65
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 line- like 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 ap- proximately 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 x b x a of such an object is given by P N (x b x a )= N n=1 d 3 Δx n 1 4πa 2 δ (|Δx n |− a) δ (3) (x b x a N n=1 Δx n ). (15.1) The last δ -function makes sure that the vectors Δx n of the chain elements add up correctly to the distance vector x b x a . 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 1/4πa 2 ensure the proper normalization of the individual one-link probabilities P 1 x)= 1 4πa 2 δ (|Δx|− a) (15.2) in the integral d 3 x b P 1 (x b x a )=1. (15.3) The same normalization holds for each N : d 3 x b P N (x b x a )=1. (15.4) 1031

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H. Kleinert, PATH INTEGRALSJanuary 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 descriptionpoint particles in spacetime. An important field of applications lies in polymerphysics where they are an ideal tool for studying the statistical fluctuations of line-like physical objects.

15.1 Polymers and Ideal Random Chains

A polymer is a long chain of many identical molecules connected with each otherat joints which allow for spatial rotations. A large class of polymers behaves ap-proximately like an idealized random chain. This is defined as a chain of N linksof a fixed length a, whose rotational angles occur all with equal probability (seeFig. 15.1). The probability distribution of the end-to-end distance vector xb−xa ofsuch an object is given by

PN(xb − xa) =N∏

n=1

[∫

d3∆xn1

4πa2δ(|∆xn| − a)

]

δ(3)(xb − xa −N∑

n=1

∆xn). (15.1)

The last δ-function makes sure that the vectors ∆xn of the chain elements add upcorrectly to the distance vector xb − xa. The δ-functions under the product enforcethe 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

4πa2δ(|∆x| − a) (15.2)

in the integral∫

d3xb P1(xb − xa) = 1. (15.3)

The same normalization holds for each N :∫

d3xb PN(xb − xa) = 1. (15.4)

1031

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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∑

n=1

∆xn

)

=∫

d3k

(2π)3eik(xb−xa)−ik

∑N

n=1∆xn , (15.5)

we see that PN (xb − xa) has the Fourier representation

PN(xb − xa) =∫

d3k

(2π)3eik(xb−xa)PN(k), (15.6)

with

PN(k) =N∏

n=1

[∫

d3∆xn1

4πa2δ(|∆xn| − a)e−ik∆xn

]

. (15.7)

Thus, the Fourier transform PN(k) factorizes into a product of N Fourier-transformed one-link probabilities:

PN(k) =[

P1(k)]N. (15.8)

These are easily calculated:

P1(k) =∫

d3∆x1

4πa2δ(|∆x| − a)e−ik∆x =

sin ka

ka. (15.9)

The desired end-to-end probability distribution is then given by the integral

PN(R) =∫

d3k

(2π)3[P1(k)]

NeikR

=1

2π2R

∫ ∞

0dkk sin kR

[

sin ka

ka

]N

, (15.10)

H. Kleinert, PATH INTEGRALS

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15.2 Moments of End-to-End Distribution 1033

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

SDaD−1δ(|∆x| − a), (15.12)

with SD being the surface of a unit sphere in D dimensions [see Eq. (1.558)].To calculate

P1(k) =∫

dD∆x1

SDaD−1δ(|∆x| − a)e−ik∆x, (15.13)

we insert for e−ik∆x = e−ik|∆x| cos∆ϑ the expansion (8.130) with (8.101), and useY0,0(x) = 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/2z) = e−iπ/2Jν(z), (15.14)

this gives

P1(k) =Γ(D/2)

(ka/2)D/2−1JD/2−1(ka), (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 rotationsso that the Fourier-transformed probability can only have even Taylor expansioncoefficients:

PN(k) = [P1(k)]N =

∞∑

l=0

PN,2l(ka)2l

(2l)!. (15.16)

The expansion coefficients provide us with a direct measure for the even momentsof the end-to-end distribution. These are defined by

〈R2l〉 ≡∫

dDRR2lPN(R). (15.17)

The relation between 〈R2l〉 and PN,2l is found by expanding the exponential underthe inverse of the Fourier integral (15.6):

PN (k) =∫

dDRe−ikRPN(R) =∞∑

n=0

dDR(−ikR)n

n!PN(R), (15.18)

1I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.406.1.

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1034 15 Path Integrals in Polymer Physics

and observing that the angular average of (kR)n is related to the average of Rn inD dimensions by

〈(kR)n〉 = kn〈Rn〉

0 , n = odd,(n− 1)!!(D − 2)!!

(D + n− 2)!!, n = even.

(15.19)

The three-dimensional result 1/(n+1) follows immediately from the angular average(1/2)

∫ 1−1 d cos θ cosn 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 distributionP

(0)N (R) = (D/2πNa2)3/2e−R2D/2Na2 . Then the expectation values of all products ofRi can be expressed in terms of the pair expectation value

〈RiRj〉(0) =1

Dδij a

2N (15.20)

via Wick’s rule (3.305). The result is

〈Ri1Ri2 · · ·Rin〉(0) =1

Dn/2δi1i2i3...in a

nNn/2, (15.21)

with the contraction tensor δi1i2i3...in of Eqs. (8.64) and (13.89), which has the re-cursive definition

δi1i2i3...in = δi1i2δi3i4...in + δi1i3δi2i4...in + . . . δi1inδi2i3...in−1 . (15.22)

A full contraction of the indices gives, for even n, the Gaussian expectation values:

〈Rn〉(0) = (D + n− 2)!!

(D − 2)!!Dn/2anNn/2 =

Γ(D/2 + n/2)

Γ(D/2)

2n/2

Dn/2anNn/2, (15.23)

for instance

R4⟩(0)

=(D + 2)

Da4N2,

R6⟩(0)

=(D + 2) (D + 4)

D2a6N3. (15.24)

By contracting (15.21) with ki1ki2 · · · kin we find

〈(kR)n〉(0) = (n− 1)!!1

Dn/2(ka)nNn/2 = kn 〈(R)n〉(0) dn, (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 particu-lar for PN(R), thus proving Eq. (15.17) for random chains. Hence, the expansioncoefficients PN,2l are related to the moments 〈R2l〉N by

PN,2l = (−1)ld2l〈R2l〉, (15.27)

H. Kleinert, PATH INTEGRALS

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15.2 Moments of End-to-End Distribution 1035

and the moment expansion (15.16) becomes

PN(k) =∞∑

l=0

(−1)l(k)2l

(2l)!d2l 〈R2l〉. (15.28)

Let us calculate the even moments 〈R2l〉 of the polymer distribution PN(R)explicitly for D = 3. We expand the logarithm of the Fourier transform PN(k) asfollows:

log PN(k) = N log P1(k) = N log

(

sin ka

ka

)

= N∞∑

l=1

22l(−1)lB2l

(2l)!2l(ka)2l, (15.29)

where Bl are the Bernoulli numbers B2 = 1/6, B4 = −1/30, . . . . Then we note thatfor a Taylor series of an arbitrary function y(x)

y(x) =∞∑

n=1

ann!xn, (15.30)

the exponential function ey(x) has the expansion

ey(x) =∞∑

n=1

bnn!xn, (15.31)

with the coefficients

bnn!

=∑

{mi}

n∏

i=0

1

mi!

(

aii!

)mi

. (15.32)

The sum over the powers mi = 0, 1, 2, . . . obeys the constraint

n =n∑

i=1

i ·mi. (15.33)

Note that the expansion coefficients an of y(x) are the cumulants of the expansioncoefficients bn of ey(x) as defined in Section 3.17. For the coefficients an of theexpansion (15.29),

an =

{

−N22l(−1)lB2l/2l for n = 2l,0 for n = 2l + 1,

(15.34)

we find, via the relation (15.27), the moments

〈R2l〉 = a2l(−1)l(2l + 1)!∑

{mi}

l∏

i=1

1

mi!

[

N22i(−1)iB2i

(2i)!2i

]mi

, (15.35)

with the sum over mi = 0, 1, 2, . . . constrained by

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1036 15 Path Integrals in Polymer Physics

l =l∑

i=1

i ·mi. (15.36)

For l = 1 and 2 we obtain the moments

〈R2〉 = a2N, 〈R4〉 = 5

3a4N2

(

1− 2

5N

)

. (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 〈R2〉 with the number of links N is characteristicfor a random chain. In the presence of interactions, there will be a different powerbehavior expressed as a so-called scaling law

〈R2〉 ∝ a2N2ν . (15.38)

The number ν is called the critical exponent of this scaling law. It is intuitivelyobvious 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 determinescompletely the shape of the distribution by an expansion

PL(R) =1

SDRD−1

∞∑

n=0

〈Rn〉 (−1)n

n!∂nRδ(R). (15.39)

This can easily be verified by calculating the integrals (15.17) using the integralsformula

dz zn∂nz δ(z) = (−1)nn! . (15.40)

15.3 Exact End-to-End Distribution in Three Dimensions

Consider the Fourier representation (15.10), rewritten as

PN(R) =i

4π2a2R

∫ ∞

−∞dη η e−iηR/a

(

sin η

η

)N

, (15.41)

with the dimensionless integration variable η ≡ ka. By expanding

sinN η =1

(2i)N

N∑

n=0

(−1)n(

N

n

)

exp [i(N − 2n)η] , (15.42)

we find the finite series

PN(R) =1

2N+2iN−1π2a2R

N∑

n=0

(−1)n(

N

n

)

IN(N − 2n− R/a), (15.43)

H. Kleinert, PATH INTEGRALS

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15.3 Exact End-to-End Distribution in Three Dimensions 1037

where IN(x) are the integrals

IN(x) ≡∫ ∞

−∞dη

eiηx

ηN−1. (15.44)

For N ≥ 2, these integrals are all singular. The singularity can be avoided by notingthat the initial integral (15.41) is perfectly regular at η = 0. We therefore replace theexpression (sin η/η)N in the integrand by [sin(η − iǫ)/(η − iǫ)]N . This regularizeseach term in the expansion (15.43) and leads to well-defined integrals:

IN(x) =∫ ∞

−∞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 lowerhalf-plane. Since the lower half-plane contains no singularity, the residue theoremshows that

IN(x) = 0, x < 0. (15.46)

For x > 0, on the other hand, an expansion of the exponential function in powersof η−iǫ produces a pole, and the residue theorem yields

IN(x) =2πiN−1

(N − 2)!xN−2, x > 0. (15.47)

Hence we arrive at the finite series

PN(R) =1

2N+1(N − 2)!πa2R

0≤n≤(N−R/a)/2

(−1)n(

N

n

)

(N − 2n− R/a)N−2. (15.48)

The distribution is displayed for various values of N in Fig. 15.2, where we haveplotted 2πR2

√NPN(R) against the rescaled distance variable ρ = R/

√Na. With

this N -dependent rescaling all curves have the same unit area. Note that theyconverge rapidly towards a universal zero-order distribution

P(0)N (R) =

3

2πNa2

3

exp

{

− 3R2

2Na2

}

→ P(0)L (R) =

D

2πaL

D

e−DR2/2aL. (15.49)

In the limit of large N , the length L will be used as a subscript rather than thediverging N . The proof of the limit is most easily given in Fourier space. Forlarge N at finite k2a2N , the Nth power of the quantity P1(k) in Eq. (15.15) can beapproximated by

[P1(k)]N ∼ e−Nk

2a2/2D. (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/

√Na.

The approach to this limit is studied analytically in the following two sections.

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1038 15 Path Integrals in Polymer Physics

Figure 15.2 End-to-end distribution PN (R) of random chain with N links. The func-

tions 2πR2√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-

bution P(0)L (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) = P(0)N (k)

[

1 +∞∑

n=1

1

NnCn(R

2/Na2)

]

, (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 k2-term and a remainder

C(k) ≡ exp

[

N∞∑

l=2

22l(−1)lB2l

(2l)!2l(k2a2)l

]

. (15.52)

Exponentiating both sides of (15.29), the end-to-end probability factorizes as

PN(k) = e−Na2k2/6C(k). (15.53)

The function C(k) is now expanded in a power series

C(k) = 1 +∑

n=1,2,...l=2n,2n+1,...

