The Cluster Expansion for the Self-Gravitating gas and the Thermodynamic Limit

0.22 MB·English

by  H. J. de Vega

The Cluster Expansion for the Self-Gravitating gas and the Thermodynamic Limit
H. J. de Vega(a) and N. G. S´anchez(b)
(a) Laboratoire de Physique Th´eorique et Hautes Energies,
Universit´e Paris VI et VII, Tour 16, 1er ´etage, 4, Place Jussieu,
75252 Paris, cedex 05, FRANCE. Laboratoire Associ´e au CNRS UMR 7589. and
(b) Observatoire de Paris, LERMA,
61, Avenue de l’Observatoire, 75014 Paris, FRANCE.
Laboratoire Associ´e au CNRS UMR 8112.
(Dated: February 2, 2008)
We develop the cluster expansion and the Mayer expansion for the self-gravitating thermal gas
and prove the existence and stability of the thermodynamic limit N,V → ∞ with N/V 31 fixed.
Theessential(dimensionless)variableishereη≡ Gm2N (whichiskeptfixedinthethermodynamic
1
V3 T
4 limit). Wesucceed in thisway toobtain theexpansion of thegrand canonical partition function in
0 powers of the fugacity. The corresponding cluster coefficients behave in the thermodynamic limit
20 as Nη j−1 cj wherecj arepurenumbers. Theyareexpressedasintegralsassociated totreecluster
diagrams. A bilinear recurrence relation for the coefficients cj is obtained from the mean field
(cid:0) (cid:1)
n equations in the Abel’s form. In this way the large j behaviour of the cj is calculated. This large
a j behaviour provides the position of the nearest singularity which corresponds to the critical point
J (collapse) oftheself-gravitating gasinthegrandcanonicalensemble. Finally,wediscusswhyother

6 attemptsto definea thermodynamic limit for theself-gravitating gas fail.
2

2
v
8 Contents
1
3
I. Introduction 1
7
0
II. The Cluster Expansion for the Self-Gravitating Gas 3
3
0
III. The Mayer Expansion for the self-gravitating gas 6
/
h
p IV. The Grand Canonical Ensemble for the self-gravitating gas 8
-
o
r V. Calculation of the Cluster coefficients cn 10
t
s
3 N
a VI. The stability of the thermodynamic limit N,V=L with L fixed (fixed η) 11
→∞
:
v
References 13
i
X
r
a I. INTRODUCTION
The self-gravitating gas has been the subject of attention since many years[1, 2, 3, 4, 5]. In refs.[2, 3] we recently
investigated the self-gravitating thermal gas using Monte Carlo simulations, mean field methods and low density
expansions[2, 3]. We have shown that the system possess a well defined infinite volume limit in the grand canonical
1
(GC), the canonical (C) and microcanonical (MC) ensembles when N,V keeping N/V 3 fixed. A relevant
→ ∞
variable here is dimensionless ratio
Gm2N
η , (1.1)
≡ V 13 T
which is kept fixed in the N,V limit. All physical quantities per particle turn out to be functions of the single
→∞
variable η and are well defined and finite in the thermodynamic limit N,V with η fixed.
→∞
InthispaperwedeveloptheclusterexpansionandMayer’sapproachtotheself-gravitatingthermalgasandprovide
a rigorous demonstration of the existence of the thermodynamic limit N,V with η fixed.
→∞
The cluster expansion is a powerful tool allowing to express the partition function in a power series of the density
forshortrangeinteractions[6]. Weapplyandadaptthismethod(theMayerexpansion)whichispurelycombinatorial
to the self-gravitating gas. We succeed in this way to obtain the coordinate partition function Q as a power series
N
2
in the thermodynamic limit. This is derived by generalizing the saddle point method used in ref.[6]. The expansion
is obtained in terms of the coefficients c which are pure numbers. More explicitly,
n
1 1 1
N≫1
logQ (η) = g(η t ) logt 1+ . (1.2)
N η η
N η − − O N
(cid:18) (cid:19)
where
+∞
g(x) c xj . (1.3)
j
≡
j=1
X
and t is the solution of the equation,
η
+∞
t g′(t η)=1 , i. e. j c (t η)j =η .
