- Email: [email protected]
Fluctuation pressure of membrane between walls
5 July 1999
Physics Letters A 257 Ž1999. 269–274
Fluctuation pressure of membrane between walls H. Kleinert
1
Freie UniÕersitat ¨ Berlin, Institut fur ¨ Theoretische Physik, Arnimallee14, D-14195 Berlin, Germany Received 2 February 1999; received in revised form 17 March 1999; accepted 22 April 1999 Communicated by P.R. Holland
Abstract For a single membrane of stiffness k fluctuating between two planar walls of distance d, we calculate analytically the pressure law
p2 ps
k B2 T 2
128 k Ž dr2 . 3
.
The prefactor p 2r128; 0.077115 . . . is in very good agreement with results from Monte Carlo simulations 0.079 " 0.002. q 1999 Elsevier Science B.V. All rights reserved.
1. A stack of n parallel, thermally fluctuating membranes exerts upon the enclosing planar walls a pressure which depends on the stiffness k and the temperature T as follows: p s an
2n
k B2 T 2
n q 1 k dr Ž n q 1 .
3
,
Ž 1.
where k B is Boltzmann’s constant and d the distance between the walls Žsee Fig. 1.. This law, first deduced from dimensional considerations by Helfrich w1x, is of fundamental importance in the statistical mechanics of membranes just as the ideal gas law pV s N k B T in the statistical
1 E-mail [email protected], www.physik.fu-berlin.der ; kleinert.
URL
http:rr
mechanics of point particles. We would therefore like to know the size of the prefactor, the stack constant a n as accurately as possible. So far, its value was determined only by extensive Monte Carlo simulations as being w2,3x
a` s 0.101 " 0.002.
Ž 2.
For a single membrane, the following value was found w4,3x:
a 1 s 0.079 " 0.002.
Ž 3.
So far, there exists no analytic theory to explain these values. The purpose of this note is to fill this gap for the constant a 1 , by calculating analytically the pressure of a single membrane between parallel walls. The
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 2 8 9 - 3
H. Kleinertr Physics Letters A 257 (1999) 269–274
270
tion of the membrane is given by the functional integral
k
½
Z s D u Ž x . exp y
H
2 k BT
Hd
2
x E 2 uŽ x .
2
5
' eyA f r k B T ,
Fig. 1. Membrane fluctuating between walls of distance d, exerting a pressure p.
theoretical tool for this has only recently become available: A strong-coupling theory developed originally in quantum mechanics w5x, was extended successfully to quantum field theories w6x, where it has been used to obtain extremely accurate values for the critical exponents of OŽ n.-symmetric scalar fields with w 4-interactions w6x. 2. Strong-coupling theory gives direct access to the large-g behavior of divergent truncated power series expansions of the type N
f N Ž g . s V a0 q
Ý ak ks1
žV /
.
q
ž
Ý ak c ks1
1yq k
b kN
/
,
Ž 5.
ž
sw 1y
d
p 2w 2 3d 2
p 4w 4 q
and add to the fluctuation energy E in the exponent of Ž7. a potential energy which keeps the membrane between ydr2 and dr2 ŽPoschl–Teller potential.: ¨ k E pot s E0pot q E ints d 2 x m4f 2 Ž u Ž x . . 2
H
k m4 s
Hd
2
½
` 2
x u Ž x. q
Ý ´k ks2
p
uŽ x . d
ls0
ž
/
,
17 90
,
31 315
,
691 14175
,
10922 467775
, ... .
,
Ž 10 .
The potential energy per area is plotted in Fig. 2. Its presence destroys the simple scaling properties of the partition function Ž7., which depends only on the
Ž 6.
is the binomial expansion of Ž1 y 1.Ž1yk q .r2 truncated after the Ž N y k .th term. Optimizing means extremizing f˜N Ž c . in c or, if an extremum does not exist, extremizing the derivative f˜NX Ž c .. 3. We apply this theory to a membrane between walls by proceeding as follows. The partition func-
5
with expansion coefficients ´ 2 , ´ 3 , ´4 , . . . :
Nyk
Ž 1 y kq . r2 Ý Ž y1. l l
2k
Ž 9.
where b kN s
/
q... ,
5d 4
Ž 8.
1 3
N
'g
arctan
p
2
f N Ž c . s g 1r q f˜N Ž c . ca0 b 0N q
pw
d
us
Ž 4.
The g ™ `-limit of f N Ž g ., to be denoted by f N) , is obtained by setting V ' cg 1r q and optimizing the function
1r q
where uŽ x . is a vertical displacement field of the membrane fluctuating between horizontal walls at u s ydr2 and dr2. The quantities A and f are the wall area and the free energy per unit area, respectively. Such a restriction of a field is hard to treat analytically. We therefore perform a transformation which maps the interval u g Žydr2,dr2. to an infinite w-axis,
k
g
Ž 7.