Cn,lNn(a2k2)l, (15.54)

H. Kleinert, PATH INTEGRALS

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15.4 Short-Distance Expansion for Long Polymer 1039

with the lowest coefficients

C1,2 = − 1

180, C1,3 = − 1

2835,

C1,4 = − 1

37800, C2,4 =

1

64800, . . . . (15.55)

For any dimension D, we factorize

PN(k) = e−Na2k2/2DC(k) (15.56)

and find C(k) by expanding (15.15) in powers of k and proceeding as before. Thisgives the coefficients

C1,2 = −1/4D2(D + 2),

C1,3 = −1/3D3(D + 2)(D + 4),

C1,4 = −(5D + 12)/8D4(D + 2)2(D + 4)(D + 6),

C2,4 = 1/32D4(D + 2)2. (15.57)

We now Fourier-transform (15.56). The leading term in C(k) yields the zero-orderdistribution (15.49) in D dimensions,

P(0)N (R) =

D

2πNa2

D

e−DR2/2Na2 , (15.58)

or, written in terms of the reduced distance variable ρ = R/Na,

P(0)N (R) =

D

2πNa2

D

e−DNρ2/2. (15.59)

To account for the corrections in C(k) we take the expansion (15.54), emphasize thedependence on k2a2 by writing

C(k) = C(k2a2),

and observe that in the Fourier transform

PN(R) =∫ d3k

(2π)3eikRPN(k) =

∫ d3k

(2π)3eikRe−Nk

2a2/2DC(k2a2), (15.60)

the series can be pulled out of the integral by replacing each power (k2a2)p by(−2D∂N )

p. The result has the form

PN(R) = C(−2D∂N)∫

d3k

(2π)3eikRe−Nk

2a2/2D = C(−2D∂N)P(0)N (R). (15.61)

Going back to coordinate space, we obtain an expansion

PN(R) =(

D

2πNa2

)D/2

e−DNρ2/2C(R). (15.62)

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1040 15 Path Integrals in Polymer Physics

For D = 3, the function C(R) is given by the series

C(R) = 1 +∞∑

n=1l=0,...,2n

Cn,lN−n(Nρ2)l, (15.63)

with the coefficients

C1,l =(

−3

4,3

2,− 9

20

)

, C2,l =(

29

160,−69

40,981

400,−1341

1400,81

800

)

. (15.64)

For any D, we find the coefficients

C1,l =

(

−D4,D

2,− D2

4(D + 2)

)

,

C2,l =

(

(3D2 − 2D + 8)D

96(D + 2),−(D2 + 2D + 8)D

8(D + 2),(3D2 + 14D + 40)D2

16(D + 2)2,

− (3D2 + 22D + 56)D3

24(D + 2)2(D + 4),

D4

32(D + 2)2

)

. (15.65)

15.5 Saddle Point Approximation to Three-Dimensional

End-to-End Distribution

Another study of the approach to the limiting distribution (15.49) proceeds via thesaddle point approximation. For this, the integral in (15.41) is rewritten as

∫ ∞

−∞dη η e−Nf(η), (15.66)

with

f(η) = iR

Naη − log

(

sin η

η

)

. (15.67)

The extremum of f lies at η = η, where η solves the equation

coth(iη)− 1

iη=

R

Na. (15.68)

The function on the left-hand side is known as the Langevin function:

L(x) ≡ coth x− 1

x. (15.69)

The extremum lies at the imaginary position η ≡ −ix with x being determined bythe equation

L(x) =R

Na. (15.70)

H. Kleinert, PATH INTEGRALS

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15.6 Path Integral for Continuous Gaussian Distribution 1041

The extremum is a minimum of f(η) since

f ′′(η) = L′(x) = − 1

sinh2 x+

1

x2> 0. (15.71)

By shifting the integration contour vertically into the complex η plane to make itrun through the minimum at −ix, we obtain

PN(R) ≈ − 1

4iπ2a2Re−Nf(η)

∫ ∞

−∞dη (−ix+ η) exp

{

−N2f ′′(η)η2

}

4π2a2R

Nf ′′(x)e−Nf(η). (15.72)

When expressed in terms of the reduced distance ρ ≡ R/Na ∈ [0, 1], this reads

PN(R) ≈ 1

(2πNa2)3/2Li(ρ)2

ρ {1− [Li(ρ)/ sinhLi(ρ)]2}1/2{

sinhLi(ρ)

Li(ρ) exp[ρLi(ρ)]

}N

. (15.73)

Here we have introduced the inverse Langevin function Li(ρ) since it allows us toexpress x as

x = 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 expandingthe right-hand side in a power series in ρ, we find

PN(R) = N(

3

2πNa2

)3/2

exp

(

− 3R2

2Na2

)(

1 +3R2

2N2a2− 9R4

20N3a4+ . . .

)

, (15.75)

with some normalization constant N . At each order of truncation, N is determinedin such a way that

d3R PN(R)= 1. As a check we take the limit ρ2 ≪ 1/N andfind 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 ampli-tude of a free particle in natural units with h = 1:

(xbτb|xaτa) =1

2π(τb − τa)/MD exp

[

−M2

(xb − xa)2

τb − τa

]

. (15.76)

We merely have to identify

xb − xa ≡ R, (15.77)

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1042 15 Path Integrals in Polymer Physics

and replace

τb − τa → Na, (15.78)

M → D/a. (15.79)

Thus we can describe a polymer with R2 ≪ Na2 by the path integral

PL(R) =∫

DDx exp

{

−D

2a

∫ L

0ds [x′(s)]2

}

=

D

2πae−DR

2/2La. (15.80)

The number of time slices is here N [in contrast to (2.66) where it was N + 1), andthe total length of the polymer is L = Na.

Let us calculate the Fourier transformation of the distribution (15.80):

PL(q) =∫

dDRe−iq·RPL(R). (15.81)

After a quadratic completion the integral yields

PL(q) = e−Laq2/2D, (15.82)

with the power series expansion

PL(q) =∞∑

l=0

(−1)lq2l

l!

(

La

2D

)l

. (15.83)

Comparison with the moments (15.23) shows that we can rewrite this as

PL(q) =∞∑

l=0

(−1)lq2l

l!

Γ(D/2)

22lΓ(D/2 + l)

R2l⟩

. (15.84)

This is a completely general relation: the expansion coefficients of the Fourier trans-form yield directly the moments of a function, up to trivial numerical factors specifiedby (15.84).

The end-to-end distribution determines rather directly the structure factor of adilute solution of polymers which is observable in static neutron and light scatteringexperiments:

S(q)=1

L2

∫ L

0ds∫ L

0ds′

eiq·[x(s)−x(s′)]⟩

. (15.85)

The average over all polymers running from x(0) to x(L) can be written, moreexplicitly, as

eiq·[x(s)−x(s′)]⟩

=∫

dDx(L)∫

dD(x(s′)−x(s))∫

dDx(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

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15.6 Path Integral for Continuous Gaussian Distribution 1043

The integrals over initial and final positions give unity due to the normalization(15.4), so that we remain with

eiq·[x(s)−x(s′)]⟩

=∫

dDRe−iq·RPs′−s(R). (15.87)

Since this depends only on L′ ≡ |s′ − s| and not on s+ s′, we decompose the doubleintegral in over s and s′ in (15.85) into 2

∫ L0 dL

′(L− L′) and obtain

S(q)=2

L2

∫ L

0dL′(L− L′)

dDReiq·R(L′)PL′(R), (15.88)

or, recalling (15.81),

S(q) =2

L2

∫ L

0dL′(L− L′)PL′(q). (15.89)

Inserting (15.82) we obtain the Debye structure factor of Gaussian random paths:

SGauss(q) =2

x2

(

x− 1 + e−x)

, x ≡ q2aL

2D. (15.90)

This function starts out like 1 − x/3 + x2/12 + . . . for small q and falls of like q−2

for q2 ≫ 2D/aL. The Taylor coefficients are determined by the moments of theend-to-end distribution. By inserting (15.84) into (15.89) we obtain:

S(q) =∞∑

l=0

(−1)lq2lΓ(D/2)

22ll!Γ(l +D/2)

2

L2

∫ L

0dL′(L− L′)

R2l⟩

. (15.91)

Although the end-to-end distribution (15.80) agrees with the true polymer dis-tribution (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, thelength of each link ∆xn is fixed. In the sliced action of the path integral Eq. (15.80),

AN = aN∑

n=1

M

2

(∆xn)2

a2, (15.92)

on the other hand, each small section fluctuates around zero with a mean square

〈(∆xn)2〉0 =

a

M=a2

D. (15.93)

Yet, if the end-to-end distance of the polymer is small compared to the completelystretched configuration, the distributions are practically the same. There exists aqualitative difference only if the polymer is almost completely stretched. While thepolymer distribution vanishes for R > Na, the path integral (15.80) gives a nonzerovalue for arbitrarily large R. Quantitatively, however, the difference is insignificantsince it is exponentially small (see Fig. 15.2).

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1044 15 Path Integrals in Polymer Physics

15.7 Stiff Polymers

The end-to-end distribution of real polymers found in nature is never the same asthat of a random chain. Usually, the joints do not allow for an equal probabilityof all spherical angles. The forward angles are often preferred and the polymer isstiff at shorter distances. Fortunately, if averaged over many links, the effects of thestiffness becomes less and less relevant. For a very long random chain with a finitestiffness one finds the same linear dependence of the square end-to-end distance onthe length L = Na as for ideal random chains which has, according to Eq. (15.23),the Gaussian expectations:

〈R2〉 = aL, 〈R2l〉 = (D + 2l − 2)!!

(D − 2)!!Dl(aL)l. (15.94)

For a stiff chain, the expectation value 〈R2〉 will increase aL to aeffL, where aeff isthe effective bond length In the limit of a very large stiffness, called the rod limit,the law (15.94) turns into

〈R2〉 ≡ L2, 〈R2l〉 ≡ L2l, (15.95)

i.e., the effective bond length length aeff increases to L. This intuitively obviousstatement can easily be found from the normalized end-to-end distribution, whichcoincides in the rod limit with the one-link expression (15.12):

P rodL (R) =

1

SDRD−1δ(R− L), (15.96)

and yields [recall (15.17)]

〈Rn〉 =∫

dDRRnP rodL (R) =

∫ ∞

0dRRn δ(R− L) = Ln. (15.97)

By expanding P rodL (R) in powers of L, we obtain the series

P rodL (R) =

1

SDRD−1

∞∑

n=0

Ln 〈Rn〉 (−1)n

n!∂nRδ(R). (15.98)

An expansion of this form holds for any stiffness: the moments of the distributionare the Taylor coefficients of the expansion of PL(R) into a series of derivatives ofδ(R)-functions.

Let us also calculate the Fourier transformation (15.81) of this distribution. Re-calling (15.15), we find

P rodL (q) = P rod(qL) ≡ Γ(D/2)

(qL/2)D/2−1JD/2−1(qL). (15.99)

For an arbitrary rotationally symmetric PL(R) = PL(R), we simply have to super-impose these distributions for all R:

PL(q) = SD

∫ ∞

0dRRD−1P rod(qL)PL(R). (15.100)

H. Kleinert, PATH INTEGRALS

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15.7 Stiff Polymers 1045

This is simply proved by decomposing and performing the Fourier transformation(15.81) on PL(R) =

∫∞0 dR′ δ(R−R′)P rod

R′ (R)=SD∫∞0 dR′R′D−1P rod

R′ (R)PL(R′), and

performing the Fourier transformation (15.81) on P rodR′ (R). In D = 3 dimensions,

(15.100) takes the simple form:

PL(q) = 4π∫ ∞

0dRR2 sin qR

qRPL(R). (15.101)

Inserting the power series expansion for the Bessel function2

Jν(z) =(

z

2

)ν ∞∑

l=0

(−1)k(z/2)2l

l!Γ(ν + l + 1)(15.102)

into (15.99), and this into (15.100), we obtain

PL(q) =∞∑

l=0

(−1)l(

q

2

)2l Γ(D/2)

l!Γ(D/2 + l)SD

∫ ∞

0dRRD−1R2lPL(R), (15.103)

in agreement with the general expansion (15.84). The same result is ob-tained by inserting into (15.103) the expansion (15.98) and using the integrals∫∞0 dRRm∂nRδ(R) = δmn(−1)nn!, which are proved by n partial integrations.