η η j η
j=1
X
Moreover, 1 logQ (η)isthe freeenergyofthe self-gravitatinggasminusthefreeenergyofanidealgasdividedby
−N N
NT. The series for g(η t ) in eqs.(1.2)-(1.3) is therefore a high temperature or low density expansion [see eq.(1.1)].
η
The coefficients c can be expressed in the thermodynamic limit as a sum of 3n-uple integrals associated to tree
n
cluster diagrams. Loop cluster diagrams are subdominant for N . The coefficients c only depend on the
n
→ ∞
geometry of the box. The first c are obtained by explicit evaluation of the cluster integrals. We have for the sphere,
n
1/3 2/3
3 4π 51 4π 4π 373
c =1 , csphere = , csphere = , csphere = .
1 2 5 3 3 70 3 4 3 315
(cid:18) (cid:19) (cid:18) (cid:19)
Moreover, we use the connection with the mean field approach to obtain a nonlinear recurrence relation for the
coefficients csphere:
n
n−1
1
cR = cR cR s2 (2n 2s+1) for n 2,
n (2n+1)(n 1) s n−s − ≥
− s=1
X
where
n−1
4π 3
csphere = cR .
n 3 n
(cid:18) (cid:19)
(The label R stands for the spherical geometry). This allows to compute systematically all cR and to find their large
n
n behaviour as
cR n=≫10.309360... n−25 1+ 1 .
n (u )n O n
GC (cid:20) (cid:18) (cid:19)(cid:21)
Here, u = 0.30034... is the radius of convergence of the series eq.(1.3). Therefore, u = u is the nearest
GC GC
singularity of g (u) in the u-plane.
R
We show below that the function g is related to the partition function (η,z) in the grand canonical ensemble
GC
Z
by
N
log (η,z)= g(η t ),
GC η
Z η
where
+∞ 3 N
Q (η) mT 2
N
(η,z)= V z .
GC
Z N=0 N! "(cid:18) 2π (cid:19) #
X
µ µ0
In addition, the fugacity (z = eT) turns out to be given by z = tη eT , where µ and µ0 are the chemical potentials
for the self-gravitating gas and for the ideal gas, respectively.
3
Thepointu =0.30034...whereg(u)issingularcorrespondstothecriticalpoint(collapse)inthegrandcanonical
GC
ensemble. Recall that the collapse point depends on the ensemble considered [2].
In conclusion, our investigationhere on the cluster expansion and the Mayer expansion for the self-gravitating gas
shows that:
The self-gravitating gas admits a consistent thermodynamic limit N,V with N fixed. In this limit,
1
• → ∞ V3
extensive thermodynamic quantities like energy, free energy, entropy are proportional to N. That is, in this
limit the energy, free energy, and entropy per particle are well defined and finite.
The cluster expansion and the mean field approach provide the same results in the thermodynamic limit
• N,V with N fixed.
1
→∞ V3
The partition function needs a small short-distance cutoff a in order to be well defined, the thermodynamic
•
limit has a finite limit for a 0. In other words, the contributions to the partition function which diverge
→
for a 0 are subdominant for N (see sec. II). For large N and fixed cutoff a the potentially divergent
→ →∞
contributions for a 0 are suppresed by a factor at least η2/N2 compared with the dominant contribution
for N . That i→s, the N,V = limit of the cutoff model (with N fixed) has a finite limit for a 0.
1
→ ∞ ∞ V3 →
Therefore, subdominant corrections in η/N can always be neglected. Realistic models of the self-gravitating
gas(interstellarmedium, galaxydistribution)requireasmallnon-zeroshortdistancecutoffsincemolecularand
atomic forces dominate over gravitational forces for short distances.
