Interaction between two rows of localized adsorption sites in a 2D one-component plasma
We compute the free energy for two rows of localized adsorption sites embedded in a two dimensional one-component plasma with neutralizing background density ρ. The interaction energy between the adsorption sites is repulsive. We also compute the average occupation number of the adsorption sites...
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Інститут фізики конденсованих систем НАН України
2005
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| Cite this: | Interaction between two rows of localized adsorption sites in a 2D one-component plasma / C.D. Santangelo, L. Blum // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 325–334. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-1196042017-06-08T03:07:51Z Interaction between two rows of localized adsorption sites in a 2D one-component plasma Santangelo, C.D. Blum, L. We compute the free energy for two rows of localized adsorption sites embedded in a two dimensional one-component plasma with neutralizing background density ρ. The interaction energy between the adsorption sites is repulsive. We also compute the average occupation number of the adsorption sites and compare it to the result for a single row of sites. The exact result indicates that the discretization does not induce charge asymmetry and no attractive forces occur. Ми розраховуємо вільну енергію для двох рядів локалізованих адсорбційних центрів, вставлених у двовимірну однокомпонентну плазму з густиною нейтралізуючого фону ρ. Енергія взаємодії між адсорбційними центрами є відштовхувальною. Ми також розраховуємо середнє число заповнення адсорбційних центрів і порівнюємо його з результатом для одного ряду центрів. Точний результат показує, що дискретизація не спричиняє зарядової асиметрії і не виникають сили притягання. 2005 Article Interaction between two rows of localized adsorption sites in a 2D one-component plasma / C.D. Santangelo, L. Blum // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 325–334. — Бібліогр.: 11 назв. — англ. 1607-324X DOI:10.5488/CMP.8.2.325 PACS: 61.20.Gy http://dspace.nbuv.gov.ua/handle/123456789/119604 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We compute the free energy for two rows of localized adsorption sites
embedded in a two dimensional one-component plasma with neutralizing
background density ρ. The interaction energy between the adsorption sites
is repulsive. We also compute the average occupation number of the adsorption
sites and compare it to the result for a single row of sites. The
exact result indicates that the discretization does not induce charge asymmetry
and no attractive forces occur. |
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Santangelo, C.D. Blum, L. |
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Santangelo, C.D. Blum, L. Interaction between two rows of localized adsorption sites in a 2D one-component plasma Condensed Matter Physics |
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Santangelo, C.D. Blum, L. |
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Santangelo, C.D. |
| title |
Interaction between two rows of localized adsorption sites in a 2D one-component plasma |
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Interaction between two rows of localized adsorption sites in a 2D one-component plasma |
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Interaction between two rows of localized adsorption sites in a 2D one-component plasma |
| title_fullStr |
Interaction between two rows of localized adsorption sites in a 2D one-component plasma |
| title_full_unstemmed |
Interaction between two rows of localized adsorption sites in a 2D one-component plasma |
| title_sort |
interaction between two rows of localized adsorption sites in a 2d one-component plasma |
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Інститут фізики конденсованих систем НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/119604 |
| citation_txt |
Interaction between two rows of localized adsorption sites in a 2D one-component plasma / C.D. Santangelo, L. Blum // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 325–334. — Бібліогр.: 11 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT santangelocd interactionbetweentworowsoflocalizedadsorptionsitesina2donecomponentplasma AT bluml interactionbetweentworowsoflocalizedadsorptionsitesina2donecomponentplasma |
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2025-07-08T16:15:11Z |
| last_indexed |
2025-07-08T16:15:11Z |
| _version_ |
1837096041888874496 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 2(42), pp. 325–334
Interaction between two rows of
localized adsorption sites in a 2D
one-component plasma
C.D.Santangelo 1 , L.Blum 2
1 Department of Physics,
University of Pennsylvania,
Philadelphia, 19104
2 Department of Physics,
Faculty of Natural Sciences,
University of Puerto Rico,
Rio Piedras, Puerto Rico 00931
Received December 2, 2004
We compute the free energy for two rows of localized adsorption sites
embedded in a two dimensional one-component plasma with neutralizing
background density ρ. The interaction energy between the adsorption sites
is repulsive. We also compute the average occupation number of the ad-
sorption sites and compare it to the result for a single row of sites. The
exact result indicates that the discretization does not induce charge asym-
metry and no attractive forces occur.