Fig. 2. Smooth Potential replacing box walls.
H. Kleinertr Physics Letters A 257 (1999) 269–274
dimensionless variable k d 2rk B T. The new partition function Z associated with the modified energy E q E pot has an additional dependence on the dimensionless variable g s p 2rm 2 d 2 . The original hard-wall system is obtained in the strong-coupling limit g ™ `. In the opposite limit where g goes to zero, the energy E q E pot becomes harmonic,
k E0 s
2
2
2
E 2 u Ž x . q m4 u 2 Ž x . ,
Hd x ½
5
Ž 11 .
s const = ey 8 m ,
Ž 12 .
leading to a partition function 1
Z0 s ey 2 Trlog Ž E
4
qm 4 .
A
2
where A is the area of the walls. For a finite distance d, the interaction energy E int is treated perturbatively order by order in g, expandint ing the exponential eyE r k B T in a power series, and each power in a sum of all pair contractions. These are pictured by loop diagrams whose lines represent the correlation function d2k
k ² uŽ x1 . uŽ x 2 . : s
kB
H T Ž 2p .
1 2
4
k q m4
e i kŽ x 1yx 2 . .
Ž 13 . The free energy density f s yk B TAy1 log Z is obtained from all connected loop diagrams. For simplicity, we shall use natural units with krk B T s 1. The lowest contribution to the free energy density comes from the expectation value of the u 4-interaction or the loop diagram 3 which is of the order 1rd 2 : m4
m4
2d
2 d2
² u4 : s 2
3² u 2 :2 ,
² u2 : s
f1 s
1 q
8
2 1 8
p2
1 q
64 m2 d 2
q a2
ž / m2 d 2
2
q...
N
p2
qa N
p2
ž /
,
m2 d 2
Ž 17 .
where a 2 , . . . ,a N are dimensionless numbers. By comparison with Ž4. we identify p s q s 1, V s m2 , g s p 2rd 2 . The function f N Ž c . of Eq. Ž5. describing the limiting large-g behavior is obtained by setting V ' cp 2r2 d 2 , and reads fN Ž c . s
p2 c d
q
2
ž
4
b 0N q 641 q
aN c Ny1
a2 c
b 2N q . . .
bNN .
/
Ž 18 .
According to the above-described strong-coupling theory, we must optimize the expression f˜N Ž c . in parentheses. Since the second term does not contain c, we separate this term out, and write c N a2 N f˜N Ž c . s 641 q D f˜N Ž c . ' 641 q b q b2 q . . . 4 0 c aN q Ny1 bNN , Ž 19 . c
ž
/
with only the remainder D f˜N Ž c . to be optimized. Let D f˜N) be ist optimal value. If we know only a 2 , we find the approximation D f 2) s 3a 2r16 . Ignoring D f N) for a moment, the first term in Ž19. yields the lowest estimate for the free energy density of the original system
(
p2 1 64 d 2
,
Ž 20 .
implying a pressure law 1 2
k 4 q m4
1 s
8 m2
.
Ž 15 .
Together with the exponent in Ž12., we thus obtain first-order free energy density m2
fN s m
f 1) s
d2k
H Ž 2p .
Continuing the perturbation expansion, yields an expansion of the general form
Ž 14 .
the line representing the pair expectation
271
p2 2
32 m d
2
.
Ž 16 .
psy
p2 1
Ef Ed
s
32 d 3
.
Ž 21 .
By comparison with the general pressure law Ž1., we identify the prefactor as being
a 1 s 12 =
p2 128
f 12 = 0.077115.
Ž 22 .
H. Kleinertr Physics Letters A 257 (1999) 269–274
272
Without the prefactor factor 1r2, this would agree perfectly with the Monte Carlo value Ž3.. Thus we expect the contribution of D f˜N) for N ™ ` to be equal or almost equal to 1r64. The calculation of the higher-order terms a2 , a3 , . . . is tedious, and will be presented in a separate detailed publication w7x. In this note we shall circumvent it by exploiting a close relationship of the present problem with a closely analogous exactly solvable one, which may be treated in precisely the same way: The euclidean version of a quantummechanical point particle in a one-dimensional box u g Žydr2,dr2.. 4. The partition function of a particle in a box is yŽ k r2 k B T . d x Ž E u . 2
H
Z s D ue
H
yA f r k B T
'e
.
Ž 23 .
The quantum-mechanical ground state energy of this system is exactly known: Ž k B Trk .p 2r2 d 2 , corresponding to a free energy density k B2 T 2 p 2
fs
. Ž 24 . k 2 d2 The path integral Ž23. may now be treated as before, i.e., we transform u to w via Ž8., and separate the field energy into a Gaussian energy Žin natural units. k 2 E0 s dx E 2 u Ž x . q m4 u 2 Ž x . Ž 25 . 2 and an interaction energy which looks the same as Ž9., except that the integration Hd 2 x runs now only over one dimension, Hdx. The first-order contribution to the free energy density is now Žin natural units with krk B T s 1.