The structure factor of a completely stiff polymer (rod limit) is obtained byinserting (15.99) into (15.89). The resulting Srod(q) depends only on qL:

Srod(qL)=4−2D

q2L2+

(

2

qL

)D/2

Γ(D/2)JD/2−2(qL)+2F (1/2; 3/2, D/2;−q2L2/4),(15.104)

where F (a; b, c; z) is the hypergeometric function (1.453). For D = 3, the integral(15.89) reduces to (2/L2)

∫ L0 dL

′ (L − L′)(sin qL′)/qL′, as in the similar equation(15.9), and the result is simply

Srod(z) =2

z2[cos z − 1 + z Si(z)] , Si(z) ≡

∫ z

0

dt

tsin t. (15.105)

This starts out like 1 − z2/36 + z4/1800 + . . . . For large z we use the limit of thesine integral3 Si(z) → π/2 to find Srod(q) → π/qL.

For of an arbitrary rotationally symmetric end-to-end distribution PL(R), thestructure factor can be expressed, by analogy with (15.100), as a superposition ofrod limits:

SL(q) = SD

∫ ∞

0dRRD−1Srod(qR)PL(R). (15.106)

When passing from long to short polymers at a given stiffness, there is a crossoverbetween the moments (15.94) and (15.95) and the behaviors of the structure func-tion. Let us study this in detail.

2I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.440.3ibid., Formulas 8.230 and 8.232.

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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 bendingenergy

ENbend =

κ

2a

N∑

n=1

(un − un−1)2, (15.107)

where un are the unit vectors specifying the directions of the links. The initial andfinal link directions of the polymer have a distribution

(ubL|ua0) =1

A

N−1∏

n=1

[

dunA

]

exp

[

− κ

2akBT

N∑

n=1

(un − un−1)2

]

, (15.108)

where A is some normalization constant, which we shall choose such that the measureof integration coincides with that of a time-sliced path integral near a unit sphere inEq. (8.151). Comparison if the bending energy (15.107) with the Euclidean action(8.152) we identify

Mr2

hǫ=

κ

akBT, (15.109)

and see that we must replace N → N − 1 and set

A =√

2πakBT/κD−1

. (15.110)

The result of the integrations in (15.108) is then known from Eq. (8.156):

(ubL|ua0) =∞∑

l=0

[

Il+D/2−1 (h)]N ∑

m

Ylm(ub)Y∗lm(ua), h ≡ κ

akBT, (15.111)

with the modified Bessel function Il+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]:

ZN =∫

duaSD

N∏

n=1

[

dunA

]

exp

[

− κ

2akBT

N∑

n=1

(un − un−1)2

]

=∫

dub

duaSD

(ubL|ua0). (15.112)

Inserting here the spectral representation (15.111) we find

ZN =[

ID/2−1

(

κ

akBT

)]N

=

[√

2πκ

akBTe−κ/akBT ID/2−1

(

κ

akBT

)

]N

. (15.113)

Knowing this we may define the normalized distribution function

PN(ub,ua) =1

ZN(ubL|ua0), (15.114)

whose integral over ua as well as over ua is unity:∫

dub PN(ub,ua) =∫

dua PN(ub,ua) = 1. (15.115)

H. Kleinert, PATH INTEGRALS

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15.7 Stiff Polymers 1047

15.7.2 Relation to Classical Heisenberg Model

The above partition function is closely related to the partition function of the one-dimensional classical Heisenberg model of ferromagnetism which is defined by

ZHeisN ≡

duaSD

N∏

n=1

[∫

dun

]

exp

[

J

kBT

N∑

n=1

un · un−1

]

, (15.116)

where J are interaction energies due to exchange integrals of electrons in a ferro-magnet. This differs from (15.112) by a trivial normalization factor, being equalto

ZHeisN =

2πJ

kBT

2−D

ID/2−1

(

J

kBT

)

N

. (15.117)

Identifying J ≡ κ/a we may use the Heisenberg partition functions for all cal-culations of stiff polymers. As an example take the correlation function betweenneighboring tangent vectors 〈un · un−1〉. In order to calculate this we observe thatthe partition function (15.116) can just as well be calculated exactly with a slightmodification that the interaction strength J of the Heisenberg model depends onthe link n. The result is the corresponding generalization of (15.117):

ZHeisN (J1, . . . , JN) =

N∏

n=1

2πJnkBT

2−D

ID/2−1

(

JnkBT

)

. (15.118)

This expression may be used as a generating function for expectation values〈un · un−1〉 which measure the degree of alignment of neighboring spin directions.Indeed, we find directly

〈un · un−1〉 = (kBT )dZHeis

N (J1, . . . , JN)

dJn

Jn≡J

=ID/2(J/kBT )

ID/2−1(J/kBT ). (15.119)

This expectation value measures directly the internal energy per link of te chain.Indeed, since the free energy is FN = −kBT logZHeis

N , we obtain [recall (1.548)]

EN = N 〈un · un−1〉 = NID/2(J/kBT )

ID/2−1(J/kBT ). (15.120)

Let us also calculate the expectation value of the angle between next-to-nearestneighbors 〈un+1 · un−1〉. We do this by considering the expectation value

〈(un+1 · un)(un · un−1)〉 = (kBT )2 d2ZHeis

N (J1, . . . , JN)

dJn+1dJn

Jn≡J

=

[

ID/2(J/kBT )

ID/2−1(J/kBT )

]2

.

(15.121)

Then we prove that the left-hand side is in fact equal to the desired expectationvalue 〈un+1 · un−1〉. For this we decompose the last vector un+1 into a component

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1048 15 Path Integrals in Polymer Physics

θn+1,n−1

θn+1,n

un−1

unun+1

θn,n−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 〈un+1 ·un−1〉 is 〈un+1 ·un−1〉 =〈(un+1·un)(un·un−1)〉+〈u⊥

n+1·un〉. Now, the energy in the Boltzmann factor dependsonly on (un+1 · un) + (un · un−1) = cos θn+1,n + cos θn,n−1, so that the integral overu⊥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 integralreceives therefore equal contributions from u⊥

n+1 and −u⊥n+1, and vanishes. This

proves that

〈un+1 · un−1〉 = 〈un+1 · un〉 〈un · un−1〉 =[

ID/2(J/kBT )

ID/2−1(J/kBT )

]2

, (15.122)

and further, by induction, that

〈ul · uk〉 =[

ID/2(J/kBT )

ID/2−1(J/kBT )

]|l−k|

. (15.123)

For the polymer, this implies an exponential falloff

〈ul · uk〉 = e−|l−k|a/ξ, (15.124)

where ξ is the persistence length

ξ = −a/ log[

ID/2(κ/akBT )

ID/2−1(κ/akBT )

]

. (15.125)

For D = 3, this is equal to

ξ = − a

log [coth(κ/akBT )− akBT/κ]. (15.126)

Knowing the correlation functions it is easy to calculate the magnetic suscepti-bility. The total magnetic moment is

M = aN∑

n=0

un, (15.127)

so that we find the total expectation value

M2⟩

= a2(N + 1)1 + e−a/ξ

1− e−a/ξ− 2a2e−a/ξ

1− e−(N+1)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

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15.7 Stiff Polymers 1049

15.7.3 End-to-End Distribution

A modification of the path integral (15.108) yields the distribution of the end-to-enddistance at given initial and final directions of the polymer links:

R = xb − xa = aN∑

n=1

un (15.129)

of the stiff polymer:

PN(ub,ua;R) =1

ZN

1

A

N−1∏

n=2

[

dunA

]

δ(D)(R− aN∑

n=1

un)

× exp

[

− κ

2akBT

N−1∑

n=1

(un+1 − un)2

]

, (15.130)

whose integral over R leads back to the distribution PN(ub,ua):

dDRPN(ub,ua;R) = PN(ub,ua). (15.131)

If we integrate in (15.130) over all final directions and average over the initialones, we obtain the physically more accessible end-to-end distribution

PN(R) =∫

dub

duaSD

PN(ub,ua;R). (15.132)

The sliced path integral, as it stands, does not yet give quite the desired prob-ability. Two small corrections are necessary, the same that brought the integralnear the surface of a sphere to the path integral on the sphere in Section 8.9. Afterincluding 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 calculationof the complete distribution is not straightforward. Moments of the distribution,however, which are defined by the integrals

〈R2l〉 =∫

dDRR2l PN(R), (15.133)

are relatively easy to find from the multiple integrals

〈R2l〉 =1

ZN

dDR1

A

dubN−1∏

n=2

[

dunA

] [

duaSD

]

δ(D)(R−N∑

n=1

aun)

× R2l exp

[

− κ

2akBT

N−1∑

n=1

(un+1 − un)2

]

. (15.134)

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1050 15 Path Integrals in Polymer Physics

Performing the integral over R gives

〈R2l〉 = 1

ZN

1

A

dubN−1∏

n=2

[

dunA

] [

duaSD

](

aN∑

n=1

un

)2l

× exp

[

− κ

2akBT

N−1∑

n=1

(un+1 − un)2

]

. (15.135)

Due to the normalization property (15.115), the trivial moment is equal to unity:

〈1〉 =∫

dDRPN(R) =∫

dub

duaSD

PN(ub,ua|L) = 1. (15.136)

15.8 Continuum Formulation

Some properties of stiff polymers are conveniently studied in the continuum limit ofthe sliced path integral (15.108), in which the bond length a goes to zero and thelink number to infinity so that L = Na stays constant. In this limit, the bendingenergy (15.107) becomes

Ebend =κ

2

∫ L

0ds (∂su)

2 , (15.137)

where

u(s) =d

dsx(s), (15.138)

is the unit tangent vector of the space curve along which the polymer runs. Theparameter 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

(ubL|ua0) =∫

Du e−(κ/2kBT )∫ L

0ds [u′(s)]2, (15.139)

This coincides with the Euclidean version of a path integral for a particle on thesurface 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 thatit does not quite agree with what we would obtain from the continuum limit of thediscrete solution (15.111) using the limiting formula (8.157), which is

P (ub,ua|L) =∞∑

l=0

exp

(

−LkBT2κ

L2

)

m

Ylm(ub)Y∗lm(ua), (15.140)

H. Kleinert, PATH INTEGRALS

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15.8 Continuum Formulation 1051

with

L2 = (D/2− 1 + l)2 − 1/4. (15.141)

For a particle on a sphere, the discrete expression (15.130) requires a correctionsince it does not contain the proper time-sliced action and measure. Accordingthe Section 8.9 the correction replaces L2 by the eigenvalues of the square angularmomentum operator in D dimensions, L2,

L2 → L2 = l(l +D − 2). (15.142)

After this replacement the expectation of the trivial moment 〈1〉 in (15.136) is equalto unity since it gives the distribution (15.140) the proper normalization:

dub P (ub,ua|L) = 1. (15.143)

This follows from the integral∫

dub∑

m

Y ∗lm(ub)Ylm(ua) = δl0 , (15.144)

which was derived in (8.250). Thus, with L2 in (15.140) instead of L2, no extranormalization factor is required. The sum over Ylm(u2)Y

∗lm(u1) may furthermore be

rewritten in terms of Gegenbauer polynomials using the addition theorem (8.126),so that we obtain

P (ub,ua|L) =∞∑

l=0

exp

(

−LkBT2κ

L2

)

1

SD

2l +D − 2

D − 2C

(D/2−1)l (u2u1). (15.145)

15.8.2 Correlation Functions and Moments

We are now ready to evaluate the expectation values of R2l. In the continuumapproximation we write

R2l =

[

∫ L

0dsu(s)

]2l

. (15.146)

The expectation value of the lowest moment 〈R2〉 is given by the double-integralover the correlation function 〈u(s2)u(s1)〉:

〈R2〉 =∫ L

0ds2

∫ L

0ds1〈u(s2)u(s1)〉 = 2

∫ L

0ds2

∫ s2

0ds1〈u(s2)u(s1)〉. (15.147)

The correlation function is calculated from the path integral via the compositionlaw as in Eq. (3.301), which yields here

〈u(s2)u(s1)〉 =∫

dub

duaSD

du2

du1

× P (ub,u2|L− s2) u2 P (u2,u1|s2 − s1) u1 P (u1,ua|s1).(15.148)

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1052 15 Path Integrals in Polymer Physics

The integrals over ua and ub remove the initial and final distributions via the nor-malization integral (15.143), leaving

〈u(s2)u(s1)〉 =∫

du2

du1

SDu2u1P (u2,u1|s2 − s1). (15.149)

Due to the manifest rotational invariance, the normalized integral over u1 can beomitted. By inserting the spectral representation (15.145) with the eigenvalues(15.142) we obtain

〈u(s2)u(s1)〉 =∫

du2 u2u1P (u2,u1|s2 − s1)

=∑

l

e−(s2−s1)kBT L2/2κ

[

du2 u2u11

SD

2l +D − 2

D − 2C

(D/2−1)l (u2u1)

]

. (15.150)

We now calculate the integral in the brackets with the help of the recursion relation(15.152) for the Gegenbauer functions4

zC(ν)l (z) =

1

2(ν + l)

[

(2ν + l − 1)C(ν)l−1(z) + (l + 1)Cν

l+1(z)]

. (15.151)

Obviously, the integral over u2 lets only the term l = 1 survive. This, in turn,involves the integral

du2 C(D/2−1)0 (cos θ) = SD. (15.152)

The l = 1 -factor D/(D − 2) in (15.150) is canceled by the first l = 1 -factor in therecursion (15.151) and we obtain the correlation function

〈u(s2)u(s1)〉 = exp

[

−(s2 − s1)kBT

2κ(D − 1)

]

, (15.153)

where D − 1 in the exponent is the eigenvalue of L2 = l(l + D − 2) at l = 1. Thecorrelation function (15.153) agrees with the sliced result (15.124) if we identify thecontinuous version of the persistence length (15.126) with

ξ ≡ 2κ/kBT (D − 1). (15.154)

Indeed, taking the limit a→ 0 in (15.125), we find with the help of the asymptoticbehavior (8.12) precisely the relation (15.154).