Key words: one-component plasma, localized adsorption, DNA attraction
PACS: 61.20.Gy
1. Introduction
The subtleties of electrostatics in condensed matter theory represent a formidable
and never ending challenge. One topic of much recent activity, has been the attrac-
tion between two macromolecules of the same charge [1]. One mechanism that has
been proposed invokes charge asymmetry related to the formation of lattices or
Wigner crystals [2,3]. One problem with this picture is that it will create a dipole
that is inconsistent with the perfect screening sum rules (Blum et al.[4]). While the
formation of Wigner crystals under special conditions is an experimental fact, the
question of the large asymmetry in the charge distribution needs clarification. As
has happened in the past the exact solution of the two dimensional Jancovici mod-
el [5] can provide an unambiguous answer to the puzzle. The interaction between
two equally charged lines ( which are charged surfaces in 2 dimensions!) has been
c© C.D.Santangelo, L.Blum 325
C.D.Santangelo, L.Blum
discussed using the exact solution, and is always repulsive [6]. Here, we study a dis-
cretized version of this problem, namely two lines of discrete adsorption sites, where
the adsorption potential is given by the Baxter [7] sticky potential. To do this we
extend the localized adsorption model of a single line [8] to the case of two lines of
discrete adsorption sites. This extension is non-trivial, and as in a similar case dis-
cussed in the past, has a simple solution for what we would call a “commensurate”
lattice [9], namely the spacing of the adsorption sites is such that the background
charge of the enclosed area corresponds to an entire number of discrete charges.
2. Formalism
2.1. Modeling the adsorption
Following Rosinberg et al. [8], the adsorption potential for a sticky site located
at the origin, given by ua(r), is modeled as
exp [−βua(r)] = 1 + λδ(r), (1)
where λ is a positive constant that measures the strength of the adsorption poten-
tial [7].
The partition function for a system of adsorption sites with locations given by
the vectors Rm is given by
ZN =
1
N !
∫
e−βV0(r1,··· ,rN )
N
∏
i=1
{[
1 + λ
M
∑
m=1
δ(ri − Rm)
]
d2ri
}
, (2)
where V0 is the potential energy of the one-component plasma in the absence of
adsorption sites. Expanding in powers of λ, it has been shown [8] that the partition
function can be written in terms of the n-point correlation functions as
ZN = Z0
N
∑
s
M
∑
m1,m2,···ms=1
λs
s!
ρ(Rm1
, · · ·Rms
), (3)
where Z0
N is the partition function of the unperturbed system.
The difference in free energy from the unperturbed system is the logarithm of
ZN/Z0
N , and is given by
∆F = −kBT
∑
s>1
λs
s!
Ts, (4)
where
Ts =
M
∑
m1,··· ,ms=1
ρT(Rm1
, · · · ,Rm2
), (5)
and where ρT gives the truncated n-body correlation functions,
ρT(r1) = ρ(r1),
ρT(r1, r2) = ρ(r1, r2) − ρ(r1)ρ(r2),
ρT(r1, r2, r3) = ρ(r1, r2, r3) − ρ(r1, r2)ρ(r3) − · · · (6)
326
Interaction between two rows of localized adsorption sites
2.2. Correlation functions of the one-component plasma
The exact solution to the one-component plasma found by Jancovici [5] for cou-
pling parameter Γ = Z2e2/(kBT ) = 2, where Z is the ion valence and e is the
elementary charge of an electron. The n-point density correlation functions are giv-
en by
ρ(r1, · · · rn) = ρn det
[
e−πρ(|zµ|2+|zγ |2)/2+πρzµz̄γ
]
∣
∣
∣
∣
µ,γ=1,··· ,n
. (7)
Here, ρ is the background charge density, z = x + iy where x and y describe coordi-
nates on the plane, and z̄ its complex conjugate. After some algebraic manipulation,
we can rewrite this expression as
ρ(r1, · · · rn) = ρn
∑
σ∈Sn
sgn(σ)
n
∏
j=1
ρ0(rj − rσ(j))ρX(rj, rσ(j)), (8)
where Sn is the group of permutations on n letters,
ρ0(r) = exp
(
−
πρ
2
r
2
)
, (9)
and ρX(r, r′) = exp [−iπρ (r × r
′) · ẑ] . (10)
The truncated correlation functions, ρT(r1, · · · , rn) are computed by restricting the
sum over σ in equation (8) to n-cycles.
