H. Kleinert, PATH INTEGRALS
January 6, 2016 (/home/kleinert/kleinert/books/pathis/pthic15.tex)
Thou com’st in such a questionable shape,
That I will speak to thee.
William Shakespeare (1564–1616), Hamlet
15
Path Integrals in Polymer Physics
The use of path integrals is not confined to the quantum-mechanical description
point particles in spacetime. An important field of applications lies in polymer
physics where they are an ideal tool for studying the statistical fluctuations of linelike physical objects.
15.1
Polymers and Ideal Random Chains
A polymer is a long chain of many identical molecules connected with each other
at joints which allow for spatial rotations. A large class of polymers behaves approximately like an idealized random chain. This is defined as a chain of N links
of a fixed length a, whose rotational angles occur all with equal probability (see
Fig. 15.1). The probability distribution of the end-to-end distance vector xb − xa of
such an object is given by
PN (xb − xa ) =
N Z
Y
d3 ∆xn
n=1
N
X
1
(3)
∆xn ). (15.1)
δ(|∆x
|
−
a)
δ
(x
−
x
−
n
b
a
4πa2
n=1
The last δ-function makes sure that the vectors ∆xn of the chain elements add up
correctly to the distance vector xb − xa . The δ-functions under the product enforce
the fixed length of the chain elements. The length a is also called the bond length
of the random chain.
The angular probabilities of the links are spherically symmetric. The factors
1/4πa2 ensure the proper normalization of the individual one-link probabilities
P1 (∆x) =
1
δ(|∆x| − a)
4πa2
(15.2)
in the integral
Z
d3 xb P1 (xb − xa ) = 1.
(15.3)
The same normalization holds for each N:
Z
d3 xb PN (xb − xa ) = 1.
1031
(15.4)
1032
15 Path Integrals in Polymer Physics
Figure 15.1 Random chain consisting of N links ∆xn of length a connecting xa = x0
and xb = xN .
If the second δ-function in (15.1) is Fourier-decomposed as
δ
(3)
xb − xa −
N
X
!
Z
∆xn =
n=1
d3 k ik(xb −xa )−ik PN ∆xn
n=1
,
e
(2π)3
(15.5)
we see that PN (xb − xa ) has the Fourier representation
PN (xb − xa ) =
Z
d3 k ik(xb −xa )
e
P̃N (k),
(2π)3
(15.6)
with
P̃N (k) =
N Z
Y
d3 ∆xn
n=1
1
δ(|∆xn | − a)e−ik∆xn .
4πa2
(15.7)
Thus, the Fourier transform P̃N (k) factorizes into a product of N Fouriertransformed one-link probabilities:
h
P̃N (k) = P̃1 (k)
.
(15.8)
1
sin ka
δ(|∆x| − a)e−ik∆x =
.
2
4πa
ka
(15.9)
These are easily calculated:
P̃1 (k) =
Z
d3 ∆x
iN
The desired end-to-end probability distribution is then given by the integral
PN (R) =
Z
d3 k
[P̃1 (k)]N eikR
(2π)3
1
=
2π 2 R
Z
0
∞
"
sin ka
dkk sin kR
ka
#N
,
(15.10)
H. Kleinert, PATH INTEGRALS
1033
15.2 Moments of End-to-End Distribution
where we have introduced the end-to-end distance vector
R ≡ xb − xa .
(15.11)
The generalization of the one-link distribution to D dimensions is
P1 (∆x) =
1
δ(|∆x| − a),
SD aD−1
(15.12)
with SD being the surface of a unit sphere in D dimensions [see Eq. (1.558)].
To calculate
P̃1 (k) =
Z
dD ∆x
1
δ(|∆x| − a)e−ik∆x ,
SD aD−1
(15.13)
we insert for√e−ik∆x = e−ik|∆x| cos ∆ϑ the expansion (8.130) with (8.101), and use
Y0,0 (x̂) = 1/ SD to perform the integral as in (8.250). Using the relation between
modified and ordinary Bessel functions Jν (z)1
Iν (e−iπ/2 z) = e−iπ/2 Jν (z),
(15.14)
this gives
P̃1 (k) =
Γ(D/2)
JD/2−1 (ka),
(ka/2)D/2−1
(15.15)
where Jµ (z) is the Bessel function.
15.2
Moments of End-to-End Distribution
The end-to-end distribution of a random chain is, of course, invariant under rotations
so that the Fourier-transformed probability can only have even Taylor expansion
coefficients:
N
P̃N (k) = [P̃1 (k)] =
∞
X
PN,2l
l=0
(ka)2l
.
(2l)!
(15.16)
The expansion coefficients provide us with a direct measure for the even moments
of the end-to-end distribution. These are defined by
hR2l i ≡
Z
dD R R2l PN (R).
(15.17)
The relation between hR2l i and PN,2l is found by expanding the exponential under
the inverse of the Fourier integral (15.6):
P̃N (k) =
1
Z
dD R e−ikR PN (R) =
∞ Z
X
n=0
dD R
(−ikR)n
PN (R),
n!
I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.406.1.
(15.18)
1034
15 Path Integrals in Polymer Physics
and observing that the angular average of (kR)n is related to the average of Rn in
D dimensions by



0
,
(n
−
1)!!(D
−
2)!!
h(kR) i = k hR i

,

(D + n − 2)!!
n
n
n
n = odd,
n = even.
(15.19)
The three-dimensional
result 1/(n+1) follows immediately from the angular average
R1
n
(1/2) −1 d cos θ cos θ being 1/(n + 1) for even n. In D dimensions, it is most easily
derived by assuming, for a moment, that the vectors R have a Gaussian distribution
2
2
(0)
PN (R) = (D/2πNa2 )3/2 e−R D/2N a . Then the expectation values of all products of
Ri can be expressed in terms of the pair expectation value
hRi Rj i(0) =
1
δij a2 N
D
(15.20)
via Wick’s rule (3.305). The result is
hRi1 Ri2 · · · Rin i(0) =
1
δi i i ...i an N n/2 ,
D n/2 1 2 3 n
(15.21)
with the contraction tensor δi1 i2 i3 ...in of Eqs. (8.64) and (13.89), which has the recursive definition
δi1 i2 i3 ...in = δi1 i2 δi3 i4 ...in + δi1 i3 δi2 i4 ...in + . . . δi1 in δi2 i3 ...in−1 .
(15.22)
A full contraction of the indices gives, for even n, the Gaussian expectation values:
hRn i(0) =
(D + n − 2)!! n n/2 Γ(D/2 + n/2) 2n/2 n n/2
a N
=
a N ,
(D − 2)!!D n/2
Γ(D/2)
D n/2
(15.23)
for instance
D
R4
E(0)
=
(D + 2) 4 2
aN ,
D
D
R6
E(0)
=
(D + 2) (D + 4) 6 3
aN .
D2
(15.24)
By contracting (15.21) with ki1 ki2 · · · kin we find
h(kR)n i(0) = (n − 1)!!
1
(ka)n N n/2 = k n h(R)n i(0) dn ,
n/2
D
(15.25)
with
dn =
(n − 1)!!(D − 2)!!
.
(D + n − 2)!!
(15.26)
Relation (15.25) holds for any rotation-invariant size distribution of R, in particular for PN (R), thus proving Eq. (15.17) for random chains. Hence, the expansion
coefficients PN,2l are related to the moments hR2l iN by
PN,2l = (−1)l d2l hR2l i,
(15.27)
H. Kleinert, PATH INTEGRALS
1035
15.2 Moments of End-to-End Distribution
and the moment expansion (15.16) becomes
P̃N (k) =
∞
X
l=0
(−1)l (k)2l
d2l hR2l i.
(2l)!
(15.28)
Let us calculate the even moments hR2l i of the polymer distribution PN (R)
explicitly for D = 3. We expand the logarithm of the Fourier transform P̃N (k) as
follows:
sin ka
log P̃N (k) = N log P̃1 (k) = N log
ka
!
=N
∞
X
22l (−1)l B2l
(ka)2l , (15.29)
(2l)!2l
l=1
where Bl are the Bernoulli numbers B2 = 1/6, B4 = −1/30, . . . . Then we note that
for a Taylor series of an arbitrary function y(x)
y(x) =
∞
X
an n
x ,
n=1 n!
(15.30)
the exponential function ey(x) has the expansion
ey(x) =
∞
X
bn n
x ,
n=1 n!
(15.31)
with the coefficients
n
X Y
bn
1 ai
=
n! {mi } i=0 mi ! i!
mi
.
(15.32)
The sum over the powers mi = 0, 1, 2, . . . obeys the constraint
n=
n
X
i=1
i · mi .
(15.33)
Note that the expansion coefficients an of y(x) are the cumulants of the expansion
coefficients bn of ey(x) as defined in Section 3.17. For the coefficients an of the
expansion (15.29),
an =
(
−N22l (−1)l B2l /2l
0
for n = 2l,
for n = 2l + 1,
(15.34)
we find, via the relation (15.27), the moments
l
X Y
1 N22i (−1)i B2i
hR2l i = a2l (−1)l (2l + 1)!
mi !
(2i)!2i
{mi } i=1
with the sum over mi = 0, 1, 2, . . .
"
constrained by
#mi
,
(15.35)
1036
15 Path Integrals in Polymer Physics
l=
l
X
i=1
i · mi .
(15.36)
For l = 1 and 2 we obtain the moments
2
2
5
.
hR i = a4 N 2 1 −
3
5N
2
4
hR i = a N,
(15.37)
In the limit of large N, the leading behavior of the moments is the same as in (15.20)
and (15.24). The linear growth of hR2 i with the number of links N is characteristic
for a random chain. In the presence of interactions, there will be a different power
behavior expressed as a so-called scaling law
hR2 i ∝ a2 N 2ν .
(15.38)
The number ν is called the critical exponent of this scaling law. It is intuitively
obvious that ν must be a number between ν = 1/2 for a random chain as in (15.37),
and ν = 1 for a completely stiff chain.
Note that the knowledge of all moments of the end-to-end distribution determines
completely the shape of the distribution by an expansion
∞
n
X
1
n (−1)
hR
i
∂Rn δ(R).
PL (R) =
D−1
SD R
n!
n=0
(15.39)
This can easily be verified by calculating the integrals (15.17) using the integrals
formula
Z
15.3
dz z n ∂zn δ(z) = (−1)n n! .
(15.40)
Exact End-to-End Distribution in Three Dimensions
Consider the Fourier representation (15.10), rewritten as
i
PN (R) = 2 2
4π a R
Z
∞
−∞
−iηR/a
dη η e
sin η
η
!N
,
(15.41)
with the dimensionless integration variable η ≡ ka. By expanding
N
1 X
N
sin η =
exp [i(N − 2n)η] ,
(−1)n
N
n
(2i) n=0
!
N
(15.42)
we find the finite series
PN (R) =
1
N
X
(−1)
2N +2 iN −1 π 2 a2 R n=0
n
!
N
IN (N − 2n − R/a),
n
(15.43)
H. Kleinert, PATH INTEGRALS
1037
15.3 Exact End-to-End Distribution in Three Dimensions
where IN (x) are the integrals
IN (x) ≡
Z
∞
−∞
dη
eiηx
.
η N −1
(15.44)
For N ≥ 2, these integrals are all singular. The singularity can be avoided by noting
that the initial integral (15.41) is perfectly regular at η = 0. We therefore replace the
expression (sin η/η)N in the integrand by [sin(η − iǫ)/(η − iǫ)]N . This regularizes
each term in the expansion (15.43) and leads to well-defined integrals:
IN (x) =
Z
∞
−∞
dη
eix(η−iǫ)
.
(η − iǫ)N −1
(15.45)
For x < 0, the contour of integration can be closed by a large semicircle in the lower
half-plane. Since the lower half-plane contains no singularity, the residue theorem
shows that
IN (x) = 0,
x < 0.
(15.46)
For x > 0, on the other hand, an expansion of the exponential function in powers
of η −iǫ produces a pole, and the residue theorem yields
IN (x) =
2πiN −1 N −2
x
,
(N − 2)!
x > 0.
(15.47)
Hence we arrive at the finite series
!
X
1
N
PN (R) = N +1
(N − 2n − R/a)N −2 . (15.48)
(−1)n
2
2
(N − 2)!πa R 0≤n≤(N −R/a)/2
n
The distribution
√ is displayed for various values of N in Fig. 15.2, where
√ we have
plotted 2πR2 N PN (R) against the rescaled distance variable ρ = R/ N a. With
this N-dependent rescaling all curves have the same unit area. Note that they
converge rapidly towards a universal zero-order distribution
(0)
PN (R)
=
s
3
3
3R2
exp
−
2πNa2
2Na2
(
)
→
(0)
PL (R)
=
s
D
D
2
e−DR /2aL . (15.49)
2πaL
In the limit of large N, the length L will be used as a subscript rather than the
diverging N. The proof of the limit is most easily given in Fourier space. For
large N at finite k 2 a2 N, the Nth power of the quantity P̃1 (k) in Eq. (15.15) can be
approximated by
[P̃1 (k)]N ∼ e−N k
2 a2 /2D
.
(15.50)
Then the Fourier transform (15.10) is performed with the zero-order result (15.49).
√
In Fig. 15.2 we see that this large-N limit is approached uniformly in ρ = R/ N a.
The approach to this limit is studied analytically in the following two sections.
1038
15 Path Integrals in Polymer Physics
Figure 15.2
√ End-to-end distribution PN (R)√of random chain with N links. The func2
tions 2πR NPN (R) are plotted against R/ N a which gives all curves the same unit
area. Note the fast convergence for growing N . The dashed curve is the continuum distri(0)
bution PL (R) of Eq. (15.49). The circles on the abscissa mark the maximal end-to-end
distance.
15.4
Short-Distance Expansion for Long Polymer
At finite N, we expect corrections which are expandable in the form
PN (R) =
(0)
PN (k)
∞
X
"
#
1
1+
C (R2 /Na2 ) ,
n n
N
n=1
(15.51)
where the functions Cn (x) are power series in x starting with x0 .
Let us derive this expansion. In three dimensions, we start from (15.29) and
separate the right-hand side into the leading k 2 -term and a remainder
"
C̃(k) ≡ exp N
∞
X
l=2
22l (−1)l B2l 2 2 l
(k a ) .
(2l)!2l
#
(15.52)
Exponentiating both sides of (15.29), the end-to-end probability factorizes as
2 k 2 /6
P̃N (k) = e−N a
C̃(k).
(15.53)
The function C̃(k) is now expanded in a power series
C̃(k) = 1 +
X
C̃n,l N n (a2 k 2 )l ,
(15.54)
n=1,2,...
l=2n,2n+1,...
H. Kleinert, PATH INTEGRALS
1039
15.4 Short-Distance Expansion for Long Polymer
with the lowest coefficients
1
,
180
1
,
= −
37800
1
,
2835
1
=
,
64800
C̃1,2 = −
C̃1,3 = −
C̃1,4
C̃2,4
... .
(15.55)
For any dimension D, we factorize
2 k 2 /2D
P̃N (k) = e−N a
C̃(k)
(15.56)
and find C̃(k) by expanding (15.15) in powers of k and proceeding as before. This
gives the coefficients
C̃1,2
C̃1,3
C̃1,4
C̃2,4
=
=
=
=
−1/4D 2 (D + 2),
−1/3D 3 (D + 2)(D + 4),
−(5D + 12)/8D 4(D + 2)2 (D + 4)(D + 6),
1/32D 4(D + 2)2 .
(15.57)
We now Fourier-transform (15.56). The leading term in C̃(k) yields the zero-order
distribution (15.49) in D dimensions,
(0)
PN (R)
=
s
D
D
2
2
e−DR /2N a ,
2
2πNa
(15.58)
or, written in terms of the reduced distance variable ρ = R/Na,
(0)
PN (R)
=
s
D
D
2
e−DN ρ /2 .
2
2πNa
(15.59)
To account for the corrections in C̃(k) we take the expansion (15.54), emphasize the
dependence on k 2 a2 by writing
C̃(k) = C̄(k 2 a2 ),
and observe that in the Fourier transform
PN (R) =
Z
Z
d3 k ikR
d3 k ikR −N k2 a2 /2D
e
P
(k)
=
e e
C̄(k 2 a2 ),
N
(2π)3
(2π)3
(15.60)
the series can be pulled out of the integral by replacing each power (k 2 a2 )p by
(−2D∂N )p . The result has the form