After performing the double-integral in (15.147) we arrive at the desired resultfor the first moment:

〈R2〉 = 2{

ξL− ξ2[

1− e−L/ξ]}

. (15.155)

4I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.933.1.

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15.8 Continuum Formulation 1053

This is valid for all D. The result may be compared with the expectation valueof the squared magnetic moment of the Heisenberg chain in Eq. (15.128), whichreduces 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

〈R2〉 = L2

1− 1

3

L

ξ+

1

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)]. Forlarge L/ξ, on the other hand, we find the small-stiffness expansion

〈R2〉 ≈ 2ξL

(

1− ξ

L

)

+ . . . , (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 =4

D − 1

κ

kBT. (15.158)

The calculation of higher expectations 〈R2l〉 becomes rapidly complicated. Take,for instance, the moment 〈R4〉, which is given by the quadruple integral over thefour-point correlation function

〈R4〉 = 8∫ L

0ds4

∫ s4

0ds3

∫ s3

0ds2

∫ s2

0ds1 δi4i3i2i1〈ui4(s4)ui3(s3)ui2(s2)ui1(s1)〉, (15.159)

with the symmetric pair contraction tensor of Eq. (15.22):

δi4i3i2i1 ≡ (δi4i3δi2i1 + δi4i2δi3i1 + δi4i1δi3i2) . (15.160)

The factor 8 and the symmetrization of the indices arise when bringing the integral

R4 =∫ L

0ds4

∫ L

0ds3

∫ L

0ds2

∫ L

0ds1 (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 evaluationof the u-integrals which proceeds by a direct extension of the previous procedure for〈R2〉. We write down the extension of expression (15.148) and perform the integralsover ua and ub which remove the initial and final distributions via the normalizationintegral (15.143), leaving δi4i3i2i1 times an integral [the extension of (15.149)]:

〈ui4(s4)ui3(s3)ui2(s2)ui1(s1)〉 =∫

du4

du3

du2

du1

SD(15.162)

× ui4ui3ui2ui1P (u4,u3|s4 − s3)P (u3,u2|s3 − s2)P (u2,u1|s2 − s1) .

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1054 15 Path Integrals in Polymer Physics

The normalized integral over u1 can again be omitted. Still, the expression is com-plicated. A somewhat tedious calculation yields

〈R4〉 = 4(D + 2)

DL2ξ2 − 8Lξ3

(

D2 + 6D − 1

D2− D − 7

D + 1e−L/ξ

)

(15.163)

+ 4ξ4[

D3 + 23D2 − 7D + 1

D3− 2

(D + 5)2

(D + 1)2e−L/ξ +

(D − 1)5

D3(D + 1)2e−2DL/(D−1)ξ

]

.

For small values of L/ξ , we find the large-stiffness expansion

〈R4〉 = L4

1− 2

3

L

ξ+

25D − 17

90(D − 1)

(

L

ξ

)2

− 47D2 − 8D + 3

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

〈R4〉 = 4D + 2

DL2ξ2

1− 2D2 + 6D − 1

D(D + 2)

ξ

L+D3 + 23D2 − 7D + 1

D2(D + 2)

(

ξ

L

)2

+ . . . ,

(15.165)where the dots denote exponentially small terms. The leading term agrees againwith the expectation 〈R4〉 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 termsare corrections caused by the stiffness of the chain.

It is possible to find a correction factor to the Gaussian distribution which main-tains the unit normalization and ensures that the moment 〈R2〉 has the small-ξ exactexpansion (15.157) whereas 〈R2〉 is equal to (15.165) up to the first correction termin ξ/L. This has the form

PL(R) =

D

4πLξ

D

e−DR2/4Lξ

{

1−2D−1

4

ξ

L+3D−1

4

R2

L2−D(4D−1)

16(D+2)

R4

ξL3

}

. (15.166)

In three dimensions, this was first written down by Daniels [3]. It is easy to matchalso the moment 〈R4〉 by adding in the curly brackets the following terms

1−7D+23D2+D3

D + 1

[

D + 2

8D

ξ2

L2

(

1 +R2

ξL

)

+1

32

R4

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 calculatethe small-stiffness expansion of all even and odd moments as follows:

〈Rn〉 = 2nΓ(D/2 + n/2)

Dn/2Γ(D/2)Lnξn

1 + A1ξ

L+ A2

(

ξ

L

)2

+ . . .

, (15.168)

where

A1 = nn− 2− 2d2 − 4d (n− 1)

4d (2 + d), A2 = n(n− 2)

1− 7d+ 23d2 + d3

8d2 (1 + d). (15.169)

H. Kleinert, PATH INTEGRALS

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15.9 Schrodinger Equation and Recursive Solution for Moments 1055

15.9 Schrodinger Equation and Recursive Solution for

End-to-End Distribution Moments

The most efficient way of calculating the moments of the end-to-end distributionproceeds by setting up a Schrodinger equation satisfied by (15.112) and solving itrecursively with similar methods as developed in 3.19 and Appendix 3C.

15.9.1 Setting up the Schrodinger Equation

In the continuum limit, we write (15.112) as a path integral [compare (15.139)]

PL(R)∝∫

dub

dua

DD−1u δ(D)

(

R−∫ L

0dsu(s)

)

e−(κ/2)∫ L

0ds [u′(s)]2, (15.170)

where we have introduced the reduced stiffness

κ =κ

kBT= (D − 1)

ξ

2, (15.171)

for brevity. After a Fourier representation of the δ-function, this becomes

PL(R) ∝∫ i∞

−i∞

dDλ

2πieκ�·R/2

dub

dua(ubL|ua 0)�, (15.172)

where

(ub L|ua 0)� ≡∫ u(L)=ub

u(0)=ua

DD−1u e−(κ/2)∫ L

0ds{[u′(s)]2+�·u(s)} (15.173)

describes a point particle of massM = κ moving on a unit sphere. In contrast to thediscussion in Section 8.7 there is now an additional external field � which prevents usfrom finding an exact solution. However, all even moments 〈Ri1Ri2 · · ·Ri2l〉 of theend-to-end distribution (15.172) can be extracted from the expansion coefficientsin powers of λi of the integral

dub∫

dua over (15.173). The presence of thesedirectional integrals permits us to assume the external electric field � to point inthe z-direction, or the Dth direction in D-dimensions. Then � = λz, and themoments

R2l⟩

are proportional to the derivatives (2/κ)2l∂2lλ∫

dub∫

dua(ubL|ua 0)�.The proportionality factors have been calculated in Eq. (15.84). It is unnecessary toknow 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 inte-gral (15.173) around the solvable case λ = 0.

In natural units with κ = 1, the path integral (15.173) solves obviously theimaginary-time Schrodinger equation

(

−1

2∆u +

1

2� · u+

d

)

(u τ |ua 0)� = 0, (15.174)

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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)� (15.175)

appears, which is a function of z = cos θ only, where θ is the angle between u andthe electric field �. For ψ(z, τ ;λ), the Schrodinger equation reads

H ψ(z, τ ;λ) = − d

dτψ(z, τ ;λ), (15.176)

with the simpler Hamiltonian operator

H ≡ H0 + λ HI = −1

2∆ + λ z

= −1

2

[

(1− z2)d2

dz2− (D − 1)z

d

dz

]

+1

2λ z . (15.177)

Now the desired moments (15.135) can be obtained from the coefficient of λ2l/(2l)! ofthe power series expansion of the z-integral over (15.175) at imaginary time τ = L:

f(L;λ) ≡∫ 1

−1dz ψ(z, L;λ). (15.178)

15.9.2 Recursive Solution of Schrodinger Equation.

The function f(L;λ) has a spectral representation

f(L;λ) =∞∑

l=0

∫ 1−1 dz ϕ

(l)†(z) exp(

−E(l)L)

∫ 1−1 dza ϕ

(l)(za)∫ 1−1 dz ϕ

(l)†(z) ϕ(l)(z), (15.179)

where ϕ(l)(z) are the solutions of the time-independent Schrodinger equationHϕ(l)(z) = E(l)ϕ(l)(z). Applying perturbation theory to this problem, we startfrom the eigenstates of the unperturbed Hamiltonian H0 = −∆/2, which are given

by the Gegenbauer polynomials CD/2−1l (z) with the eigenvalues E

(l)0 = l(l+D−2)/2.

Following the methods explained in 3.19 and Appendix 3C we now set up a recur-sion scheme for the perturbation expansion of the eigenvalues and eigenfunctions[4].Starting point is the expansion of energy eigenvalues and wave-functions in pow-ers of the coupling constant λ:

E(l) =∞∑

j=0

ǫ(l)j λj, |ϕ(l)〉 =

∞∑

l′,i=0

γ(l)l′,i λ

i αl′ |l′〉 . (15.180)

The wave functions ϕ(l)(z) are the scalar products 〈z|ϕ(l)(λ)〉. We have insertedextra normalization constants αl′ for convenience which will be fixed soon. Theunperturbed state vectors |l〉 are normalized to unity, but the state vectors |ϕ(l)〉 of

H. Kleinert, PATH INTEGRALS

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15.9 Schrodinger Equation and Recursive Solution for Moments 1057

the interacting system will be normalized in such a way, that 〈ϕ(l)|l〉 = 1 holds toall orders, implying that

γ(l)l,i = δi,0 γ

(l)k,0 = δl,k . (15.181)

Inserting the above expansions into the Schrodinger equation, projecting the re-sult onto the base vector 〈k|αk, and extracting the coefficient of λj, we obtain theequation

γ(l)k,iǫ

(k)0 +

∞∑

j=0

αjαkVk,j γ

(l)j,i−1 =

i∑

j=0

ǫ(l)j γ

(l)k,i−j , (15.182)

where Vk,j = λ〈k|z|j〉 are the matrix elements of the interaction between unper-turbed states. For i = 0, Eq. (15.182) is satisfied identically. For i > 0, it leads tothe following two recursion relations, one for k = l:

ǫ(l)i =

n=±1

γ(l)l+n,i−1W

(l)n , (15.183)

the other one for k 6= l:

γ(l)k,i =

i−1∑

j=1

ǫ(l)j γ

(l)k,i−j −

n=±1

γ(l)k+n,i−1W

(l)n

ǫ(k)0 − ǫ

(l)0

, (15.184)

where only n = −1 and n = 1 contribute to the sums over n since

W (l)n ≡ αl+n

αl〈l| z |l + n〉 = 0, for n 6= ±1. (15.185)

The vanishing of W (l)n for n 6= ±1 is due to the band-diagonal form of the matrix of

the interaction z in the unperturbed basis |n〉. It is this property which makes thesums in (15.183) and (15.184) finite and leads to recursion relations with a finite

number of terms for all ǫ(l)i and γ

(l)k,i. To calculate W (l)

n , it is convenient to express〈l| z |l + n〉 as matrix elements between unnormalized noninteracting states |n} as

〈l| z |l + n〉 = {l|z|l + n}√

{l|l}{l + n|l + n}, (15.186)

where expectation values are defined by the integrals

{k|F (z)|l}≡∫ 1

−1CD/2−1k (z)F (z)C

D/2−1l (z)(1− z2)(D−3)/2 dz, (15.187)

from which we find5

{l|l} =24−D Γ(l +D − 2) π

l! (2l +D − 2) Γ(D/2− 1)2. (15.188)

5I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 7.313.1 and 7.313.2.

