Journal of Statistical Physics

, Volume 175, Issue 6, pp 1066–1079 | Cite as

Fourth Moment of the Charge Density Induced Around a Guest Charge in Two-Dimensional Jellium

  • Ladislav ŠamajEmail author


The model under consideration is the classical two-dimensional one-component plasma (jellium) of pointlike particles with charge e, interacting pairwisely via the logarithmic Coulomb potential and immersed in a uniform neutralizing background charge density. The system is in thermal equilibrium at the inverse temperature \(\beta \), its thermodynamics depends only on the coupling constant \({\varGamma }=\beta e^2\). We put into an infinite (homogeneous and translationally invariant) plasma a guest particle of charge Ze with \(Z>-2/{\varGamma }\) in order to prevent from the collapse of the jellium charges onto it. The guest particle induces a screening cloud (the excess charge density) in the plasma. The zeroth and second moments of this screening cloud were derived previously for any fluid value of \({\varGamma }\). In this paper, we propose a formula for the fourth moment of the screening cloud. The derivation is based on the assumption that the fourth moment is, similarly as the second moment, analytic in Z around \(Z=0\). An exact treatment of the limit \(Z\rightarrow \infty \) shows that it is a finite (cube) polynomial in Z. The \({\varGamma }\)-dependence of the polynomial coefficients is determined uniquely by considering the limits \(Z\rightarrow 0\) and \(Z\rightarrow \infty \), and the compressibility sum rule for \(Z=1\). The formula for the fourth moment of screening cloud is checked in the leading and first correction orders of the Debye–Hückel limit \({\varGamma }\rightarrow 0\) and at the exactly solvable free-fermion point \({\varGamma }=2\). Sufficient conditions for sign oscillations of the induced charge density which follow from the second-moment and fourth-moment sum rules are discussed.


Coulomb fluids Jellium Logarithmic interaction Sum rules 



The support received from Grant VEGA No. 2/0003/18 is acknowledged.


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Authors and Affiliations

  1. 1.Institute of PhysicsSlovak Academy of SciencesBratislavaSlovakia

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