d
a
y
x
Figure 1. Two infinite lines of adsorption sites separated by a distance d and
having periodicity a.
We suppose that the two lines of sticky sites with periodicity a are separated
by a distance d (see figure 1). We describe the sticky site locations by introducing
integer variables ni and Ising variables δi that take on a value of either 0 or 1. Then
any Ri can be written as
Ri = anix̂ + dδiŷ. (11)
Since we are calculating the sums over all the positions of the particles, all n-
cycles are equivalent by a suitable relabeling of the summation indices. This leads
to the general expression
Ts = ρs(s − 1)!(−1)s
∑
n1,···
∑
δ1,···
ρ0(Rm1,δ1 − Rm2,δ2)ρ0(Rm2,δ2 − Rm3
, δ3) · · ·
×ρ0(Rms,δs
−Rm1,δ1)ρX(Rm1,δ1,Rm2,δ2)ρX(Rm2,δ2 ,Rm3,δ3) · · ·ρX(Rms,δs
,Rm1,δ1).
(12)
327
C.D.Santangelo, L.Blum
3. Free energy
We first consider ρX(R1,R2), and substitute in the adsorption site positions from
equation (11). This gives
ρX(R1,R2) = e−iπρad(n1δ2−δ1n2). (13)
For the particular choice of background charge density
ρ = 2m/(ad), (14)
where m is a positive integer, ρX = 1 when evaluated on the adsorption sites. We
therefore specialize to densities where this simplification occurs.
We also find that (R1 − R2)
2 = a2(n1 − n2)
2 + d2(δ1 − δ2)
2, leading to
Ts = ρs(s − 1)!(−1)s
∑
n1,···
e−t[(n1−n2)2+···+(ns−n1)2]
∑
δ1,···
e−t′[(δ1−δ2)2+···+(δs−δ1)2], (15)
where t = πρa2/2 and t′ = πρd2/2. Since the sum over the ni and the sum over the
δi decompose, we can evaluate the sum over δi using a transfer matrix. We define
the transfer matrix, T , to have the components
T =
(
1 e−πρd2/2
e−πρd2/2 1
)
. (16)
Then
Ts = ρs(s − 1)!(−1)str(T s)
∑
n1,···
e−t[(n1−n2)2+···+(ns−n1)2]. (17)
Diagonalizing T gives the eigenvalues λ± = 1±e−πρd2/2, allowing us to take the trace
easily. Notice that the decoupling of the Ising variables, δi and integer variables, ni
only decouple at densities given by equation (14). At other densities, these additional
couplings between the ni and δi complicate the evaluation of the transfer matrix
trace.