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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|z|l} =l + 1

2l +D − 2{l + 1|l + 1}, (15.190)

{l − 1|z|l} =l +D − 3

2l +D − 2{l − 1|l − 1} . (15.191)

Inserting these together with (15.188) into (15.186) gives

〈l|z|l − 1〉 =√

l(l +D − 3)

(2l +D − 2)(2l +D − 4), (15.192)

and a corresponding result for 〈l|z|l + 1〉. We now fix the normalization constantsαl′ by setting

W(l)1 =

αl+1

αl〈l| z |l + 1〉 = 1 (15.193)

for all l, which determines the ratios

αlαl+1

= 〈l| z |l + 1〉 =√

(l + 1) (l +D − 2)

(2l +D) (2l +D − 2). (15.194)

Setting further α1 = 1, we obtain

αl =

l∏

j=1

(2l +D − 2)(2l +D − 4)

l(l +D − 3)

1/2

. (15.195)

Using this we find from (15.185) the remaining nonzero W (l)n for n = −1:

W(l)−1 =

l(l +D − 3)

(2l +D − 2)(2l +D − 4). (15.196)

We are now ready to solve the recursion relations of (15.183) and (15.184) γ(l)k,i and

ǫ(l)i order by order in i. For the initial order i = 0, the values of the γ

(l)k,i are

given by Eq. (15.181). The coefficients ǫ(l)i are equal to the unperturbed energies

ǫ(l)0 = E

(l)0 = l(l +D − 2)/2. For each i = 1, 2, 3, . . . , there is only a finite number

of non-vanishing γ(l)k,j and ǫ

(l)j with j < i on the right-hand sides of (15.183) and

(15.184) which allows us to calculate γ(l)k,i and ǫ

(l)i on the left-hand sides. In this way

it is easy to find the perturbation expansions for the energy and the wave functionsto high orders.

H. Kleinert, PATH INTEGRALS

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15.9 Schrodinger Equation and Recursive Solution for Moments 1059

Inserting the resulting expansions (15.180) into Eq. (15.179), only the totallysymmetric parts in ϕ(l)(z) will survive the integration in the numerators, i.e., wemay insert only

ϕ(l)symm(z) = 〈z|ϕ(l)

symm〉 =∞∑

i=0

γ(l)0,i λ

i 〈z|0〉 . (15.197)

The denominators of (15.179) become explicitly∑

l′,i |γ(l)l′,i αl′ |2 λ2i, where the sumover i is limited by power of λ2 up to which we want to carry the perturbationseries; also l′ is restricted to a finite number of terms only, because of the band-diagonal structure of the γ

(l)l′,i.

Extracting the coefficients of the power expansion in λ from (15.179) we obtainall desired moments of the end-to-end distribution, in particular the second andfourth moments (15.155) and (15.164). Higher even moments are easily found withthe help of aMathematica 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 〈Rn〉 in powers of L/ξ to find a general large-stiffnessexpansion valid for all even and odd n:

〈Rn〉Ln

= 1− n

6

L

ξ+n (−13− n + 5D (1 + n))

360 (D − 1)

L2

ξ2− a3

L3

ξ3+ a4

L4

ξ4+. . . , (15.198)

where

a3 = n444−63n+15n2+7D2 (4+15n+5n2)+2D (−124−141n+7n2)

45360(D − 1)2,

a4 =n

5443200(d− 1)3

(

D0 +D1d+D2d2 +D3d

3)

, (15.199)

with

D0=3(

−5610+2921n−822n2+67n3)

, D1 = 8490+12103n−3426n2+461n3,

D2=45(

−2−187n−46n2+7n3)

, D4 = 35(

−6+31n+30n2+5n3)

. (15.200)

The lowest odd moments are, up to order l4,

〈R 〉L

= 1 − l

6+

5D−7

180(D−1)l2 − 33 − 43D + 14D2

3780(D − 1)2 l3 − 861 − 1469D + 855D2 − 175D3

453600 (D − 1)3 l4 . . . ,

R3⟩

L3= 1 − l

2+

5D−4

30 (D−1)l2 − 195−484D+329D2

7560 (−1 + d)2 l3 − 609−2201D+2955D2−1435D3

151200 (D−1)3 l4 . . . .

15.9.3 From Moments to End-to-End Distribution for D=3

We now use the recursively calculated moments to calculate the end-to-end distri-bution itself. It can be parameterized by an analytic function of r = R/L [4]:

PL(R) ∝ rk(1− rβ)m, (15.201)

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1060 15 Path Integrals in Polymer Physics

whose moments are exactly calculable:

r2l⟩

=

Γ

(

3 + k + 2 l

β

)

Γ

(

3 + k

β+ m + 1

)

Γ(

3+kβ

)

Γ

(

3 + k + 2 l

β+ m + 1

) . (15.202)

We now adjust the three parameters k, β, and m to fit the three most importantmoments of this distribution to the exact values, ignoring all others. If the dis-tances were distributed uniformly over the interval r ∈ [0, 1], the moments would

be 〈r2l〉unif = 1/(2l + 2). Comparing our exact moments⟨

r2l⟩

(ξ) with those of

the uniform distribution we find that 〈r2l〉(ξ)/〈r2l〉unif has a maximum for n close tonmax(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 〈r2〉, 〈r4〉,and 〈r6〉. In particular, we have fitted 〈r2〉, 〈r4〉 and 〈r6〉 for small persistence lengthξ < L/2. For ξ = L/2, we have started with 〈r4〉, for ξ = L with 〈r8〉 and for ξ = 2Lwith 〈r16〉, including always the following two higher even moments. After these ad-justments, whose results are shown in Fig. 15.4, we obtain the distributions shown inFig.15.6 for various persistence lengths ξ. They are in excellent agreement with theMonte Carlo data (symbols) and better than the one-loop perturbative results (thincurves) 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.

0.5 1 1.5 2

2.5

5

7.5

10

12.5

15

17.5

0.5 1 1.5 2

20

40

60

80

0.5 1 1.5 2

12

14

18

20

22k

ξ/L

m

ξ/L

β

ξ/L

6

16

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 ap-proximated by Gaussian random chain distributions on a lattice with lattice constantaeff = 2ξ, i.e., PL(R) → e−3R2/4Lξ [recall (15.75)]. This ensures that the lowest mo-ment 〈R2〉 = aeffL is properly fitted. In fact, we can easily check that our fittingprogram 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 thecorrect 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)−5/2e−1/4ξ(1−r), although the distribution at ξ = 2 is fitted numerically extremelywell.

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15.9 Schrodinger Equation and Recursive Solution for Moments 1061

The distribution functions can be inserted into Eq. (15.89) to calculate the struc-ture factors shown in Fig. 15.5. They interpolate smoothly between the Debye limit(15.90) and the stiff limit (15.105).

5 10 15 20 25 30

0.2

0.4

0.6

0.8

1

S(q)

ξ/L = 2

��✠ ξ/L = 1

��✠ ξ/L = 1/2

��✠ ξ/L = 1/5, . . . , 1/400��✠

q√ξ

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 distribu-

tions 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/q2 [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). InEq. (3.233) we have shown that a harmonic path integral including the integralsover the end points can be found, up to a trivial factor, by summing over all pathswith 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 +∞∑

n=1

un cos νns, νn = nπ/L. (15.203)

Let us parametrize the unit vectors u in D dimensions in terms of the first D − 1-dimensional coordinates uµ ≡ qµ with µ = 1, . . . , D − 1. The Dth component isthen given by a power series

σ ≡√

1− q2 ≈ 1− q2/2− (q2)2/8 + . . . . (15.204)

The we approximate the action harmonically as follows:

A = A(0) +Aint =κ

2

∫ L

0ds [u′(s)]2 +

1

2δ(0) log(1− q2)

≈ κ

2

∫ L

0ds [q′(s)]2 − 1

2δ(0)

∫ L

0ds q2. (15.205)

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1062 15 Path Integrals in Polymer Physics

The last term comes from the invariant measure of integration dD−1q/√1− q2 [recall

(10.636) and (10.641)].Assuming, as before, that R points into the z, or Dth, direction we factorize

δ(D)

(

R−∫ L

0dsu(s)

)

= δ

(

R− L+∫ L

0ds{

1

2q2(s) +

1

8[q2(s)]2 + . . .

}

)

× δ(D−1)

(

∫ L

0ds q(s)

)

, (15.206)

where R ≡ |R|. The second δ-function on the right-hand side enforces

q = L−1∫ L

0ds qµ(s) = 0 , µ = 1, . . . , d− 1 , (15.207)

and thus the vanishing of the zero-frequency parts qµ0 in the first D− 1 componentsof the Fourier decomposition (15.203).

It was shown in Eqs. (10.632) and (10.642) that the last δ-function has a dis-torting effect upon the measure of path integration which must be compensated bya Faddeev-Popov action

AFPe =

D − 1

2L

∫ L

0ds q2, (15.208)

where the number D of dimensions of qµ-space (10.642) has been replaced by thepresent number D − 1.

In the large-stiffness limit we have to take only the first harmonic term in theaction (15.205) into account, so that the path integral (15.170) becomes simply

PL(R) ∝∫

NBCD ′D−1q δ

(

R− L+∫ L

0ds

1

2q2(s)

)

e−(κ/2)∫ L

0ds [q′(s)]2. (15.209)

The subscript of the integral emphasizes the Neumann boundary conditions. Theprime on the measure of the path integral indicates the absence of the zero-frequencycomponent of qµ(s) in the Fourier decomposition due to (15.207). Representing theremaining δ-function in (15.209) by a Fourier integral, we obtain

PL(R) ∝ κ∫ i∞

−i∞

dω2

2πieκω

2(L−R)∫

NBCD ′D−1q exp

[

− κ2

∫ L

0ds(

q′2 + ω2q2)

]

. (15.210)

The integral over all paths with Neumann boundary conditions is known fromEq. (2.456). At zero average path, the result is

NBCD ′D−1q exp

[

− κ2

∫ L

0ds(

q′2 + ω2q2)

]

∝(

ωL

sinhωL

)(D−1)/2

, (15.211)

so that

PL(R) ∝ κ∫ i∞

−i∞

dω2

2πieκω

2(L−R)(

ωL

sinhωL

)(D−1)/2

. (15.212)

H. Kleinert, PATH INTEGRALS

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15.9 Schrodinger Equation and Recursive Solution for Moments 1063

The integral can easily be done in D = 3 dimensions using the original productrepresentation (2.455) of ωL/ sinhωL, where

PL(R) ∝ κ∫ i∞

−i∞

dω2

2πieκω

2(L−R)∞∏

n=1

(

1 +ω2

ν2n

)−1

. (15.213)

If we shift the contour of integration to the left we run through poles at ω2 = −ν2kwith residues

∞∑

k=1

ν2k

∞∏

n(6=k)=1

(

1− k2

n2

)−1

. (15.214)

The product is evaluated by the limit procedure for small ǫ:

∞∏

n=1

[

1− (k + ǫ)2

n2

]−1

[

1− (k + ǫ)2

k2

]

→ (k + ǫ)π

sin(k + ǫ)π

−2ǫ

k→ −2ǫπ

sin(k + ǫ)π

→ − 2ǫπ

cos kπ sin ǫπ→ 2(−1)k+1. (15.215)

Hence we obtain [6]

PL(R) ∝∞∑

k=1

2(−1)k+1κ ν2k e−κν2k(L−R). (15.216)

It is now convenient to introduce the reduced end-to-end distance r ≡ R/L and theflexibility of the polymer l ≡ L/ξ, and replace κ ν2k (L− R) → k2π2(1− r)/l so that(15.216) becomes

PL(R) = NL(R) = N∞∑

k=1

(−1)k+1k2π2e−k2π2(1−r)/l, (15.217)

where N is a normalization factor determined to satisfy∫

d3RPL(R) = 4πL3∫ ∞

0dr r2 PL(R) = 1. (15.218)

The sum must be evaluated numerically, and leads to the distributions shown inFig. 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

sinhωL

)(D−1)/2

= (ωL)(D−1)/2∞∑

k=0

(−1)k(

−(D − 1)/2

k

)

e−(2k+(D−1)/2)ωL, (15.219)

and obtain from (15.212):

PL(R) ∝∞∑

k=0

(−1)k(

−(D − 1)/2

k

)

Ik(R/L), (15.220)

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1064 15 Path Integrals in Polymer Physics

ξ/L=

4πr2PL(R)

Figure 15.6 Normalized end-to-end distribution of stiff polymer accord-

ing 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 〈R2〉 = aeffL 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) ≡∫ i∞

−i∞

2πiω(D+1)/2e−[2k+(D−1)/2]ω+(D−1)ω2(1−r)/2l , (15.221)

where ω is the dimensionless variable ωL. The integrals are evaluated with the helpof the formula6

∫ i∞

−i∞

dx

2πixνeβx

2/2−qx =1√

2πβ(ν+1)/2e−q

2/4βDν

(

q/√

β)

, (15.222)

where Dν(z) is the parabolic cylinder functions which for integer ν are proportionalto Hermite polynomials:7

Dn(z) =1√2n e

−z2/4Hn(z/√2). (15.223)

Thus we find

6I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formulas 3.462.3 and 3.462.4.7ibid., Formula 9.253.