PN (R) = C̄(−2D∂N )
Z
d3 k ikR −N k2 a2 /2D
(0)
e e
= C̄(−2D∂N )PN (R).
3
(2π)
(15.61)
Going back to coordinate space, we obtain an expansion
D
PN (R) =
2πNa2
D/2
2 /2
e−DN ρ
C(R).
(15.62)
1040
15 Path Integrals in Polymer Physics
For D = 3, the function C(R) is given by the series
∞
X
C(R) = 1 +
Cn,l N −n (Nρ2 )l ,
(15.63)
n=1
l=0,...,2n
with the coefficients
C1,l
3 3
9
= − , ,−
,
4 2 20
C2,l
29
69 981 1341 81
=
.
,− ,
,−
,
160 40 400 1400 800
(15.64)
For any D, we find the coefficients
C1,l =
C2,l =
15.5
D2
D D
,
− , ,−
4 2
4(D + 2)
(3D 2 − 2D + 8)D (D 2 + 2D + 8)D (3D 2 + 14D + 40)D 2
,−
,
,
96(D + 2)
8(D + 2)
16(D + 2)2
!
D4
(3D 2 + 22D + 56)D 3
.
,
−
24(D + 2)2 (D + 4) 32(D + 2)2
!
(15.65)
Saddle Point Approximation to Three-Dimensional
End-to-End Distribution
Another study of the approach to the limiting distribution (15.49) proceeds via the
saddle point approximation. For this, the integral in (15.41) is rewritten as
Z
∞
−∞
dη η e−N f (η) ,
(15.66)
with
!
R
sin η
f (η) = i
.
η − log
Na
η
(15.67)
The extremum of f lies at η = η̄, where η̄ solves the equation
coth(iη̄) −
R
1
=
.
iη̄
Na
(15.68)
The function on the left-hand side is known as the Langevin function:
1
L(x) ≡ coth x − .
x
(15.69)
The extremum lies at the imaginary position η̄ ≡ −ix̄ with x̄ being determined by
the equation
L(x̄) =
R
.
Na
(15.70)
H. Kleinert, PATH INTEGRALS
1041
15.6 Path Integral for Continuous Gaussian Distribution
The extremum is a minimum of f (η) since
f ′′ (η̄) = L′ (x̄) = −
1
1
+ 2 > 0.
2
sinh x̄ x̄
(15.71)
By shifting the integration contour vertically into the complex η plane to make it
run through the minimum at −ix̄, we obtain
∞
1
N
−N f (η̄)
e
dη (−ix̄ + η) exp − f ′′ (η̄)η 2
2
2
4iπ a R
2
−∞
s
2π
η̄
e−N f (η̄) .
=
2
2
4π a R Nf ′′ (x̄)
Z
PN (R) ≈ −
(15.72)
When expressed in terms of the reduced distance ρ ≡ R/Na ∈ [0, 1], this reads
1
Li (ρ)2
PN (R) ≈
(2πNa2 )3/2 ρ {1 − [Li (ρ)/ sinh Li (ρ)]2 }1/2
(
sinh Li (ρ)
Li (ρ) exp[ρLi (ρ)]
)N
. (15.73)
Here we have introduced the inverse Langevin function Li (ρ) since it allows us to
express x̄ as
x̄ = Li (ρ)
(15.74)
[inverting Eq. (15.70)]. The result (15.73) is valid in the entire interval ρ ∈ [0, 1]
corresponding to R ∈ [0, Na]; it ignores corrections of the order 1/N. By expanding
the right-hand side in a power series in ρ, we find
PN (R) = N
3
2πNa2
3/2
3R2
exp −
2Na2
!
3R2
9R4
1+
−
+ . . . , (15.75)
2N 2 a2 20N 3 a4
!
with some normalization
constant N . At each order of truncation, N is determined
R 3
in such a way that d R PN (R)= 1. As a check we take the limit ρ2 ≪ 1/N and
find powers in ρ which agree with those in (15.62), for D = 3, with the expansion
(15.63) of the correction factor.
15.6
Path Integral for Continuous Gaussian Distribution
The limiting end-to-end distribution (15.49) is equal to the imaginary-time amplitude of a free particle in natural units with h̄ = 1:
1
(xb τb |xa τa ) = q
D
2π(τb − τa )/M
M (xb − xa )2
exp −
.
2 τb − τa
"
#
(15.76)
We merely have to identify
xb − xa ≡ R,
(15.77)
1042
15 Path Integrals in Polymer Physics
and replace
τb − τa → Na,
M → D/a.
(15.78)
(15.79)
Thus we can describe a polymer with R2 ≪ Na2 by the path integral
PL (R) =
Z
(
D
D D x exp −
2a
L
Z
0
)
ds [x′ (s)]2 =
s
D −DR2 /2La
e
.
2πa
(15.80)
The number of time slices is here N [in contrast to (2.66) where it was N + 1), and
the total length of the polymer is L = Na.
Let us calculate the Fourier transformation of the distribution (15.80):
P̃L (q) =
Z
dD R e−i q·R PL (R).
(15.81)
After a quadratic completion the integral yields
P̃L (q) = e−Laq
2 /2D
,
(15.82)
with the power series expansion
P̃L (q) =
∞
X
(−1)
lq
l=0
2l
l!
La
2D
l
.
(15.83)
Comparison with the moments (15.23) shows that we can rewrite this as
P̃L (q) =
∞
X
(−1)l
l=0
E
D
q 2l
Γ(D/2)
2l
.
R
l! 22l Γ(D/2 + l)
(15.84)
This is a completely general relation: the expansion coefficients of the Fourier transform yield directly the moments of a function, up to trivial numerical factors specified
by (15.84).
The end-to-end distribution determines rather directly the structure factor of a
dilute solution of polymers which is observable in static neutron and light scattering
experiments:
1
S(q) = 2
L
Z
L
0
ds
Z
0
L
D
′
E
ds′ eiq·[x(s)−x(s )] .
(15.85)
The average over all polymers running from x(0) to x(L) can be written, more
explicitly, as
D
′
E
eiq·[x(s)−x(s )] =
Z
dD x(L)
Z
dD (x(s′ )−x(s))
′
Z
dD x(0)
(15.86)
×PL−s′ (x(L)−x(s′ ))e−iq·x(s ) Ps′ −s (x(s′ )−x(s))eiq·x(s) Ps−0 (x(s)−x(0)).
H. Kleinert, PATH INTEGRALS
1043
15.6 Path Integral for Continuous Gaussian Distribution
The integrals over initial and final positions give unity due to the normalization
(15.4), so that we remain with
D
iq·[x(s)−x(s′ )]
e
E
=
Z
dD R e−iq·R Ps′ −s (R).
(15.87)
Since this depends only on L′ ≡ |s′ − s| and
not on s + s′ , we decompose the double
R
L
integral in over s and s′ in (15.85) into 2 0 dL′ (L − L′ ) and obtain
Z
2 ZL ′
′
′
dL (L − L ) dD R eiq·R(L ) PL′ (R),
S(q) = 2
L 0
(15.88)
or, recalling (15.81),
2 ZL ′
dL (L − L′ )P̃L′ (q).
L2 0
S(q) =
(15.89)
Inserting (15.82) we obtain the Debye structure factor of Gaussian random paths:
S Gauss (q) =
2 −x
,
x
−
1
+
e
x2
x≡
q 2 aL
.
2D
(15.90)
This function starts out like 1 − x/3 + x2 /12 + . . . for small q and falls of like q −2
for q 2 ≫ 2D/aL. The Taylor coefficients are determined by the moments of the
end-to-end distribution. By inserting (15.84) into (15.89) we obtain:
∞
X
Γ(D/2)
2
S(q) = (−1) q 2l
2 l!Γ(l + D/2) L2
l=0
l 2l
Z
0
L
D
E
dL′ (L − L′ ) R2l .
(15.91)
Although the end-to-end
√ distribution (15.80) agrees with the true polymer distribution (15.1) for R ≪ Na, it is important to realize that the nature of the
fluctuations in the two expressions is quite different. In the polymer expression, the
length of each link ∆xn is fixed. In the sliced action of the path integral Eq. (15.80),
AN = a
N
X
M (∆xn )2
,
a2
n=1 2
(15.92)
on the other hand, each small section fluctuates around zero with a mean square
h(∆xn )2 i0 =
a
a2
= .
M
D
(15.93)
Yet, if the end-to-end distance of the polymer is small compared to the completely
stretched configuration, the distributions are practically the same. There exists a
qualitative difference only if the polymer is almost completely stretched. While the
polymer distribution vanishes for R > Na, the path integral (15.80) gives a nonzero
value for arbitrarily large R. Quantitatively, however, the difference is insignificant
since it is exponentially small (see Fig. 15.2).
1044
15.7
15 Path Integrals in Polymer Physics
Stiff Polymers
The end-to-end distribution of real polymers found in nature is never the same as
that of a random chain. Usually, the joints do not allow for an equal probability
of all spherical angles. The forward angles are often preferred and the polymer is
stiff at shorter distances. Fortunately, if averaged over many links, the effects of the
stiffness becomes less and less relevant. For a very long random chain with a finite
stiffness one finds the same linear dependence of the square end-to-end distance on
the length L = Na as for ideal random chains which has, according to Eq. (15.23),
the Gaussian expectations:
hR2 i = aL,
hR2l i =
(D + 2l − 2)!!
(aL)l .
l
(D − 2)!!D
(15.94)
For a stiff chain, the expectation value hR2 i will increase aL to aeff L, where aeff is
the effective bond length In the limit of a very large stiffness, called the rod limit,
the law (15.94) turns into
hR2 i ≡ L2 ,
hR2l i ≡ L2l ,
(15.95)
i.e., the effective bond length length aeff increases to L. This intuitively obvious
statement can easily be found from the normalized end-to-end distribution, which
coincides in the rod limit with the one-link expression (15.12):
PLrod (R) =
1
δ(R − L),
SD RD−1
(15.96)
and yields [recall (15.17)]
n
hR i =
Z
D
d RR
n
PLrod(R)
=
Z
0
∞
dR Rn δ(R − L) = Ln .
(15.97)
By expanding PLrod(R) in powers of L, we obtain the series
PLrod (R) =
∞
n
X
1
n
n (−1)
L
hR
i
∂ n δ(R).
SD RD−1 n=0
n! R
(15.98)
An expansion of this form holds for any stiffness: the moments of the distribution
are the Taylor coefficients of the expansion of PL (R) into a series of derivatives of
δ(R)-functions.
Let us also calculate the Fourier transformation (15.81) of this distribution. Recalling (15.15), we find
P̃Lrod (q) = P̃ rod (qL) ≡
Γ(D/2)
JD/2−1 (qL).
(qL/2)D/2−1
(15.99)
For an arbitrary rotationally symmetric PL (R) = PL (R), we simply have to superimpose these distributions for all R:
P̃L (q) = SD
Z
0
∞
dR RD−1 P̃ rod(qL)PL (R).
(15.100)
H. Kleinert, PATH INTEGRALS
1045
15.7 Stiff Polymers
This is simply proved
by decomposing and performing
the Fourier transformation
R∞
R∞
′
′
rod
′
(15.81) on PL (R) = 0 dR δ(R − R )PR′ (R)=SD 0 dR′ R′D−1 PRrod
′ (R)PL (R ), and
performing the Fourier transformation (15.81) on PRrod
In D = 3 dimensions,
′ (R).
(15.100) takes the simple form:
P̃L (q) = 4π
Z
∞
0
dR R2
sin qR
PL (R).
qR
(15.101)
Inserting the power series expansion for the Bessel function2
Jν (z) =
ν X
∞
z
2
l=0
(−1)k (z/2)2l
l!Γ(ν + l + 1)
(15.102)
into (15.99), and this into (15.100), we obtain
P̃L (q) =
∞
X
(−1)
l=0
l
2l
q
2
Γ(D/2)
SD
l!Γ(D/2 + l)
Z
0
∞
dR RD−1 R2l PL (R),
(15.103)
in agreement with the general expansion (15.84).
The same result is obtained
by inserting into (15.103) the expansion (15.98) and using the integrals
R∞
m n
n
0 dR R ∂R δ(R) = δmn (−1) n!, which are proved by n partial integrations.
The structure factor of a completely stiff polymer (rod limit) is obtained by
inserting (15.99) into (15.89). The resulting S rod (q) depends only on qL:
S
rod
!D/2
2
4−2D
(qL)= 2 2 +
q L
qL
Γ(D/2)JD/2−2 (qL) + 2F (1/2; 3/2, D/2;−q 2L2/4),(15.104)
where F (a; b, c; z) is the hypergeometric function (1.453). For D = 3, the integral
R
(15.89) reduces to (2/L2 ) 0L dL′ (L − L′ )(sin qL′ )/qL′ , as in the similar equation
(15.9), and the result is simply
S
rod
2
(z) = 2 [cos z − 1 + z Si(z)] ,
z
Si(z) ≡
Z
0
z
dt
sin t.
t
(15.105)
This starts out like 1 − z 2 /36 + z 4 /1800 + . . . . For large z we use the limit of the
sine integral3 Si(z) → π/2 to find S rod (q) → π/qL.
For of an arbitrary rotationally symmetric end-to-end distribution PL (R), the
structure factor can be expressed, by analogy with (15.100), as a superposition of
rod limits:
SL (q) = SD
Z
0
∞
dR RD−1 S rod (qR)PL (R).
(15.106)
When passing from long to short polymers at a given stiffness, there is a crossover
between the moments (15.94) and (15.95) and the behaviors of the structure function. Let us study this in detail.
2
3
I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.440.
ibid., Formulas 8.230 and 8.232.
1046
15 Path Integrals in Polymer Physics
15.7.1
Sliced Path Integral
The stiffness of a polymer pictured in Fig. 15.1 may be parameterized by the bending
energy
N
κ X
(un − un−1 )2 ,
2a n=1
N
Ebend
=
(15.107)
where un are the unit vectors specifying the directions of the links. The initial and
final link directions of the polymer have a distribution
−1
1 NY
(ub L|ua 0) =
A n=1
N
κ X
dun
exp −
(un − un−1 )2 ,
A
2akB T n=1
"Z
#
#
"
(15.108)
where A is some normalization constant, which we shall choose such that the measure
of integration coincides with that of a time-sliced path integral near a unit sphere in
Eq. (8.151). Comparison if the bending energy (15.107) with the Euclidean action
(8.152) we identify
Mr 2
κ
=
,
h̄ǫ
akB T
and see that we must replace N → N − 1 and set
A=
D−1
q
2πakB T /κ
(15.109)
.
(15.110)
The result of the integrations in (15.108) is then known from Eq. (8.156):
(ub L|ua 0) =
∞ h
X
I˜l+D/2−1 (h)
iN X
∗
Ylm (ub )Ylm
(ua ),
m
l=0
h≡
κ
,
akB T
(15.111)
with the modified Bessel function I˜l+D/2−1 (z) of Eq. (8.11).
The partition function of the polymer is obtained by integrating over all final
and averaging over all initial link directions [1]:
Z
N
N
κ X
dun
d ua Y
exp −
(un − un−1 )2
ZN =
SD n=1
A
2akB T n=1
Z
Z
d ua
= d ub
(ub L|ua 0).
SD
Inserting here the spectral representation (15.111) we find
"
Z
ZN = I˜D/2−1
κ
akB T
N
#
=
"s
"
2πκ −κ/akB T
κ
e
ID/2−1
akB T
akB T
Knowing this we may define the normalized distribution function
1
PN (ub , ua ) =
(ub L|ua 0),
ZN
whose integral over ua as well as over ua is unity:
Z
dub PN (ub , ua ) =
Z
dua PN (ub , ua ) = 1.
#
#N
(15.112)
.
(15.113)
(15.114)
(15.115)
H. Kleinert, PATH INTEGRALS
1047
15.7 Stiff Polymers
15.7.2
Relation to Classical Heisenberg Model
The above partition function is closely related to the partition function of the onedimensional classical Heisenberg model of ferromagnetism which is defined by
ZNHeis
≡
Z
N Z
N
J X
d ua Y
dun exp
un · un−1 ,
SD n=1
kB T n=1
"
#
(15.116)
where J are interaction energies due to exchange integrals of electrons in a ferromagnet. This differs from (15.112) by a trivial normalization factor, being equal
to
s
ZNHeis = 
2πJ
kB T
2−D
ID/2−1
J
kB T