II. THE CLUSTER EXPANSION FOR THE SELF-GRAVITATING GAS
We investigate in this section the self-gravitating gas in thermal equilibrium at temperature T β−1. That is, we
≡
work in the canonical ensemble where the system of N particles is in contact with a thermal bath at temperature T.
We assume the gas being on a cubic box of side L.
The partition function of the system can be written as
1 N d3p d3q
(N,T)= ... l l e−βHN , (2.1)
ZC N! (2π)3
Z Z l=1
Y
where
N p2 1
H = l Gm2 . (2.2)
N
2m − ~q ~q
l j A
l=1 1≤l<j≤N | − |
X X
G is Newton’s gravitationalconstant.
At short distances, the particle interaction for the self-gravitating gas in physical situations is not gravitational.
Its exactnature depends onthe problemunder consideration(opacitylimit, Vander Waals forcesfor molecules etc.).
We shall just assume a repulsive short distance potential, that is,
1 for ~q ~q A
1 −|q~l−q~j| | l− j|≥
v (~q ~q )= = (2.3)
A | l− j| − ~q ~q 
| l− j|A  +A1 for|~ql−~qj|≤A
where A<<L is the short distance cut-off. 
The integrals over the momenta p , (1 l N) in eq.(2.1) can be computed immediately.
l
≤ ≤
It is convenient to introduce the dimensionless variables~r , 1 l N making explicit the volume dependence as
l
≤ ≤
~q = L~r , ~r =(x ,y ,z ),
l l l l l l
0 x ,y ,z 1. (2.4)
l l l
≤ ≤
That is, in the new coordinates the gas is inside a cube of unit volume.
4
The partition function takes now the form,
3N
1 mTL2 2
(N,T)= Q (η), (2.5)
C N
Z N! 2π
(cid:18) (cid:19)
where
1 1 N
Q (η) ... d3r eηu(~r1,...,~rN) , (2.6)
N l
≡
Z0 Z0 l=1
Y
η is the dimensionless variable [2]
Gm2N
η , (2.7)
≡ LT
and u(~r ,...,~r ) is defined by
1 N
1 1
u(~r ,...,~r ) , a A/L 1 (2.8)
1 N
≡ N ~r ~r ≡ ≪
l j a
1≤l<j≤N | − |
X
In this way all dependence on the volume V =L3 is buried in the variable η.
The coordinate partition function Q (η) can be written as
N
1 1 N
η
QN(η)= ... d3rl eN|~rl−~rj|a . (2.9)
Z0 Z0 l=1 1≤l<j≤N
Y Y
We are now interested to expand Q (η) in powers of η. In order to do this is convenient to define:
N N
η
flj eN|~rl−~rj|a 1.
≡ −
For small η and fixed a>0 we have,
N
η η 2
f = + , (2.10)
lj
N ~r ~r O N
| l− j|a (cid:20)(cid:16) (cid:17) (cid:21)
All the integrals over r in eq.(2.6) are finite provided we keep a>0.
l
We can now multiply out the products of f’s in the coordinate partition function Q (η),
N
(1+f )=1+ f + f f +...
lj ij ij kl
1≤l<j≤N
Y X X
Thus, by introducing the f functions the effects of interparticle forces are better exhibited.
ij
A systematic treatment of such sum of products can be found in ref.[6]. The outcome is that a general term in the
partition function Q (η) eq.(2.9), can be factorized as the product of severalintegrals over the coordinates~r . Each
N j
integral corresponds to a ‘cluster’ of particles and is called b (η,N) where
j
1 1 1 j
b (η,N)= ... d3r S , (2.11)
j l 1,2,...,j
j!
Z Z0 Z0 l=1
Y
with [6],
S =1
j
S =f
12 12
S =f f +f f +f f +f f f
123 12 23 12 13 13 23 12 13 23
5
S =f f f +nineteen permutations+
1234 12 23 34
+f f f f +fourteen permutations+
12 13 24 34
+f f f f f +five permutations+f f f f f f . (2.12)
12 13 14 24 34 12 13 14 23 24 34
The main result is that Q (η) can be expressed as the infinite sum, the so called ‘cluster expansion’ as:
N
N [b (η,N)]mj
j
Q (η)=N! , (2.13)
N
m !
j
m, Nj=X1(jmj)=N jY=1
P
where e
m (m ,...,m ).
