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On the Yang-Lee distribution of roots
Hiis Hauge, E. Hemmer, P. C.
Physica 29 1338-1344
1963
ON THE YANG-LEE DISTRIBUTION OF ROOTS by E. HIlS HAUGE and P . C. HEMMER Instltutt for teor ctisk fysikk, Norges Tekniske Hags kole, Trond heirn, Norge
Synopsis The Yang-Lee distribution of roots for two examples of repulsive potentials, viz. one-dimensional hard rods and weak long range repulsion, is calculated. In both cases the root distribution is located on part of the negative real axis.
1. Irdmduction: Yang and Lee l ) have shown that the equation of state of a classical gas, condensation included, is completely determined by the distribution of roots of the grand partition function as function of the fugacity z, in the limit of an infinite volume. However, the knowledge of the properties of th ese distributions of roots for real gases seems t o be extremely meager. In this article the exact limiting distribution of roots for two extreme cases of purely r epulsive forces is determined. The first example is a gas of one-dimensional hard rods, and in the second example the particles interact with very weak repulsive forces of long range . In both cases the limiting root distribution falls on part of the negative real axis.
2. The electrostatic analogue. Let us recall the connect ion between the roots of the grand partition function Zg and the equation of state. Under the assumptions that the pair potential contained a hard core, was bounded from below and had a finite range, Yang and Lee l) proved the existence of X(z, T) = lim V-lIn Zg(z, V, T).
v_co
The equation of state is then given implicitly by th e two relations
P/kT =
(2)
X(z)
(3)
p = ZX' (z),
where p = l /v is the number density. The existence of the hard core d makes Zg a polynomial, of degree M(V) say. One can then express x(.~) in terms of the roots Zi, of Zg. X(z)
= lim - 1 lIl(V 1: )In ( 1 - - Z) = V-oo
V
i~l
Jg(s) ln (1 -
Zi
S
a -
1338 -
Z) ds.
-
(4)
ON THE YANG-LEE DISTRIBUTION OF ROOTS
1339
The last form arises if in the limit the zeros coalesce into lines C in the complex plane. Vg(s)ds is the number of roots in the line interval ds at z = s. The normalization is (5) J g(s)ds = M(V)IV, o
Lee and Yang pointed out that X(z) as given by (4) is the complex logarithmic potential of an electrostatic charge distribution on the lines C with the line charge density equal to g(s) per unit length. Separating the real and imaginary part of the potential 2)
x(x
+ iy)
= et« y)
+ i1Jl(x, y),
(6)
the lines f/>(x, y) -:- canst. are the equipotentials and the lines 1JI(x, y) = canst. are the lines of force. Using the Cauchy-Riemann relation f)(]>18y = = - 8Pl8x we deduce 8lJf
E y = - 1m (pjz) = - ----,
(7)
ax
Because the roots Zi outside the real axis must occur in conjugate pairs, the real axis is a line of symmetry of the charge distribution: f/>(x, -y) = (]>(x, y) 1JI(x, -y) = _lP(X, y).
(8)
In particular, if P(x, 0+) does not vanish for some intervals of x, it is different from P(x, 0-), and it follows that the corresponding part of the real axis is charged. The charge distribution can be evaluated as the line integral of the electric force, Q = (1/2n) ~ Eds, and we find g(x) = _1_ [Ey(x, 0+) _ Ey(x, 0-)] = _ ~
~ n
81J1(x, 0+) , b
(9)
by (7) and (8),
3. Hard rods. A gas of one-dimensional particles, interacting with a hard core of diameter d, obeys the well-known Tonks' equation of state-')
P=
kT v - d
=
kTp pd .
1-
(10)
Insertion of the equations (2) and (3) for the pressure and the density yields z(1
+ Xd)X' =
X'
(11 )
with solution x(z) eX(Z) 'd =
Z.