N

.
(15.117)
Identifying J ≡ κ/a we may use the Heisenberg partition functions for all calculations of stiff polymers. As an example take the correlation function between
neighboring tangent vectors hun · un−1 i. In order to calculate this we observe that
the partition function (15.116) can just as well be calculated exactly with a slight
modification that the interaction strength J of the Heisenberg model depends on
the link n. The result is the corresponding generalization of (15.117):
N
Y
s
2πJn

ZNHeis (J1 , . . . , JN ) =
kB T
n=1
2−D
ID/2−1

Jn 
.
kB T
(15.118)
This expression may be used as a generating function for expectation values
hun · un−1 i which measure the degree of alignment of neighboring spin directions.
Indeed, we find directly
ID/2 (J/kB T )
dZNHeis (J1 , . . . , JN ) =
.
hun · un−1 i = (kB T )
dJn
ID/2−1 (J/kB T )
Jn ≡J
(15.119)
This expectation value measures directly the internal energy per link of te chain.
Indeed, since the free energy is FN = −kB T log ZNHeis , we obtain [recall (1.548)]
EN = N hun · un−1 i = N
ID/2 (J/kB T )
.
ID/2−1 (J/kB T )
(15.120)
Let us also calculate the expectation value of the angle between next-to-nearest
neighbors hun+1 · un−1 i. We do this by considering the expectation value
ID/2 (J/kB T )
d2 ZNHeis (J1 , . . . , JN ) =
h(un+1 · un )(un · un−1 )i = (kB T )2
dJn+1 dJn
ID/2−1 (J/kB T )
Jn ≡J
"
#2
.
(15.121)
Then we prove that the left-hand side is in fact equal to the desired expectation
value hun+1 · un−1 i. For this we decompose the last vector un+1 into a component
1048
15 Path Integrals in Polymer Physics
θn+1,n
un+1
θn+1,n−1
un
θn,n−1
un−1
Figure 15.3 Neighboring links for the calculation of expectation values.
parallel to un and a component perpendicular to it: un+1 = (un+1 ·un ) un +u⊥
n+1 . The
corresponding decomposition of the expectation value hun+1 ·un−1 i is hun+1 ·un−1 i =
h(un+1 ·un )(un ·un−1 )i +hu⊥
n+1 ·un i. Now, the energy in the Boltzmann factor depends
only on (un+1 · un ) + (un · un−1 ) = cos θn+1,n + cos θn,n−1 , so that the integral over
u⊥
n+1 runs over a surface of a sphere of radius sin θn+1,n in D − 1 dimensions [recall
(8.117)] and the Boltzmann factor does not depend on the angles. The integral
⊥
receives therefore equal contributions from u⊥
n+1 and −un+1 , and vanishes. This
proves that
"
ID/2 (J/kB T )
hun+1 · un−1 i = hun+1 · un i hun · un−1 i =
ID/2−1 (J/kB T )
#2
,
(15.122)
and further, by induction, that
"
ID/2 (J/kB T )
hul · uk i =
ID/2−1 (J/kB T )
#|l−k|
.
(15.123)
For the polymer, this implies an exponential falloff
hul · uk i = e−|l−k|a/ξ ,
(15.124)
where ξ is the persistence length
"
#
ID/2 (κ/akB T )
ξ = −a/ log
.
ID/2−1 (κ/akB T )
(15.125)
For D = 3, this is equal to
ξ=−
a
log [coth(κ/akB T ) − akB T /κ]
.
(15.126)
Knowing the correlation functions it is easy to calculate the magnetic susceptibility. The total magnetic moment is
M=a
N
X
un ,
(15.127)
n=0
so that we find the total expectation value
D
E
M2 = a2 (N + 1)
−(N +1)a/ξ
1 + e−a/ξ
2 −a/ξ 1 − e
−
2a
e
.
1 − e−a/ξ
(1 − e−a/ξ )2
(15.128)
The susceptibility is directly proportional to this. For more details see [2].
H. Kleinert, PATH INTEGRALS
1049
15.7 Stiff Polymers
15.7.3
End-to-End Distribution
A modification of the path integral (15.108) yields the distribution of the end-to-end
distance at given initial and final directions of the polymer links:
R = xb − xa = a
N
X
(15.129)
un
n=1
of the stiff polymer:
−1
1 1 NY
PN (ub , ua ; R) =
ZN A n=2
N
X
dun (D)
un )
δ (R − a
A
n=1
"Z
#
−1
κ NX
(un+1 − un )2 ,
× exp −
2akB T n=1
#
"
(15.130)
whose integral over R leads back to the distribution PN (ub , ua ):
Z
dD R PN (ub , ua ; R) = PN (ub , ua ).
(15.131)
If we integrate in (15.130) over all final directions and average over the initial
ones, we obtain the physically more accessible end-to-end distribution
PN (R) =
Z
dub
Z
dua
PN (ub , ua ; R).
SD
(15.132)
The sliced path integral, as it stands, does not yet give quite the desired probability. Two small corrections are necessary, the same that brought the integral
near the surface of a sphere to the path integral on the sphere in Section 8.9. After
including these, we obtain a proper overall normalization of PN (ub , ua ; R).
15.7.4
Moments of End-to-End Distribution
Since the δ (D) -function in (15.130) contains vectors un of unit length, the calculation
of the complete distribution is not straightforward. Moments of the distribution,
however, which are defined by the integrals
2l
hR i =
Z
dD R R2l PN (R),
(15.133)
are relatively easy to find from the multiple integrals
1
hR i =
ZN
2l
× R
2l
Z
1
d R
A
D
"
Z
dub
κ
exp −
2akB T
NY
−1
n=2
N
−1
X
n=1
"Z
dun
A
N
X
dua (D)
aun )
δ (R −
SD
n=1
# "Z
(un+1 − un )
2
#
#
.
(15.134)
1050
15 Path Integrals in Polymer Physics
Performing the integral over R gives
1 1
hR2l i =
ZN A
Z
dub
NY
−1
n=2
"Z
dun
A
# "Z
dua
SD
#
a
N
X
n=1
un
!2l
−1
κ NX
× exp −
(un+1 − un )2 .
2akB T n=1
"
#
(15.135)
Due to the normalization property (15.115), the trivial moment is equal to unity:
h1i =
15.8
Z
D
d R PN (R) =
Z
dub
Z
dua
PN (ub , ua |L) = 1.
SD
(15.136)
Continuum Formulation
Some properties of stiff polymers are conveniently studied in the continuum limit of
the sliced path integral (15.108), in which the bond length a goes to zero and the
link number to infinity so that L = Na stays constant. In this limit, the bending
energy (15.107) becomes
Ebend
κ
=
2
Z
L
0
ds (∂s u)2 ,
(15.137)
where
u(s) =
d
x(s),
ds
(15.138)
is the unit tangent vector of the space curve along which the
√ polymer runs. The
parameter s is the arc length of the line elements, i.e., ds = dx2 .
15.8.1
Path Integral
If the continuum limit is taken purely formally on the product of integrals (15.108),
we obtain a path integral
(ub L|ua 0) =
Z
Du e−(κ/2kB T )
RL
0
ds [u′ (s)]2
,
(15.139)
This coincides with the Euclidean version of a path integral for a particle on the
surface of a sphere. It is a nonlinear σ-model (recall p. 746).
The result of the integration has been given in Section 8.9 where we found that
it does not quite agree with what we would obtain from the continuum limit of the
discrete solution (15.111) using the limiting formula (8.157), which is
∞
X
!
X
kB T
∗
P (ub , ua |L) =
exp −L
L2
Ylm (ub )Ylm
(ua ),
2κ
m
l=0
(15.140)
H. Kleinert, PATH INTEGRALS
1051
15.8 Continuum Formulation
with
L2 = (D/2 − 1 + l)2 − 1/4.
(15.141)
For a particle on a sphere, the discrete expression (15.130) requires a correction
since it does not contain the proper time-sliced action and measure. According
the Section 8.9 the correction replaces L2 by the eigenvalues of the square angular
momentum operator in D dimensions, L̂2 ,
L2 → L̂2 = l(l + D − 2).
(15.142)
After this replacement the expectation of the trivial moment h1i in (15.136) is equal
to unity since it gives the distribution (15.140) the proper normalization:
Z
dub P (ub , ua |L) = 1.
(15.143)
This follows from the integral
Z
dub
X
∗
Ylm
(ub )Ylm (ua ) = δl0 ,
(15.144)
m
which was derived in (8.250). Thus, with L̂2 in (15.140) instead of L2 , no extra
∗
normalization factor is required. The sum over Ylm (u2 )Ylm
(u1 ) may furthermore be
rewritten in terms of Gegenbauer polynomials using the addition theorem (8.126),
so that we obtain
∞
X
!
kB T 2 1 2l + D − 2 (D/2−1)
P (ub , ua |L) =
exp −L
L̂
Cl
(u2 u1 ).
2κ
SD D − 2
l=0
15.8.2
(15.145)
Correlation Functions and Moments
We are now ready to evaluate the expectation values of R2l . In the continuum
approximation we write
2l
R =
"Z
0
L
ds u(s)
#2l
.
(15.146)
The expectation value of the lowest moment hR2 i is given by the double-integral
over the correlation function hu(s2 )u(s1 )i:
hR2 i =
Z
0
L
ds2
Z
L
0
ds1 hu(s2 )u(s1 )i = 2
Z
0
L
ds2
Z
0
s2
ds1 hu(s2 )u(s1 )i.
(15.147)
The correlation function is calculated from the path integral via the composition
law as in Eq. (3.301), which yields here
dua
du2 du1
SD
× P (ub , u2 |L − s2 ) u2 P (u2 , u1 |s2 − s1 ) u1 P (u1 , ua |s1 ). (15.148)
hu(s2 )u(s1 )i =
Z
dub
Z
Z
Z
1052
15 Path Integrals in Polymer Physics
The integrals over ua and ub remove the initial and final distributions via the normalization integral (15.143), leaving
hu(s2 )u(s1 )i =
Z
du2
Z
du1
u2 u1 P (u2 , u1 |s2 − s1 ).
SD
(15.149)
Due to the manifest rotational invariance, the normalized integral over u1 can be
omitted. By inserting the spectral representation (15.145) with the eigenvalues
(15.142) we obtain
hu(s2 )u(s1 )i =
=
X
Z
du2 u2 u1 P (u2 , u1 |s2 − s1 )
−(s2 −s1 )kB T L̂2 /2κ
e
l
"Z
#
1 2l + D − 2 (D/2−1)
Cl
(u2 u1 ) . (15.150)
du2 u2 u1
SD D − 2
We now calculate the integral in the brackets with the help of the recursion relation
(15.152) for the Gegenbauer functions4
(ν)
zCl (z) =
h
i
1
(ν)
ν
(2ν + l − 1)Cl−1 (z) + (l + 1)Cl+1
(z) .
2(ν + l)
(15.151)
Obviously, the integral over u2 lets only the term l = 1 survive. This, in turn,
involves the integral
Z
(D/2−1)
du2 C0
(cos θ) = SD .
(15.152)
The l = 1 -factor D/(D − 2) in (15.150) is canceled by the first l = 1 -factor in the
recursion (15.151) and we obtain the correlation function
"
#
kB T
hu(s2 )u(s1 )i = exp −(s2 − s1 )
(D − 1) ,
2κ
(15.153)
where D − 1 in the exponent is the eigenvalue of L̂2 = l(l + D − 2) at l = 1. The
correlation function (15.153) agrees with the sliced result (15.124) if we identify the
continuous version of the persistence length (15.126) with
ξ ≡ 2κ/kB T (D − 1).
(15.154)
Indeed, taking the limit a → 0 in (15.125), we find with the help of the asymptotic
behavior (8.12) precisely the relation (15.154).
After performing the double-integral in (15.147) we arrive at the desired result
for the first moment:
n
h
hR2 i = 2 ξL − ξ 2 1 − e−L/ξ
4
io
.
(15.155)
I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.933.1.
H. Kleinert, PATH INTEGRALS
1053
15.8 Continuum Formulation
This is valid for all D. The result may be compared with the expectation value
of the squared magnetic moment of the Heisenberg chain in Eq. (15.128), which
reduces to this in the limit a → 0 at fixed L = (N + 1)a.
For small L/ξ, the second moment (15.155) has the large-stiffness expansion

1
1L
+
hR2 i = L2 1 −
3ξ
12
L
ξ
!2
1
−
60
L
ξ
!3

+ . . . ,
(15.156)
the first term being characteristic for a completely stiff chain [see Eq. (15.95)]. For
large L/ξ, on the other hand, we find the small-stiffness expansion
!
ξ
+ ... ,
hR i ≈ 2ξL 1 −
L
2
(15.157)
where the dots denote exponentially small terms. The first term agrees with relation
(15.94) for a random chain with an effective bond length
aeff = 2ξ =
κ
4
.
D − 1 kB T
(15.158)
The calculation of higher expectations hR2l i becomes rapidly complicated. Take,
for instance, the moment hR4 i, which is given by the quadruple integral over the
four-point correlation function
4
hR i = 8
Z
0
L
ds4
Z
s4
0
Z
ds3
0
s3
ds2
Z
s2
0
ds1 δi4 i3 i2 i1 hui4 (s4 )ui3 (s3 )ui2 (s2 )ui1 (s1 )i, (15.159)
with the symmetric pair contraction tensor of Eq. (15.22):
δi4 i3 i2 i1 ≡ (δi4 i3 δi2 i1 + δi4 i2 δi3 i1 + δi4 i1 δi3 i2 ) .
(15.160)
The factor 8 and the symmetrization of the indices arise when bringing the integral
R4 =
Z
0
L
ds4
Z
0
L
ds3
Z
L
0
ds2
Z
L
0
ds1 (u(s4 )u(s3 ))(u(s2 )u(s1 ))
(15.161)
to the s-ordered form in (15.159). This form is needed for the s-ordered evaluation
of the u-integrals which proceeds by a direct extension of the previous procedure for
hR2 i. We write down the extension of expression (15.148) and perform the integrals
over ua and ub which remove the initial and final distributions via the normalization
integral (15.143), leaving δi4 i3 i2 i1 times an integral [the extension of (15.149)]:
du1
(15.162)
SD
× ui4 ui3 ui2 ui1 P (u4 , u3 |s4 − s3 )P (u3 , u2 |s3 − s2 )P (u2 , u1 |s2 − s1 ) .
hui4 (s4 )ui3 (s3 )ui2 (s2 )ui1 (s1 )i =
Z
du4
Z
du3
Z
du2
Z
1054
15 Path Integrals in Polymer Physics
The normalized integral over u1 can again be omitted. Still, the expression is complicated. A somewhat tedious calculation yields
D 2 + 6D − 1 D − 7 −L/ξ
4(D + 2) 2 2
(15.163)
L ξ − 8Lξ 3
−
e
hR i =
D
D2
D+1
"
#
3
2
(D + 5)2 −L/ξ
(D − 1)5 −2DL/(D−1)ξ
4 D + 23D − 7D + 1
+ 4ξ
.
−2
e
+ 3
e
D3
(D + 1)2
D (D + 1)2
!
4
For small values of L/ξ , we find the large-stiffness expansion

25D − 17
2L
+
hR4 i = L4 1 −
3ξ
90(D − 1)
L
ξ
!2
7D 2 − 8D + 3
−4
315(D − 1)2
L
ξ
!3

+ . . .  ,(15.164)
the leading term being equal to (15.95) for a completely stiff chain.
In the opposite limit of large L/ξ, the small-stiffness expansion is
hR4 i = 4

2
3
2
D+2 2 2
D + 6D − 1 ξ
D + 23D − 7D + 1
L ξ 1−2
+
D
D(D + 2) L
D 2 (D + 2)
ξ
L
!2 
 + ...
,
(15.165)
where the dots denote exponentially small terms. The leading term agrees again
with the expectation hR4 i of Eq. (15.94) for a random chain whose distribution is
(15.49) with an effective link length aeff = 2ξ of Eq. (15.158). The remaining terms
are corrections caused by the stiffness of the chain.
It is possible to find a correction factor to the Gaussian distribution which maintains the unit normalization and ensures that the moment hR2 i has the small-ξ exact
expansion (15.157) whereas hR2 i is equal to (15.165) up to the first correction term
in ξ/L. This has the form
PL (R) =
s
D
D
2D−1 ξ 3D−1 R2 D(4D−1) R4
2
e−DR /4Lξ 1−
+
−
. (15.166)
4πLξ
4 L
4 L2 16(D+2) ξL3
(
)
In three dimensions, this was first written down by Daniels [3]. It is easy to match
also the moment hR4 i by adding in the curly brackets the following terms
1 R4
R2
1−7D+23D 2 +D 3 D + 2 ξ 2
+
1
+
.
D+1
8D L2
ξL
32 L4
"
!
#
(15.167)
These terms do not, however, improve the fits to Monte Carlo data for ξ > 1/10L,
since the expansion is strongly divergent.
From the approximation (15.166) with the additional term (15.167) we calculate
the small-stiffness expansion of all even and odd moments as follows:

where
A1 = n
2n Γ(D/2 + n/2) n n 
ξ
ξ
hRn i =
L
ξ
1
+
A
+
A
1
2
D n/2 Γ(D/2)
L
L
n − 2 − 2d2 − 4d (n − 1)
,
4d (2 + d)
A2 = n(n − 2)
!2

+ ... ,
1 − 7d + 23d2 + d3
.
8d2 (1 + d)
(15.168)
(15.169)
H. Kleinert, PATH INTEGRALS
1055
15.9 Schrödinger Equation and Recursive Solution for Moments
15.9
Schrödinger Equation and Recursive Solution for
End-to-End Distribution Moments
The most efficient way of calculating the moments of the end-to-end distribution
proceeds by setting up a Schrödinger equation satisfied by (15.112) and solving it
recursively with similar methods as developed in 3.19 and Appendix 3C.
15.9.1
Setting up the Schrödinger Equation
In the continuum limit, we write (15.112) as a path integral [compare (15.139)]
Z
Z
Z
PL (R) ∝ dub dua D
D−1
uδ
(D)
Z
R−
0
L
!
ds u(s) e−(κ̄/2)
RL
0
ds [u′ (s)]2
,
(15.170)
where we have introduced the reduced stiffness
κ̄ =
κ
ξ
= (D − 1) ,
kB T
2
(15.171)
for brevity. After a Fourier representation of the δ-function, this becomes
PL (R) ∝
Z
i∞
−i∞
dD λ κ̄ ·R/2
e
2πi
Z
dub
Z
dua (ub L|ua 0) ,
(15.172)
where
(ub L|ua 0) ≡
Z
u(L)=ub
u(0)=ua
D D−1u e−(κ̄/2)
RL
0
ds {[u′ (s)]2 + ·u(s)}
(15.173)
describes a point particle of mass M = κ̄ moving on a unit sphere. In contrast to the
discussion in Section 8.7 there is now an additional external field which prevents us
from finding an exact solution. However, all even moments hRi1 Ri2 · · · Ri2l i of the
end-to-end distribution (15.172) Rcan be
extracted from the expansion coefficients
R
in powers of λi of the integral dub dua over (15.173). The presence of these
directional integrals permits us to assume the external electric field to point in
R = λẑ, and the
the z-direction,
E or the Dth direction in D-dimensions. Then
D
R
moments R2l are proportional to the derivatives (2/κ̄)2l ∂λ2l dub dua (ub L|ua 0) .
The proportionality factors have been calculated in Eq. (15.84). It is unnecessary to
know these since we can always use the rod limit (15.95) to normalize the moments.
To find these derivatives, we perform a perturbation expansion of the path integral (15.173) around the solvable case λ = 0.
In natural units with κ̄ = 1, the path integral (15.173) solves obviously the
imaginary-time Schrödinger equation
1
d
1
− ∆u + · u +
2
2
dτ
!
(u τ |ua 0) = 0,
(15.174)
1056
15 Path Integrals in Polymer Physics
where ∆u is the Laplacian on a unit sphere. In the probability distribution (15.211),
only the integrated expression
ψ(z, τ ; λ) ≡
dua (u τ |ua 0)
Z
(15.175)
appears, which is a function of z = cos θ only, where θ is the angle between u and
the electric field . For ψ(z, τ ; λ), the Schrödinger equation reads
Ĥ ψ(z, τ ; λ) = −
d
ψ(z, τ ; λ),
dτ
(15.176)
with the simpler Hamiltonian operator
1
Ĥ ≡ Ĥ0 + λ ĤI = − ∆ + λ z
2
"
#
2
1
1
d
2 d
= − (1 − z ) 2 − (D − 1)z
+ λz.
2
dz
dz
2
(15.177)
Now the desired moments (15.135) can be obtained from the coefficient of λ2l /(2l)! of
the power series expansion of the z-integral over (15.175) at imaginary time τ = L:
f (L; λ) ≡
15.9.2
1
Z
−1
dz ψ(z, L; λ).
(15.178)
Recursive Solution of Schrödinger Equation.
The function f (L; λ) has a spectral representation
f (L; λ) =
∞
X
l=0
R1
R
(l)†
(z) exp −E (l) L
−1 dz ϕ
R1
−1
1
−1
dza ϕ(l) (za )
dz ϕ(l)† (z) ϕ(l) (z)
,
(15.179)
where ϕ(l) (z) are the solutions of the time-independent Schrödinger equation
Ĥϕ(l) (z) = E (l) ϕ(l) (z). Applying perturbation theory to this problem, we start
from the eigenstates of the unperturbed Hamiltonian Ĥ0 = −∆/2, which are given
D/2−1
(l)
by the Gegenbauer polynomials Cl
(z) with the eigenvalues E0 = l(l+D −2)/2.
Following the methods explained in 3.19 and Appendix 3C we now set up a recursion scheme for the perturbation expansion of the eigenvalues and eigenfunctions
[4].Starting point is the expansion of energy eigenvalues and wave-functions in powers of the coupling constant λ:
E
(l)
=
∞
X
j=0
(l)
ǫj
j
λ,
(l)
|ϕ i =
∞
X
l′ ,i=0
(l)
γl′,i λi αl′ |l′ i .
(15.180)
The wave functions ϕ(l) (z) are the scalar products hz|ϕ(l) (λ)i. We have inserted
extra normalization constants αl′ for convenience which will be fixed soon. The
unperturbed state vectors |li are normalized to unity, but the state vectors |ϕ(l) i of
H. Kleinert, PATH INTEGRALS
15.9 Schrödinger Equation and Recursive Solution for Moments
1057
the interacting system will be normalized in such a way, that hϕ(l) |li = 1 holds to
all orders, implying that
(l)
(l)
γl,i = δi,0
γk,0 = δl,k .
(15.181)
Inserting the above expansions into the Schrödinger equation, projecting the result onto the base vector hk|αk , and extracting the coefficient of λj , we obtain the
equation
(l) (k)
γk,i ǫ0 +
i
X
αj
(l) (l)
(l)
ǫj γk,i−j ,
Vk,j γj,i−1 =
α
j=0
j=0 k
∞
X
(15.182)
where Vk,j = λhk|z|ji are the matrix elements of the interaction between unperturbed states. For i = 0, Eq. (15.182) is satisfied identically. For i > 0, it leads to
the following two recursion relations, one for k = l:
(l)
ǫi =
X
(l)
γl+n,i−1Wn(l) ,
(15.183)
n=±1
the other one for k 6= l:
(l)
γk,i =
i−1
X
j=1
(l) (l)
ǫj γk,i−j −
(k)
ǫ0
X
(l)
γk+n,i−1Wn(l)
n=±1
(l)
− ǫ0
,
(15.184)
where only n = −1 and n = 1 contribute to the sums over n since
Wn(l) ≡
αl+n
hl| z |l + ni = 0, for n 6= ±1.
αl
(15.185)
The vanishing of Wn(l) for n 6= ±1 is due to the band-diagonal form of the matrix of
the interaction z in the unperturbed basis |ni. It is this property which makes the
sums in (15.183) and (15.184) finite and leads to recursion relations with a finite
(l)
(l)
number of terms for all ǫi and γk,i . To calculate Wn(l) , it is convenient to express
hl| z |l + ni as matrix elements between unnormalized noninteracting states |n} as
{l|z|l + n}
hl| z |l + ni = q
,
{l|l}{l + n|l + n}
(15.186)
where expectation values are defined by the integrals
Z
{k|F (z)|l} ≡
1
−1
D/2−1
Ck
D/2−1
(z) F (z) Cl
(z)(1 − z 2 )(D−3)/2 dz,
(15.187)
from which we find5
{l|l} =
5
24−D Γ(l + D − 2) π
.
l! (2l + D − 2) Γ(D/2 − 1)2
I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 7.313.1 and 7.313.2.
(15.188)
1058
15 Path Integrals in Polymer Physics
Expanding the numerator of (15.186) with the help of the recursion relation (15.151)
for the Gegenbauer polynomials written now in the form
(l + 1)|l + 1} = (2l + D − 2) z |l} − (l + D − 3)|l − 1},
(15.189)
we find the only non-vanishing matrix elements to be
l+1
{l + 1|l + 1},
2l + D − 2
l+D−3
{l − 1|z|l} =
{l − 1|l − 1} .
2l + D − 2
{l + 1|z|l} =
(15.190)
(15.191)
Inserting these together with (15.188) into (15.186) gives
hl|z|l − 1i =
v
u
u
t
l(l + D − 3)
,
(2l + D − 2)(2l + D − 4)
(15.192)
and a corresponding result for hl|z|l + 1i. We now fix the normalization constants
αl′ by setting
(l)
W1 =
αl+1
hl| z |l + 1i = 1
αl
(15.193)
for all l, which determines the ratios
v
u
u (l + 1) (l + D − 2)
αl
= hl| z |l + 1i = t
.
αl+1
(2l + D) (2l + D − 2)
(15.194)
Setting further α1 = 1, we obtain