H ½
m4
²u 2
4:
p2 fN Ž c . s
3² u 2
2 :2
,
Ž 26 .
2d 2d with the pair expectation dq 1 1 ² u2 : s s , 2 4 2p q q m 2 m2
d2
f˜N Ž c . ,
Ž 30 .
with f˜N Ž c . s 14 q D f˜N Ž c . ' 14 q aN
ž
c 4
b 0N q
a2 c
b 2N q . . .
bN . Ž 31 . c Ny1 N Here the first term yields the lowest approximation q
/
1 p2
f1 s
, Ž 32 . 4 d2 which is precisely half the exact result. Thus we conclude that the optimal value of the neglected expression D f N Ž c . must be once more equal to 1r4 in the limit N ™ `. In order to see how this happens, we extend the Bender–Wu recursion relation for the perturbation coefficients of the anharmonic oscillator w8x. It yields for the ground state energy an expansion 1 2
3p 2 q
4 d2
p4 ´4 y
p6 q
5
m4 s
From this we find the function f N Ž c . defined in Eq. Ž5. governing the strong-coupling limit d ™ 0 by setting V s m2 ' cp 2r2 d 2 :
16 d 6
Ž 21 ´42 y 15´6 .
Ž 333´43 y 360 ´4 ´6 q 105´ 8 .
p8 y
8 d4
128d 8
Ž 30885´44 y 44880 ´42´6 q 6990 ´62
q1512 ´ 4 ´ 8 q 3780 ´ 10 . q . . . .
Ž 33.
Inserting the coefficients Ž10. we find a 2 ,a4 ,a6 , . . . : 1 16
1 ,y 256 ,
1 2048
5 , y 65536 ,
7 524288
21 , y 8388608 , ... ,
Ž 34 .
whereas the odd coefficients a 3 ,a5 ,a 7 , . . . vanish.
Ž 27 .
H
leading to a first-order free energy density f1 s
m2
1 p2
, 2 2 d2 and a full perturbation expansion of the form f s m2
q
ž
1 2
1 p2 q
2 m2 d 2
p4 q a2
m4 d 4
/
q... .
Ž 28 .
Ž 29.
Fig. 3. Plots of the functions D f˜N Ž c . of Eq. Ž35., all being optimal exactly at cs1 with D f˜N) s1r4.
H. Kleinertr Physics Letters A 257 (1999) 269–274 Table 1 The functions D f˜N Ž c . of Eq. Ž35., and their optimal values D f˜N) N
D f˜N Ž c .
1
c 4 3c 16 5c 32 35 c 256 63 512
D f˜N)
y1
q c16 c
2
q
3
q
4 5
3 c y1 cy 3 32 256 y1 15c 5c 3 cy 5 128 512 2048 y1 3 35c 35c 7c y 5 5 256 3048 4096 65536 315c y1 105c 3 63 c y 5 45c y 7 2048 4096 16384 131072
1 4 1 4 1 4 1 4 1 4
y
y
q
q
y
q
q
y
y
q
y
y9
7c q 524288
To account for this fact, we resum the series containing only the even terms c a2 a4 a2 N D f˜N s q b 1N q 3 b 2N q 5 bNN , Ž 35 . 4 c c c taking the coefficients biN of Eq. Ž6. with the parameters p s 1, q s 2. This yields the functions D f˜N Ž c . plotted in Fig. 3 and listed in Table 1. For all D f˜N Ž c ., optimization yields a strong-coupling value D f˜N) equal to 1r4, thus raising the initial value 1r4 in Ž31. to the correct final value 1r2. 5. To exploit this property of a particle in a box for the system at hand, the membrane between walls, we make the following crucial observation: The Feynman integrals determining the first two terms in the free energy densities in Eq. Ž16. for a membrane and in Eq. Ž28. for a particle are related to each other by a simple transformation of the integration variables. The membrane integrals d2k
H Ž 2p .
log Ž k 4 q m 4 . s 2
d2k
H Ž 2p .