(12)
The integration constant is determined by the infinite dilution limit:
1340
E. HIlS HAUGE AND P. C. HEMMER
xlz -
1 when z --? O. Putting z = r exp (i~) and X = ep elimination of r]J: r = P cosec (f - Pd) e'l'd eotg (~- 'i'd)
+ ilfJ, we find
by
(13)
In fig. I the stream lines lff = constant are drawn. The interval O<':Psn/d covers the upper half plane, and -n/d < P < 0 the lower. (One has to take that interval of ':P which gives lJI = 0 at z = 0).
Fig. 1. Lines of forces from the charge distribution g(z) for hard rods.
The figure shows convincingly that the charge is distributed along part of the negative real axis. On the neg. real axis eq. (13) reduces to
x=
-
P e-wcotg qI/sin P
(14)
with
x = xd. P 5: n corresponds to x s - lie. P
=
lJId and
(15)
The interval 0 < The density there is given by eq. (9) gh'l'.(x)
= _ ~ 8P n
8x
=
~ ;It
sins P sin2 P - P sin 2 P
+ p2
e1f7cotg 1p (16)
One can easily assure oneself that (16) yields the complete root distribu-
1341
ON THE YANG-LEE DISTRIBUTION OF ROOTS
tion (of finite measure). It follows simply from -lied.
"fd
f
gh.r·(X) dx =
f
~
dP
= u«,
(17)
0
-co
in agreement with the normalization (5) for this case. The two equations (14) and (16) determine g(x) as function of x, and in fig. 2 this root distribution is shown. It diverges at x = -lIed as gh.r·(x) '""" (-1 - xed)-:. g(.)
1Ji
lO
I I I
I \
I
I I I I
0.5
I I
, I I
II I
-3
-2
-1
o.d
-1/0'
Fig. 2. The distribution of roots for hard rods.
By means of equation (4) one can verify that the resulting root distribution gives the correct equation of state. One has -lied.
X(z)
f
= gh.r·(x)ln
"
(1 - :) dx
=
~d
f
dqJln (
1-
X(~)) =
o
where we have used the equations (9) and (14). The value of the last integral follows from _1_ ,( t- k - 1 e k t dt = kk/k!
2ni j
(19)
,
(0+)
if .one chooses the path of integration as ItI = lJI/sin P, P
=
arg t, The
1342
E. HIlS HAUGE AND P. C. HEMMER
result is x(Z)
1
00
kk-l
00
d
k=l
kl
k=l
Zk [
= - - L --- (- Zd)k = L -
dk-1
-h! dX k- 1
e- kxrl
]
.
x=o
(20)
Appealing to the theorem of Lagr a nges), we conclude that X satisfies the equation in agreement with the result (12) obtained directly from the equation of state.
4. Weak and long range repulsion. Consider an intermolecular repulsion of the form ~(r) = y3 t(yr) (22)
t
where y is a very small parameter. We assume to be a positive integrable function. In the limit y -l>- 0, the equation of state corresponding to this potential approaches a P/kT = p + _p2, (23) 2
where a=
k~ Jd(l') ¢(r)
(24)
is a positive, y-independent constant. This is most readily deduced from the virial theorem 00
p=
J kTp -:3 2n
def>(r)
r3~1h(r;p) dr =
o 00
2n = kTp - 3
J
S3
o
dj(s) --ii2(S/y; p) ds. ds
(25)
We have put s = yr. Let now y -? O. At infinite separation ii2 -l>- p2, and by a partial integration the result (23) follows. In this case the density p as a function of the fugacity z is given by (26)
Comparing with equation (12) we see that the function p(z) = ZX' (z) in this case equals Xh.r·(z) for hard rods, with d replaced by a. Hence the function X(z) in this case is analytic everywhere except for :x s - 1lea on the real axis. The density g(x, a) is related to the density gh.r·(x, d) in a simple way.
ON THE YANG-LEE DISTRIBUTION OF ROOTS
1343
Using eq. (4) we have -llea
p(z)
=
zX'(z) =
Jgh.r·(x, a) In (1 - :) dx.
(27)
An integration by part gives
f+J
-llea
x(z)
=
x
gh.l',
-00
(~, a) d~ In (I
- : )ctx.