αl = 
l
Y
j=1
1/2
(2l + D − 2)(2l + D − 4) 
l(l + D − 3)
.
(15.195)
Using this we find from (15.185) the remaining nonzero Wn(l) for n = −1:
(l)
W−1 =
l(l + D − 3)
.
(2l + D − 2)(2l + D − 4)
(15.196)
(l)
We are now ready to solve the recursion relations of (15.183) and (15.184) γk,i and
(l)
(l)
ǫi order by order in i. For the initial order i = 0, the values of the γk,i are
(l)
given by Eq. (15.181). The coefficients ǫi are equal to the unperturbed energies
(l)
(l)
ǫ0 = E0 = l(l + D − 2)/2. For each i = 1, 2, 3, . . . , there is only a finite number
(l)
(l)
of non-vanishing γk,j and ǫj with j < i on the right-hand sides of (15.183) and
(l)
(l)
(15.184) which allows us to calculate γk,i and ǫi on the left-hand sides. In this way
it is easy to find the perturbation expansions for the energy and the wave functions
to high orders.
H. Kleinert, PATH INTEGRALS
1059
15.9 Schrödinger Equation and Recursive Solution for Moments
Inserting the resulting expansions (15.180) into Eq. (15.179), only the totally
symmetric parts in ϕ(l) (z) will survive the integration in the numerators, i.e., we
may insert only
(l)
ϕsymm
(z)
=
(l)
hz|ϕsymm
i
=
∞
X
i=0
(l)
γ0,i λi hz|0i .
(15.197)
(l)
The denominators of (15.179) become explicitly l′ ,i |γl′ ,i αl′ |2 λ2i , where the sum
over i is limited by power of λ2 up to which we want to carry the perturbation
series; also l′ is restricted to a finite number of terms only, because of the band(l)
diagonal structure of the γl′ ,i .
Extracting the coefficients of the power expansion in λ from (15.179) we obtain
all desired moments of the end-to-end distribution, in particular the second and
fourth moments (15.155) and (15.164). Higher even moments are easily found with
the help of a Mathematica program, which is available for download in notebook form
[5]. The expressions are too lengthy to be written down here. We may, however,
expand the even moments hRn i in powers of L/ξ to find a general large-stiffness
expansion valid for all even and odd n:
P
hRn i
n L n (−13 − n + 5D (1 + n)) L2
L3
L4
=
1
−
+
−
a
+
a
+. . . ,
3
4
Ln
6ξ
360 (D − 1)
ξ2
ξ3
ξ4
(15.198)
where
444−63n+15n2 +7D 2 (4+15n+5n2)+2D (−124−141n+7n2)
,
45360(D − 1)2
n
2
3
a4 =
,
3 D0 + D1 d + D2 d + D3 d
5443200(d − 1)
a3 = n
(15.199)
with
D0 = 3 −5610+2921n−822n2 +67n3 , D1 = 8490+12103n−3426n2 +461n3 ,
D2 = 45 −2−187n−46n2 +7n3 ,
The lowest odd moments are, up to order l4 ,
hR i
= 1−
L
3
R
= 1−
L3
15.9.3
D4 = 35 −6+31n+30n2 +5n3 .
(15.200)
l
5D−7 2 33 − 43D + 14D2 3 861 − 1469D + 855D2 − 175D3 4
+
l −
l −
l ... ,
2
3
6 180(D−1)
3780(D − 1)
453600 (D − 1)
l
5D−4 2 195−484D+329D2 3 609−2201D+2955D2 −1435D3 4
l −
l ... .
+
l −
2
3
2 30 (D−1)
7560 (−1 + d)
151200 (D−1)
From Moments to End-to-End Distribution for D=3
We now use the recursively calculated moments to calculate the end-to-end distribution itself. It can be parameterized by an analytic function of r = R/L [4]:
PL (R) ∝ r k (1 − r β )m ,
(15.201)
1060
15 Path Integrals in Polymer Physics
whose moments are exactly calculable:
3+k
3+ k +2l
Γ
Γ
+m+1
2l β
β
.
r =
3+k +2l
Γ
Γ 3+k
+
m
+
1
β
β
(15.202)
We now adjust the three parameters k, β, and m to fit the three most important
moments of this distribution to the exact values, ignoring all others. If the distances were distributed uniformly over the interval r ∈ [0, 1],
E moments would
D the
2l unif
2l
be hr i
= 1/(2l + 2). Comparing our exact moments r (ξ) with those of
the uniform distribution we find that hr 2l i(ξ)/hr 2l iunif has a maximum for n close to
nmax (ξ) ≡ 4ξ/L. We identify the most important moments as those with n = nmax (ξ)
and n = nmax (ξ) ± 1. If nmax (ξ) ≤ 1, we choose the lowest even moments hr 2 i, hr 4 i,
and hr 6 i. In particular, we have fitted hr 2 i, hr 4 i and hr 6 i for small persistence length
ξ < L/2. For ξ = L/2, we have started with hr 4 i, for ξ = L with hr 8 i and for ξ = 2L
with hr 16 i, including always the following two higher even moments. After these adjustments, whose results are shown in Fig. 15.4, we obtain the distributions shown in
Fig.15.6 for various persistence lengths ξ. They are in excellent agreement with the
Monte Carlo data (symbols) and better than the one-loop perturbative results (thin
curves) of Ref. [6], which are good only for very stiff polymers. The Mathematica
program to do these fits are available from the internet address given in Footnote 5.
k
17.5
15
12.5
10
7.5
5
2.5
80
22
20
18
16
14
12
β
660
40
20
0.5
1
ξ/L
1.5
2
0.5
1
ξ/L
1.5
2
m
0.5
1
1.5
ξ/L
2
Figure 15.4 Paramters k, β, and m for a best fit of end-to-end distribution (15.201).
For small persistence lengths ξ/L = 1/400, 1/100, 1/30, the curves are well approximated by Gaussian random chain distributions on a lattice with lattice constant
2
aeff = 2ξ, i.e., PL (R) → e−3R /4Lξ [recall (15.75)]. This ensures that the lowest moment hR2 i = aeff L is properly fitted. In fact, we can easily check that our fitting
program yields for the parameters k, β, m in the end-to-end distribution (15.201)
the ξ → 0 behavior: k → −ξ, β → 2 + 2ξ, m → 3/4ξ, so that (15.201) tends to the
correct Gaussian behavior.
In the opposite limit of large ξ, we find that k → 10ξ−7/2, β → 40ξ+5, m → 10,
which has no obvious analytic approach to the exact limiting behavior PL (R) → (1−
r)−5/2 e−1/4ξ(1−r) , although the distribution at ξ = 2 is fitted numerically extremely
well.
H. Kleinert, PATH INTEGRALS
15.9 Schrödinger Equation and Recursive Solution for Moments
1061
The distribution functions can be inserted into Eq. (15.89) to calculate the structure factors shown in Fig. 15.5. They interpolate smoothly between the Debye limit
(15.90) and the stiff limit (15.105).
1
0.8
0.6
S(q)
0.4
0.2
5
ξ/L = 2
✠ ξ/L = 1
✠ ξ/L = 1/2
✠
ξ/L = 1/5, . . . , 1/400
✠
√
q ξ
10
15
20
25
30
Figure 15.5 Structure functions for different persistence lengths ξ/L = 1/400, 1/100,
1/30, 1/10, 1/5, 1/2, 1, 2, (from bottom to top) following from the end-to-end distributions in Fig. 15.6. The curves with low ξ almost coincide in this plot over the ξ-dependent
absissa. The very stiff curves fall off like 1/q, the soft ones like 1/q 2 [see Eqs. (15.105) and
(15.90)].
15.9.4
Large-Stiffness Approximation to End-to-End
Distribution
The full end-to-end distribution (15.132) cannot be calculated exactly. It is, however,
quite easy to find a satisfactory approximation for large stiffness [6].
We start with the expression (15.170) for the end-to-end distribution PL (R). In
Eq. (3.233) we have shown that a harmonic path integral including the integrals
over the end points can be found, up to a trivial factor, by summing over all paths
with Neumann boundary conditions. These are satisfied if we expand the fields u(s)
into a Fourier series of the form (2.452):
u(s) = u0 + (s) = u0 +
∞
X
un cos νn s,
νn = nπ/L.
(15.203)
n=1
Let us parametrize the unit vectors u in D dimensions in terms of the first D − 1
-dimensional coordinates uµ ≡ q µ with µ = 1, . . . , D − 1. The Dth component is
then given by a power series
σ≡
q
1 − q 2 ≈ 1 − q 2 /2 − (q 2 )2 /8 + . . . .
(15.204)
The we approximate the action harmonically as follows:
A = A
κ̄
≈
2
(0)
Z
0
+A
L
int
κ̄
=
2
Z
0
L
1
ds [u′ (s)]2 + δ(0) log(1 − q 2 )
2
1
ds [q ′(s)]2 − δ(0)
2
Z
0
L
ds q 2.
(15.205)
1062
15 Path Integrals in Polymer Physics
√
The last term comes from the invariant measure of integration dD−1 q/ 1 − q 2 [recall
(10.636) and (10.641)].
Assuming, as before, that R points into the z, or Dth, direction we factorize
δ
(D)
R−
Z
L
0
ds u(s)
!
= δ R−L+
Z
× δ (D−1)
L
0
L
Z
0
ds
!
1 2
1
q (s) + [q 2 (s)]2 + . . .
2
8
!
ds q(s) ,
(15.206)
where R ≡ |R|. The second δ-function on the right-hand side enforces
q̄ = L−1
Z
L
0
ds q µ(s) = 0 ,
µ = 1, . . . , d − 1 ,
(15.207)
and thus the vanishing of the zero-frequency parts q0µ in the first D − 1 components
of the Fourier decomposition (15.203).
It was shown in Eqs. (10.632) and (10.642) that the last δ-function has a distorting effect upon the measure of path integration which must be compensated by
a Faddeev-Popov action
AFP
e =
D−1
2L
Z
L
0
ds q 2,
(15.208)
where the number D of dimensions of q µ -space (10.642) has been replaced by the
present number D − 1.
In the large-stiffness limit we have to take only the first harmonic term in the
action (15.205) into account, so that the path integral (15.170) becomes simply
PL (R) ∝
Z
NBC
D ′ D−1 q δ R − L +
Z
L
0
!
RL
′
2
1
ds q 2 (s) e−(κ̄/2) 0 ds [q (s)] .
2
(15.209)
The subscript of the integral emphasizes the Neumann boundary conditions. The
prime on the measure of the path integral indicates the absence of the zero-frequency
component of q µ (s) in the Fourier decomposition due to (15.207). Representing the
remaining δ-function in (15.209) by a Fourier integral, we obtain
PL (R) ∝ κ̄
Z
i∞
−i∞
dω 2 κ̄ω2 (L−R)
e
2πi
Z
NBC
D
′ D−1
"
κ̄
q exp −
2
Z
0
L
′2
2 2
ds q + ω q
#
. (15.210)
The integral over all paths with Neumann boundary conditions is known from
Eq. (2.456). At zero average path, the result is
Z
NBC
D
′ D−1
"
κ̄
q exp −
2
Z
L
0
′2
2 2
ds q + ω q
#
ωL
∝
sinh ωL
(D−1)/2
,
(15.211)
so that
PL (R) ∝ κ̄
Z
i∞
−i∞
dω 2 κ̄ω2 (L−R)
e
2πi
ωL
sinh ωL
(D−1)/2
.
(15.212)
H. Kleinert, PATH INTEGRALS
1063
15.9 Schrödinger Equation and Recursive Solution for Moments
The integral can easily be done in D = 3 dimensions using the original product
representation (2.455) of ωL/ sinh ωL, where
PL (R) ∝ κ̄
i∞
Z
−i∞
∞
ω2
dω 2 κ̄ω2 (L−R) Y
1+ 2
e
2πi
νn
n=1
!−1
.
(15.213)
If we shift the contour of integration to the left we run through poles at ω 2 = −νk2
with residues
∞
X
∞
Y
νk2
k=1
k2
1− 2
n
n(6=k)=1
!−1
.
(15.214)
The product is evaluated by the limit procedure for small ǫ:

"
∞
Y

n=1
(k + ǫ)2
1−
n2
#−1  "

(k + ǫ)2
1−

k2
#
→
(k + ǫ)π −2ǫ
−2ǫπ
→
sin(k + ǫ)π k
sin(k + ǫ)π
→ −
2ǫπ
→ 2(−1)k+1. (15.215)
cos kπ sin ǫπ
Hence we obtain [6]
PL (R) ∝
∞
X
2
2(−1)k+1κ̄ νk2 e−κ̄νk (L−R) .
(15.216)
k=1
It is now convenient to introduce the reduced end-to-end distance r ≡ R/L and the
flexibility of the polymer l ≡ L/ξ, and replace κ̄ νk2 (L − R) → k 2 π 2 (1 − r)/l so that
(15.216) becomes
PL (R) = N L(R) = N
∞
X
(−1)k+1 k 2 π 2 e−k
2 π 2 (1−r)/l
,
(15.217)
dr r 2 PL (R) = 1.
(15.218)
k=1
where N is a normalization factor determined to satisfy
Z
d3 R PL (R) = 4πL3
Z
∞
0
The sum must be evaluated numerically, and leads to the distributions shown in
Fig. 15.6.
The above method is inconvenient if D 6= 3, since the simple pole structure of
(15.213) is no longer there. For general D, we expand
ωL
sinh ωL
(D−1)/2
= (ωL)
(D−1)/2
∞
X
(−1)
k=0
k
!
−(D − 1)/2 −(2k+(D−1)/2)ωL
e
, (15.219)
k
and obtain from (15.212):
PL (R) ∝
∞
X
(−1)k
k=0
!
−(D − 1)/2
Ik (R/L),
k
(15.220)
1064
15 Path Integrals in Polymer Physics
4πr2 PL (R)
ξ/L=
Figure 15.6
Normalized end-to-end distribution of stiff polymer according to our analytic formula (15.201) plotted for persistence lengths ξ/L =
1/400, 1/100, 1/30, 1/10, 1/5, 1/2, 1, 2 (fat curves). They are compared with the Monte
Carlo calculations (symbols) and with the large-stiffness approximation (15.217) (thin
curves) of Ref. [6] which fits well for ξ/L = 2 and 1 but becomes bad small ξ/L < 1.
For very small values such as ξ/L = 1/400, 1/100, 1/30, our theoretical curves are well
approximated by Gaussian random chain distributions on a lattice with lattice constant
aeff = 2ξ of Eq. (15.158) which ensures that the lowest moments hR2 i = aeff L are properly
fitted. The Daniels approximation (15.166) fits our theoretical curves well up to larger
ξ/L ≈ 1/10.
with the integrals
Ik (r) ≡
Z
i∞
−i∞
dω̄ (D+1)/2 −[2k+(D−1)/2]ω̄+(D−1)ω̄2 (1−r)/2l
ω̄
e
,
2πi
(15.221)
where ω̄ is the dimensionless variable ωL. The integrals are evaluated with the help
of the formula6
Z
i∞
−i∞
q
dx ν βx2 /2−qx
1
−q 2 /4β
β ,
x e
=√
e
D
q/
ν
2πi
2πβ (ν+1)/2
(15.222)
where Dν (z) is the parabolic cylinder functions which for integer ν are proportional
to Hermite polynomials:7
√
1
2
Dn (z) = √ n e−z /4 Hn (z/ 2).
2
(15.223)
Thus we find
6
7
I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 3.462.3 and 3.462.4.
ibid., Formula 9.253.
H. Kleinert, PATH INTEGRALS
1065
15.9 Schrödinger Equation and Recursive Solution for Moments
"
1
l
Ik (r) = √
2π (D−1)(1−r)
#(D+3)/4
[2k+(D−1)/2]2
− 4(D−1)(1−r)/l
e