1 N
≡
Itmustbestressedthateqs.(2.11)-(2.13)arepurelycombinatorialandtheyapplybothfortheshortrangeinteractions
e
considered in ref.[6] as well as the long range Newton potential.
Since f = η for large N, and S contains at least a product of j factors f , we see that
lj O N 1,2,...,j il
(cid:0) (cid:1)
η j−1
S = , for N j .
1,2,...,j
O N ≫
(cid:20)(cid:16) (cid:17) (cid:21)
Therefore, the cluster integrals eq.(2.11) take the form
η j−1 η
b (η,N)= c 1+ , for N j . (2.14)
j j
N O N ≫
(cid:16) (cid:17) h (cid:16) (cid:17)i
Here the coefficients c are positive numbers which only depend on the geometry of the box. As shown in fig. 1 the
j
dominant terms for largeN are cluster diagrams with a tree structure. In the largeN limit, cluster diagrams with a
loop structure as the last term in eq.(2.12) are subdominant.
From eqs.(2.10)-(2.12) and (2.14) we find,
c =1,
1
1 1 1 d3r d3r
1 2
c = ,
2
2 ~r ~r
Z0 Z0 | 1− 2|
1 1 1 1 d3r d3r d3r
1 2 3
c = . (2.15)
3
2 ~r ~r ~r ~r
Z0 Z0 Z0 | 1− 2|| 2− 3|
For a sphere of unit volume we find [2]
1/3 2/3
3 4π 51 4π
csphere = =0.967195171... , csphere = =1.893206013... (2.16)
2 5 3 3 70 3
(cid:18) (cid:19) (cid:18) (cid:19)
For the cubic geometry chosen, it takes the value [2]
1 1 1 (1 z)dz
ccube =4 (1 x)dx (1 y)dy − =0.94116...
2 Z0 − Z0 − Z0 x2+y2+z2
Furthermore, we find for the coefficient c , p
4
1 1 1 1 1 d3r d3r d3r d3r 1 1 1 1 1 d3r d3r d3r d3r
1 2 3 4 1 2 3 4
c = + . (2.17)
4
2 ~r ~r ~r ~r ~r ~r 6 ~r ~r ~r ~r ~r ~r
Z0 Z0 Z0 Z0 | 1− 2|| 2− 3|| 3− 4| Z0 Z0 Z0 Z0 | 1− 2|| 2− 3|| 2− 4|
The combinatorialfactors 1 and 1 take into account the symmetries of the respective cluster diagrams.
2 6
6
Inserting here the expansion in spherical harmonics,
1 ∞ 1 rl m=+l
=4π < Y¯ (rˆ′)Y (rˆ),
~r ~r′ 2l+1 rl+1 lm lm
| − | l=0 > m=−l
X X
where r max(r,r′)andr min(r,r′), the angularintegralsandthen the radialintegralscanbeperformedwith
> <
≡ ≡
the result,
1 1 1 1 d3r d3r d3r d3r b b 4 r2 dr 4π 62
1 2 3 4 =(4π)4 ... l=1 l l = .
~r ~r ~r ~r ~r ~r r> r> r> 3 35
Z0 Z0 Z0 Z0 | 1− 2|| 2− 3|| 3− 4| Z0 Z0 Q1,2 2,3 3,4
and
1 1 1 1 d3r d3r d3r d3r b b 4 r2 dr 4π 188
1 2 3 4 =(4π)4 ... l=1 l l = .