The sum over the ni can be expressed in terms of Jacobi theta functions [8],
where the Jacobi theta function is defined as
θ3(ζ, t) =
∞
∑
n=−∞
e−tn2
e2iπnζ . (18)
First, notice that
∫
dζ θs
3(ζ, t) =
∑
n1,··· ,ns−1
e−t[n2
1
+n2
2
+···+(n1+n2+··· )2] (19)
for s > 1 and
∫
dζ θ3(ζ, t) = 1. This leads to the expression
Ts = ρs(s− 1)!(−1)s
[
(1 + e−πρd2/2)s + (1 − e−πρd2/2)s
]
∫ 1
0
dζ θ3
(
ζ, πρa2/2
)s
, (20)
328
Interaction between two rows of localized adsorption sites
Substituting this into equation (4) leads to a sum of two series, both of which are
absolutely summable when |λρ(1+e−πρd2/2)θ(0, t)| < 1. By analytic continuation, we
extend this sum to the full range of parameters, leading to the free energy difference
between the OCP with and without adsorption sites given by
∆f = −
kBT
a2
∫ 1
0
dζ ln
[
{
1 + λρe−(πρd2/2)θ3
(
ζ, πρa2/2
)
}2
− λ2ρ2e−πρd2
θ2
3
(
ζ, πρa2/2
)
]
. (21)
In the limit that d → ∞, we expect the free energy to be a sum of the free
energies of two independent lines of sticky sites. Indeed, this limit yields
∆fd→∞ = −2
kBT
a2
∫ 1
0
dζ ln
[
1 + λρθ3(ζ, πρa2/2)
]
. (22)
In the opposite limit, d → 0, we expect the free energy to agree with that of a single
line of adsorption sites with a potential given by 2λ. It is easy to see that the free
energy in this limit is
∆fd→0 = −
kBT
a2
∫ 1
0
dζ ln
[
1 + 2λρθ3(ζ, πρa2/2)
]
. (23)
Written in terms of the dimensionless constants t = πρa2/2 = πa/d, t′ =
πρd2/2 = πd/a and Λ = λρ = 2λ/(ad), we find the change in free energy as the
adsorption sites approach each other, ∆F = ∆f − ∆fd→∞, to be given by
∆F = −
kBT
a2
∫ 1
0
dζ ln
[
1 + Λ(1 + e−t′)θ3(ζ, t)
1 + Λθ3(ζ, t)
]
−
kBT
a2
∫ 1
0
dζ ln
[
1 + Λ(1 − e−t′)θ3(ζ, t)
1 + Λθ3(ζ, t)
]
. (24)
Recall that this free energy is only valid when ρad/2 = m for any positive integer
m. Thus, we can compare the free energy of states with the same lattice constant
a and background charge density ρ, only for integer multiples of some specific valid
separation d. To be more specific, suppose we have the free energy at some density
such that ρad/2 = 1 and separation d, then at separation md in the same background
density, we have ρad/2 = m. Thus, the free energy formula equation (21) will be
valid only for integer multiples of d.
In figure 2, we plot ∆F as a function of d for two values of the background density,
ρ, given by ρad0/2 = 1 and ρad0/2 = 5, where d0 = 0.1a is the smallest value of
d. The lines of adsorption sites are always repulsive. For larger d, ∆F becomes zero
quickly. In figure 3, we plot ∆F as a function of d for different values of λa2. As
the depth of the potential increases, the repulsive strength of the interaction also
increases. This is indicative of the adsorption sites pinning charge on them, leading
to their repulsion.
329
C.D.Santangelo, L.Blum
0.002 0.004 0.006 0.008 0.01
d
2
4
6
8
10
D
F
ê
H
k
B
T
L
Figure 2. A typical set of free energies ∆f − ∆fd→∞ with background density
ρad0/2 = 1 (circles, solid lines) and ρad0/2 = 5 (stars, dotted lines). Here, d is
measured in units of a and is an integer multiple of 0.1a. We further set λa2 = 100.
The lines are guides to the eye.
0.002 0.004 0.006 0.008 0.01
d
2
4
6
8
10
D
F
ê
H
k
B
T
L
Figure 3. A typical set of free energies ∆f − ∆fd→∞ with background density
ρad0/2 = 1 for values of λa2 = 0.001 (squares, dashed line), 0.1 (stars, dotted
line), 10 (circles, solid line). Here, d is measured in units of a and is an integer
multiple of 0.1a. The lines are guides to the eye.
Finally, we compute the average occupation number of a site. This is given by [8]
〈n〉 = −a2λ
∂
∂λ
∆f
kBT
. (25)
This is plotted as a function of λa2 for in figure 4 for two lines with separation
d = 0.1a and ρad/2 = 1 (solid line) and for a single line of adsorption sites (dashed
line).