2k+(D−1)/2 
,
D(D+1)/2  q
(D−1)(1−r)/l
(15.224)
which becomes for D = 3


(2k+1)2
1
1
2k+1 
− 4(1−r)/l
Ik (r) = √ q
.
e
H2  q
3
2 2π 2(1−r)/l
2 (1−r)/l
(15.225)
If the sum (15.220) is performed numerically for D = 3, and the integral over
PL (R) is normalized to satisfy (15.218), the resulting curves fall on top of those
in Fig. 15.6 which were calculated from (15.217). In contrast to (15.217), which
converges rapidly for small r, the sum (15.220) convergent rapidly for r close to
unity.
Let us compare the low moments of the above distribution with the exact moments in Eqs. (15.156) and (15.164). in the large-stiffness expansion. We set ω̄ ≡ ωL
and expand
2
f (ω̄ ) ≡
s
ω̄
sinh ω̄
D−1
(15.226)
in a power series
D − 1 2 (D−1)(5D−1) 4 (D−1)(15+14D+35D 2) 6
ω̄ −
ω̄ +. . . .(15.227)
f (ω̄ )=1− 2 ω̄ +
2 ·3
25 ·32 ·5
27 ·34 ·5·7
Under the integral (15.212), each power of ω̄ 2 may be replaced by a differential
operator
2
2
ˆ ≡−
ω̄ 2 → ω̄
L d
2l d
=−
.
κ̄ dr
D − 1 dr
(15.228)
ˆ 2 ) can then be pulled out of theintegral,
The expansion f (ω̄
which by itself yields a
2
ˆ δ(r − 1), which is a series of
δ-function, so that we obtain PL (R) in the form f ω̄
derivatives of δ-functions of r − 1 staring with
(−1+5 D) l2 d2
(15+14 D+35 D 2) l3 d3
l d
+
+
+ . . . δ(r − 1).
PL (R) ∝ 1+
6 dr
360 (D−1) dr 2
dr 3
45360 (D−1)2
(15.229)
#
"
From this we find easily the moments
m
hR i =
Z
D
m
d RR PL (R) ∝
Z
0
∞
dr r D−1 r m PL (R).
(15.230)
Let us introduce
auxiliary expectation values with respect to the simple integrals
R
hf (r)i1 ∝ dr f (r)PL , rather than to D-dimensional volume integrals. The unnormalized moments hr m i are then given by hr D−1+m i1 . Within these one-dimensional
expectations, the moments of z = r − 1 are
h(r − 1)n i1 ∝
Z
∞
0
dr (r − 1)n PL (R).
(15.231)
1066
15 Path Integrals in Polymer Physics
The moments hr m i h[1+(r −1)]m i = h[1+(r −1)]D−1+mRi1 are obtained by expanding
the binomial in powers of r −1 and using the integrals dz z m δ (n) (z) = (−1)m m! δmn
to find, up to the third power in L/ξ = l,
D−1
(5D−1)(D−2) 2
(35D3 +14D+15)(D−2)(D−3) 3
1−
l +
l
−
l ,
6
360
45360(D − 1)
2
2
D+1
(5D−1)D(D+1) 2
(35D +14D+ 15)D(D+1)
R = N L2 1 −
l+
l −
l3 ,
6
360(D − 1)
45360(D − 1)
4
D+3
(5D−1)(D+2)(D+3) 2
R = N L4 1 −
l+
l
6
360(D − 1)
(35D2 +14D+15)(D+1)(D+2)(D+3) 3
−
.
l
45360(D − 1)2
0
R =N
The zeroth moment determines the normalization factor N to ensure that hR0 i = 1.
Dividing this out of the other moments yields
"
1
R2 = L2 1 − l +
3
"
D E
2
R4 = L4 1 − l +
3
D
E
#
13D−9 2
8
l −
l3 + . . . ,
180(D−1)
945
#
23D−11 2 123D 2 −98D+39 3
l −
l + ... .
90(D−1)
1890(D−1)2
(15.232)
(15.233)
These agree up to the l-terms with the exact expansions (15.156) and (15.164) [or
with the general formula (15.198)]. For D = 3, these expansions become
1
R = L 1− l+
3
D E
2
R4 = L4 1 − l +
3
D
2
E
2
1 2
l −
12
29 2
l −
90
8 3
l + ... ,
945
71 3
l + ... .
630
(15.234)
(15.235)
Remarkably, these happen to agree in one more term with the exact expansions
(15.156) and (15.164) than expected, as will be understood after Eq. (15.295).
15.9.5
Higher Loop Corrections
Let us calculate perturbative corrections to the large-stiffness limit. For this we replace the harmonic path integral (15.210) by the full expression
!
Z
Z L p
tot
cor
−1
(15.236)
P (r; L) = SD
D ′ D−1 q(s)δ r − L−1
ds 1 − q 2 (s) e−A [q]−A [qb ,qa ] ,
NBC
0
where
1
Atot [q] =
2ε
Z
0
L
ds [ gµν (q) q̇ µ (s)q̇ ν (s)−εδ(s, s) log g(q(s))] + AFP − εL
R
.
8
(15.237)
For convenience, we have introduced here the parameter ε = kB T /κ = 1/κ̄, the reduced inverse
stiffness, related to the flexibility l ≡ L/ξ by l = εL(d − 1)/2. We also have added the correction
term εL R/8 to have a unit normalization of the partition function
Z ∞
Z = SD
dr rD−1 P (r; L) = 1.
(15.238)
0
H. Kleinert, PATH INTEGRALS
1067
15.9 Schrödinger Equation and Recursive Solution for Moments
The Faddeev-Popov action, whose harmonic approximation was used in (15.208), is now
!
Z L p
FP
−1
A [q] = −(D − 1) log L
ds 1 − q 2 (s) ,
(15.239)
0
To perform higher-order calculations we have added an extra action which corrects for the
omission of fluctuations of the velocities at the endpoints when restricting the paths to Neumann
boundary conditions with zero end-point velocities. The extra action contains, of course, only at
the endpoints and reads [7]
Acor [qb , qa ] = − log J[qb , qa ] = −[q 2 (0) + q 2 (L)]/4.
(15.240)
Thus we represent the partition function (15.238) by the path integral with Neumann boundary
conditions
Z
Z=
D ′ D−1 q(s) exp −Atot [q] − Acor [qb , qa ] − AFP [q] .
(15.241)
NBC
A similar path integral can be set up for the moments of the distribution. We express the
square distance R2 in coordinates (15.207) as
2
R =
Z
L
ds
0
Z
L
0
′
′
ds u(s) · u(s ) =
Z
0
L
p
ds 1 − q 2 (s)
!2
= R2 ,
(15.242)
we find immediately the following representation for all, even and odd, moments [compare (15.147)]
2 n
h (R ) i =
* "Z
L
0
Z
0
L
′
′
ds ds u(s) · u(s )
#n +
(15.243)
in the form
h Rn i =
Z
NBC
D ′ D−1 q(s) exp −Atot [q] − Acor [qb , qa ] − AFP
n [q] .
(15.244)
This differs from Eq. (15.241) only by in the Faddeev-Popov action for these moments, which is
!
Z L p
FP
−1
(15.245)
An [q] = −(n + D − 1) log L
ds 1 − q 2 (s) ,
0
rather than (15.239).
There is no need to divide the path integral (15.244) by Z since this has unit normalization, as
will be verified order by order in the perturbation expansion. The Green function of the operator
d2 /ds2 with these boundary conditions has the form
∆′N (s, s′ ) =
L
3
−
| s − s′ | (s + s′ ) (s2 + s′2 )
−
+
.
2
2
2L
(15.246)
The zero temporal average (15.207) manifests itself in the property
Z
0
L
ds ∆′N (s, s′ )
= 0.
In the following we shall simply write ∆(s, s′ ) for ∆′N (s, s′ ), for brevity.
(15.247)
1068
15 Path Integrals in Polymer Physics
Partition Function and Moments Up to Four Loops
We are now prepared to perform the explicit perturbative calculation of the partition function and
all even moments in powers of the inverse stiffness ε up to order ε2 ∝ l2 . This requires evaluating
Feynman diagrams up to four loops. The associated integrals will contain products of distributions,
which will be calculated unambiguously with the help of our simple formulas in Chapter 10.
For a systematic treatment of the expansion parameter ε, we rescale the coordinates q µ → εq µ ,
and rewrite the path integral (15.244) as
Z
D ′ D−1 q(s) exp {−Atot,n [q; ε].} ,
(15.248)
h Rn i =
NBC
with the total action
L
(q q̇)2
1
1
2
2
q̇ + ε
+ δ(0) log(1 − εq )
Atot,n [q; ε] =
ds
2
1 − εq 2
2
0
" Z
#
1 L p
R
ε 2
− σn log
q (0) + q 2 (L) − εL .
ds 1 − εq 2 −
L 0
4
8
Z
(15.249)
The constant σn is an abbreviation for
σn ≡ n + (d − 1).
(15.250)
For n = 0, the path integral (15.248) must yield the normalized partition function Z = 1.
For the perturbation expansion we separate
Atot,n [q; ε] = A(0) [q] + Aint
n [q; ε],
(15.251)
with a free action
(0)
A
1
[q] =
2
Z
L
ds q̇ 2 (s),
(15.252)
0
and a large-stiffness expansion of the interaction
int1
2 int2
Aint
n [q; ε] = εAn [q] + ε An [q] + . . . .
(15.253)
The free part of the path integral (15.248) is normalized to unity:
Z
Z
RL 2
(0)
−(1/2)
ds q̇ (s)
0
D ′ D−1 q(s) e
= 1.
D ′ D−1 q(s) e−A [q] =
Z (0) ≡
NBC
(15.254)
NBC
The first expansion term of the interaction (15.253) is
Z
1 L R
int1
An [q] =
ds [q(s)q̇(s)]2 − ρn (s) q 2 (s) − L ,
2 0
8
(15.255)
where
ρn (s) ≡ δn + [δ(s) + δ(s − L)] /2,
δn ≡ δ(0) − σn /L.
(15.256)
The second term in ρn (s) represents the end point terms in the action (15.249) and is important
for canceling singularities in the expansion.
The next expansion term in (15.253) reads
Z
σn i 2
1h
1 L
2
int2
δ(0) −
q (s) q 2 (s)
ds [q(s)q̇(s)] −
An [q] =
2 0
2
2L
Z L Z L
σn
+
ds
ds′ q 2 (s)q 2 (s′ ).
(15.257)
8L2 0
0
H. Kleinert, PATH INTEGRALS
15.9 Schrödinger Equation and Recursive Solution for Moments
1069
The perturbation expansion of the partition function in powers of ε consists of expectation
values of the interaction and its powers to be calculated with the free partition function (15.254).
For an arbitrary functional of q(s), these expectation values will be denoted by
Z
RL 2
−(1/2)
ds q̇ (s)
0
D ′ D−1 q(s) F [q] e
.
(15.258)
h F [q] i0 ≡
NBC
With this notation, the perturbative expansion of the path integral (15.248) reads
1 int
2
h Rn i/Ln = 1 − hAint
n [q; ε]i0 + hAn [q; ε] i0 − . . .
2 1 int1 2
int2
int1
2
= 1 − ε hAn [q]i0 + ε −hAn [q]i0 + hAn [q] i0 − . . . .
2
(15.259)
For the evaluation of the expectation values we must perform all possible Wick contractions with
the basic propagator
h q µ (s)q ν (s′ ) i0 = δ µν ∆(s, s′ ) ,
(15.260)
where ∆(s, s′ ) is the Green function (15.246) of the unperturbed action (15.252). The relevant
loop integrals Ii and Hi are calculated using the dimensional regularization rules of Chapter 10.
They are listed in Appendix 15A and Appendix 15B.
We now state the results for various terms in the expansion (15.259):
(D−1) σn
1
1
R
(D−1)n
int1
hAn [q]i0 =
I1 + DI2 − ∆(0, 0) − ∆(L, L) − L = L
,
2
L
2
2
8
12
i (D−1)σ σn (D2 −1) h
n
(D−1)I12 + 2I5
δ(0) +
I3 +2(D+2)I4 +
hAint2
n [q]i0 =
4
2L
8L2
(D2 − 1)
(D−1) = L3
(25D2 +36D+23) + n(11D+5) ,
δ(0) + L2
120
1440
2
D
2
R
L2 (D−1)L
1 int1 2
δ(0) +
− δn +
−
hA [q] i0 =
2 n
2
12
2L
L
8
(D−1) n
+
[H1 − 2(H2n + H3n − H5 ) + H6 − 4D(H4n − H7 − H10 )
4
(D−1)
+ H11 + 2D2 (H8 + H9 ) +
[DH12 + 2(D + 2)H13 + DH14 ]
4
2
(D−1)
L2 (D−1)n
+ L3
δ(0)
=
2
12
120
(D−1) + L2
(25D2 − 22D + 25) + 4(n + 4D − 2)
1440
(D−1)
(D−1)D
δ(0) + L2
(29D − 1) .
(15.261)
+ L3
120
720
Inserting these results into Eq. (15.259), we find all even and odd moments up to order ε2 ∝ l2
h Rn i/Ln
(D−1)n
(D−1)2 n2
(D−1)(4n + 5D − 13)n
− O(ε3 )
+ ε2 L 2
+
12
288
1440
2
n
(4n + 5D − 13)n 2
n
= 1− l+
l − O(l3 ).
(15.262)
+
6
72
360(D−1)
= 1 − εL
For n = 0 this gives the properly normalized partition function Z = 1. For all n it reproduces the
large-stiffness expansion (15.198) up to order l4 .
1070
15 Path Integrals in Polymer Physics
Correlation Function Up to Four Loops
As an important test of the correctness of our perturbation theory we calculating the correlation
function up to four loops and verify that it yields the simple expression (15.153), which reads in
the present units
′
′
G(s, s′ ) = e−|s−s |/ξ = e−|s−s |l/L .
(15.263)
Starting point is the path integral representation with Neumann boundary conditions for the twopoint correlation function
Z
n
o
(0)
D ′ D−1 q(s) f (s, s′ ) exp −Atot [q; ε] ,
(15.264)
G(s, s′ ) = hu(s) · u(s′ )i =
NBC
with the action of Eq. (15.249) for n = 0. The function in the integrand f (s, s′ ) ≡ f (q(s), q(s′ )) is
an abbreviation for the scalar product u(s) · u(s′ ) expressed in terms of independent coordinates
q µ (s):
p
p
(15.265)
f (q(s), q(s′ )) ≡ u(s) · u(s′ ) = 1 − q 2 (s) 1 − q 2 (s′ ) + q(s) q(s′ ).
√
Rescaling the coordinates q → ε q, and expanding in powers of ε yields:
f (q(s), q(s′ )) = 1 + εf1 (q(s), q(s′ )) + ε2 f2 (q(s), q(s′ )) + . . . ,
(15.266)
where
1 2
1
q (s) − q 2 (s′ ),
2
2
1 2
1
1 2
2 ′
q (s) q (s ) − [q (s)]2 − [q 2 (s′ )]2 .
4
8
8
f1 (q(s), q(s′ ))
= q(s) q(s′ ) −
(15.267)
f2 (q(s), q(s′ ))
=
(15.268)
We shall attribute the integrand f (q(s), q(s′ )) to an interaction Af [q; ε] defined by
f
f (q(s), q(s′ )) ≡ e−A
[q;ε]
,
(15.269)
which has the ε-expansion
Af [q; ε] = − log f (q(s), q(s′ ))
1
= −εf1 (q(s), q(s′ ))+ε2 −f2 (q(s), q(s′ ))+ f12 (s, s′ ) −. . . ,
2
(15.270)
to be added to the interaction (15.253) with n = 0. Thus we obtain the perturbation expansion of
the path integral (15.264)
E
D
E
1 D int
f
(A0 [q; ε] + Af [q; ε])2 − . . . .
G(s, s′ ) = 1− (Aint
+
0 [q; ε] + A [q; ε])
2
0
0
(15.271)
Inserting the interaction terms (15.253) and (15.271), we obtain
G(s, s′ ) = 1+ εhf1 (q(s), q(s′ ))i0+ε2 hf2 (q(s), q(s′ ))i0 −hf1 (q(s), q(s′ )) Aint1
0 [q]i0 +. . . ,
(15.272)
and the expectation values can now be calculated using the propagator (15.260) with the Green
function (15.246).
In going through this calculation we observe that because of translational invariance in the
pseudotime, s → s + s0 , the Green function ∆(s, s′ ) + C is just as good a Green function satisfying
Neumann boundary conditions as ∆(s, s′ ). We may demonstrate this explicitly by setting C =
H. Kleinert, PATH INTEGRALS
15.9 Schrödinger Equation and Recursive Solution for Moments
1071
L(a−1)/3 with an arbitrary constant a, and calculating the expectation values in Eq. (15.272) using
the modified Green function. Details are given in Appendix 15C [see Eq. (15C.1)], where we list
various expressions and integrals appearing in the Wick contractions of the expansion Eq. (15.272).
Using these results we find the a-independent terms up to second order in ε:
(D − 1)
| s − s′ |,
(15.273)
2
hf2 (q(s), q(s′ ))i0 − hf1 (q(s), q(s′ )) Aint1
0 [q]i0
1
(D − 1)
1
1
1
K3 + K4 + K5
= (D − 1) D1 − (D + 1)D22 − K1 − DK2 −
2
8
L
2
2
1
= (D − 1)2 (s − s′ )2 .
(15.274)
8
hf1 (q(s), q(s′ ))i0 = −
This leads indeed to the correct large-stiffness expansion of the exact two-point correlation function
(15.263):
G(s, s′ ) = 1−ε
(D−1)2
| s−s′ | (s−s′ )2
D−1
− . . . . (15.275)
| s−s′ | + ε2
(s−s′ )2 +. . .= 1−
+
2
8
ξ
2ξ 2
Radial Distribution up to Four Loops
We now turn to the most important quantity characterizing a polymer, the radial distribution function. We eliminate the δ-function in Eq. (15.236) enforcing the end-to-end distance by considering
the Fourier transform
Z
P (k; L) = dr eik(r−1) P (r; L) .
(15.276)
This is calculated from the path integral with Neumann boundary conditions
Z
P (k; L) =
D ′ D−1 q(s) exp −Atot
k [q; ε] ,
(15.277)
NBC
where the action Atot
k [q; ε] reads, with the same rescaled coordinates as in (15.249),
L
1 2
(q q̇)2
1
ik p
2
2−1
1
−
εq
q̇ + ε
+
δ(0)
log(1
−
εq
)
−
2
1 − εq 2
2
L
0
1 2
R
−
ε q (0) + q 2 (L) − εL ≡ A0 [q] + Aint
(15.278)
k [q; ε].
4
8
Atot
k [q; ε] =
Z
ds
As before in Eq. (15.253), we expand the interaction in powers of the coupling constant ε. The
first term coincides with Eq. (15.255), except that σn is replaced by
ρk (s) = δk + [δ(s) + δ(s − L)] /2,
δk = δ(0) − ik/L,
(15.279)
so that
Aint1
k [q]
=
Z
0
L
ds
o
R
1n
2
[q(s)q̇(s)] − ρk (s) q 2 (s) − L .
2
8
(15.280)
The second expansion term Aint2
k [q] is simpler than the previous (15.257) by not containing the
last nonlocal term:
Z L
1
ik
1
2
2
int2
[q(s)q̇(s)] −
δ(0) −
q (s) q 2 (s).
(15.281)
Ak [q] =
ds
2
2
2L
0
1072
15 Path Integrals in Polymer Physics
Apart from that, the perturbation expansion of (15.277) has the same general form as in (15.259):
1 int1 2
int2
int1
2
P (k; L) = 1 − ε hAk [q]i0 + ε −hAk [q]i0 + hAk [q] i0 − . . . .
(15.282)
2
The expectation values can be expressed in terms of the same integrals listed in Appendix 15A
and Appendix 15B as follows:
1
1
R
(D − 1) ik
int1
I1 + DI2 − ∆(0, 0) − ∆(L, L) − L
hA, k [q]i0 =
2
L
2
2
8
(D − 1) [(D − 1) − ik]
,
(15.283)
= −L
12
ik
(D2 − 1)
δ(0) +
I3 + 2(D + 2)I4
hAint2
=
, k [q]i0
4
2L
(D2 − 1)
(D2 − 1) [7(D + 2) + 3ik]
= L3
δ(0) + L2
,
(15.284)
120
720
2
1 int1 2
D
2
R
L2 (D − 1)L
δ(0) +
− δk +
−
hA, k [q] i0 =
2
2
12
2L
L
8
(D − 1) k
+
H1 − 2(H2k + H3k − H5 ) + H6 − 4D(H4k − H7 − H10 )
4
(D − 1)
[DH12 + 2(D + 2)H13 + DH14 ]
+ H11 + 2D2 (H8 + H9 ) +
4
2
(D − 1)2 [(D − 1) − ik]
= L2
2 · 122
(D − 1) (D − 1)
δ(0) + L2
(13D2 − 6D + 21) + 4ik(2D + ik)
+ L3
120
1440
2 (D − 1)
3 (D − 1)D
δ(0) + L
(29D − 1) .
(15.285)
+ L
120
720
In this way we find the large-stiffness expansion up to order ε2 :
(D − 1)
(D − 1)
P (k; L) = 1 + εL
[(D − 1) − ik] + ε2 L2
(15.286)
12
1440
× (ik)2 (5D−1)−2ik(5D2 −11D+8)+(D−1)(5D2 −11D+14) +O(ε3 ).
This can also be rewritten as
(D−1)
(5D2 −11D+14) 2
(D−3)
P (k; L) = P1 loop (k; L) 1+
l +O(l3 ) ,
l+
ik+
6
180(D−1)
360
(15.287)
where the prefactor P1 loop (k; L) has the expansion
P1 loop (k; L) = 1 − εL
(D − 1)
(D − 1)(5D − 1)
(ik) + ε2 L2
(ik)2 − . . . .
22 · 3
25 · 32 · 5
(15.288)
With the identification ω̄ 2 = ikεL, this is the expansion of one-loop functional determinant in
(15.227). By Fourier-transforming (15.286), we obtain the radial distribution function
l2
l ′
[δ (r−1) + (d − 1) δ(r−1)] +
[(5d − 1) δ ′′ (r−1)
6
360(d − 1)
+ 2(5d2 −11d+8) δ ′(r−1) + (d−1)(5d2 −11d+14)δ(r−1) + O(l3 ).
(15.289)
P (r; l) = δ(r−1) +
H. Kleinert, PATH INTEGRALS
15.9 Schrödinger Equation and Recursive Solution for Moments
1073
As a crosscheck, we can calculate from this expansion once more the even and odd moments
Z
h Rn i = Ln dr rn+(D−1) P (r; l),
(15.290)
and find that they agree with Eq. (15.262).
Using the higher-order expansion of the moments in (15.198) we can easily extend the distribution (15.289) to arbitrarily high orders in l. Keeping only the terms up to order l4 , we find that
the one-loop end-to-end distribution function (15.212) receives a correction factor:
Z ∞
dω̂ 2 −iω̂2 (r−1)(D−1)/2l ω̄ (D−1)/2 −V (l,ω̂2 )
P (r; l) ∝
e
,
(15.291)
e
sinh ω̄
−∞ 2π
with
V (l, ω̂ 2 ) ≡ V0 (l) + V̄ (l, ω̂ 2 ) = V0 (l) + V1 (l)
ω̂ 6
ω̂ 4
ω̂ 2
+ V2 (l) 2 + V3 (l) 3 + . . . .
l
l
l
(15.292)
The first term
d−1
d − 9 2 (d − 1) 32 − 13 d + 5 d2 3
V0 (l) = −
l+
l +
l
6
360
6480
2
3
4
34 − 272 d + 259 d − 110 d + 25 d 4
l + ...
−
259200
(15.293)
contributes only to the normalization of P (r; l), and can be omitted in (15.291). The remainder
has the expansion coefficients
−455 + 431 d + 91 d2 + 5 d3 4
d−3 2
(−5 + 9 d) 3
V1 (l) = −
l + ... ,
l +
l +
2
360
7560 (−1 + d)
907200 (d − 1)
−31 + 42 d + 25 d2 l4
(5 − 3 d) l3
−
+ ... ,
(15.294)
V2 (l) = −
7560
907200 (d − 1)
(d − 1) l4
V3 (l) = −
+ ... .
18900
In the physical most interesting case of three dimensions, the first nonzero correction arises to
order l3 . This explains the remarkable agreement of the moments in (15.234) and (15.234) up to
order l2 .
The correction terms V̄ (l, ω̂ 2 ) may be included perturbatively into the sum over k in
Eq. (15.220) by noting that the expectation value of powers of ω̂ 2 /l within the ω̂-integral (15.221)
are
2 2k + (D − 1)/2 2 2
, ω̂ /l = 3a4k ,
ω̂ /l = a2k ≡
(D − 1)(1 − r)
so that we obtain an extra factor e−fk
fk =
V1 (l)a2k
+
where up to order l4 :
3V2 (l)−V12 (l)
a4k
2 3
ω̂ /l = 15a6k ,
4 3
+ 15V3 (l)−12V1(l)V2 (l)+ V1 (l) a6k + . . . ,
3
(15.295)
(15.296)
3D − 5 3 156 − 231D − 26D2 − 7D3 4
l +
l + ... ,
2520
907200 (D − 1)
D−1 4
4
l + ... .
(15.297)
15V3 (l)−12V1 (l)V2 (l)+ V13 (l) = −
3
1260
3V2 (l)−V12 (l) =
1074
15.10
15 Path Integrals in Polymer Physics
Excluded-Volume Effects
A significant modification of these properties is brought about by the interactions
between the chain elements. If two of them come close to each other, the molecular
forces prevent them from occupying the same place. This is called the excludedvolume effect. In less than four dimensions, it gives rise to a scaling law for the
expectation value hR2 i as a function of L:
hR2 i ∝ L2ν ,
(15.298)
as stated in (15.38). The critical exponent ν is a number between the random-chain
value ν = 1/2 and the stiff-chain value ν = 1.
To derive this behavior we consider the polymer in the limiting path integral
approximation (15.80) to a random chain which was derived for R2 /La ≪ 1 and
which is very accurate whenever the probability distribution is sizable. Thus we
start with the time-sliced expression
1
PN (R) = q
D
2πa/M
NY
−1
n=1
with the action
AN = a