~r ~r ~r ~r ~r ~r r> r> r> 3 105
Z0 Z0 Z0 Z0 | 1− 2|| 2− 3|| 2− 4| Z0 Z0 Q1,2 2,3 2,4
Therefore,
4π 373
csphere = . (2.18)
4 3 315
Notice that all these ‘tree’ cluster diagrams eqs.(2.15) and (2.17) have a finite limit for zero cutoff a.
Divergent pieces for a 0 are subdominant as η2 for large N. The leading divergent contribution for a 0 to
→ N2 →
Q (η) to the nth. order in η takes the form[2]
N
ηn a3−n for n>3,
ηn 1 d3r d3r n!Nn−2
1 2 a→0
N(N 1) (2.19)
n!Nn2 − Z |~r1−~r2|na ∼  η3 loga for n=3.
N
This gives for the physical quantities (see next section) contributions of the order
ηn η3
a3−n forn>3 and loga for n=3.
n!Nn−1 N2
These contributions are clearly negligible in the N limit with fixed short-distance cutoff.
→∞
III. THE MAYER EXPANSION FOR THE SELF-GRAVITATING GAS
It is convenient to consider the generating function [6],
+∞
X(η,z) b (η,N)zj , (3.1)
j
≡
j=1
X
It can be shown that [6]
+∞
Q (η)
N zN =eX(η,z) , (3.2)
N!
N=0
X
where z is an auxiliary variable whose physical meaning (the fugacity) will appear in sec. IV.
Hence, we can compute the coefficients Q (η) from eq.(3.2) by contour integration,
N
Q (η) dz eX(η,z)
N
= .
N! 2πi zN+1
I
We choose as contour a cercle of radius r,
z =r eiθ , 0 θ 2π .
≤ ≤
7
Integrating over θ yields,
Q (η) +π dθ
N = eX(η,reiθ)−N logr−iN θ .
N! 2π
Z−π
For large N, we can use eqs.(2.14) to express the b (η,N) in X(η,z) [see eq.(3.1)] and we find,
j
+∞
N z η j N z η
N≫1
X(η,z) = c = g , (3.3)
j
η N η N
Xj=1 (cid:16) (cid:17) (cid:16) (cid:17)
where
+∞
g(x) c xj . (3.4)
j
≡
j=1
X
Thus,
QN(η) N=≫1 +π dθ eN η1 g reNiθη −logr−iθ = +π dθ eNΨ(r,θ) . (3.5)
N! 2π h (cid:16) (cid:17) i 2π
Z−π Z−π
We can now apply the steepest descent method to this integral for large N since the integrand has the structure
eNΨ(r,θ) where,
1 r eiθ η
Ψ(r,θ)= g logr iθ ,
η N − −
(cid:18) (cid:19)
∂Ψ reiθ r eiθ η
(r,θ)=i g′ i,
∂θ N N −
(cid:18) (cid:19)
∂Ψ eiθ r eiθ η 1
(r,θ)= g′ . (3.6)
∂r N N − r
(cid:18) (cid:19)
The saddle point, solution of ∂Ψ(r,θ)=0= ∂Ψ(r,θ) is thus found at
∂θ ∂r
(r,θ) =(N t,0),
saddle
where t is N-independent and is a solution of the equation,
tg′(tη)=1. (3.7)
That is, t is a function of η defined by the constraint eq.(3.7) or more explicitly,
+∞
j c (tη)j =η .
j
j=1
X
Choosing r=N t as integration path in eq.(3.5) and expanding the integrand around θ =0 yields,
QN(η) N=≫1eNΨ(Nt,0) +∞ dθ e−12N θ2ξ(η) = eNΨ(Nt,0) 1+ 1 ,
N! 2π 2π N ξ(η) O N
Z−∞ (cid:20) (cid:18) (cid:19)(cid:21)
where, p
1
Ψ(Nt,0)= g(η t ) log[N t ]
η η
η −
∂2Ψ +∞
ξ(η) (Nt,0)= j2 c tj >0. (3.8)
≡−∂θ2 j
j=1
X
where t is a function of η defined by eq.(3.7).