It is clear from figure 4 that the repulsion inhibits the adsorption of the ions.
However, the separation d = 0.1a is very small. At separations on the order of the
site spacing, there is no appreciable difference in the fraction of occupied sites as a
function of λ.
330
Interaction between two rows of localized adsorption sites
0.02 0.04 0.06 0.08 0.1
0.2
0.4
0.6
0.8
1
<
n
>
l a 2
Figure 4. The fraction of occupied sites as a function of λa2 for two lines separated
by a distance d = 0.1a with ρad/2 = 1 (solid). This is compared to the fraction of
occupied sites for a single line of adsorption sites at the same background density
(dashed line).
4. Average density
In this section, we will compute the average counterion density at an arbitrary
point R0. We can find the density directly by fixing the position of one of the ions.
This will require the computation of the quantity
Ts(R0) = ρs+1
∑
R1
· · ·
∑
Rs
s
∏
j=0
ρ0(Rj ,−Rσ(j))ρX(Rj,Rσ(j)), (26)
where the sum is over all s-cycles σ. Then the average density is given by [8]
〈ρ(x0, y0)〉 − ρ =
∞
∑
s=1
λs
s!
Ts(R0), (27)
where x0 and y0 are the components of R0.
For πρad = 2πn, n an integer, and for Rj = anj x̂ + dδj ŷ, equation (26) decom-
poses into the product
Ts = ρs+1s!(−1)sT (1)
s T (2)
s , (28)
where
T (1)
s =
∑
n1···ns
e−πρx2
0
/2e−(πρa2/2)[n2
1
+(n1−n2)2+···+(ns−1−ns)2+n2
s ]e−πρax0(n1+ns)+iπρay0(n1−ns)
(29)
and
T (2)
s =
∑
δ1···δs
e−πρy2
0
/2e−(πρd2/2)[δ2
1
+(δ1−δ2)2+···+δ2
s ]e−πρdy0(δ1+δs)eiπρd(δ1−δs). (30)
331
C.D.Santangelo, L.Blum
Defining z = x0 + iy0 and z̄ its complex conjugate, we find that
T (1)
s =
∑
n1···ns
e−πρx2
0
/2e−t[n2
1
+···+n2
s+(n1+···ns)2]e−πρa(n1z−nsz̄). (31)
Using the transfer matrix T and defining a new diagonal matrix,
M =
(
1 0
0 e−πρd2/2+πρdy0+iπρdx0
)
, (32)
and its complex conjugate M̄, we find
T (2)
s = e−πρy2
0
/2tr(T s−1M̄TM) (33)
for s > 1 and T
(2)
0 = e−πρy2
0
/2. Evaluating T
(2)
s gives
T (2)
s =
[
λs−1
+ G+(z) + λs−1
− G−(z)
]
, (34)
where
4G+(z) = exp
(
−πρy2
0/2
) (
λ+|1 + A|2 + λ−|1 − A|2
)
and
4G−(z) = exp
(
−πρy2
0/2
) (
λ−|1 + A|2 + λ+|1 − A|2
)
,
and
A = exp
(
−πρd2/2 + πρdy0 + iπρdx0
)
.
Using the results of Rosinberg et al. [8], we can compute T
(2)
s in terms of
F (ζ, z, t) =
1
2
{
θ3
[
ζ +
iz̄t
πa
, t
]
θ3
[
ζ +
izt
πa
, t
]
+ θ3
[
ζ −
iz̄t
πa
, t
]
θ3
[
ζ −
izt
πa
, t
]}
.