dD xn
Z
q
2πa/M

D  exp

−AN /h̄ ,
N
X
M (∆xn )2
,
a2
n=1 2
(15.299)
(15.300)
and the mass parameter (15.79). In the sequel we use natural units in which energies
are measured in units of kB T , and write down all expressions in the continuum limit.
The probability (15.299) is then written as
PL (R) =
Z
L [x]
D D x e−A
,
(15.301)
where we have used the label L = Na rather than N. From the discussion in
the previous section we know that although this path integral represents an ideal
random chain, we can also account for a finite stiffness by interpreting the number
a as an effective length parameter aeff given by (15.158). The total Euclidean time
in the path integral τb − τa = h̄/kB T corresponds to the total length of the polymer
L.
We now assume that the molecules of the polymer repel each other with a twobody potential V (x, x′ ). Then the action in the path integral (15.301) has to be
supplemented by an interaction
Aint =
1
2
Z
0
L
dτ
Z
0
L
dτ ′ V (x(τ ), x(τ ′ )).
(15.302)
Note that the interaction is of a purely spatial nature and does not depend on the
parameters τ , τ ′ , i.e., it does not matter which two molecules in the chain come
close to each other.
H. Kleinert, PATH INTEGRALS
1075
15.10 Excluded-Volume Effects
The effects of an interaction of this type are most elegantly calculated by making use of a Hubbard-Stratonovich transformation. Generalizing the procedure in
Subsection 7.15.1, we introduce an auxiliary fluctuating field variable ϕ(x) at every
space point x and replace Aint by
Aϕint
=
Z
1Z D D ′
d xd x ϕ(x)V −1 (x, x′ )ϕ(x′ ).
dτ ϕ(x(τ )) −
2
L
0
(15.303)
Here V −1 (x, x′ ) denotes the inverse of V (x, x′ ) under functional multiplication, defined by the integral equation
Z
dD x′ V −1 (x, x′ )V (x′ , x′′ ) = δ (D) (x − x′′ ).
(15.304)
To see the equivalence of the action (15.303) with (15.302), we rewrite (15.303) as
Aϕint =
Z
dD x ρ(x)ϕ(x) −
1Z D D ′
d xd x ϕ(x)V −1 (x, x′ )ϕ(x′ ),
2
(15.305)
where ρ(x) is the particle density
ρ(x) ≡
Z
L
0
dτ δ (D) (x − x(τ )).
(15.306)
Then we perform a quadratic completion to
Aϕint = −
i
1Z D D ′h ′
d xd x ϕ (x)V −1 (x, x′ )ϕ′ (x′ ) − ρ(x)V (x, x′ )ρ(x′ ) ,
2
(15.307)
with the shifted field
ϕ′ (x) ≡ ϕ(x) −
Z
dD x′ V (x, x′ )ρ(x′ ).
(15.308)
Now we perform the functional integral
Z
ϕ
Dϕ(x) e−Aint
(15.309)
integrating ϕ(x) at each point x from −i∞ to i∞ along the imaginary field axis.
−1/2
The result is a constant functional determinant [det V −1 (x, x′ )]
. This can be
ignored since we shall ultimately normalize the end-to-end distribution to unity.
Inserting (15.306) into the surviving second term in (15.307), we obtain precisely
the original interaction (15.302).
Thus we may study the excluded-volume problem by means of the equivalent
path integral
PL (R) ∝
Z
D D x(τ )
Z
Dϕ(x) e−A ,
(15.310)
where the action A is given by the sum
A = AL [x, ẋ, ϕ] + A[ϕ],
(15.311)
1076
15 Path Integrals in Polymer Physics
of the line and field actions
L
A [x, ϕ] ≡
Z
L
M 2
dτ
ẋ + ϕ(x(τ )) ,
2
(15.312)
1
2
Z
(15.313)
0
A[ϕ] ≡ −
dD xdD x′ ϕ(x)V −1 (x, x′ )ϕ(x′ ),
respectively. The path integral (15.310) over x(τ ) and ϕ(x) has the following physical interpretation. The line action (15.312) describes the orbit of a particle in a
space-dependent random potential ϕ(x). The path integral over x(τ ) yields the endto-end distribution of the fluctuating polymer in this potential. The path integral
over all potentials ϕ(x) with the weight e−A[ϕ] accounts for the repulsive cloud of
the fluctuating chain elements. To be convergent, all ϕ(x) integrations in (15.310)
have to run along the imaginary field axis.
To evaluate the path integrals (15.310), it is useful to separate x(τ )- and ϕ(x)integrations and to write end-to-end distributions as an average over ϕ-fluctuations
PL (R) ∝
Z
Dϕ(x) e−A[ϕ] PLϕ (R, 0),
(15.314)
where
PLϕ (R, 0) =
Z
L [x,ϕ]
D D x(τ ) e−A
(15.315)
is the end-to-end distribution of a random chain moving in a fixed external potential
ϕ(x). The presence of this potential destroys the translational invariance of PLϕ . This
is why we have recorded the initial and final points 0 and R. In the final distribution
PL (R) of (15.314), the invariance is of course restored by the integration over all
ϕ(x).
It is possible to express the distribution PLϕ (R, 0) in terms of solutions of an
associated Schrödinger equation. With the action (15.312), this equation is obviously
"
#
∂
1
−
∂R 2 + ϕ(R) PLϕ (R, 0) = δ (D) (R − 0)δ(L).
∂L 2M
(15.316)
If ψEϕ (R) denotes the time-independent solutions of the Hamiltonian operator
Ĥ ϕ = −
1
∂R 2 + ϕ(R),
2M
(15.317)
the probability PLϕ (R) has a spectral representation of the form
PLϕ (R, 0) =
Z
dEe−EL ψEϕ (R)ψEϕ ∗ (0),
L > 0.
(15.318)
From now on, we assume the interaction to be dominated by the simplest possible
repulsive potential proportional to a δ-function:
V (x, x′ ) = vaD δ (D) (x − x′ ).
(15.319)
H. Kleinert, PATH INTEGRALS
1077
15.10 Excluded-Volume Effects
Then the functional inverse is
V −1 (x, x′ ) = v −1 a−D δ (D) (x − x′ ),
(15.320)
and the ϕ-action (15.312) reduces to
v −1 a−D Z D 2
A[ϕ] = −
d x ϕ (x).
2
(15.321)
The path integrals (15.314), (15.315) can be solved approximately by applying the
semiclassical methods of Chapter 4 to both the x(τ )- and the ϕ(x)-path integrals.
These are dominated by the extrema of the action and evaluated via the leading
saddle point approximation. In the ϕ(x)-integral, the saddle point is given by the
equation
v −1 a−D ϕ(x) =
δ
log PLϕ (R, 0).
δϕ(x)
(15.322)
This is the semiclassical approximation to the exact equation
−1 −D
v a
hϕ(x)i = hρ(x)i ≡
*Z
0
L
dτ δ
(D)
(x − x(τ ))
+
,
(15.323)
x
where h. . .ix is the average over all line fluctuations calculated with the help of the
probability distribution (15.315).
The exact equation (15.323) follows from a functional differentiation of the path
integral for PLϕ with respect to ϕ(x):
δ
PLϕ (R) =
δϕ(x)
Z
δ
Dϕ
δϕ(x)
Z
L [x,ϕ]−A[ϕ]
D D x e−A
= 0.
(15.324)
By anchoring one end of the polymer at the origin and carrying the path integral
from there to x(τ ), and further on to R, the right-hand side of (15.323) can be
expressed as a convolution integral over two end-to-end distributions:
*Z
0
L
dτ δ
(D)
(x − x(τ ))
+
=
x
Z
0
L
ϕ
dL′ PLϕ′ (x)PL−L
′ (R − x).
(15.325)
With (15.323), this becomes
v −1 a−D hϕ(x)ix =
Z
0
L
ϕ
dL′ PLϕ′ (x)PL−L
′ (R − x),
(15.326)
which is the same as (15.323).
According to Eq. (15.322), the extremal ϕ(x) depends really on two variables, x
and R. This makes the solution difficult, even at the semiclassical level. It becomes
simple only for R = 0, i.e., for a closed polymer. Then only the variable x remains
and, by rotational symmetry, ϕ(x) can depend only on r = |x|. For R 6= 0, on the
1078
15 Path Integrals in Polymer Physics
other hand, the rotational symmetry is distorted to an ellipsoidal geometry, in which
a closed-form solution of the problem is hard to find. As an approximation, we may
use a rotationally symmetric ansatz ϕ(x) ≈ ϕ(r) also for R 6= 0 and calculate the
end-to-end probability distribution PL (R) via the semiclassical approximation to
the two path integrals in Eq. (15.310).
The saddle point in the path integral over ϕ(x) gives the formula [compare
(15.314)]
PL (R) ∼
PLϕ (R, 0)
=
Z
( Z
D
D x exp −
L
0
)
M 2
dτ
ẋ + ϕ(r(τ )) .
2
(15.327)
Thereby it is hoped that for moderate R, the error is small enough to justify this
approximation. Anyhow, the analytic results supply a convenient starting point for
better approximations.
Neglecting the ellipsoidal distortion, it is easy to calculate the path integral over
x(τ ) for PLϕ (R, 0) in the saddle point approximation. At an arbitrary given ϕ(r),
we must find the classical orbits. The Euler-Lagrange equation has the first integral
of motion
M 2
ẋ − ϕ(r) = E = const.
(15.328)
2
At fixed L, we have to find the classical solutions for all energies E and all angular
momenta l. The path integral reduces an ordinary double integral over E and
l which, in turn, is evaluated in the saddle point approximation. In a rotationally
symmetric potential ϕ(r), the leading saddle point has the angular momentum l = 0
corresponding to a symmetric polymer distribution. Then Eq. (15.328) turns into a
purely radial differential equation
dr
dτ = q
.
2[E + ϕ(r)]/M
(15.329)
For a polymer running from the origin to R we calculate
L=
Z
dr
R
q
2[E + ϕ(r)]/M
0
.
(15.330)
This determines the energy E as a function of L. It is a functional of the yet
unknown field ϕ(r):
E = EL [ϕ].
(15.331)
The classical action for such an orbit can be expressed in the form
M 2
ẋ + ϕ(r(τ ))
2
0
Z L
Z L
M 2
ẋ − ϕ(r(τ )) +
dτ M ẋ2
= −
dτ
2
0
0
Acl [x, ϕ] =
Z
L
dτ
= −EL +
Z
0
R
q
dr 2M[E + ϕ(r)].
(15.332)
H. Kleinert, PATH INTEGRALS
1079
15.10 Excluded-Volume Effects
In this expression, we may consider E as an independent variational parameter.
The relation (15.330) between E, L, R, ϕ(r), by which E is fixed, reemerges when
extremizing the classical expression Acl [x, ϕ]:
∂
Acl [x, ẋ, ϕ] = 0.
∂E
(15.333)
The classical approximation to the entire action A[x, ϕ] + A[ϕ] is then
Acl = −EL +
Z
0
r
dr
′
q
2M[E +
ϕ(r ′ )]
1
− v −1 a−D
2
Z
dD xϕ2 (r).
(15.334)
This action is now extremized independently in ϕ(r), E. The extremum in ϕ(r) is
obviously given by the algebraic equation
ϕ(r ′ ) =
(
0
q
−1 ′ 1−D
MvaD SD
r
/ 2M[E + ϕ(r ′)]
for
r ′ > r,
r ′ < r,
(15.335)
which is easily solved. We rewrite it as
E + ϕ(r) = ξ 3 ϕ−2 (r),
(15.336)
ξ 3 = αr −2δ ,
(15.337)
δ ≡D−1>0
(15.338)
with the abbreviation
where
and
α≡
M 2 2D −2
v a SD .
2
(15.339)
For large ξ ≫ 1/E, i.e., small r ≪ α2/δ E −6/δ , we expand the solution as follows
ϕ(r) = ξ −
E E2
+
+ ... .
3
9
(15.340)
This expansion is reinserted into the classical action (15.334), making it a power
series in E. A further extremization in E yields E = E(L, r). The extremal value of
the action yields an approximate distribution function of a monomer in the closed
polymer (which runs through the origin):
PL (R) ∝ e−Acl (L,R) .
(15.341)
To see how this happens consider first the noninteracting limit where v = 0.
Then the solution of (15.335) is ϕ(r) ≡ 0, and the classical action (15.334) becomes
√
Acl = −EL + 2MER.
(15.342)
1080
15 Path Integrals in Polymer Physics
The extremization in E gives
M R2
,
E=
2 L2
(15.343)
M R2
.
2 L
(15.344)
yielding the extremal action
Acl =
The approximate distribution is therefore
2 /2L
PN (R) ∝ e−Acl = e−M R
.
(15.345)
The interacting case is now treated in the same way. Using (15.336), the classical
action (15.334) can be written as
Acl = −EL +
s
M
2
Z
R
0
dr
′
#
"
q
1
1
.
ϕ + E − ξ 3/2
2
ϕ+E
(15.346)
By expanding this action in a power series in E [after having inserted (15.340) for
ϕ], we obtain with ǫ(r) ≡ E/ξ = Eα−1/3 r 2δ/3
Acl = −EL +
s
M 1/6
α
2
Z
R
0
′ ′−δ/3
dr r
1
3
+ ǫ(r ′ ) − ǫ2 (r ′ ) + . . . .
2
6
(15.347)
As long as δ < 3, i.e., for
D < 4,
(15.348)
the integral exists and yields an expansion
1
Acl = −EL + a0 (R) + a1 (R)E − a2 (R)E 2 + . . . ,
2
(15.349)
with the coefficients
M
a0 (R) = −
2
a1 (R) = 3
s
1
a2 (R) =
3
s
9 1−δ/3 1/6 1
R
α
,
2
δ−3
M 1+δ/3 −1/6 1
R
α
,
2
δ+3
s
M 1+δ −1/2 1
R α
.
2
δ+1
(15.350)
Extremizing Acl in E gives the action
Acl = a0 (R) +
1
[L − a1 (R)]2 + . . . .
2a2 (R)
(15.351)
H. Kleinert, PATH INTEGRALS
1081
15.11 Flory’s Argument
The approximate end-to-end distribution function is therefore
)
(
1
PL (R) ≈ N exp −a0 (R) −
[L − a1 (R)]2 ,
2a2 (R)
(15.352)
where N is an appropriate normalization factor. The distribution is peaked around
L=3
s
M 1+δ/3 −1/6 1
R
α
.
2
δ+3
(15.353)
This shows the most important consequence of the excluded-volume effect: The
average value of R2 grows like

