning

confidence: 99%

“…The preliminary material and the basics of the monomial expansion method will be described in section 3. Although, the generalities of the method may be also found in [15,16,17] this section has been included in order to introduce the notation used along the document. The statistical average of the quadratic contribution to the energy (the quadratic sum introduced by the parabolic confining potential in Eqs.…”

Section: Introductionmentioning

confidence: 99%

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Exact Energy Expansion of the two-dimensional Dyson Gas for Odd Values of $Γ/2$

Salazar,

Téllez

2017

Preprint

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Using the expansion on monomial functions of the Vandermonde determinant to the power Γ = Q 2 /(kBT ), a way to find the excess energy Uexc of the two dimensional one component plasma 2dOCP on the hard and soft disk (or Dyson Gas) for odd values of Γ/2 is provided. At Γ = 2, the current study not only corroborates the result for the particle-particle energy contribution of the Dyson gas found by Shakirov by using an alternative approach but also provides the exact N -finite expansion of the excess energy of the 2dOCP on the hard disk. The excess energy is fitted to an ansatz of the formto study the finite-size corrections with K i coefficients and N the number of particles. In particular, the bulk term of the excess energy is in agreement with the well known result of Jancovici for the hard disk in the thermodynamic limit. Finally, an expression is found for the pair correlation function which still keeps a link with the random matrix theory via the kernel in the Ginibre Ensemble for odd values of Γ/2. A comparison between analytical 2-body density function and histograms obtained with Monte Carlo simulations for small systems and Γ = 2, 6, 10, . . . shows that the approach described in this work may be used to study analytically the crossover behaviour from a disordered system to small crystals.

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Exact Energy Computation of the One Component Plasma on a Sphere for Even Values of the Coupling Parameter

Cited by 11 publications

References 28 publications

Finite-Size Effects in Non-neutral Two-Dimensional Coulomb Fluids

Finite-Size Effects in Non-neutral Two-Dimensional Coulomb Fluids

Logarithmic Finite-Size Correction in Non-neutral Two-Component Plasma on Sphere

Exact Energy Expansion of the two-dimensional Dyson Gas for Odd Values of $Γ/2$

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