η
8
Using now Stirling’s formula for the N! factorial we find,
1 1 1
logQ (η)= g(η t ) logt 1+ . (3.9)
N η η
N η − − O N
(cid:18) (cid:19)
Therefore, the free energy can be written as,
F F 1
0
− = g(η t )+logt +1,
η η
NT −η
where F stands for the free energy of the ideal gas,
0
3
eV mT 2
F = NT log . (3.10)
0
− "N (cid:18) 2π (cid:19) #
The pressure (at the surface) follows from the thermodynamic relation
∂F
p= ,
− ∂V
(cid:18) (cid:19)T
we find
pV 2 g(η t ) 1 η 1 f(x)
η
f(η) = + and logQ (η)=3 dx − . (3.11)
N
≡ NT 3 3η N x
Z0
The dimensionless ratio pV was called f(η) in refs.[2, 3]. We can express all physical quantities in terms of the
NT
function f(η). We find from eqs.(3.9) and (3.11),
g(η t ) 1 η 1 f(x)
η
3f(η)=2+ and g(η t )+logt +1= 3 dx − . (3.12)
η η
η − η − x
Z0
That is,
1 η 1 f(x)
logt =f(η) 1 dx − . (3.13)
η
3 − − x
Z0
We can solve eq.(3.7) for t in powers of η with the result,
η
g(η t )=η t +c (η t )2+c (η t )3+ (η4),
η η 2 η 3 η
O
t =1 2c η+ 8(c )2 3c η2+ (η3). (3.14)
η 2 2 3
− − O
and from eq.(3.12) we find for f(η) and g(η t ), (cid:2) (cid:3)
η
c 2
f(η)=1 2 η+ 2(c )2 c η2+ (η3),
2 3
− 3 3 − O
η 1 f(x) (cid:2) (cid:3)
3 dx − = c η+ 2(c )2 c η2+ (η3)
2 2 3
− x − − O
Z0
g(η t )=η 1 c η+2 2(c (cid:2))2 c η2 (cid:3)+ (η4). (3.15)
η 2 2 3
− − O
It must be noticed that the function f(η) ((cid:8)and hence all(cid:2)physical qua(cid:3)ntit(cid:9)ies) have the same expression whether we
compute it from the mean field approach [2, 3] or from the saddle point eqs.(3.5), (3.9) and (3.11).
IV. THE GRAND CANONICAL ENSEMBLE FOR THE SELF-GRAVITATING GAS
The definition of the variable z through eq.(3.2) suggests that z is related to the fugacity. To be more precise, the
grand partition function is defined in terms of the canonical partition function as,
+∞ +∞ 3 N
Q (η) mT 2
(η,z)= (N,T)zN = N V z (4.1)
GC C
Z N=0Z N=0 N! "(cid:18) 2π (cid:19) #
X X
9
where we used eq.(2.5).
3
Therefore, z must be multiplied by the ideal gas factor mT 2 V.
2π
(cid:0) (cid:1)
At the saddle point we have after this renormalization
3 3
mT 2 mT 2
V z =N t and X η, V z =log (η,z). (4.2)
η GC
(cid:18) 2π (cid:19) (cid:18) 2π (cid:19) ! Z
Now, we know from ref.[2] that the chemical potential µ takes the form
µ µ η 1 f(x)
0
− = 3 dx − 3[1 f(η)] (4.3)
T − x − −
Z0
where µ stands for the chemical potential of the ideal gas,
0
3
V mT 2
µ = T log . (4.4)
0
− "N (cid:18) 2π (cid:19) #
From eq.(3.13) and (4.3) we have,
µ µ
0
− =logt . (4.5)
η
T
Therefore, from eqs.(4.2)-(4.5),
tη =e−µT0 z , eTµ =z , (4.6)
that is, we can identify z with the fugacity.
We have found in ref.[2] that f(η) obeys in the spherical case the first order differential equation of Abel’s type,
ηR[3f(ηR) 1]f′(ηR)+[3f(ηR) 3+ηR]f(ηR)=0. (4.7)
− −
where the variable ηR is defined as,
1/3
4π
ηR η =1.61199... η .