(28)
-ilea
Appealing again to eq. (4), we conclude that x
g(x) =
~
J
gh.r. (g, a) dg.
(29)
-Ilea
This result can also be written in the parametric form (see eq. (9)) g(x) =
-P/nx,
(30)
where P as function of x = xa is given by eq. (14). This relation follows also directly from the first equality in eq. (7), together with p = Xh . !·•• g(x)
ID
0.5
-3
-2
-1
-I/~
0
xa
Fig. 3. The limiting distribution of roots for weak and long range repulsion.
The function g(x) is everywhere finite and is shown in fig. 3. It is not normalizable, however, because it behaves like s]» for x -+ - 00. This is in accordance with the normalization (5), since in the absence of a hard core, the number M(V) of particles that can be contained in a volume V is infinite. 5. Conclusions. For two contrasting cases of repulsive potentials the limiting Yang-Lee distribution of roots of the grand partition function has been found, and in both cases part of the negative axis is the singular line. It is easy to speculate that this is always the case for repulsive potentials,
1344
ON THE YANG-LEE DISTRIBUTION OF ROOTS
and if so, the so-called Kirkwood phase transition in a gas of hard spheres would be ruled out. This, however, as well as the Kirkwood transition itself, is an open question. We want to emphasize that in the second example of weak repulsion of long range, the Yang-Lee theory does not apply, strictly speaking, because the potential has no hard core. Still there exists a function g(z) , related in the usual way (4) to the equation of state. It seems therefore that the YangLee theory could possibly be generalized to cases where the grand partition function belongs to a wider class of functions than polynomials. Received 22-3-63
REFERENCES 1) 2) 3) 4)
Yang, C. N. and Lee, T. D., Phys. Rev. 87 (1952) 404. Lee, T. D. and Yang, C. N., Phys. Rev. 87 (1952) 410. Tonks, L., Phys. Rev. 50 (1936) 955. Compo Whittaker, E. T. and Watson, G. N., A course in Modern Analysis (Cambridge 1958), p. 132.
Physica 29 1338-1344
1963
ON THE YANG-LEE DISTRIBUTION OF ROOTS by E. HIlS HAUGE and P . C. HEMMER Instltutt for teor ctisk fysikk, Norges Tekniske Hags kole, Trond heirn, Norge
Synopsis The Yang-Lee distribution of roots for two examples of repulsive potentials, viz. one-dimensional hard rods and weak long range repulsion, is calculated. In both cases the root distribution is located on part of the negative real axis.
1. Irdmduction: Yang and Lee l ) have shown that the equation of state of a classical gas, condensation included, is completely determined by the distribution of roots of the grand partition function as function of the fugacity z, in the limit of an infinite volume. However, the knowledge of the properties of th ese distributions of roots for real gases seems t o be extremely meager. In this article the exact limiting distribution of roots for two extreme cases of purely r epulsive forces is determined. The first example is a gas of one-dimensional hard rods, and in the second example the particles interact with very weak repulsive forces of long range . In both cases the limiting root distribution falls on part of the negative real axis.
2. The electrostatic analogue. Let us recall the connect ion between the roots of the grand partition function Zg and the equation of state. Under the assumptions that the pair potential contained a hard core, was bounded from below and had a finite range, Yang and Lee l) proved the existence of X(z, T) = lim V-lIn Zg(z, V, T).
v_co
The equation of state is then given implicitly by th e two relations
P/kT =
(2)
X(z)
(3)
p = ZX' (z),
where p = l /v is the number density. The existence of the hard core d makes Zg a polynomial, of degree M(V) say. One can then express x(.~) in terms of the roots Zi, of Zg. X(z)
= lim - 1 lIl(V 1: )In ( 1 - - Z) = V-oo
V
i~l
Jg(s) ln (1 -
Zi
S
a -
1338 -
Z) ds.
