7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Kinetic closure with a charged disk modelFrancesco Pegoraro1, Philip J. Morrison2,
1 Department of Physics, University of Pisa, Italy
2 Physics Department, The University of Texas at Austin, TX, USAe-mail (speaker): francesco.pegoraro@unipi.it
Motivation
The BBGKY hierarchy is the starting
point for deriving the Vlasov equation with a collision
operator. With a large plasma parameter g −1one justifies
eliminating 2-point correlations in terms of the 1-point
function. Numerically testing the validity of theseassumptions is prohibitive. because of high dimensionality, In this presentation, we will present a 1-D modelcomposed of interacting aligned charged disks so as toaddress the validity of the Bogoliubov assumption in acomputable model. A system of N classical particles
obeys the Liouville equation for the distribution function(d.f.) in 6N-dimensional phase space∂F (N )N

∂F (N )N

∂F (N )

pi+ℱ=0
∑ mi ∂q i ∑ i ∂pi
∂t
i=1
i=1
with ℱi the total force acting on the i th particle. By+

integration over the variables of a subset of particles, theLiouville equation can be transformed into a chain ofequations connecting the evolution of s -particle d.f. F (s)with the (s + 1) -d.f. F (s+1) . By truncating this chain byneglecting terms of order g 2 we obtain a closed system
involving F (1) and F (2) . Bogolyubov’s approach
postulates that F (2) rapidly reaches an asymptotic form,
uniquely determined by the instantaneous form of F (1).
In 3-D space the equation for the time evolution of F (2)is13-dimensional (one time plus the 12-dimensional twoparticle phase space). In 1-D space the equation for F (2)(actually the two-foil d.f.) is 5-dimensional (time plus 4dimensional phase space). However in 1-D the electricfield generated by a charge foil does not decay with
distance so that the two-foil interaction energy diverges
at infinity. This makes it unclear how to define a finite
correlation length. A possible compromise is to introduce
an interaction potential that behaves as that of a 1-D foil
at short distances and decays as the inverse of thedistance at large distances. Any interaction potential ofthis type introduces by necessity a length scale notpresent in the Coulomb interaction. A possible choice[1] is to consider ta system of aligned charged disks
where the characteristic length is the radius b of the disk.
In this case the electrostatic interaction energy is finite
both at zero and at infinite distance between disks.
Model The interaction energy between two uniformly
charged thin disks of radius b and charge Q1,2 is given
by [2] (4/b)Q1Q2V (ξ ), with ξ = x /bt | x1 − x 2 | /b andV (ξ ) = − (ξ /2){1 − 1/(3π)[(4 − ξ 2 )E (−4/ξ 2 )+(4 + ξ 2 )K(−4/ξ 2 )]}, with E and K elliptic integralsand V (ξ = 0) = 4/(3π), V (ξ ) = 1/(4ξ ) for ξ → ∞.
The spatial spectrum of V (ξ ) differs from that of theCoulomb potential at large wavelengths k b ≪ 1 , wherei t s F. T. V̂ (k b) = (1/8)[Co n s t − 2 ln[(k b)2 ] − . ] . .
while for (k b)2 ≫ 1, V̂ (k b) = 1/(k b)2. The logarithmic
term at small wave-numbers arises in 1-D from the 1/ξdependence of the interaction potential at large distances.
Defining the densities ni,e(x, t ) (disks per unit length)
and the charge density ρ (x) = Q [ni − ne(x, t )] andtaking ion disks immobile, the cold fluid equations,∂t ne + ∂x (neue ) = 0 , Me ne (∂t + ue ∂x) ue = neℱe .

4Qe d +∞
d x′ ρ (x′)V ( | x − x′| /b) ,b d x ∫−∞2 [(k b)2 V̂ (k b)]
give the dispersion relation ω 2 = ωDpe
2where ωDpe
= (4Q 2 no )/(Me b 2 ) . For k b ≫ 1 the diskℱe(x) = −plasma waves become Langmuir waves with frequency
ωDpe. For k b ≪ 1 we find (logarithmically corrected)``cold electron-disk sound'' waves of the form
ω 2 = k 2[C1 + C2 ln(k b)2 ]vD2 , where C1,2 are constants
2and bvD2 = b 2 ωDpe
= 4Q 2 no /Me. The phase velocity ofLangmuir waves is smaller than vD while for old soundwaves we have ω 2 /k 2 ∼ [ln(k b)]2 vD2 > > vD2 .
The disk screened potential at thermodynamic
equilibrium exhibits an exponential-type behaviour forsmall x /b while it decreases as (x /b)−s with s ⪆ 1 forlarge x /b . This tail originates from the logarithmic
contribution to V̂ (k b) in the limit of small k b.
The derivation of the Vlasov equation follows standardlines with the particle density as the independent variable∂fj (x, v, t )+v∂fj (x, v, t )+ℱj (x, t ) ∂fj (x, v, t )∂t∂xMjwith fj (x, v, t) s the d.f. of the j species.

∂v= 0,Present conclusions In order to model correlation
effects in a 3-D plasma using a reduced dimensionality
configuration, it may be convenient to define an effective
interaction energy constructed so as to retain the physicaleffects that we deem must be included in order to obtainan informative model. A numerical project is under way
in order to investigate the kinetic properties of this 1-D
plasma model and to integrate numerically the equationfor F (2) with given initial conditions.
References[1] F. Pegoraro, P.J.Morrison, 2022, arXiv/2210.04254.
[2] Orion Ciftja, Results in Physics, 15, 102684 (2019).