H. Kleinert, PATH INTEGRALS

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15.9 Schrodinger Equation and Recursive Solution for Moments 1065

Ik(r)=1√2π

[

l

(D−1)(1−r)

](D+3)/4

e−

[2k+(D−1)/2]2

4(D−1)(1−r)/lD(D+1)/2

2k+(D−1)/2√

(D−1)(1−r)/l

,

(15.224)which becomes for D = 3

Ik(r)=1

2√2π

1√

2(1−r)/l3 e

−(2k+1)2

4(1−r)/lH2

2k+1

2√

(1−r)/l

. (15.225)

If the sum (15.220) is performed numerically for D = 3, and the integral overPL(R) is normalized to satisfy (15.218), the resulting curves fall on top of thosein Fig. 15.6 which were calculated from (15.217). In contrast to (15.217), whichconverges rapidly for small r, the sum (15.220) convergent rapidly for r close tounity.

Let us compare the low moments of the above distribution with the exact mo-ments in Eqs. (15.156) and (15.164). in the large-stiffness expansion. We set ω ≡ ωLand expand

f(ω2)≡√

ω

sinh ω

D−1

(15.226)

in a power series

f(ω2)=1−D − 1

22 ·3 ω2+(D−1)(5D−1)

25 ·32 ·5 ω4− (D−1)(15+14D+35D2)

27 ·34 ·5·7 ω6+. . . .(15.227)

Under the integral (15.212), each power of ω2 may be replaced by a differentialoperator

ω2 → ˆω2 ≡ −L

κ

d

dr= − 2l

D − 1

d

dr. (15.228)

The expansion f(ˆω2) can then be pulled out of the integral, which by itself yields a

δ-function, so that we obtain PL(R) in the form f(

ˆω2)

δ(r− 1), which is a series ofderivatives of δ-functions of r − 1 staring with

PL(R) ∝[

1+l

6

d

dr+

(−1+5D) l2

360 (D−1)

d2

dr2+

(15+14D+35D2) l3

45360 (D−1)2d3

dr3+ . . .

]

δ(r − 1).

(15.229)

From this we find easily the moments

〈Rm〉 =∫

dDRRm PL(R) ∝∫ ∞

0dr rD−1 rmPL(R). (15.230)

Let us introduce auxiliary expectation values with respect to the simple integrals〈f(r)〉1 ∝

dr f(r)PL, rather than to D-dimensional volume integrals. The unnor-malized moments 〈rm〉 are then given by 〈rD−1+m〉1. Within these one-dimensionalexpectations, the moments of z = r − 1 are

〈(r − 1)n〉1 ∝∫ ∞

0dr (r − 1)nPL(R). (15.231)

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1066 15 Path Integrals in Polymer Physics

The moments 〈rm〉 〈[1+(r−1)]m〉 = 〈[1+(r−1)]D−1+m〉1 are obtained by expandingthe binomial in powers of r−1 and using the integrals

dz zmδ(n)(z) = (−1)mm! δmnto find, up to the third power in L/ξ = l,

R0⟩

=N[

1 − D−1

6l +

(5D−1)(D−2)

360l2 − (35D3+14D+15)(D−2)(D−3)

45360(D− 1)l3]

,

R2⟩

=NL2

[

1 − D + 1

6l +

(5D−1)D(D+1)

360(D− 1)l2 − (35D2+14D+ 15)D(D+1)

45360(D− 1)l3]

,

R4⟩

=NL4

[

1 − D + 3

6l +

(5D−1)(D+2)(D+3)

360(D− 1)l2

− (35D2+14D+15)(D+1)(D+2)(D+3)

45360(D− 1)2l3]

.

The zeroth moment determines the normalization factor N to ensure that 〈R0〉 = 1.Dividing this out of the other moments yields

R2⟩

= L2

[

1− 1

3l +

13D−9

180(D−1)l2− 8

945l3+ . . .

]

, (15.232)

R4⟩

= L4

[

1− 2

3l +

23D−11

90(D−1)l2− 123D2−98D+39

1890(D−1)2l3+ . . .

]

. (15.233)

These agree up to the l-terms with the exact expansions (15.156) and (15.164) [orwith the general formula (15.198)]. For D = 3, these expansions become

R2⟩

= L2[

1− 1

3l +

1

12l2 − 8

945l3 + . . .

]

, (15.234)

R4⟩

= L4[

1− 2

3l +

29

90l2 − 71

630l3 + . . .

]

. (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 har-monic path integral (15.210) by the full expression

P (r;L)=S−1D

NBC

D ′D−1q(s)δ

(

r − L−1

∫ L

0

ds√

1 − q2(s)

)

e−Atot[q]−Acor[qb,qa], (15.236)

where

Atot[q]=1

∫ L

0

ds [ gµν(q) qµ(s)qν(s)−εδ(s, s) log g(q(s))] + AFP− εL

R

8. (15.237)

For convenience, we have introduced here the parameter ε = kBT/κ = 1/κ, the reduced inverse

stiffness, related to the flexibility l ≡ L/ξ by l = εL(d− 1)/2. We also have added the correctionterm εLR/8 to have a unit normalization of the partition function

Z = SD

∫ ∞

0

dr rD−1 P (r;L) = 1. (15.238)

H. Kleinert, PATH INTEGRALS

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15.9 Schrodinger Equation and Recursive Solution for Moments 1067

The Faddeev-Popov action, whose harmonic approximation was used in (15.208), is now

AFP[q] = −(D − 1) log

(

L−1

∫ L

0

ds√

1 − q2(s)

)

, (15.239)

To perform higher-order calculations we have added an extra action which corrects for theomission of fluctuations of the velocities at the endpoints when restricting the paths to Neumannboundary conditions with zero end-point velocities. The extra action contains, of course, only atthe endpoints and reads [7]

Acor[qb, qa] = − log J [qb, qa] = −[q2(0) + q2(L)]/4. (15.240)

Thus we represent the partition function (15.238) by the path integral with Neumann boundaryconditions

Z =

NBC

D ′D−1q(s) exp{

−Atot[q] −Acor[qb, qa] −AFP[q]}

. (15.241)

A similar path integral can be set up for the moments of the distribution. We express thesquare distance R2 in coordinates (15.207) as

R2 =

∫ L

0

ds

∫ L

0

ds′ u(s) · u(s′) =

(

∫ L

0

ds√

1 − q2(s)

)2

= R2 , (15.242)

we find immediately the following representation for all, even and odd, moments [compare (15.147)]

〈 (R2)n 〉 =

⟨[

∫ L

0

∫ L

0

ds ds′ u(s) · u(s′)

]n⟩

(15.243)

in the form

〈Rn 〉 =

NBC

D ′D−1q(s) exp{

−Atot[q] −Acor[qb, qa] −AFPn [q]

}

. (15.244)

This differs from Eq. (15.241) only by in the Faddeev-Popov action for these moments, which is

AFPn [q] = −(n + D − 1) log

(

L−1

∫ L

0

ds√

1 − q2(s)

)

, (15.245)

rather than (15.239).

There is no need to divide the path integral (15.244) by Z since this has unit normalization, aswill be verified order by order in the perturbation expansion. The Green function of the operatord2/ds2 with these boundary conditions has the form

∆′N(s, s′) =

L

3− | s− s′ |

2− (s + s′)

2+

(s2 + s′2)

2L. (15.246)

The zero temporal average (15.207) manifests itself in the property

∫ L

0

ds∆′N(s, s′) = 0 . (15.247)

In the following we shall simply write ∆(s, s′) for ∆′N(s, s′), for brevity.

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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 andall even moments in powers of the inverse stiffness ε up to order ε2 ∝ l2. This requires evaluatingFeynman 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 qµ → εqµ,and rewrite the path integral (15.244) as

〈Rn 〉 =

NBC

D ′D−1q(s) exp {−Atot,n[q; ε].} , (15.248)

with the total action

Atot,n[q; ε] =

∫ L

0

ds

[

1

2

(

q2 + ε(qq)2

1 − εq2

)

+1

2δ(0) log(1 − εq2)

]

− σn log

[

1

L

∫ L

0

ds√

1 − εq2

]

− ε

4

[

q2(0) + q2(L)]

− εLR

8. (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] + Aintn [q; ε], (15.251)

with a free action

A(0)[q] =1

2

∫ L

0

ds q2(s), (15.252)

and a large-stiffness expansion of the interaction

Aintn [q; ε] = εAint1

n [q] + ε2Aint2n [q] + . . . . (15.253)

The free part of the path integral (15.248) is normalized to unity:

Z(0) ≡∫

NBC

D ′D−1q(s) e−A(0)[q] =

NBC

D ′D−1q(s) e−(1/2)

L

0ds q2(s)

= 1. (15.254)

The first expansion term of the interaction (15.253) is

Aint1n [q] =

1

2

∫ L

0

ds{

[q(s)q(s)]2 − ρn(s) q2(s)}

− LR

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 importantfor canceling singularities in the expansion.

The next expansion term in (15.253) reads

Aint2n [q] =

1

2

∫ L

0

ds

{

[q(s)q(s)]2 − 1

2

[

δ(0) − σn

2L

]

q2(s)

}

q2(s)

+σn

8L2

∫ L

0

ds

∫ L

0

ds′ q2(s)q2(s′). (15.257)

H. Kleinert, PATH INTEGRALS

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15.9 Schrodinger Equation and Recursive Solution for Moments 1069

The perturbation expansion of the partition function in powers of ε consists of expectationvalues 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

〈F [q] 〉0 ≡∫

NBC

D ′D−1q(s)F [q] e−(1/2)

L

0ds q2(s)

. (15.258)

With this notation, the perturbative expansion of the path integral (15.248) reads

〈Rn 〉/Ln = 1 − 〈Aintn [q; ε]〉0 +

1

2〈Aint

n [q; ε]2〉0 − . . .