-
(4)
ON THE YANG-LEE DISTRIBUTION OF ROOTS
1339
The last form arises if in the limit the zeros coalesce into lines C in the complex plane. Vg(s)ds is the number of roots in the line interval ds at z = s. The normalization is (5) J g(s)ds = M(V)IV, o
Lee and Yang pointed out that X(z) as given by (4) is the complex logarithmic potential of an electrostatic charge distribution on the lines C with the line charge density equal to g(s) per unit length. Separating the real and imaginary part of the potential 2)
x(x
+ iy)
= et« y)
+ i1Jl(x, y),
(6)
the lines f/>(x, y) -:- canst. are the equipotentials and the lines 1JI(x, y) = canst. are the lines of force. Using the Cauchy-Riemann relation f)(]>18y = = - 8Pl8x we deduce 8lJf
E y = - 1m (pjz) = - ----,
(7)
ax
Because the roots Zi outside the real axis must occur in conjugate pairs, the real axis is a line of symmetry of the charge distribution: f/>(x, -y) = (]>(x, y) 1JI(x, -y) = _lP(X, y).
(8)
In particular, if P(x, 0+) does not vanish for some intervals of x, it is different from P(x, 0-), and it follows that the corresponding part of the real axis is charged. The charge distribution can be evaluated as the line integral of the electric force, Q = (1/2n) ~ Eds, and we find g(x) = _1_ [Ey(x, 0+) _ Ey(x, 0-)] = _ ~
~ n
81J1(x, 0+) , b
(9)
by (7) and (8),
3. Hard rods. A gas of one-dimensional particles, interacting with a hard core of diameter d, obeys the well-known Tonks' equation of state-')
P=
kT v - d
=
kTp pd .
1-
(10)
Insertion of the equations (2) and (3) for the pressure and the density yields z(1
+ Xd)X' =
X'
(11 )
with solution x(z) eX(Z) 'd =
Z.
(12)
The integration constant is determined by the infinite dilution limit:
1340
E. HIlS HAUGE AND P. C. HEMMER
xlz -
1 when z --? O. Putting z = r exp (i~) and X = ep elimination of r]J: r = P cosec (f - Pd) e'l'd eotg (~- 'i'd)
+ ilfJ, we find
by
(13)
In fig. I the stream lines lff = constant are drawn. The interval O<':Psn/d covers the upper half plane, and -n/d < P < 0 the lower. (One has to take that interval of ':P which gives lJI = 0 at z = 0).
Fig. 1. Lines of forces from the charge distribution g(z) for hard rods.
The figure shows convincingly that the charge is distributed along part of the negative real axis. On the neg. real axis eq. (13) reduces to
x=
-
P e-wcotg qI/sin P
(14)
with
x = xd. P 5: n corresponds to x s - lie. P
=
lJId and
(15)
The interval 0 < The density there is given by eq. (9) gh'l'.(x)
= _ ~ 8P n
8x
=
~ ;It
sins P sin2 P - P sin 2 P
+ p2
e1f7cotg 1p (16)
One can easily assure oneself that (16) yields the complete root distribu-
1341
ON THE YANG-LEE DISTRIBUTION OF ROOTS
tion (of finite measure). It follows simply from -lied.
"fd
f
gh.r·(X) dx =
f
~
dP
= u«,
(17)
0
-co
in agreement with the normalization (5) for this case. The two equations (14) and (16) determine g(x) as function of x, and in fig. 2 this root distribution is shown. It diverges at x = -lIed as gh.r·(x) '""" (-1 - xed)-:. g(.)
1Ji
lO
I I I
I \
I
I I I I
0.5
I I
, I I
II I
-3
-2
-1
o.d
-1/0'
Fig. 2. The distribution of roots for hard rods.
By means of equation (4) one can verify that the resulting root distribution gives the correct equation of state. One has -lied.
X(z)
f
= gh.r·(x)ln
"
(1 - :) dx
=
~d
f
dqJln (
1-
X(~)) =
o
where we have used the equations (9) and (14). The value of the last integral follows from _1_ ,( t- k - 1 e k t dt = kk/k!