This gives
Ts = (−1)ss!ρs+1
[
λs−1
+ G+(z) + λs−1
− G−(z)
]
e−πx2
0
/2
∫ 1
0
dζ θs−1
3 (ζ, t)F (ζ, z, t). (35)
Using equation (27), we find the average density
〈ρ(z)〉 − ρ = −λG+(z)ρ2e−πρx2
0
∫ 1
0
dζ
F (ζ, z, t)
1 + λρλ+θ3(ζ, t)
− λG−(z)ρ2e−πρx2
0
∫ 1
0
dζ
F (ζ, z, t)
1 + λρλ−θ3(ζ, t)
. (36)
Finally, we note that
4G+(x0, y0) = λ+
{
e−πρy2
0 + e−πρ(y0−d)2 + 2e−πρ[y2
0
+(y0−d)2]/2 cos(πρdx0)
}
+ λ−
{
e−πρy2
0 + e−πρ(y0−d)2 − 2e−πρ[y2
0
+(y0−d)2]/2 cos(πρdx0)
}
, (37)
332
Interaction between two rows of localized adsorption sites
-1 -0.5 0.5 1
x
0
-40
-30
-20
-10
<r>-r
Figure 5. 〈ρ〉 − ρ as a function of y0 for x0 = 0 in units of a. Here ρad = 2n and
λρ = 8. We show d = 0.25a (solid), d = 0.5a (dashed), and d = 1a (dotted).
and
4G−(x0, y0) = λ−
{
e−πρy2
0 + e−πρ(y0−d)2 + 2e−πρ[y2
0
+(y0−d)2]/2 cos(πρdx0)
}
+ λ+
{
e−πρy2
0 + e−πρ(y0−d)2 − 2e−πρ[y2
0
+(y0−d)2]/2 cos(πρdx0)
}
. (38)
Notice that, for πρad = 2πn, the periodicity of G±(x0, y0) in x0 is always com-
mensurate with the lattice spacing a. In figure 5, we plot 〈ρ〉 − ρ as a function of y0
for x0 = 0, ρad = 2n, and λρ = 8 and for a variety of different spacings. Notice that
the density has two peaks for large separations but a single peak as the separation
d becomes smaller than the periodicity a. As a function of x0, the density is always
periodic with period a.
5. Discussion
The main conclusion of our calculation is that for the geometry that we have
chosen no attractive forces are induced by the discrete structure of the charged line.
The derivation of ∆F is valid for d taking values that are integer multiples of
2/(aρ). Further, equation (21) gives the correct free energy for d = 0 (as we have
already seen). It is conceivable, then, that equation (21) is correct for all values of
ρ and d.
One of the features of this model that makes an exact evaluation possible is that
a density ρ can be found such that the Ising variables δi and the integer variables
ni are uncoupled. When ρad 6= 2m for an integer m, a coupling does indeed arise
that makes the computation of the free energy more difficult. Additionally, if the
adsorption sites are not aligned, an additional x̂ δi ∆a component arises in Ri, where
∆a measures the degree of misalignment. This component introduces a coupling
between the ni and δi that will be discussed in future work.
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C.D.Santangelo, L.Blum
6. Acknowledgements
L.B. wants to acknowledge the warm hospitality of Fyl Pincus at the MRL of the
University of California in Santa Barbara, where this project was started. The idea
of this research was inspired during discussions with Fyl. Support from NSF through
grant DMR02–03755 and DOE grant DE–FG02–03ER 15422 is also acknowledged.
References
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Взаємодія між двома рядами локалізованих
адсорбційних центрів у двовимірній
однокомпонентній плазмі.
К.Д.Сантанджело 1 , Л.Блюм 2
1 Університет Пенсільванії, Філадельфія 19104, США
2 Університет Пуерто Ріко, Пуерто Ріко
Отримано 2 грудня 2004 р.
Ми розраховуємо вільну енергію для двох рядів локалізованих
адсорбційних центрів, вставлених у двовимірну однокомпонентну
плазму з густиною нейтралізуючого фону ρ. Енергія взаємодії між
адсорбційними центрами є відштовхувальною. Ми також розрахов-
уємо середнє число заповнення адсорбційних центрів і порівнюємо
його з результатом для одного ряду центрів. Точний результат
показує, що дискретизація не спричиняє зарядової асиметрії і не
виникають сили притягання.
Ключові слова: однокомпонентна плазма, локалізована адсорбція
PACS: 61.20.Gy
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