≡ 3
(cid:18) (cid:19)
Using eqs.(4.7) and (3.13) yields,
logt =logf(ηR) ηR , i. e. t =e−ηR f(ηR). (4.8)
η η
−
Eq.(4.1) provides the grand partition function. The above results showing that (η,z) is dominated by the
GC
Z
canonical ensemble, together with eq.(4.6) prove that the canonical and grand canonical ensembles are equivalent in
their common region of validity as stated in refs.[2, 3].
In the thermodynamic limit, (η,z) is given by eqs.(3.3) and (4.2) as
GC
Z
3
N z η mT 2 N
log (η,z)= g V = g(η t ). (4.9)
GC η
Z η N (cid:18) 2π (cid:19) ! η
Using here eq.(3.12) gives,
log (η,z)=N[3f(η) 2],
GC
Z −
whichexactlycoincideswiththeexpressionfoundinref.[2]forthegrandpartitionfunctioninthemeanfieldapproach.
10
V. CALCULATION OF THE CLUSTER COEFFICIENTS cn
We compute in this section the cluster coefficients c for the sphere by using a non-linear differential equation for
n
the function g(u).
We first find from eq.(3.7)
4π 1/3
ug′ (u)=ηR with u t ηR and g (u) g(u) (5.1)
R ≡ η R ≡ 3
(cid:18) (cid:19)
Then, combining eqs.(4.7) and (5.1) yields the second order differential equation for g (u),
R
u2g′′(u)+ug′ (u) [2ug′ (u)+g (u)] 2u2g′′(u) ug′ (u)+g (u)=0. (5.2)
R R R R − R − R R
Alternatively, if we(cid:2)choose ηR = ug′ (u(cid:3)) as variable [see eq.(5.1)], we find from eq.(4.7) the first order non-linear
R
differential equation
dg
ηR+g R +ηR 2ηR+g 2 =0.
R dηR R−
(cid:2) (cid:3) (cid:2) (cid:3)
That is, the invariance of eq.(5.2) under the rescaling of the variable u allows to reduce by one the order of the
differential equation.
Eq.(5.1) has as regular solution around u=0,
∞ n−1
4π 3
g (u)= cR un with csphere = cR , (5.3)
R n n 3 n
n=1 (cid:18) (cid:19)
X
and we can choose cR =1.
1
By inserting eq.(5.3) into eq.(5.2) the following nonlinear recurrence relation for the cR coefficients is obtained:
n
n−1
1
cR = cR cR s2 (2n 2s+1) for n 2. (5.4)
n (2n+1)(n 1) s n−s − ≥
− s=1
X
We find,
3 51 373 14911 2047
cR =1, cR = , cR = , cR = , cR = , cR = ,...
1 2 5 3 70 4 315 5 6600 6 429
in agreement with eqs. (2.15), (2.16) and (2.18).
By evaluating numerically the cR from the recurrence relation eq.(5.4), we find for large n:
n
cR n=≫10.309360... n−52 1+ 1 . (5.5)
n (u )n O n
GC (cid:20) (cid:18) (cid:19)(cid:21)
Here, u = 0.30034... is the radius of convergence of the series eq.(5.3). Therefore, u = u is the nearest
GC GC
singularity of g (u) in the u-plane.
R
As we see from eq.(4.9), g (u) gives the grand canonical partition function as a function of u = t ηR. Since
R η
u=t ηR is proportionalto the fugacity[see eq.(4.6)], u must be relatedto the criticalpointofthe selfgravitating
η GC
gas in the grand canonical ensemble. Indeed, using the critical value for ηR in the grand canonical ensemble [2],
2
ηR =0.79735... and f(ηR )= ,
GC GC 3ηR
GC
from eq.(4.8) we obtain,
2
uGC =ηGRC tηGRC =ηGRC e−ηGRC f(ηGRC)= 3 e−ηGRC =0.30034... ,