1 2
4
k qm
m2 4
s
Thus, if we multiply each loop integral by a factor 1r4, we find immediately the free energy density f 1 of the membrane in Eq. Ž16. from that of the particle in the box in Eq. Ž28.. But the analogy carries further: By differentiation Ž36. and Ž37. with respect to m2 , we see that also all Feynman integrals Hd 2 krŽ2p . 2 Ž k 4 q m 4 . n are related to the HdqrŽ2p .Ž q 2 q m4 . n by the same factor 1r4. This property has the consequence that most of the connected loop diagrams contributing to the perturbation expansion of the free energy density, shown in Fig. 4 up to five loops, are related by a factor Ž1r4. L , where L is the number of loops. In particular, all such diagrams coincide which are usually summed in the Hartree–Fock approximation Žchain diagrams, daisy diagrams, etc... Only the topological more involved diagrams 3–1, 4–1, 4–2, 4–5, 5– 2, 5–3, 5–5, 5–6, 5–7, 5–11, 5–12, 5–15 in Fig. 4 do not follow this pattern. For a particle in a box, we can easily calculate the associated Feynman integrals in x-space as described in Chapter 3 of Ref. w5x, and find that they contribute less than 5% to the sum of all diagrams at each loop level. This implies that the corresponding results for the membrane between walls will differ at most by this relative amount from those for the particle in the box. We therefore conclude that since the optimal value of D f˜N Ž c . in Eq. Ž31. doubles the initial value for N ™ `, the analogous function for the membrane between walls in Eq. Ž19. will double approximately. For the quantitative deviations see the forthcoming publication w7x. A precise doubling of the result Ž22. leads to a very good agreement with the Monte Carlo number Ž2..
,
1 4
Ž 36 .
8 m2
go over into those of the particle in the box dq
H 2p log Ž q
2
dq
q m4 . s m2 ,
H 2p q
1 2
qm
1 4
s
2 m2 Ž 37 .
by the transformation 2
k ™ q,
d2k
H Ž 2p .
2
™ 14
`
dq
Hy` 2p .
273
Fig. 4. Vacuum diagrams up to five loops.
H. Kleinertr Physics Letters A 257 (1999) 269–274
274
6. The alert reader will have noted that the field transformation Ž8. is rather special. We may, for instance, chose any mapping w us
w 1q8p 2w 2 r3d 2 q w4 w 4 r d 4 q . . . q Ž2 w r d . n x
s w y 23
p2 d2
w3q...q OŽw5.,
1r n
Ž 38 .
which has a doubled coefficient of w 3 with respect to the expansion Ž8.. As a consequence, the functions f˜N Ž c . in Ž19. and Ž31. would have a doubled first term. Since this would be the correct final value, the remaining functions D f˜N Ž c . would have to converge to a vanishing optimal value for N ™ ` Žin the particle case exactly, in the membrane case approximately.. To reach this goal, the coefficients w4 ,w6 , . . . in Ž38. can be chosen rather arbitrarily, although there are a few convenient ways for which the speed of convergence is fast. A preferred choice is one in which all coefficients a 2 ,a 3 ,a 3 , . . . of the perturbation expansion vanishes for a particle in a box. This and other possibilities will be studied separately w9,10x.
Acknowledgements The author is grateful to Prof. A. Chervyakov, Prof. B. Hamprecht, Dr. A. Pelster, and M. Bachmann for numerous useful discussions.
References w1x W. Helfrich, Z. Naturforsch. A 33 Ž1978. 305; W. Helfrich, R.M. Servuss, Nuovo Cimento D 3 Ž1984. 137. w2x W. Janke, H. Kleinert, Phys. Lett. 58 Ž1987. 144. w3x W. Janke, H. Kleinert, M. Meinhardt, Phys. Lett. B 217 Ž1989. 525. The simulations were repeated by G. Gompper, D.M. Kroll, Europhys. Lett. 9 Ž1989. 58 with the results a 1 f 0.0798, a` f Ž4r3. a 1 f 0.106. A further simulation of a` was recently reported by R.R. Netz, R. Lipowski, Europhys. Lett. 29 Ž1995. 345, with the result a` f Ž3r2.p 2 r128f 0.119, which is half as big as Helfrich’s original estimate 3p 2 r128 in Ref. w1x. Another estimate a` f 0.0810 was given by F. David, J. de Phys. 51 Ž1990. C7-115. w4x W. Janke, H. Kleinert, Phys. Lett. A 117 Ž1986. 353. w5x H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, World Scientific, Singapore, 1995, sec. ed., pp. 1–850. w6x H. Kleinert, Phys. Rev. D 57 Ž1998. 2264 Žaps1997jun25_001., Addendum Žcond.matr9803268.; Phys. Lett. B 434 Ž1998. 74 Žcond-matr9801167.. w7x M. Bachmann, H. Kleinert, A. Pelster, Strong-Coupling Calculation of Pressure for Fluctuating Membrane between Walls, Berlin preprint Žin preparation.. w8x C.M. Bender, T.T. Wu, Phys. Rev. 184 Ž1969. 1231; Phys. Rev. D 7 Ž1973. 1620. w9x H. Kleinert, A. Chervyakov, B. Hamprecht, Variational Perturbation Expansion for Particle in a Box, Berlin preprint Žin preparation.. w10x H. Kleinert, A. Chervyakov, Inverse Variational Perturbation Theory for Hard-Wall Systems, Berlin preprint Žin preparation..