D+2
hR2 i ≈ α1/(D+2) 
3
s
6/(D+2)
2 
L
M
.
(15.354)
Thus we have found a scaling law of the form (15.298) with the critical exponent
ν=
3
.
D+2
(15.355)
The repulsion between the chain elements makes the excluded-volume chain reach
out further into space than a random chain [although less than a completely stiff
chain, which is always reached by the solution (15.354) for D = 1].
The restriction D < 4 in (15.348) is quite important. The value
D uc = 4
(15.356)
is called the upper critical dimension. Above it, the set of all possible intersections
of a random chain has the measure zero and any short-range repulsion becomes
irrelevant. In fact, for D > D uc it is possible to show that the polymer behaves like
a random chain without any excluded-volume effect satisfying hR2 i ∝ L.
15.11
Flory’s Argument
Once we expect a power-like scaling law of the form
hR2 i ∝ L2ν ,
(15.357)
the critical exponent (15.355) can be derived from a very simple dimensional argument due to Flory. We take the action
A=
Z
0
L
dτ
M 2 vaD
ẋ −
2
2
Z
0
L
dτ
Z
0
L
dτ ′ δ (D) (x(τ ) − x(τ ′ )),
(15.358)
with M = D/a, and replace the two terms by their dimensional content, L for the
τ -variable and R- for each x-component. Then
A∼
M R2 vaD L2
L
−
.
2 L2
2 RD
(15.359)
1082
15 Path Integrals in Polymer Physics
Extremizing this expression in R at fixed L gives
R
∼ R−D−1 L2 ,
L
(15.360)
R2 ∼ L6/(D+2) ,
(15.361)
implying
and thus the critical exponent (15.355).
15.12
Polymer Field Theory
There exists an alternative approach to finding the power laws caused by the
excluded-volume effects in polymers which is superior to the previous one. It is
based on an intimate relationship of polymer fluctuations with field fluctuations in
a certain somewhat artificial and unphysical limit. This limit happens to be accessible to approximate analytic methods developed in recent years in quantum field
theory. According to Chapter 7, the statistical mechanics of a many-particle ensemble can be described by a single fluctuating field. Each particle in such an ensemble
moves through spacetime along a fluctuating orbit in the same way as a random
chain does in the approximation (15.80) to a polymer in Section 15.6. Thus we can
immediately conclude that ensembles of polymers may also be described by a single
fluctuating field. But how about a single polymer? Is it possible to project out
a single polymer of the ensemble in the field-theoretic description? The answer is
positive. We start with the result of the last section. The end-to-end distribution
of the polymer in the excluded-volume problem is rewritten as an integral over the
fluctuating field ϕ(x):
PL (xb , xa ) =
Z
Dϕ e−A[ϕ] PLϕ (xb , xa ),
(15.362)
with an action for the auxiliary ϕ(x) field [see (15.312)]
1
A[ϕ] = −
2
Z
dD xdD x′ ϕ(x)V −1 (x, x′ )ϕ(x′ ),
(15.363)
and an end-to-end distribution [see (15.315)]
PLϕ (xb , xa )
=
Z
(
Dx exp −
Z
0
L
M 2
ẋ + ϕ(x(τ ))
dτ
2
)
,
(15.364)
which satisfies the Schrödinger equation [see (15.316)]
"
#
∂
1
−
∇2 + ϕ(x) PLϕ (x, x′ ) = δ (3) (x − x′ )δ(L).
∂L 2M
(15.365)
H. Kleinert, PATH INTEGRALS
1083
15.12 Polymer Field Theory
Since PL and PLϕ vanish for L < 0, it is convenient to go over to the Laplace
transforms
∞
1
2
Pm2 (x, x ) =
dLe−Lm /2M PL (x, x′ ),
2M 0
Z ∞
1
2
ϕ
′
dLe−Lm /2M PLϕ (x, x′ ).
Pm2 (x, x ) =
2M 0
Z
′
(15.366)
(15.367)
The latter satisfies the L-independent equation
[−∇2 + m2 + 2Mϕ(x)]Pmϕ2 (x, x′ ) = δ (3) (x − x′ ).
(15.368)
The quantity m2 /2M is, of course, just the negative energy variable −E in (15.332):
−E ≡
m2
.
2M
(15.369)
The distributions Pmϕ2 (x, x′ ) describe the probability of a polymer of any length run2
ning from x′ to x, with a Boltzmann-like factor e−Lm /2M governing the distribution
of lengths. Thus m2 /2M plays the role of a chemical potential.
We now observe that the solution of Eq. (15.368) can be considered as the correlation function of an auxiliary fluctuating complex field ψ(x):
Pmϕ2 (x, x′ ) = Gϕ0 (x, x′ ) = hψ ∗ (x)ψ(x′ )iϕ
R
Dψ ∗ (x)Dψ(x) ψ ∗ (x)ψ(x′ ) exp {−A[ψ ∗ , ψ, ϕ]}
R
,
≡
Dψ ∗ (x)Dψ(x) exp {−A[ψ ∗ , ψ, ϕ]}
(15.370)
with a field action
Z
n
o
A[ψ ∗ , ψ, ϕ] = dD x ∇ψ ∗ (x)∇ψ(x) + m2 ψ ∗ (x)ψ(x) + 2Mϕ(x)ψ ∗ (x)ψ(x) .(15.371)
The second part of Eq. (15.370) defines the expectations h. . .iψ . In this way, we
express the Laplace-transformed distribution Pm2 (xb , xa ) in (15.366) in the purely
field-theoretic form
Pm2 (x, x′ ) =
Z
Dϕ exp {−A[ϕ]} hψ ∗ (x)ψ(x′ )iϕ
1
dD ydD y′ ϕ(y)V −1 (y, y′)ϕ(y′ )
2
R
Dψ ∗ Dψ ψ ∗ (x)ψ(x′ ) exp {−A[ψ ∗ , ψ, ϕ]}
R
R
×
.
Dψ ∗ Dψ exp {−A[ψ ∗ , ψ, ϕ]}
=
Z
Dϕ exp
Z
(15.372)
It involves only a fluctuating field which contains all information on the path fluctuations. The field ψ(x) is, of course, the analog of the second-quantized field in
Chapter 7.
Consider now the probability distribution of a single monomer in a closed polymer chain. Inserting the polymer density function
ρ(R) ≡
Z
0
L
dτ δ (D) (R − x(τ ))
(15.373)
1084
15 Path Integrals in Polymer Physics
into the original path integral for a closed polymer
PL (R) =
Z
L
0
dτ
Z
D
D x
Z
Dϕ exp {−AL − A[ϕ]} δ (D) (R − x(τ )),
(15.374)
the δ-function splits the path integral into two parts
PL (R) =
Z
Dϕ exp {−A[ϕ]}
Z
0
L
ϕ
dτ PL−τ
(0, R)Pτϕ (R, 0).
(15.375)
When going to the Laplace transform, the convolution integral factorizes, yielding
Pm2 (R) =
Z
Dϕ(x) exp {−A[ϕ]} Pmϕ2 (0, R)Pmϕ2 (R, 0).
(15.376)
With the help of the field-theoretic expression for Pm2 (R) in Eq. (15.370), the product of the correlation functions can be rewritten as
Pmϕ2 (0, R)Pmϕ2 (0, R) = hψ ∗ (R)ψ(0)iϕ hψ ∗ (0)ψ(R)iϕ .
(15.377)
We now observe that the field ψ appears only quadratically in the action A[ψ ∗ , ψ, ϕ].
The product of correlation functions in (15.377) can therefore be viewed as a term
in the Wick expansion (recall Section 3.10) of the four-field correlation function
hψ ∗ (R)ψ(R)ψ ∗ (0)ψ(0)iϕ .
(15.378)
This would be equal to the sum of pair contractions
hψ ∗ (R)ψ(R)iϕ hψ ∗ (0)ψ(0)iϕ + hψ ∗ (R)ψ(0)iϕ hψ ∗ (0)ψ(R)iϕ .
(15.379)
There are no contributions containing expectations of two ψ or two ψ ∗ fields which
could, in general, appear in this expansion. This allows the right-hand side of
(15.377) to be expressed as a difference between (15.378) and the first term of
(15.379):
Pmϕ2 (0, R)Pmϕ2 (0, R) = hψ ∗ (R)ψ(R)ψ ∗(0)ψ(0)iϕ − hψ ∗ (R)ψ(R)iϕ hψ ∗ (0)ψ(0)iϕ .
(15.380)
The right-hand side only contains correlation functions of a collective field, the
density field [8]
ρ(R) = ψ ∗ (R)ψ(R),
(15.381)
Pmϕ2 (0, R)Pmϕ2 (0, R) = hρ(R)ρ(0)iϕ − hρ(R)iϕ hρ(0)iϕ .
(15.382)
in terms of which
Now, the right-hand side is the connected correlation function of the density field
ρ(R):
hρ(R)ρ(0)iϕ,c ≡ hρ(R)ρ(0)iϕ − hρ(R)iϕ hρ(0)iϕ .
(15.383)
H. Kleinert, PATH INTEGRALS
1085
15.12 Polymer Field Theory
In Section 3.10 we have shown how to generate all connected correlation functions: The action A[ψ, ψ ∗ , ϕ] is extended by a source term in the density field ρ(x)
∗
Asource [ψ , ψ, K] = −
Z
D
d xK(x)ρ(x) =
Z
dD xK(x)ψ ∗ (x)ψ(x)),
(15.384)
and one considers the partition function
Z[K, ϕ] ≡
Z
DψDψ ∗ exp {−A [ψ ∗ , ψ, ϕ] − Asource [ψ ∗ , ψ, K]} .
(15.385)
This is the generating functional of all correlation functions of the density field
ρ(R) = ψ ∗ (R)ψ(R) at a fixed ϕ(x). They are obtained from the functional derivatives
hρ(x1 ) · · · ρ(xn )iϕ = Z[K, ϕ]−1
δ
δ
···
Z[K, ϕ]
.
K=0
δK(x1 )
δK(xn )
(15.386)
Recalling Eq. (3.559), the connected correlation functions of ρ(x) are obtained similarly from the logarithm of Z[K, ϕ]:
hρ(x1 ) · · · ρ(xn )iϕ,c =
δ
δ
.
···
log Z[K, ϕ]
K=0
δK(x1 )
δK(xn )
(15.387)
For n = 2, the connectedness is seen directly by performing the differentiations
according to the chain rule:
δ
δ
log Z[K, ϕ]
K=0
δK(R) δK(0)
δ
δ
Z −1 [K, ϕ]
Z[K, ϕ]
=
K=0
δK(R)
δK(0)
= hρ(R)ρ(0)iϕ − hρ(R)iϕ hρ(0)iϕ .
hρ(R)ρ(0)iϕ,c =
(15.388)
This agrees indeed with (15.383). We can therefore rewrite the product of Laplacetransformed distributions (15.382) at a fixed ϕ(x) as
Pmϕ2 (0, R)Pmϕ2 (0, R) =
δ
δ
log Z[K, ϕ]
.
K=0
δK(R) δK(0)
(15.389)
The Laplace-transformed monomer distribution (15.376) is then obtained by averaging over ϕ(x), i.e., by the path integral
Pm2 (R) =
δ
δ
δK(R) δK(0)
Z
Dϕ(x) exp {−A[ϕ]} log Z[K, ϕ]
K=0
.
(15.390)
1086
15 Path Integrals in Polymer Physics
Were it not for the logarithm in front of Z, this would be a standard calculation of
correlation functions within the combined ψ, ϕ field theory whose action is
Atot [ψ ∗ , ψ, ϕ] = A[ψ ∗ , ψ, ϕ] + A[ϕ]
Z
=
1
−
2
dD x ∇ψ ∗ ∇ψ + m2 ψ ∗ ψ + 2Mϕψ ∗ ψ
Z
dD xdD x′ ϕ(x)V −1 (x, x′ )ϕ(x′ ).
(15.391)
To account for the logarithm we introduce a simple mathematical device called the
replica trick [9]. We consider log Z[K, ϕ] in (15.388)–(15.390) as the limit
1 n
(Z − 1) ,
n→0 n
(15.392)
log Z = lim
and observe that the nth power of the generating functional, Z n , can be thought
of as arising from a field theory in which every field ψ occurs n times, i.e., with n
identical replica. Thus we add an extra internal symmetry label α = 1, . . . , n to the
fields ψ(x) and calculate Z n formally as
Z n [K, ϕ] =
Z
Dψα∗ Dψα exp {−A[ψα∗ , ψα , ϕ]−A[ϕ] −Asource [ψα∗ , ψα , K]} ,
(15.393)
with the replica field action
A[ψα , ψα∗ , ϕ] =
Z
dD x ∇ψα∗ ∇ψα + m2 ψα∗ ψα + 2Mϕ ψα∗ ψα ,
(15.394)
and the source term
Asource [ψα∗ , ψα , K]
=−
Z
dD xψα∗ (x)ψα (x)K(x).
(15.395)
A sum is implied over repeated indices α. By construction, the action is symmetric
under the group U(n) of all unitary transformations of the replica fields ψα .
In the partition function (15.393), it is now easy to integrate out the ϕ(x)fluctuations. This gives
Z n [K, ϕ] =
Z
Dψα∗ Dψα exp {−An [ψα∗ , ψα ] − Asource [ψα∗ , ψα , K]} ,
(15.396)
with the action
A
n
[ψα∗ , ψα ]
=
Z
dD x ∇ψα∗ ∇ψα + m2 ψα∗ ψα
Z
1
2
dD xdD x′ ψα ∗ (x)ψα (x)V (x, x′ )ψβ ∗ (x′ )ψβ (x′ ). (15.397)
(2M)
+
2
It describes a self-interacting field theory with an additional U(n) symmetry.
H. Kleinert, PATH INTEGRALS
1087
15.12 Polymer Field Theory
In the special case of a local repulsive potential V (x, x′ ) of Eq. (15.319), the
second term becomes simply
1
Aint [ψα∗ , ψα ] = (2M)2 vaD
2
Z
dD x [ψα ∗ (x)ψα (x)]2 .
(15.398)
Using this action, we can find log Z[K, ϕ] via (15.392) from the functional integral
log Z[K, ϕ] ≡ lim
n→0
1
n
Z
Dψα∗ Dψα exp {−An [ψα∗ , ψα ] − Asource [ψα∗ , ψα , K]} − 1 .
(15.399)
This is the generating functional of the Laplace-transformed distribution (15.390)
which we wanted to calculate.
A polymer can run along the same line in two orientations. In the above description with complex replica fields it was assumed that the two orientations can
be distinguished. If they are indistinguishable, the polymer fields Ψα (x) have to be
taken as real.
Such a field-theoretic description of a fluctuating polymer has an important
advantage over the initial path integral formulation based on the analogy with a
particle orbit. It allows us to establish contact with the well-developed theory of
critical phenomena in field theory. The end-to-end distribution of long polymers
at large L is determined by the small-E regime in Eqs. (15.330)–(15.349), which
corresponds to the small-m2 limit of the system here [see (15.369)]. This is precisely
the regime studied in the quantum field-theoretic approach to critical phenomena
in many-body systems [10, 11]. It can be shown that for D larger than the upper
critical dimension D uc = 4, the behavior for m2 → 0 of all Green functions coincides
with the free-field behavior. For D = D uc , this behavior can be deduced from scale
invariance arguments of the action, using naive dimensional counting arguments.
The fluctuations turn out to cause only logarithmic corrections to the scale-invariant
power laws. One of the main developments in quantum field theory in recent years
was the discovery that the scaling powers for D < D uc can be calculated via an
expansion of all quantities in powers of
ǫ = D uc − D,
(15.400)
the so-called ǫ-expansion. The ǫ-expansion for the critical exponent ν which rules
the relation between R2 and the length of a polymer L, hR2 i ∝ L2ν , can be derived
from a real φ4 -theory with n replica as follows [12]:
ν −1 = 2 +
ǫ2
(n+2)ǫ
n+8
n
−1 −
ǫ
2(n+8)2 (13n
+ 44)
+ 8(n+8)4 [3n3 − 452n2 − 2672n − 5312
ǫ3
+ 32(n+8)
6
+ ζ(3)(n + 8) · 96(5n + 22)]
[3n5 + 398n4 − 12900n3 − 81552n2 − 219968n − 357120
+ ζ(3)(n + 8) · 16(3n4 − 194n3 + 148n2 + 9472n + 19488)
+ ζ(4)(n + 8)3 · 288(5n + 22)
1088
15 Path Integrals in Polymer Physics
− ζ(5)(n + 8)2 · 1280(2n2 + 55n + 186)]
4
ǫ
7
6
5
4
+ 128(n+8)
8 [3n − 1198n − 27484n − 1055344n
−5242112n3 − 5256704n2 + 6999040n − 626688
− ζ(3)(n + 8) · 16(13n6 − 310n5 + 19004n4 + 102400n3
− 381536n2 − 2792576n − 4240640)
− ζ 2 (3)(n + 8)2 · 1024(2n4 + 18n3 + 981n2 + 6994n + 11688)
+ ζ(4)(n + 8)3 · 48(3n4 − 194n3 + 148n2 + 9472n + 19488)
+ ζ(5)(n + 8)2 · 256(155n4 + 3026n3 + 989n2 − 66018n − 130608)
− ζ(6)(n + 8)4 · 6400(2n2 + 55n + 186)
o
+ ζ(7)(n + 8)3 · 56448(14n2 + 189n + 526)] ,
(15.401)
where ζ(x) is Riemann’s zeta function (2.521). As shown above, the single-polymer
properties must emerge in the limit n → 0. There, ν −1 has the ǫ-expansion
ν −1 = 2 −
ǫ
11 2
−
ǫ + 0.114 425 ǫ3 − 0.287 512 ǫ4 + 0.956 133 ǫ5.
4 128
(15.402)
This is to be compared with the much simpler result of the last section
ν −1 =
ǫ
D+2
=2− .
3
3
(15.403)
A term-by-term comparison is meaningless since the field-theoretic ǫ-expansion has
a grave problem: The coefficients of the ǫn -terms grow, for large n, like n!, so that
the series does not converge anywhere! Fortunately, the signs are alternating and
the series can be resummed [13]. A simple first approximation used in ǫ-expansions
is to re-express the series (15.402) as a ratio of two polynomials of roughly equal
degree
ν
−1
|rat
2. + 1.023 606 ǫ − 0.225 661 ǫ2
=
,
1. + 0.636 803 ǫ − 0.011 746 ǫ2 + 0.002 677 ǫ3
(15.404)
called a Padé approximation. Its Taylor coefficients up to ǫ5 coincide with those of
the initial series (15.402). It can be shown that this approximation would converge
with increasing orders in ǫ towards the exact function represented by the divergent
series. In Fig. 15.7, we plot the three functions (15.403), (15.402), and (15.404), the
last one giving the most reliable approximation
ν −1 ≈ 0.585.
(15.405)
Note that the simple Flory curve lies very close to the Padé curve whose calculation
