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A test for the first moment sum rule in the two-dimensional one-component plasma
15 January 1981
CHEMICAL PHYSICS LETTERS
Volume 85, number 3
COhfWENT A TEST FOR THE FIRST MOMENT SUM RULE IN THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA L BLUhl* Dcparrmcrrr
of P~I_ISICS Umersrf_v of P~rrrro RICO RIO hedras.
Puerto RICO 00931,
USA
Jlld
D HENDERSON * IRJI Rcscarch Lahoraror_l . San Rrccwrd
29 Scptcmber
198
Jose Cahfonrra
95193.
US-1
I
The recent cuct solution oi lhc one-component two-dunenslonal mhomogencous conswent wth the sum rule for the iis. moment oi the polr correlation funcllon
III J rcant fld
conununtcat~un
[ I ] WCdcrtvcd n suni rule for the
pair
correlation
plasma by Jmcovm IS shown to be
function
of an electrolyte
near a
ch.rrged tntcrf.rcc
Thus sum rule. which IS tn contrJdtcIton to the ones dcrtved by Gruber et al of long-raugcd corrcldttons m the mhomogcneous regton nc.rr the flat interface. aP,(=I -
1
aE
171. and so tmpltes the extstcncc 1s
,,I =Pp,(‘I),~
e,J&
-
:l)P,(=&(l
2) %
wltcrc p,(zt)
IS tllc smglct densrty 01 tome spcctes
I at 3 dtstancc zl from the charged wall. E
clcctnc
P = I/knT
fxtor.
kld.
corrclatton For
IS
the Boltzntann
thermal
e,
IS
IS the
ektcrnal
the electrtc charge of tons]. and Ir,,(1.2)
appltcd
IS the pair
Functton
IIIC
~nl~o~~~ogcncous
tltng bxhground.
one-component
plasma tn two dtmenstons (with
w1t11 bulk density pb = I/a.
a logartthmic
interactton)
on a neutral-
and plasma coupltng constant
r=flez=z_
(3
13111~sfound that
ktncovic:
p(z)
satdies
eucrly
(3) whcrc o qs = -oe
IS
the surface chxge
m electron
umts. or
(4
= E/2x.
wnh qs IIIC surface charge per umt area and a(r) l
Supported ta part by NST Cnnt
374
No CHE80-01969
ts the error function,
CHEMICAL PHYSICS LETTERS
Volume 85. number 3
The parr correlatron
function
IS January 1981
IS
PT(l.7_)=P(Z&(Z2)II(l.2)
= ev[-(zt
- z2)21(41n3)
Constdcr now 111~M-t-hand
(6) stdc of (I )
P= j= dz2 r P~(I.~_)(:~ -=l)d_v2 -Em 0
=_-
:3[
(=2 - =I)
e\p[-(z2
(7)
- z1)2]
dz2_[r,z,,
eap{--f7 - 7_-“2(=t +=d”} x _<,I2 R(I - 1 +@(r2)
cxp(-[rt
- 2”2(zt
- c,p[_7’,q,.7(,, dt2 / _a2
I + aqf,)
+ =2)1’3
dr,
_ r,), d,,, _
w
But _Y ClPl
_= so Ill31
Y,,,(t,- _
- tz)j d_v2= 7%6(t1 -
- ‘a)
(9)
(10)
Exchangmg _f
-is-l12
the order m the mtcgratton. c-S2
and usmg
&=1+@(t).
(I I)
--I we get exp(-
P=e[I lntegrattng
p=_l
+ .I’?- [ I + +(t)](t
- 3”rz,)}
dr
_
+*(t)]”
by parts
112 exp[--2(an
() L
H 2n
+ zt)2]
1 + @(-21/2,,)
whtch satisfies eq. (1). one [2],
tz + Z3/‘z,) Ic_t2
I ap =_-AZ___. 4s aa
the sum rule of Blum et al
e ap
(13)
2 aE
[ 11. We therefore
conclude
that thts sum rule. and not the cariter
Is correct. 375
Volume
85
numbs
CHEMICAL
3
We thank. Professors B Jancovtc~, J.L. Lebowttz
PHYSICS
15 January 1981
LE-ITERS
and hi S. Wcrthetm
for very useful comments.
References [ 11 L Ulum. D Henderson, J L Lebowtz, Ch Grubcr and Ph A blxtm, J. Chcm [2] Ch Grubcr J L Lcboalrrand Ph A Martm. J Chem Phys. 75 (1981) 94-I [ 3 1 B Jancovw J Phys 0’~s) 42 (198 1) L223. prwatc commumalion
Phys ,
to be
pubhshed.
ERRATUM
W L Faust, L S Goldberg,
B B Cratg and R G W~ss, Ttme-resohed
emtsston from short-pulse
tton tnschantsms.
Chem
fragmentatton
Phys
Letters
Cz Swan
of CO- ekldence for two Cz forma-
83 (1981)
7-65
in the process of pubhcatton. the cttatton of authors and affiltattons was modtfied front the form presented m the ongmal manuscrtpt The correct cttatton to be restored IS
B B CRAIG Nod
* Dzpartrtret~r
316
*. W L FAUST,
Rcseorclr Loborarory
L S GOLDBERG
IVash:r~gro~r.DC 10375
and R G WEISS * US-I
of ChenrrsrryGcorgerowrz U~II ersrr), I~tdmg~orr.DC
XlOS7.