Physics Letters A 257 Ž1999. 269–274
Fluctuation pressure of membrane between walls H. Kleinert
1
Freie UniÕersitat ¨ Berlin, Institut fur ¨ Theoretische Physik, Arnimallee14, D-14195 Berlin, Germany Received 2 February 1999; received in revised form 17 March 1999; accepted 22 April 1999 Communicated by P.R. Holland
Abstract For a single membrane of stiffness k fluctuating between two planar walls of distance d, we calculate analytically the pressure law
p2 ps
k B2 T 2
128 k Ž dr2 . 3
.
The prefactor p 2r128; 0.077115 . . . is in very good agreement with results from Monte Carlo simulations 0.079 " 0.002. q 1999 Elsevier Science B.V. All rights reserved.
1. A stack of n parallel, thermally fluctuating membranes exerts upon the enclosing planar walls a pressure which depends on the stiffness k and the temperature T as follows: p s an
2n
k B2 T 2
n q 1 k dr Ž n q 1 .
3
,
Ž 1.
where k B is Boltzmann’s constant and d the distance between the walls Žsee Fig. 1.. This law, first deduced from dimensional considerations by Helfrich w1x, is of fundamental importance in the statistical mechanics of membranes just as the ideal gas law pV s N k B T in the statistical
1 E-mail [email protected], www.physik.fu-berlin.der ; kleinert.
URL
http:rr
mechanics of point particles. We would therefore like to know the size of the prefactor, the stack constant a n as accurately as possible. So far, its value was determined only by extensive Monte Carlo simulations as being w2,3x
a` s 0.101 " 0.002.
Ž 2.
For a single membrane, the following value was found w4,3x:
a 1 s 0.079 " 0.002.
Ž 3.
So far, there exists no analytic theory to explain these values. The purpose of this note is to fill this gap for the constant a 1 , by calculating analytically the pressure of a single membrane between parallel walls. The
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 2 8 9 - 3
H. Kleinertr Physics Letters A 257 (1999) 269–274
270
tion of the membrane is given by the functional integral
k
½
Z s D u Ž x . exp y
H
2 k BT
Hd
2
x E 2 uŽ x .
2
5
' eyA f r k B T ,
Fig. 1. Membrane fluctuating between walls of distance d, exerting a pressure p.
theoretical tool for this has only recently become available: A strong-coupling theory developed originally in quantum mechanics w5x, was extended successfully to quantum field theories w6x, where it has been used to obtain extremely accurate values for the critical exponents of OŽ n.-symmetric scalar fields with w 4-interactions w6x. 2. Strong-coupling theory gives direct access to the large-g behavior of divergent truncated power series expansions of the type N
f N Ž g . s V a0 q
Ý ak ks1
žV /
.
q
ž
Ý ak c ks1
1yq k
b kN
/
,
Ž 5.
ž
sw 1y
d
p 2w 2 3d 2
p 4w 4 q
and add to the fluctuation energy E in the exponent of Ž7. a potential energy which keeps the membrane between ydr2 and dr2 ŽPoschl–Teller potential.: ¨ k E pot s E0pot q E ints d 2 x m4f 2 Ž u Ž x . . 2
H
k m4 s
Hd
2
½
` 2
x u Ž x. q
Ý ´k ks2
p
uŽ x . d
ls0
ž
/
,
17 90
,
31 315
,
691 14175
,
10922 467775
, ... .
,
Ž 10 .
The potential energy per area is plotted in Fig. 2. Its presence destroys the simple scaling properties of the partition function Ž7., which depends only on the
Ž 6.
is the binomial expansion of Ž1 y 1.Ž1yk q .r2 truncated after the Ž N y k .th term. Optimizing means extremizing f˜N Ž c . in c or, if an extremum does not exist, extremizing the derivative f˜NX Ž c .. 3. We apply this theory to a membrane between walls by proceeding as follows. The partition func-
5
with expansion coefficients ´ 2 , ´ 3 , ´4 , . . . :
Nyk
Ž 1 y kq . r2 Ý Ž y1. l l
2k
Ž 9.
where b kN s
/
q... ,
5d 4
Ž 8.
1 3
N
'g
arctan
p
2
f N Ž c . s g 1r q f˜N Ž c . ca0 b 0N q
pw
d
us
Ž 4.
The g ™ `-limit of f N Ž g ., to be denoted by f N) , is obtained by setting V ' cg 1r q and optimizing the function
1r q
where uŽ x . is a vertical displacement field of the membrane fluctuating between horizontal walls at u s ydr2 and dr2. The quantities A and f are the wall area and the free energy per unit area, respectively. Such a restriction of a field is hard to treat analytically. We therefore perform a transformation which maps the interval u g Žydr2,dr2. to an infinite w-axis,
k
g
Ž 7.