= 1 − ε 〈Aint1n [q]〉0 + ε2

(

−〈Aint2n [q]〉0 +

1

2〈Aint1

n [q]2〉0)

− . . . . (15.259)

For the evaluation of the expectation values we must perform all possible Wick contractions withthe basic propagator

〈 qµ(s)qν(s′) 〉0 = δµν∆(s, s′) , (15.260)

where ∆(s, s′) is the Green function (15.246) of the unperturbed action (15.252). The relevantloop 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):

〈Aint1n [q]〉0 =

(D−1)

2

[

σn

LI1 + DI2 −

1

2∆(0, 0) − 1

2∆(L,L)

]

− LR

8= L

(D−1)n

12,

〈Aint2n [q]〉0 =

(D2−1)

4

[(

δ(0) +σn

2L

)

I3+2(D+2)I4

]

+(D−1)σn

8L2

[

(D−1)I21 + 2I5]

= L3 (D2 − 1)

120δ(0) + L2 (D−1)

1440

[

(25D2+36D+23) + n(11D+5)]

,

1

2〈Aint1

n [q]2〉0 =L2

2

{

(D−1)L

12

[(

δ(0) +D

2L

)

−(

δn +2

L

)]

− R

8

}2

+(D−1)

4[Hn

1 − 2(Hn2 + Hn

3 −H5) + H6 − 4D(Hn4 −H7 −H10)

+ H11 + 2D2(H8 + H9)]

+(D−1)

4[DH12 + 2(D + 2)H13 + DH14]

=L2

2

[

(D−1)n

12

]2

+ L3 (D−1)

120δ(0)

+ L2 (D−1)

1440

[

(25D2 − 22D + 25) + 4(n + 4D − 2)]

+ L3 (D−1)D

120δ(0) + L2 (D−1)

720(29D− 1) . (15.261)

Inserting these results into Eq. (15.259), we find all even and odd moments up to order ε2 ∝ l2

〈Rn 〉/Ln = 1 − εL(D−1)n

12+ ε2L2

[

(D−1)2n2

288+

(D−1)(4n+ 5D − 13)n

1440

]

−O(ε3)

= 1 − n

6l +

[

n2

72+

(4n + 5D − 13)n

360(D−1)

]

l2 −O(l3). (15.262)

For n = 0 this gives the properly normalized partition function Z = 1. For all n it reproduces thelarge-stiffness expansion (15.198) up to order l4.

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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 correlationfunction up to four loops and verify that it yields the simple expression (15.153), which reads inthe 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 two-point correlation function

G(s, s′) = 〈u(s) · u(s′)〉 =

NBC

D ′D−1q(s) f(s, s′) exp{

−A(0)tot[q; ε]

}

, (15.264)

with the action of Eq. (15.249) for n = 0. The function in the integrand f(s, s′) ≡ f(q(s), q(s′)) isan abbreviation for the scalar product u(s) · u(s′) expressed in terms of independent coordinatesqµ(s):

f(q(s), q(s′)) ≡ u(s) · u(s′) =√

1 − q2(s)√

1 − q2(s′) + q(s) q(s′). (15.265)

Rescaling the coordinates q → √ε q, and expanding in powers of ε yields:

f(q(s), q(s′)) = 1 + εf1(q(s), q(s′)) + ε2f2(q(s), q(s

′)) + . . . , (15.266)

where

f1(q(s), q(s′)) = q(s) q(s′) − 1

2q2(s) − 1

2q2(s′), (15.267)

f2(q(s), q(s′)) =

1

4q2(s) q2(s′) − 1

8[q2(s)]2 − 1

8[q2(s′)]2. (15.268)

We shall attribute the integrand f(q(s), q(s′)) to an interaction Af [q; ε] defined by

f(q(s), q(s′)) ≡ e−Af [q;ε], (15.269)

which has the ε-expansion

Af [q; ε] = − log f(q(s), q(s′))

= −εf1(q(s), q(s′))+ε2

[

−f2(q(s), q(s′))+

1

2f21 (s, s′)

]

−. . . , (15.270)

to be added to the interaction (15.253) with n = 0. Thus we obtain the perturbation expansion ofthe path integral (15.264)

G(s, s′) = 1−⟨

(Aint0 [q; ε] + Af [q; ε])

0+

1

2

(Aint0 [q; ε] + Af [q; ε])2

0− . . . . (15.271)

Inserting the interaction terms (15.253) and (15.271), we obtain

G(s, s′) = 1+ε〈f1(q(s), q(s′))〉0+ε2[

〈f2(q(s), q(s′))〉0−〈f1(q(s), q(s′))Aint10 [q]〉0

]

+. . . ,

(15.272)

and the expectation values can now be calculated using the propagator (15.260) with the Greenfunction (15.246).

In going through this calculation we observe that because of translational invariance in thepseudotime, s → s+ s0, the Green function ∆(s, s′)+C is just as good a Green function satisfyingNeumann boundary conditions as ∆(s, s′). We may demonstrate this explicitly by setting C =

H. Kleinert, PATH INTEGRALS

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15.9 Schrodinger 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) usingthe modified Green function. Details are given in Appendix 15C [see Eq. (15C.1)], where we listvarious 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:

〈f1(q(s), q(s′))〉0 = − (D − 1)

2|s− s′ |, (15.273)

〈f2(q(s), q(s′))〉0 − 〈f1(q(s), q(s′))Aint10 [q]〉0

= (D − 1)

[

1

2D1 −

1

8(D + 1)D2

2 −K1 −DK2 −(D − 1)

LK3 +

1

2K4 +

1

2K5

]

=1

8(D − 1)2(s− s′)2. (15.274)

This leads indeed to the correct large-stiffness expansion of the exact two-point correlation function(15.263):

G(s, s′) = 1−εD−1

2|s−s′ | + ε2

(D−1)2

8(s−s′)2+. . .=1− |s−s′ |

ξ+

(s−s′)2

2ξ2− . . . . (15.275)

Radial Distribution up to Four Loops

We now turn to the most important quantity characterizing a polymer, the radial distribution func-tion. We eliminate the δ-function in Eq. (15.236) enforcing the end-to-end distance by consideringthe Fourier transform

P (k;L) =

dr eik(r−1) P (r;L) . (15.276)

This is calculated from the path integral with Neumann boundary conditions

P (k;L) =

NBC

D ′D−1q(s) exp{

−Atotk [q; ε]

}

, (15.277)

where the action Atotk [q; ε] reads, with the same rescaled coordinates as in (15.249),

Atotk [q; ε] =

∫ L

0

ds

{

1

2

[

q2 + ε(qq)2

1 − εq2

]

+1

2δ(0) log(1 − εq2) − ik

L

(

1 − εq2 − 1)

}

− 1

4ε[

q2(0) + q2(L)]

− εLR

8≡ A0[q] + Aint

k [q; ε]. (15.278)

As before in Eq. (15.253), we expand the interaction in powers of the coupling constant ε. Thefirst 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

Aint1k [q] =

∫ L

0

ds1

2

{

[q(s)q(s)]2 − ρk(s) q

2(s)}

− LR

8. (15.280)

The second expansion term Aint2k [q] is simpler than the previous (15.257) by not containing the

last nonlocal term:

Aint2k [q] =

∫ L

0

ds1

2

{

[q(s)q(s)]2 − 1

2

(

δ(0) − ik

2L

)

q2(s)

}

q2(s). (15.281)

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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):

P (k;L) = 1 − ε 〈Aint1k [q]〉0 + ε2

(

−〈Aint2k [q]〉0 +

1

2〈Aint1

k [q]2〉0)

− . . . . (15.282)

The expectation values can be expressed in terms of the same integrals listed in Appendix 15Aand Appendix 15B as follows:

〈Aint1, k [q]〉0 =

(D − 1)

2

[

ik

LI1 + DI2 −

1

2∆(0, 0) − 1

2∆(L,L)

]

− LR

8

= −L(D − 1) [(D − 1) − ik]

12, (15.283)

〈Aint2, k [q]〉0 =

(D2 − 1)

4

[(

δ(0) +ik

2L

)

I3 + 2(D + 2)I4

]

= L3 (D2 − 1)

120δ(0) + L2 (D2 − 1) [7(D + 2) + 3ik]

720, (15.284)

1

2〈Aint1

, k [q]2〉0 =L2

2

{

(D − 1)L

12

[(

δ(0) +D

2L

)

−(

δk +2

L

)]

− R

8

}2

+(D − 1)

4

[

Hk1 − 2(Hk

2 + Hk3 −H5) + H6 − 4D(Hk

4 −H7 −H10)

+ H11 + 2D2(H8 + H9)]

+(D − 1)

4[DH12 + 2(D + 2)H13 + DH14]

= L2 (D − 1)2 [(D − 1) − ik]2

2 · 122

+ L3 (D − 1)

120δ(0) + L2 (D − 1)

1440

[

(13D2 − 6D + 21) + 4ik(2D + ik)]

+ L3 (D − 1)D

120δ(0) + L2 (D − 1)

720(29D− 1) . (15.285)

In this way we find the large-stiffness expansion up to order ε2:

P (k;L) = 1 + εL(D − 1)

12[(D − 1) − ik] + ε2L2 (D − 1)

1440(15.286)

×[

(ik)2(5D−1)−2ik(5D2−11D+8)+(D−1)(5D2−11D+14)]

+O(ε3).

This can also be rewritten as

P (k;L)=P1 loop(k;L)

{

1+(D−1)

6l+

[

(D−3)

180(D−1)ik+

(5D2−11D+14)

360

]

l2+O(l3)

}

,

(15.287)

where the prefactor P1 loop(k;L) has the expansion

P1 loop(k;L) = 1 − εL(D − 1)

22 · 3 (ik) + ε2L2 (D − 1)(5D − 1)

25 · 32 · 5 (ik)2 − . . . . (15.288)

With the identification ω2 = ikεL, this is the expansion of one-loop functional determinant in(15.227). By Fourier-transforming (15.286), we obtain the radial distribution function

P (r; l) = δ(r−1) +l

6[δ′(r−1) + (d− 1) δ(r−1)] +

l2

360(d− 1)[(5d− 1) δ′′(r−1)

+ 2(5d2−11d+8) δ′(r−1) + (d−1)(5d2−11d+14)δ(r−1)]

+ O(l3). (15.289)

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15.9 Schrodinger Equation and Recursive Solution for Moments 1073

As a crosscheck, we can calculate from this expansion once more the even and odd moments

〈Rn 〉 = 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 distri-

bution (15.289) to arbitrarily high orders in l. Keeping only the terms up to order l4, we find thatthe one-loop end-to-end distribution function (15.212) receives a correction factor:

P (r; l) ∝∫ ∞

−∞

dω2

2πe−iω2(r−1)(D−1)/2l

( ω

sinh ω

)(D−1)/2

e−V (l,ω2), (15.291)

with

V (l, ω2) ≡ V0(l) + V (l, ω2) = V0(l) + V1(l)ω2

l+ V2(l)

ω4

l2+ V3(l)

ω6

l3+ . . . . (15.292)

The first term

V0(l) = −d− 1

6l +

d− 9

360l2 +

(d− 1)(

32 − 13 d + 5 d2)

6480l3

− 34 − 272 d+ 259 d2 − 110 d3 + 25 d4

259200l4 + . . . (15.293)

contributes only to the normalization of P (r; l), and can be omitted in (15.291). The remainderhas the expansion coefficients

V1(l) = −d− 3

360l2 +

(−5 + 9 d)

7560 (−1 + d)l3 +

(

−455 + 431 d+ 91 d2 + 5 d3)

907200 (d− 1)2 l4 + . . . ,

V2(l) = − (5 − 3 d) l3

7560−(

−31 + 42 d+ 25 d2)

l4

907200 (d− 1)+ . . . , (15.294)

V3(l) = − (d− 1) l4

18900+ . . . .

In the physical most interesting case of three dimensions, the first nonzero correction arises toorder l3. This explains the remarkable agreement of the moments in (15.234) and (15.234) up toorder l2.

The correction terms V (l, ω2) may be included perturbatively into the sum over k inEq. (15.220) by noting that the expectation value of powers of ω2/l within the ω-integral (15.221)are

ω2/l⟩

= a2k ≡ 2k + (D − 1)/2

(D − 1)(1 − r),⟨

ω2/l⟩2

= 3a4k,⟨

ω2/l⟩3

= 15a6k, (15.295)

so that we obtain an extra factor e−fk

fk = V1(l)a2k +

[

3V2(l)−V 21 (l)

]

a4k +

[

15V3(l)−12V1(l)V2(l)+4

3V 31 (l)

]

a6k + . . . , (15.296)

where up to order l4:

3V2(l)−V 21 (l) =

3D − 5

2520l3 +

156 − 231D− 26D2 − 7D3

907200 (D − 1)l4 + . . . ,

15V3(l)−12V1(l)V2(l)+4

3V 31 (l) = −D − 1

1260l4 + . . . . (15.297)

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1074 15 Path Integrals in Polymer Physics

15.10 Excluded-Volume Effects

A significant modification of these properties is brought about by the interactionsbetween the chain elements. If two of them come close to each other, the molecularforces prevent them from occupying the same place. This is called the excluded-

volume effect . In less than four dimensions, it gives rise to a scaling law for theexpectation value 〈R2〉 as a function of L:

〈R2〉 ∝ L2ν , (15.298)

as stated in (15.38). The critical exponent ν is a number between the random-chainvalue ν = 1/2 and the stiff-chain value ν = 1.