2ni j
(19)
,
(0+)
if .one chooses the path of integration as ItI = lJI/sin P, P
=
arg t, The
1342
E. HIlS HAUGE AND P. C. HEMMER
result is x(Z)
1
00
kk-l
00
d
k=l
kl
k=l
Zk [
= - - L --- (- Zd)k = L -
dk-1
-h! dX k- 1
e- kxrl
]
.
x=o
(20)
Appealing to the theorem of Lagr a nges), we conclude that X satisfies the equation in agreement with the result (12) obtained directly from the equation of state.
4. Weak and long range repulsion. Consider an intermolecular repulsion of the form ~(r) = y3 t(yr) (22)
t
where y is a very small parameter. We assume to be a positive integrable function. In the limit y -l>- 0, the equation of state corresponding to this potential approaches a P/kT = p + _p2, (23) 2
where a=
k~ Jd(l') ¢(r)
(24)
is a positive, y-independent constant. This is most readily deduced from the virial theorem 00
p=
J kTp -:3 2n
def>(r)
r3~1h(r;p) dr =
o 00
2n = kTp - 3
J
S3
o
dj(s) --ii2(S/y; p) ds. ds
(25)
We have put s = yr. Let now y -? O. At infinite separation ii2 -l>- p2, and by a partial integration the result (23) follows. In this case the density p as a function of the fugacity z is given by (26)
Comparing with equation (12) we see that the function p(z) = ZX' (z) in this case equals Xh.r·(z) for hard rods, with d replaced by a. Hence the function X(z) in this case is analytic everywhere except for :x s - 1lea on the real axis. The density g(x, a) is related to the density gh.r·(x, d) in a simple way.
ON THE YANG-LEE DISTRIBUTION OF ROOTS
1343
Using eq. (4) we have -llea
p(z)
=
zX'(z) =
Jgh.r·(x, a) In (1 - :) dx.
(27)
An integration by part gives
f+J
-llea
x(z)
=
x
gh.l',
-00
(~, a) d~ In (I
- : )ctx.
(28)
-ilea
Appealing again to eq. (4), we conclude that x
g(x) =
~
J
gh.r. (g, a) dg.
(29)
-Ilea
This result can also be written in the parametric form (see eq. (9)) g(x) =
-P/nx,
(30)
where P as function of x = xa is given by eq. (14). This relation follows also directly from the first equality in eq. (7), together with p = Xh . !·•• g(x)
ID
0.5
-3
-2
-1
-I/~
0
xa
Fig. 3. The limiting distribution of roots for weak and long range repulsion.
The function g(x) is everywhere finite and is shown in fig. 3. It is not normalizable, however, because it behaves like s]» for x -+ - 00. This is in accordance with the normalization (5), since in the absence of a hard core, the number M(V) of particles that can be contained in a volume V is infinite. 5. Conclusions. For two contrasting cases of repulsive potentials the limiting Yang-Lee distribution of roots of the grand partition function has been found, and in both cases part of the negative axis is the singular line. It is easy to speculate that this is always the case for repulsive potentials,
1344
ON THE YANG-LEE DISTRIBUTION OF ROOTS
and if so, the so-called Kirkwood phase transition in a gas of hard spheres would be ruled out. This, however, as well as the Kirkwood transition itself, is an open question. We want to emphasize that in the second example of weak repulsion of long range, the Yang-Lee theory does not apply, strictly speaking, because the potential has no hard core. Still there exists a function g(z) , related in the usual way (4) to the equation of state. It seems therefore that the YangLee theory could possibly be generalized to cases where the grand partition function belongs to a wider class of functions than polynomials. Received 22-3-63
REFERENCES 1) 2) 3) 4)
Yang, C. N. and Lee, T. D., Phys. Rev. 87 (1952) 404. Lee, T. D. and Yang, C. N., Phys. Rev. 87 (1952) 410. Tonks, L., Phys. Rev. 50 (1936) 955. Compo Whittaker, E. T. and Watson, G. N., A course in Modern Analysis (Cambridge 1958), p. 132.