Fig. 2. Smooth Potential replacing box walls.
H. Kleinertr Physics Letters A 257 (1999) 269–274
dimensionless variable k d 2rk B T. The new partition function Z associated with the modified energy E q E pot has an additional dependence on the dimensionless variable g s p 2rm 2 d 2 . The original hard-wall system is obtained in the strong-coupling limit g ™ `. In the opposite limit where g goes to zero, the energy E q E pot becomes harmonic,
k E0 s
2
2
2
E 2 u Ž x . q m4 u 2 Ž x . ,
Hd x ½
5
Ž 11 .
s const = ey 8 m ,
Ž 12 .
leading to a partition function 1
Z0 s ey 2 Trlog Ž E
4
qm 4 .
A
2
where A is the area of the walls. For a finite distance d, the interaction energy E int is treated perturbatively order by order in g, expandint ing the exponential eyE r k B T in a power series, and each power in a sum of all pair contractions. These are pictured by loop diagrams whose lines represent the correlation function d2k
k ² uŽ x1 . uŽ x 2 . : s
kB
H T Ž 2p .
1 2
4
k q m4
e i kŽ x 1yx 2 . .
Ž 13 . The free energy density f s yk B TAy1 log Z is obtained from all connected loop diagrams. For simplicity, we shall use natural units with krk B T s 1. The lowest contribution to the free energy density comes from the expectation value of the u 4-interaction or the loop diagram 3 which is of the order 1rd 2 : m4
m4
2d
2 d2
² u4 : s 2
3² u 2 :2 ,
² u2 : s
f1 s
1 q
8
2 1 8
p2
1 q
64 m2 d 2
q a2
ž / m2 d 2
2
q...
N
p2
qa N
p2
ž /
,
m2 d 2
Ž 17 .
where a 2 , . . . ,a N are dimensionless numbers. By comparison with Ž4. we identify p s q s 1, V s m2 , g s p 2rd 2 . The function f N Ž c . of Eq. Ž5. describing the limiting large-g behavior is obtained by setting V ' cp 2r2 d 2 , and reads fN Ž c . s
p2 c d
q
2
ž
4
b 0N q 641 q
aN c Ny1
a2 c
b 2N q . . .
bNN .
/
Ž 18 .
According to the above-described strong-coupling theory, we must optimize the expression f˜N Ž c . in parentheses. Since the second term does not contain c, we separate this term out, and write c N a2 N f˜N Ž c . s 641 q D f˜N Ž c . ' 641 q b q b2 q . . . 4 0 c aN q Ny1 bNN , Ž 19 . c
ž
/
with only the remainder D f˜N Ž c . to be optimized. Let D f˜N) be ist optimal value. If we know only a 2 , we find the approximation D f 2) s 3a 2r16 . Ignoring D f N) for a moment, the first term in Ž19. yields the lowest estimate for the free energy density of the original system
(
p2 1 64 d 2
,
Ž 20 .
implying a pressure law 1 2
k 4 q m4
1 s
8 m2
.
Ž 15 .
Together with the exponent in Ž12., we thus obtain first-order free energy density m2
fN s m
f 1) s
d2k
H Ž 2p .
Continuing the perturbation expansion, yields an expansion of the general form
Ž 14 .
the line representing the pair expectation
271
p2 2
32 m d
2
.
Ž 16 .
psy
p2 1
Ef Ed
s
32 d 3
.
Ž 21 .
By comparison with the general pressure law Ž1., we identify the prefactor as being
a 1 s 12 =
p2 128
f 12 = 0.077115.
Ž 22 .
H. Kleinertr Physics Letters A 257 (1999) 269–274
272
Without the prefactor factor 1r2, this would agree perfectly with the Monte Carlo value Ž3.. Thus we expect the contribution of D f˜N) for N ™ ` to be equal or almost equal to 1r64. The calculation of the higher-order terms a2 , a3 , . . . is tedious, and will be presented in a separate detailed publication w7x. In this note we shall circumvent it by exploiting a close relationship of the present problem with a closely analogous exactly solvable one, which may be treated in precisely the same way: The euclidean version of a quantummechanical point particle in a one-dimensional box u g Žydr2,dr2.. 4. The partition function of a particle in a box is yŽ k r2 k B T . d x Ž E u . 2
H
Z s D ue
H
yA f r k B T
'e
.
Ž 23 .
The quantum-mechanical ground state energy of this system is exactly known: Ž k B Trk .p 2r2 d 2 , corresponding to a free energy density k B2 T 2 p 2
fs
. Ž 24 . k 2 d2 The path integral Ž23. may now be treated as before, i.e., we transform u to w via Ž8., and separate the field energy into a Gaussian energy Žin natural units. k 2 E0 s dx E 2 u Ž x . q m4 u 2 Ž x . Ž 25 . 2 and an interaction energy which looks the same as Ž9., except that the integration Hd 2 x runs now only over one dimension, Hdx. The first-order contribution to the free energy density is now Žin natural units with krk B T s 1.
