Notes on a 1-dimensional electrostatic plasma model F. Pegoraro1 P.J. Morrison2 1Department of Physics University of Pisa Pisa Italy

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Notes on a 1-dimensional electrostatic plasma model

F. Pegoraro1, P.J. Morrison 2

1Department of Physics, University of Pisa, Pisa, Italy

2Physics Department, University of Texas at Austin, Austin, TX

Abstract

A starting point for deriving the Vlasov equation is the BBGKY hierarchy that describes

the dynamics of coupled marginal distribution functions. With a large value of the plasma

parameter one can justify eliminating 2-point correlations in terms of the 1-point function

in order to derive the Vlasov Landau Lenard Balescu (VLLB) theory. Because of the high

dimensionality of the problem, numerically testing the assumptions of the VLLB theory is

prohibitive. In these notes we propose a physically reasonable interaction model that low-

ers the dimensionality of the problem and may bring such computations within reach. We

introduce a 1-dimensional (1-D) electrostatic plasma model formulated in terms of the interac-

tion of parallelly-aligned charged disks. This model combines 1-dimensional features at short

distances and 3-dimensional features at large distances.

1 Introduction

The construction of a charged-disks electrostatic model is motivated by the aim [1] to verify nu-

merically the validity of the procedure which is adopted, see e.g. Ref.[2], in order to solve the

BBGKY hierarchy [3] in a weakly coupled plasma when deriving the kinetic plasma equation, i.e.

the Vlasov equation. Such a veriﬁcation might be based on the numerical integration of the time

evolution of the two-point particle distribution function as obtained from a proper Hamiltonian

truncation of the the BBGKY hierarchy at the level of the three-point distribution function [4].

In 3-D space the equation for the time evolution of this two-point distribution function would be

13-dimensional (one time dimension plus the 12-dimensional two-particle phase space). In 1-D

space the equation for the two-point (actually two foil) distribution would be 5-dimensional (time

plus 4-dimensional two-foil phase space). However in 1-D the electric ﬁeld generated by a charge

foil does not decay with distance so that the two-foil interaction energy diverges at inﬁnity. This

makes it impossible to deﬁne a ﬁnite correlation length. A possible compromise is to consider the

interaction of aligned, uniformly charged disks, a sort of a single-row abacus made of thin disk-

shaped beads, where we can introduce a characteristic length b, the radius of the disk, which we

may identify with the Debye radius λd(same for the electrons and ions). In this case the electro-

static interaction energy between two charged disks is ﬁnite both at zero and at inﬁnite distance ξ

from the source and its absolute value is a decreasing function of ξ.

The present notes are dedicated to the derivation of the main properties of such a disk-plasma

in order to clarify where and how the dynamics of such a model system can mimic the dynamics of

a real plasma, at least as long as the longitudinal electric ﬁeld limit is concerned. In analogy with

a real particle plasma, in the derivation which follows we will introduce concepts such as positive

arXiv:2210.04254v1 [physics.plasm-ph] 9 Oct 2022

and negative charged disk densities, collective electric ﬁeld shielding, disk waves etc., but we will

not use a Poisson type equation with the charge disk density as the source. It will be enough to

refer to the expression of the interaction energy between the disks derived in Ref.[5] that in the

present model will play the role of the inter-particle Coulomb energy in a real 3-D plasma.

2 Disk Model

The “disk model” describes a globally neutral system consisting of a large number (as calculated

inside intervals with length of the order of the disk radius b) of aligned, inﬁnitely thin disks with

equal radius, as sketched in Fig.1, equal or opposite electric charge and different masses (ion-disks,

electron-disks) that are free to move along the xaxis (the planes of the disks are orthogonal to the

xaxis) and to pass through each other.

Figure 1: Schematic view, not in scale, of the disk conﬁguration with the different colours repre-

senting different charges.

In the following, for the sake of simplicity, the ion-disks are taken to be immobile. We denote

the mass of an electron disk by Meand its charge by Qe=−Qiwith Qithe charge of an ion disk.

2.1 Discrete disks: interaction energy

The interaction energy W(x1, x2)between two uniformly charged inﬁnitely thin disks of radius b

located at x1and x2respectively can be written in c.g.s. units as [5].

W(x1, x2) = 4Q1Q2

bV(ξ) = Q1ϕ2=Q2ϕ1,(1)

where

V(ξ) = −ξ

21−1

3π(4 −ξ2)E(−4/ξ2) + (4 + ξ2)K(−4/ξ2).(2)

and ϕ1,ϕ2are the potentials generated by the disks 1and 2respectively. Here ξ(x1, x2) =

|x1−x2|/b,Q1=πb2σ1and Q2=πb2σ2are the (ﬁxed) charges of the two disks, σ1,2are their

surface density and E(−4/ξ2)and K(−4/ξ2)are elliptic integrals [6]. The “disks-electrostatics”

is illustrated in more detail in Appendix A. A plot of V(ξ)is given in Fig.2 .

012345

0.1

0.2

0.3

0.4

V(ξ)

Figure 2: Plot of the potential V(ξ)versus ξ. Note the 1/(4ξ)-behavior for large ξand the ﬁnite

value 4/(3π)at ξ= 0.

The corresponding electric force FQ2acting on the disk 1 due to the disk 2 is given by

FQ2(x1) = −FQ1(x2) = −(4Q1Q2/b2)V0(ξ) sign (x1−x2),where (3)

V0(ξ) = dV (ξ)

dξ =1

2+1

2πξ2E(−4/ξ2)−(4 + ξ2)K(−4/ξ2).

A plot of −V0(ξ)is given in Fig.3

2 4 6 8 10

0.02

0.04

0.06

0.08

0.10

0.12

0.14

-V′(ξ)

0.2 0.4 0.6 0.8 1.0

0.2

0.3

0.4

0.5

-V′(ξ)

Figure 3: Plot of −V0(ξ)versus ξ. Note the 1/(4ξ2)behavior for large ξ(left frame) and the value

−V0(0) = 1/2(right frame).

3 Continuous limit

Deﬁne an electron-disk density ne(x, t)(number of disks per unit length) and an ion-disk density

ni(x, t) = ni(for simplicity stationary and homogeneous). This deﬁnes a disk-charge density

(charge per unit length)

ρ(x) = Q(ni−ne(x, t)),(4)

with Q=Qi=−Qe.

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Notesona1-dimensionalelectrostaticplasmamodelF.Pegoraro1,P.J.Morrison21DepartmentofPhysics,UniversityofPisa,Pisa,Italy2PhysicsDepartment,UniversityofTexasatAustin,Austin,TXAbstractAstartingpointforderivingtheVlasovequationistheBBGKYhierarchythatdescribesthedynamicsofcoupledmarginaldistributionfuncti...

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Notes on a 1-dimensional electrostatic plasma model F. Pegoraro1 P.J. Morrison2 1Department of Physics University of Pisa Pisa Italy

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