USA
CHEMICAL PHYSICS LETTERS
Volume 85, number 3
COhfWENT A TEST FOR THE FIRST MOMENT SUM RULE IN THE TWO-DIMENSIONAL ONE-COMPONENT PLASMA L BLUhl* Dcparrmcrrr
of P~I_ISICS Umersrf_v of P~rrrro RICO RIO hedras.
Puerto RICO 00931,
USA
Jlld
D HENDERSON * IRJI Rcscarch Lahoraror_l . San Rrccwrd
29 Scptcmber
198
Jose Cahfonrra
95193.
US-1
I
The recent cuct solution oi lhc one-component two-dunenslonal mhomogencous conswent wth the sum rule for the iis. moment oi the polr correlation funcllon
III J rcant fld
conununtcat~un
[ I ] WCdcrtvcd n suni rule for the
pair
correlation
plasma by Jmcovm IS shown to be
function
of an electrolyte
near a
ch.rrged tntcrf.rcc
Thus sum rule. which IS tn contrJdtcIton to the ones dcrtved by Gruber et al of long-raugcd corrcldttons m the mhomogcneous regton nc.rr the flat interface. aP,(=I -
1
aE
171. and so tmpltes the extstcncc 1s
,,I =Pp,(‘I),~
e,J&
-
:l)P,(=&(l
2) %
wltcrc p,(zt)
IS tllc smglct densrty 01 tome spcctes
I at 3 dtstancc zl from the charged wall. E
clcctnc
P = I/knT
fxtor.
kld.
corrclatton For
IS
the Boltzntann
thermal
e,
IS
IS the
ektcrnal
the electrtc charge of tons]. and Ir,,(1.2)
appltcd
IS the pair
Functton
IIIC
~nl~o~~~ogcncous
tltng bxhground.
one-component
plasma tn two dtmenstons (with
w1t11 bulk density pb = I/a.
a logartthmic
interactton)
on a neutral-
and plasma coupltng constant
r=flez=z_
(3
13111~sfound that
ktncovic:
p(z)
satdies
eucrly
(3) whcrc o qs = -oe
IS
the surface chxge
m electron
umts. or
(4
= E/2x.
wnh qs IIIC surface charge per umt area and a(r) l
Supported ta part by NST Cnnt
374
No CHE80-01969
ts the error function,
CHEMICAL PHYSICS LETTERS
Volume 85. number 3
The parr correlatron
function
IS January 1981
IS
PT(l.7_)=P(Z&(Z2)II(l.2)
= ev[-(zt
- z2)21(41n3)
Constdcr now 111~M-t-hand
(6) stdc of (I )
P= j= dz2 r P~(I.~_)(:~ -=l)d_v2 -Em 0
=_-
:3[
(=2 - =I)
e\p[-(z2
(7)
- z1)2]
dz2_[r,z,,
eap{--f7 - 7_-“2(=t +=d”} x _<,I2 R(I - 1 +@(r2)
cxp(-[rt
- 2”2(zt
- c,p[_7’,q,.7(,, dt2 / _a2
I + aqf,)
+ =2)1’3
dr,
_ r,), d,,, _
w
But _Y ClPl
_= so Ill31
Y,,,(t,- _
- tz)j d_v2= 7%6(t1 -
- ‘a)
(9)
(10)
Exchangmg _f
-is-l12
the order m the mtcgratton. c-S2
and usmg
&=1+@(t).
(I I)
--I we get exp(-
P=e[I lntegrattng
p=_l
+ .I’?- [ I + +(t)](t
- 3”rz,)}
dr
_
+*(t)]”
by parts
112 exp[--2(an
() L
H 2n
+ zt)2]
1 + @(-21/2,,)
whtch satisfies eq. (1). one [2],
tz + Z3/‘z,) Ic_t2
I ap =_-AZ___. 4s aa
the sum rule of Blum et al
e ap
(13)
2 aE
[ 11. We therefore
conclude
that thts sum rule. and not the cariter
Is correct. 375
Volume
85
numbs
CHEMICAL
3
We thank. Professors B Jancovtc~, J.L. Lebowttz
PHYSICS
15 January 1981
LE-ITERS
and hi S. Wcrthetm
for very useful comments.
References [ 11 L Ulum. D Henderson, J L Lebowtz, Ch Grubcr and Ph A blxtm, J. Chcm [2] Ch Grubcr J L Lcboalrrand Ph A Martm. J Chem Phys. 75 (1981) 94-I [ 3 1 B Jancovw J Phys 0’~s) 42 (198 1) L223. prwatc commumalion
Phys ,
to be
pubhshed.
ERRATUM
W L Faust, L S Goldberg,
B B Cratg and R G W~ss, Ttme-resohed
emtsston from short-pulse
tton tnschantsms.
Chem
fragmentatton
Phys
Letters
Cz Swan
of CO- ekldence for two Cz forma-
83 (1981)
7-65
in the process of pubhcatton. the cttatton of authors and affiltattons was modtfied front the form presented m the ongmal manuscrtpt The correct cttatton to be restored IS
B B CRAIG Nod
* Dzpartrtret~r
316
*. W L FAUST,
Rcseorclr Loborarory
L S GOLDBERG
IVash:r~gro~r.DC 10375
and R G WEISS * US-I
of ChenrrsrryGcorgerowrz U~II ersrr), I~tdmg~orr.DC
XlOS7.
USA