H ½
m4
²u 2
4:
p2 fN Ž c . s
3² u 2
2 :2
,
Ž 26 .
2d 2d with the pair expectation dq 1 1 ² u2 : s s , 2 4 2p q q m 2 m2
d2
f˜N Ž c . ,
Ž 30 .
with f˜N Ž c . s 14 q D f˜N Ž c . ' 14 q aN
ž
c 4
b 0N q
a2 c
b 2N q . . .
bN . Ž 31 . c Ny1 N Here the first term yields the lowest approximation q
/
1 p2
f1 s
, Ž 32 . 4 d2 which is precisely half the exact result. Thus we conclude that the optimal value of the neglected expression D f N Ž c . must be once more equal to 1r4 in the limit N ™ `. In order to see how this happens, we extend the Bender–Wu recursion relation for the perturbation coefficients of the anharmonic oscillator w8x. It yields for the ground state energy an expansion 1 2
3p 2 q
4 d2
p4 ´4 y
p6 q
5
m4 s
From this we find the function f N Ž c . defined in Eq. Ž5. governing the strong-coupling limit d ™ 0 by setting V s m2 ' cp 2r2 d 2 :
16 d 6
Ž 21 ´42 y 15´6 .
Ž 333´43 y 360 ´4 ´6 q 105´ 8 .
p8 y
8 d4
128d 8
Ž 30885´44 y 44880 ´42´6 q 6990 ´62
q1512 ´ 4 ´ 8 q 3780 ´ 10 . q . . . .
Ž 33.
Inserting the coefficients Ž10. we find a 2 ,a4 ,a6 , . . . : 1 16
1 ,y 256 ,
1 2048
5 , y 65536 ,
7 524288
21 , y 8388608 , ... ,
Ž 34 .
whereas the odd coefficients a 3 ,a5 ,a 7 , . . . vanish.
Ž 27 .
H
leading to a first-order free energy density f1 s
m2
1 p2
, 2 2 d2 and a full perturbation expansion of the form f s m2
q
ž
1 2
1 p2 q
2 m2 d 2
p4 q a2
m4 d 4
/
q... .
Ž 28 .
Ž 29.
Fig. 3. Plots of the functions D f˜N Ž c . of Eq. Ž35., all being optimal exactly at cs1 with D f˜N) s1r4.
H. Kleinertr Physics Letters A 257 (1999) 269–274 Table 1 The functions D f˜N Ž c . of Eq. Ž35., and their optimal values D f˜N) N
D f˜N Ž c .
1
c 4 3c 16 5c 32 35 c 256 63 512
D f˜N)
y1
q c16 c
2
q
3
q
4 5
3 c y1 cy 3 32 256 y1 15c 5c 3 cy 5 128 512 2048 y1 3 35c 35c 7c y 5 5 256 3048 4096 65536 315c y1 105c 3 63 c y 5 45c y 7 2048 4096 16384 131072
1 4 1 4 1 4 1 4 1 4
y
y
q
q
y
q
q
y
y
q
y
y9
7c q 524288
To account for this fact, we resum the series containing only the even terms c a2 a4 a2 N D f˜N s q b 1N q 3 b 2N q 5 bNN , Ž 35 . 4 c c c taking the coefficients biN of Eq. Ž6. with the parameters p s 1, q s 2. This yields the functions D f˜N Ž c . plotted in Fig. 3 and listed in Table 1. For all D f˜N Ž c ., optimization yields a strong-coupling value D f˜N) equal to 1r4, thus raising the initial value 1r4 in Ž31. to the correct final value 1r2. 5. To exploit this property of a particle in a box for the system at hand, the membrane between walls, we make the following crucial observation: The Feynman integrals determining the first two terms in the free energy densities in Eq. Ž16. for a membrane and in Eq. Ž28. for a particle are related to each other by a simple transformation of the integration variables. The membrane integrals d2k
H Ž 2p .
log Ž k 4 q m 4 . s 2
d2k
H Ž 2p .