requires a great amount of work.
There exists a general scaling relation between the exponent ν and another exponent appearing in the total number of polymer configurations of length L which
behaves like
S = Lα−2 .
(15.406)
H. Kleinert, PATH INTEGRALS
1089
15.13 Fermi Fields for Self-Avoiding Lines
Figure 15.7 Comparison of critical exponent ν in Flory approximation (dashed line)
with result of divergent ǫ-expansion obtained from quantum field theory (dotted line) and
its Padé resummation (solid line). The value of the latter gives the best approximation
ν ≈ 0.585 at ǫ = 1.
The relation is
α = 2 − Dν.
(15.407)
Direct enumeration studies of random chains on a computer suggest a number
1
α∼ ,
4
(15.408)
corresponding to ν = 7/12 ≈ 0.583, very close to (15.405).
The Flory estimate for the exponent α reads, incidentally,
α=
4−D
.
D+2
(15.409)
In three dimensions, this yields α = 1/5, not too far from (15.408).
The discrepancies arise from inaccuracies in both treatments. In the first treatment, they are due to the use of the saddle point approximation and the fact that
the δ-function does not completely rule out the crossing of the lines, as required by
the true self-avoidance of the polymer. The field theoretic ǫ-expansion, on the other
hand, which in principle can give arbitrarily accurate results, has the problem of
being divergent for any ǫ. Resummation procedures are needed and the order of the
expansion must be quite large (≈ ǫ5 ) to extract reliable numbers.
15.13
Fermi Fields for Self-Avoiding Lines
There exists another way of enforcing the self-avoiding property of random lines
[14]. It is based on the observation that for a polymer field theory with n fluctuating
complex fields Ψα and a U(n)-symmetric fourth-order self-interaction as in the action
1090
15 Path Integrals in Polymer Physics
(15.397), the symmetric incorporation of a set of m anticommuting Grassmann fields
removes the effect of m of the Bose fields. For free fields this observation is trivial
since the functional determinant of Bose and Fermi fields are inverse to one-another.
In the presence of a fourth-order self-interaction, where the replica action has the
form (15.397), we can always go back, by a Hubbard-Stratonovich transformation,
to the action involving the auxiliary field ϕ(x) in the exponent of (15.393). This
exponent is purely quadratic in the replica field, and each path integral over a Fermi
field cancels a functional determinant coming from the Bose field.
This boson-destructive effect of fermions allows us to study theories with a negative number of replica. We simply have to use more Fermi than Bose fields. Moreover, we may conclude that a theory with n = −2 has necessarily trivial critical
exponents. From the above arguments it is equivalent to a single complex Fermi
field theory with fourth-order self-interaction. However, for anticommuting Grassmann fields, such an interaction vanishes:
(θ† θ)2 = [(θ1 − iθ2 )(θ1 + iθ2 )]2 = [2iθ1 θ2 ]2 = 0.
(15.410)
Looking back at the ǫ-expansion for the critical exponent ν in Eq. (15.401) we can
verify that up to the power ǫ5 all powers in ǫ do indeed vanish and ν takes the
mean-field value 1/2.
Appendix 15A
Basic Integrals
∆(0, 0) = ∆(L, L) = L/3,
Z L
I1 =
ds ∆(s, s) = L2 /6,
(15A.1)
(15A.2)
0
I2
=
Z
L
ds ˙∆ 2 (s, s) = L/12,
(15A.3)
ds ∆2 (s, s) = L3 /30,
(15A.4)
ds ∆(s, s) ˙∆ 2 (s, s) = 7L2 /360,
(15A.5)
0
I3
=
Z
L
0
I4
=
Z
L
0
I5
=
Z
L
ds
0
Z
L
ds′ ∆2 (s, s′ ) = L4 /90,
(15A.6)
0
∆(0, L) = ∆(L, 0) = −L/6,
Z L
ds ∆2 (s, 0) + ∆2 (s, L) = 2L3 /45,
I6 =
(15A.7)
(15A.8)
0
I7
=
Z
L
ds
=
Z
L
I9
=
=
Z
(15A.9)
Z
L
(15A.10)
ds ∆(s, s) ˙∆ 2 (s, 0) + ˙∆ 2 (s, L) = 11L2 /90,
(15A.11)
0
L
0
I10
ds′ ∆(s, s)˙∆ 2 (s, s′ ) = L3 /45,
ds′ ˙∆ (s, s)∆(s, s′ )˙∆ (s, s′ ) = L3 /180,
ds
0
Z
L
0
0
I8
Z
L
ds ˙∆ (s, s) [∆(s, 0)˙∆ (s, 0) + ∆(s, L)˙∆ (s, L)] = 17L2 /360, (15A.12)
0
H. Kleinert, PATH INTEGRALS
Appendix 15B
Loop Integrals
˙∆ (0, 0) = −˙∆ (L, L) = −1/2,
Z L Z L
ds′ ∆(s, s) ∆˙(s, s′ )˙∆ (s′ , s′ ) = L3 /360,
ds
I11 =
Z
=
Appendix 15B
(15A.13)
(15A.14)
0
0
I12
1091
L
ds
0
Z
L
ds′ ∆(s, s′ )˙∆ 2 (s, s′ ) = L3 /90.
(15A.15)
0
Loop Integrals
We list here the Feynman integrals evaluated with dimensional regularization rules whenever necessary. Depending whether they occur in the calculation of the moments from the expectations
(15.261)–(15.261) of from the expectations (15.283)–(15.285) we encounter the integrals depending
on ρn (s) ≡ δn + [δ(s) + δ(s − L)] /2 with δn = δ(0) − σn /L or ρk (s) = δk + [δ(s) + δ(s − L)] /2
with δk = δ(0) − ik/L:
Z L Z L
n(k)
H1
=
ds
ds′ ρn(k) (s)ρn(k) (s′ )∆2 (s, s′ ),
0
1 2
+ δn(k) I6 +
∆ (0, 0) + ∆2 (0, L) ,
2
Z L Z L
I9
=
ds
ds′ ρn(k) (s′ )∆(s, s)˙∆ 2 (s, s′ ) = δn(k) I7 + ,
2
0
0
Z L Z L
I6
′
′
2
′
=
ds
ds ρn(k) (s )˙∆ d(s, s)∆ (s, s ) = δn(k) I5 +
δ(0),
2
0
0
Z L Z L
I10
.
=
ds
ds′ ρn(k) (s′ )˙∆ (s, s)˙∆ (s, s′ )∆(s, s′ ) = δn(k) I8 +
2
0
0
=
n(k)
H2
n(k)
H3
n(k)
H4
0
2
I5
δn(k)
(15B.1)
(15B.2)
(15B.3)
(15B.4)
When calculating h Rn i, we need to insert here δn = δ(0) − σn /L, thus obtaining
L4 2
L3
L2 (45−24D+4D2)−4n(6−2D−n) ,
δ (0)+
(3−D−n) δ(0)+
90
45
360
L2
L3
δ(0) +
(15−4D−4n),
H2n =
45
180
L3
L4 2
δ (0)+
(3−D−n)δ(0),
H3n =
90
90
L3
L2
H4n =
δ(0)+
(21−4D−4n) ,
180
720
H1n =
(15B.5)
(15B.6)
(15B.7)
(15B.8)
where the values for n = 0 correspond to the partition function Z = h R0 i. The substitution
δk = δ(0) − ik/L required for the calculation of P (k; L) yields
L3
L2
5L2
L4 2
δ (0) +
[2 + (−ik)] δ(0) +
(−ik) [4 + (−ik)] +
,
90
45
90
72
L2
L3
δ(0) +
[11 + 4(−ik)] ,
H2k =
45
180
L3
L4 2
δ (0) +
[2 + (−ik)] δ(0),
H3k =
90
90
3
2
L
L
H4k =
δ(0) +
[17 + 4(−ik)] .
180
720
H1k =
The other loop integrals are
Z L Z L
L3
δ(0),
H5 =
ds
ds′ ∆(s, s)˙∆ 2 (s, s′ )˙∆ d(s′ , s′ ) = δ(0)I7 =
45
0
0
(15B.9)
(15B.10)
(15B.11)
(15B.12)
(15B.13)
1092
15 Path Integrals in Polymer Physics
H6 =
H7 =
H8 =
H9 =
Z
Z
Z
Z
H10 =
H11 =
L
ds
0
ds
0
L
Z
L4 2
δ (0),
90
ds′ ˙∆ (s, s)˙∆ (s, s′ )∆(s, s′ )˙∆ d(s′ , s′ ) = δ(0)I8 =
0
L
ds
0
L
Z
0
L
ds
L
ds
0
ds′ ˙∆ (s, s)∆(s, s′ )˙∆ d(s, s′ )˙∆ (s′ , s′ ) =
Z
L
ds′ ∆(s, s)˙∆ (s, s′ )˙∆ d(s, s′ )˙∆ (s′ , s′ ) =
0
L
ds
0
L
Z
ds′ ˙∆ (s, s)˙∆ (s, s′ ) ∆˙(s, s′ )˙∆ (s′ , s′ ) = −
0
0
Z
ds′ ˙∆ d(s, s)∆2 (s, s′ )˙∆ d(s′ , s′ ) = δ 2 (0)I5 =
0
L
Z
L
Z
Z
L
L3
δ(0),
180
L2
,
720
(15B.14)
(15B.15)
(15B.16)
I10
I8
7L2
−
− H8 =
,
2
L
360
I7
7L2
I9
−
=
,
4
2L
360
(15B.17)
(15B.18)
I1 2 2(I3 − I11 )
) −
− 2I4 ,
L
L
L3
L2
=
δ(0) +
,
(15B.19)
30
9
ds′ ∆(s, s)˙∆ d2 (s, s′ )∆(s′ , s′ ) = δ(0)I3 + (
0
+ 2 ∆2 (L, L) ∆˙(L, L) − ∆2 (0, 0) ∆˙(0, 0) − 2H10
Z L Z L
L2
,
(15B.20)
H12 =
ds
ds′ ˙∆ 2 (s, s′ ) ∆˙ 2 (s, s′ ) =
90
0
0
Z L Z L
I4
I12
H12
L2
H13 =
ds
ds′ ∆(s, s′ )˙∆ (s, s′ ) ∆˙(s, s′ )˙∆ d(s′ , s) =
−
−
=−
, (15B.21)
2
2L
2
720
0
0
Z L Z L
2(I3 − I12 )
I5
H14 =
ds
ds′ ∆2 (s, s′ )˙∆ d2 (s, s′ ) = δ(0)I3 −
− 2I4 + 2 ,
L
L
0
0
2
11L2
L3
δ(0) +
.
(15B.22)
+ 2 ∆ (L, L) ∆˙(L, L) − ∆2 (0, 0) ∆˙(0, 0) − 2H13 =
30
72
Appendix 15C
Integrals Involving Modified Green Function
To demonstrate the translational invariance of results showed in the main text we use the slightly
modified Green function
∆(s, s′ ) =
L
| s − s′ | (s + s′ ) (s2 + s′2 )
a−
−
+
,
3
2
2
2L
(15C.1)
containing an arbitrary constant a. The following combination yields the standard Feynman propagator for the infinite interval [compare (3.249)]
1
1
1
∆F (s, s′ ) = ∆F (s−s′ ) = ∆(s, s′ )− ∆(s, s)− ∆(s′ , s′ ) = − (s−s′ ).
2
2
2
Other useful relations fulfilled by the Green function (15C.1) are, assuming s ≥ s′ ,
(s−s′ )(s+s′ )2 La
′
2
′
′ ′
′ s(L−s)
D1 (s, s ) = ∆ (s, s )−∆(s, s)∆(s , s ) = (s−s )
,
−
+
L
4L2
3
(s−s′ )(s+s′ −L)
.
D2 (s, s′ ) = ∆(s, s)−∆(s′ , s′ ) =
L
(15C.2)
(15C.3)
(15C.4)
The following integrals are needed:
Z L
(s4 +s′4 ) (2s3 +s′3 )
J1 (s, s′ ) =
dt ∆(t, t)˙∆ (t, s)˙∆ (t, s′ ) =
−
,
4L2
3L
0
(20a−9) 2
((a+3)s2 +as′2 ) sa
− L+
L ,
+
6
3
180
(15C.5)
H. Kleinert, PATH INTEGRALS
1093
Notes and References
L
(−2s4 +6s2 s′2 +3s′4 )
,
24L2
0
(3s3 −3s2 s′ −6ss′2 −s′3 ) (5s2 −12ss′ −4as′2 ) s′ a
(20a−3) 2
+
−
−
L+
L ,
12L
24
6
720
J2 (s, s′ ) =
Z
dt ˙∆ (t, t)∆(t, s)˙∆ (t, s′ ) =
(15C.6)
L
4
2 ′2
′4
2
′2
(s +6s s +s ) (s +3s )s
+
,
dt ∆(t, s)∆(t, s′ ) = −
60L
6
0
(5a2 −10a+6) 3
(s2 +s′2 )
L+
L .
−
6
45
J3 (s, s′ ) =
Z
These are the building blocks for other relations:
(d−1)
(d+1) 2
(d−1)(s−s′)
′
′
′
hf2 (q(s), q(s ))i =
D1 (s, s )−
D2 (s, s ) = −
2
4
4
2La (d−3)s−(d+1)s′ (d−1)s2 −(d+1)s′2 d(s−s′ )(s+s′ )2
,
+
−
+
×
3
2
L
2L2
(15C.7)
(15C.8)
and
1
1
K1 (s, s′ ) = J1 (s, s′ )− J1 (s, s)− J1 (s′ , s′ ),
2
2
′
(s+s
) (s2 +ss′ +s′2 )
La
′
,
−
+
= − (s−s )
6
4
6L
K2 (s, s′ ) = J2 (s, s′ )+J2 (s′ , s)−J2 (s, s)−J2 (s′ , s′ ),
1 (5s+7s′) (s+s′ )2
= − (s−s′ )2
,
−
+
4
12L
4L2
(s+2s′ ) (s+s′ )2
1
1
,
−
K3 (s, s′ ) = J3 (s, s′ )− J3 (s, s)− J3 (s′ , s′ ) = −(s−s′ )2
2
2
6
8L
1
1
(s−s′)2 (s+s′ −2L)2
K4 (s, s′ ) = ∆(0, s)∆(0, s′ )− ∆2 (0, s)− ∆2 (0, s′ ) = −
,
2
2
8L2
1
1
(s2 −s′2 )2
K5 (s, s′ ) = ∆(L, s)∆(L, s′ )− ∆2 (L, s)− ∆2 (L, s′ ) = −
.
2
2
8L2
(15C.9)
(15C.10)
(15C.11)
(15C.12)
(15C.13)
Notes and References
Random chains were first considered by
K. Pearson, Nature 77, 294 (1905)
who studied the related problem of a random walk of a drunken sailor.
A.A. Markoff, Wahrscheinlichkeitsrechnung, Leipzig 1912;
L. Rayleigh, Phil. Mag. 37, 3221 (1919);
S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
The exact solution of PN (R) was found by
L.R.G. Treloar, Trans. Faraday Soc. 42, 77 (1946);
K. Nagai, J. Phys. Soc. Japan, 5, 201 (1950);
M.C. Wang and E. Guth, J. Chem. Phys. 20, 1144 (1952);
C. Hsiung, A. Hsiung, and A. Gordus, J. Chem. Phys. 34, 535 (1961).
The path integral approach to polymer physics has been advanced by
S.F. Edwards, Proc. Phys. Soc. Lond. 85, 613 (1965); 88, 265 (1966); 91, 513 (1967); 92, 9 (1967).
See also
H.P. Gilles, K.F. Freed, J. Chem. Phys. 63, 852 (1975)
and the comprehensive studies by
K.F. Freed, Adv. Chem. Phys. 22,1 (1972);
1094
15 Path Integrals in Polymer Physics
K. Itō and H.P. McKean, Diffusion Processes and Their Simple Paths, Springer, Berlin, 1965.
An alternative model to stimulate stiffness was formulated by
A. Kholodenko, Ann. Phys. 202, 186 (1990); J. Chem. Phys. 96, 700 (1991)
exploiting the statistical properties of a Fermi field.
Although the physical properties of a stiff polymer are slightly misrepresented by this model, the
distribution functions agree quite well with those of the Kratky-Porod chain. The advantage of
this model is that many properties can be calculated analytically.
The important role of the upper critical dimension Duc = 4 for polymers was first pointed out by
R.J. Rubin, J. Chem. Phys. 21, 2073 (1953).
The simple scaling law hR2 i ∝ L6/(D+2) , D < 4 was first found by
P.J. Flory, J. Chem. Phys. 17, 303 (1949) on dimensional grounds.
The critical exponent ν has been deduced from computer enumerations of all self-avoiding polymer
configurations by
C. Domb, Adv. Chem. Phys. 15, 229 (1969);
D.S. Kenzie, Phys. Rep. 27, 35 (1976).
For a general discussion of the entire subject, in particular the field-theoretic aspects, see
P.G. DeGennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, N.Y., 1979.
The relation between ensembles of random chains and field theory is derived in detail in
H. Kleinert, Gauge Fields in Condensed Matter , World Scientific, Singapore, 1989,
Vol. I, Part I; Fluctuating Fields and Random Chains, World Scientific, Singapore, 1989
(http://www.physik.fu-berlin.de/~kleinert/b1).
The particular citations in this chapter refer to the publications
[1] The path integral (15.112) for stiff polymers was proposed by
O. Kratky and G. Porod, Rec. Trav. Chim. 68, 1106 (1949).
The lowest moments were calculated by
J.J. Hermans and R. Ullman, Physica 18, 951 (1952);
N. Saito, K. Takahashi, and Y. Yunoki, J. Phys. Soc. Japan 22, 219 (1967),
and up to order 6 by
R. Koyama, J. Phys. Soc. Japan, 34, 1029 (1973).
Still higher moments are found numerically by
H. Yamakawa and M. Fujii, J. Chem. Phys. 50, 6641 (1974),
and in a large-stiffness expansion in
T. Norisuye, H. Murakama, and H. Fujita, Macromolecules 11, 966 (1978).
The last two papers also give the end-to-end distribution for large stiffness.
[2] H.E. Stanley, Phys. Rev. 179, 570 (1969).
[3] H.E. Daniels, Proc. Roy. Soc. Edinburgh 63A, 29 (1952);
W. Gobush, H. Yamakawa, W.H. Stockmayer, and W.S. Magee, J. Chem. Phys. 57, 2839
(1972).
[4] B. Hamprecht and H. Kleinert, J. Phys. A: Math. Gen. 37, 8561 (2004) (ibid.http/345).
Mathematica program can be downloaded from ibid.http/b5/prm15.
[5] The Mathematica notebook can be
from ibid.http/b5/pgm15. The program needs
obtained
less than 2 minutes to calculate R32 .
[6] J. Wilhelm and E. Frey, Phys. Rev. Lett. 77, 2581 (1996).
See also:
A. Dhar and D. Chaudhuri, Phys. Rev. Lett 89, 065502 (2002);
S. Stepanow and M. Schuetz, Europhys. Lett. 60, 546 (2002);
J. Samuel and S. Sinha, Phys. Rev. E 66 050801(R) (2002).
H. Kleinert, PATH INTEGRALS
Notes and References
1095
[7] H. Kleinert and A. Chervyakov, Perturbation Theory for Path Integrals of Stiff Polymers,
Berlin Preprint 2005 (ibid.http/358).
[8] For a detailed theory of such fields with applications to superconductors and superfluids see
H. Kleinert, Collective Quantum Fields, Fortschr. Phys. 26, 565 (1978) (ibid.http/55).
[9] S.F. Edwards and P.W. Anderson, J. Phys. F: Metal Phys. 965 (1975).
Applications of replica field theory to the large-L behavior of hR2 i and other critical exponents were given by
P.G. DeGennes, Phys. Lett. A 38, 339 (1972);
J. des Cloizeaux, Phys. Rev. 10, 1665 (1974); J. Phys. (Paris) Lett. 39 L151 (1978).
[10] See D.J. Amit, Renormalization Group and Critical Phenomena, World Scientific, Singapore, 1984; G. Parisi, Statistical Field Theory, Addison-Wesley, Reading, Mass., 1988.
[11] H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4 -Theories, World Scientific,
Singapore, 2000 (ibid.http/b8).
[12] For the calculation of such quantities within the φ4 -theory in 4 − ǫ dimensions up to order
ǫ5 , see
H. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin, and S.A. Larin, Phys. Lett. B
272, 39 (1991) (hep-th/9503230).
[13] For a comprehensive list of the many references on the resummation of divergent ǫexpansions obtained in the field-theoretic approach to critical phenomena, see the textbook
[11].
[14] A.J. McKane, Phys. Lett. A 76, 22 (1980).