1 2
4
k qm
m2 4
s
Thus, if we multiply each loop integral by a factor 1r4, we find immediately the free energy density f 1 of the membrane in Eq. Ž16. from that of the particle in the box in Eq. Ž28.. But the analogy carries further: By differentiation Ž36. and Ž37. with respect to m2 , we see that also all Feynman integrals Hd 2 krŽ2p . 2 Ž k 4 q m 4 . n are related to the HdqrŽ2p .Ž q 2 q m4 . n by the same factor 1r4. This property has the consequence that most of the connected loop diagrams contributing to the perturbation expansion of the free energy density, shown in Fig. 4 up to five loops, are related by a factor Ž1r4. L , where L is the number of loops. In particular, all such diagrams coincide which are usually summed in the Hartree–Fock approximation Žchain diagrams, daisy diagrams, etc... Only the topological more involved diagrams 3–1, 4–1, 4–2, 4–5, 5– 2, 5–3, 5–5, 5–6, 5–7, 5–11, 5–12, 5–15 in Fig. 4 do not follow this pattern. For a particle in a box, we can easily calculate the associated Feynman integrals in x-space as described in Chapter 3 of Ref. w5x, and find that they contribute less than 5% to the sum of all diagrams at each loop level. This implies that the corresponding results for the membrane between walls will differ at most by this relative amount from those for the particle in the box. We therefore conclude that since the optimal value of D f˜N Ž c . in Eq. Ž31. doubles the initial value for N ™ `, the analogous function for the membrane between walls in Eq. Ž19. will double approximately. For the quantitative deviations see the forthcoming publication w7x. A precise doubling of the result Ž22. leads to a very good agreement with the Monte Carlo number Ž2..
,
1 4
Ž 36 .
8 m2
go over into those of the particle in the box dq
H 2p log Ž q
2
dq
q m4 . s m2 ,
H 2p q
1 2
qm
1 4
s
2 m2 Ž 37 .
by the transformation 2
k ™ q,
d2k
H Ž 2p .
2
™ 14
`
dq
Hy` 2p .
273
Fig. 4. Vacuum diagrams up to five loops.
H. Kleinertr Physics Letters A 257 (1999) 269–274
274
6. The alert reader will have noted that the field transformation Ž8. is rather special. We may, for instance, chose any mapping w us
w 1q8p 2w 2 r3d 2 q w4 w 4 r d 4 q . . . q Ž2 w r d . n x
s w y 23
p2 d2
w3q...q OŽw5.,
1r n
Ž 38 .
which has a doubled coefficient of w 3 with respect to the expansion Ž8.. As a consequence, the functions f˜N Ž c . in Ž19. and Ž31. would have a doubled first term. Since this would be the correct final value, the remaining functions D f˜N Ž c . would have to converge to a vanishing optimal value for N ™ ` Žin the particle case exactly, in the membrane case approximately.. To reach this goal, the coefficients w4 ,w6 , . . . in Ž38. can be chosen rather arbitrarily, although there are a few convenient ways for which the speed of convergence is fast. A preferred choice is one in which all coefficients a 2 ,a 3 ,a 3 , . . . of the perturbation expansion vanishes for a particle in a box. This and other possibilities will be studied separately w9,10x.
Acknowledgements The author is grateful to Prof. A. Chervyakov, Prof. B. Hamprecht, Dr. A. Pelster, and M. Bachmann for numerous useful discussions.
References w1x W. Helfrich, Z. Naturforsch. A 33 Ž1978. 305; W. Helfrich, R.M. Servuss, Nuovo Cimento D 3 Ž1984. 137. w2x W. Janke, H. Kleinert, Phys. Lett. 58 Ž1987. 144. w3x W. Janke, H. Kleinert, M. Meinhardt, Phys. Lett. B 217 Ž1989. 525. The simulations were repeated by G. Gompper, D.M. Kroll, Europhys. Lett. 9 Ž1989. 58 with the results a 1 f 0.0798, a` f Ž4r3. a 1 f 0.106. A further simulation of a` was recently reported by R.R. Netz, R. Lipowski, Europhys. Lett. 29 Ž1995. 345, with the result a` f Ž3r2.p 2 r128f 0.119, which is half as big as Helfrich’s original estimate 3p 2 r128 in Ref. w1x. Another estimate a` f 0.0810 was given by F. David, J. de Phys. 51 Ž1990. C7-115. w4x W. Janke, H. Kleinert, Phys. Lett. A 117 Ž1986. 353. w5x H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, World Scientific, Singapore, 1995, sec. ed., pp. 1–850. w6x H. Kleinert, Phys. Rev. D 57 Ž1998. 2264 Žaps1997jun25_001., Addendum Žcond.matr9803268.; Phys. Lett. B 434 Ž1998. 74 Žcond-matr9801167.. w7x M. Bachmann, H. Kleinert, A. Pelster, Strong-Coupling Calculation of Pressure for Fluctuating Membrane between Walls, Berlin preprint Žin preparation.. w8x C.M. Bender, T.T. Wu, Phys. Rev. 184 Ž1969. 1231; Phys. Rev. D 7 Ž1973. 1620. w9x H. Kleinert, A. Chervyakov, B. Hamprecht, Variational Perturbation Expansion for Particle in a Box, Berlin preprint Žin preparation.. w10x H. Kleinert, A. Chervyakov, Inverse Variational Perturbation Theory for Hard-Wall Systems, Berlin preprint Žin preparation..