To derive this behavior we consider the polymer in the limiting path integralapproximation (15.80) to a random chain which was derived for R2/La ≪ 1 andwhich is very accurate whenever the probability distribution is sizable. Thus westart with the time-sliced expression

PN(R) =1

2πa/MD

N−1∏

n=1

dDxn√

2πa/MD

exp(

−AN/h)

, (15.299)

with the action

AN = aN∑

n=1

M

2

(∆xn)2

a2, (15.300)

and the mass parameter (15.79). In the sequel we use natural units in which energiesare measured in units of kBT , and write down all expressions in the continuum limit.The probability (15.299) is then written as

PL(R) =∫

DDx e−AL[x], (15.301)

where we have used the label L = Na rather than N . From the discussion inthe previous section we know that although this path integral represents an idealrandom chain, we can also account for a finite stiffness by interpreting the numbera as an effective length parameter aeff given by (15.158). The total Euclidean timein the path integral τb− τa = h/kBT corresponds to the total length of the polymerL.

We now assume that the molecules of the polymer repel each other with a two-body potential V (x,x′). Then the action in the path integral (15.301) has to besupplemented by an interaction

Aint =1

2

∫ L

0dτ∫ L

0dτ ′ V (x(τ),x(τ ′)). (15.302)

Note that the interaction is of a purely spatial nature and does not depend on theparameters τ , τ ′, i.e., it does not matter which two molecules in the chain comeclose to each other.

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15.10 Excluded-Volume Effects 1075

The effects of an interaction of this type are most elegantly calculated by mak-ing use of a Hubbard-Stratonovich transformation. Generalizing the procedure inSubsection 7.15.1, we introduce an auxiliary fluctuating field variable ϕ(x) at everyspace point x and replace Aint by

Aϕint =

∫ L

0dτ ϕ(x(τ))− 1

2

dDxdDx′ ϕ(x)V −1(x,x′)ϕ(x′). (15.303)

Here V −1(x,x′) denotes the inverse of V (x,x′) under functional multiplication, de-fined by the integral equation

dDx′ 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 =

dDx ρ(x)ϕ(x)− 1

2

dDxdDx′ ϕ(x)V −1(x,x′)ϕ(x′), (15.305)

where ρ(x) is the particle density

ρ(x) ≡∫ L

0dτ δ(D)(x− x(τ)). (15.306)

Then we perform a quadratic completion to

Aϕint = −1

2

dDxdDx′[

ϕ′(x)V −1(x,x′)ϕ′(x′)− ρ(x)V (x,x′)ρ(x′)]

, (15.307)

with the shifted field

ϕ′(x) ≡ ϕ(x)−∫

dDx′ V (x,x′)ρ(x′). (15.308)

Now we perform the functional integral∫

Dϕ(x) e−Aϕint (15.309)

integrating ϕ(x) at each point x from −i∞ to i∞ along the imaginary field axis.

The result is a constant functional determinant [det V −1(x,x′)]−1/2

. This can beignored 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 preciselythe original interaction (15.302).

Thus we may study the excluded-volume problem by means of the equivalentpath integral

PL(R) ∝∫

DDx(τ)∫

Dϕ(x) e−A, (15.310)

where the action A is given by the sum

A = AL[x, x, ϕ] +A[ϕ], (15.311)

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1076 15 Path Integrals in Polymer Physics

of the line and field actions

AL[x, ϕ] ≡∫ L

0dτ[

M

2x2 + ϕ(x(τ))

]

, (15.312)

A[ϕ] ≡ −1

2

dDxdDx′ ϕ(x)V −1(x,x′)ϕ(x′), (15.313)

respectively. The path integral (15.310) over x(τ) and ϕ(x) has the following phys-ical interpretation. The line action (15.312) describes the orbit of a particle in aspace-dependent random potential ϕ(x). The path integral over x(τ) yields the end-to-end distribution of the fluctuating polymer in this potential. The path integralover all potentials ϕ(x) with the weight e−A[ϕ] accounts for the repulsive cloud ofthe 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) ∝∫

Dϕ(x) e−A[ϕ]P ϕL (R, 0), (15.314)

where

P ϕL (R, 0) =

DDx(τ) e−AL[x,ϕ] (15.315)

is the end-to-end distribution of a random chain moving in a fixed external potentialϕ(x). The presence of this potential destroys the translational invariance of P ϕ

L . Thisis why we have recorded the initial and final points 0 and R. In the final distributionPL(R) of (15.314), the invariance is of course restored by the integration over allϕ(x).

It is possible to express the distribution P ϕL (R, 0) in terms of solutions of an

associated Schrodinger equation. With the action (15.312), this equation is obviously

[

∂L− 1

2M∂R

2 + ϕ(R)

]

P ϕL (R, 0) = δ(D)(R− 0)δ(L). (15.316)

If ψϕE(R) denotes the time-independent solutions of the Hamiltonian operator

Hϕ = − 1

2M∂R

2 + ϕ(R), (15.317)

the probability P ϕL (R) has a spectral representation of the form

P ϕL (R, 0) =

dEe−ELψϕE(R)ψϕ ∗E (0), L > 0. (15.318)

From now on, we assume the interaction to be dominated by the simplest possiblerepulsive potential proportional to a δ-function:

V (x,x′) = vaDδ(D)(x− x′). (15.319)

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15.10 Excluded-Volume Effects 1077

Then the functional inverse is

V −1(x,x′) = v−1a−Dδ(D)(x− x′), (15.320)

and the ϕ-action (15.312) reduces to

A[ϕ] = −v−1a−D

2

dDxϕ2(x). (15.321)

The path integrals (15.314), (15.315) can be solved approximately by applying thesemiclassical 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 leadingsaddle point approximation. In the ϕ(x)-integral, the saddle point is given by theequation

v−1a−Dϕ(x) =δ

δϕ(x)logP ϕ

L (R, 0). (15.322)

This is the semiclassical approximation to the exact equation

v−1a−D〈ϕ(x)〉 = 〈ρ(x)〉 ≡⟨

∫ L

0dτ δ(D)(x− x(τ))

x

, (15.323)

where 〈. . .〉x is the average over all line fluctuations calculated with the help of theprobability distribution (15.315).

The exact equation (15.323) follows from a functional differentiation of the pathintegral for P ϕ

L with respect to ϕ(x):

δ

δϕ(x)P ϕL (R) =

Dϕ δ

δϕ(x)

DDx e−AL[x,ϕ]−A[ϕ] = 0. (15.324)

By anchoring one end of the polymer at the origin and carrying the path integralfrom there to x(τ), and further on to R, the right-hand side of (15.323) can beexpressed as a convolution integral over two end-to-end distributions:

∫ L

0dτ δ(D)(x− x(τ))

x

=∫ L

0dL′ P ϕ

L′(x)PϕL−L′(R− x). (15.325)

With (15.323), this becomes

v−1a−D〈ϕ(x)〉x =∫ L

0dL′ P ϕ

L′(x)PϕL−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 becomessimple only for R = 0, i.e., for a closed polymer. Then only the variable x remainsand, by rotational symmetry, ϕ(x) can depend only on r = |x|. For R 6= 0, on the

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1078 15 Path Integrals in Polymer Physics

other hand, the rotational symmetry is distorted to an ellipsoidal geometry, in whicha closed-form solution of the problem is hard to find. As an approximation, we mayuse a rotationally symmetric ansatz ϕ(x) ≈ ϕ(r) also for R 6= 0 and calculate theend-to-end probability distribution PL(R) via the semiclassical approximation tothe 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) ∼ P ϕL (R, 0) =

DDx exp

{

−∫ L

0dτ[

M

2x2 + ϕ(r(τ))

]

}

. (15.327)

Thereby it is hoped that for moderate R, the error is small enough to justify thisapproximation. Anyhow, the analytic results supply a convenient starting point forbetter approximations.

Neglecting the ellipsoidal distortion, it is easy to calculate the path integral overx(τ) for P ϕ

L (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 integralof motion

M

2x2 − ϕ(r) = E = const. (15.328)

At fixed L, we have to find the classical solutions for all energies E and all angularmomenta l. The path integral reduces an ordinary double integral over E andl which, in turn, is evaluated in the saddle point approximation. In a rotationallysymmetric potential ϕ(r), the leading saddle point has the angular momentum l = 0corresponding to a symmetric polymer distribution. Then Eq. (15.328) turns into apurely radial differential equation

dτ =dr

2[E + ϕ(r)]/M. (15.329)

For a polymer running from the origin to R we calculate

L =∫ R

0

dr√

2[E + ϕ(r)]/M. (15.330)

This determines the energy E as a function of L. It is a functional of the yetunknown field ϕ(r):

E = EL[ϕ]. (15.331)

The classical action for such an orbit can be expressed in the form

Acl[x, ϕ] =∫ L

0dτ[

M

2x2 + ϕ(r(τ))

]

= −∫ L

0dτ[

M

2x2 − ϕ(r(τ))

]

+∫ L

0dτM x2

= −EL+∫ R

0dr√

2M [E + ϕ(r)]. (15.332)

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15.10 Excluded-Volume Effects 1079

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 whenextremizing the classical expression Acl[x, ϕ]:

∂EAcl[x, x, ϕ] = 0. (15.333)

The classical approximation to the entire action A[x, ϕ] +A[ϕ] is then

Acl = −EL+∫ r

0dr′√

2M [E + ϕ(r′)]− 1

2v−1a−D

dDxϕ2(r). (15.334)

This action is now extremized independently in ϕ(r), E. The extremum in ϕ(r) isobviously given by the algebraic equation

ϕ(r′) =

{

0

MvaDS−1D r′ 1−D/

2M [E + ϕ(r′)]for

r′ > r,r′ < r,

(15.335)

which is easily solved. We rewrite it as

E + ϕ(r) = ξ3ϕ−2(r), (15.336)

with the abbreviation

ξ3 = αr−2δ, (15.337)

where

δ ≡ D − 1 > 0 (15.338)

and

α ≡ M

2v2a2DS−2

D . (15.339)

For large ξ ≫ 1/E, i.e., small r ≪ α2/δE−6/δ, we expand the solution as follows

ϕ(r) = ξ − E

3+E2

9+ . . . . (15.340)

This expansion is reinserted into the classical action (15.334), making it a powerseries in E. A further extremization in E yields E = E(L, r). The extremal value ofthe action yields an approximate distribution function of a monomer in the closedpolymer (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)

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1080 15 Path Integrals in Polymer Physics

The extremization in E gives

E =M

2

R2

L2, (15.343)

yielding the extremal action

Acl =M

2

R2

L. (15.344)

The approximate distribution is therefore

PN(R) ∝ e−Acl = e−MR2/2L. (15.345)

The interacting case is now treated in the same way. Using (15.336), the classicalaction (15.334) can be written as

Acl = −EL+

M

2

∫ R

0dr′

[

ϕ+ E − 1

2ξ3/2

1

ϕ+ E

]

. (15.346)

By expanding this action in a power series in E [after having inserted (15.340) forϕ], we obtain with ǫ(r) ≡ E/ξ = Eα−1/3r2δ/3

Acl = −EL+

M

2α1/6

∫ R

0dr′r′−δ/3

[

3

2+ ǫ(r′)− 1

6ǫ2(r′) + . . .

]

. (15.347)

As long as δ < 3, i.e., for

D < 4, (15.348)

the integral exists and yields an expansion

Acl = −EL+ a0(R) + a1(R)E − 1

2a2(R)E

2 + . . . , (15.349)

with the coefficients

a0(R) = −M2

9

2R1−δ/3α1/6 1

δ − 3,

a1(R) = 3

M

2R1+δ/3α−1/6 1

δ + 3,

a2(R) =1

3

M

2R1+δα−1/2 1

δ + 1. (15.350)

Extremizing Acl in E gives the action

Acl = a0(R) +1

2a2(R)[L− a1(R)]

2 + . . . . (15.351)

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15.11 Flory’s Argument 1081

The approximate end-to-end distribution function is therefore

PL(R) ≈ N exp

{

−a0(R)−1

2a2(R)[L− a1(R)]

2

}

, (15.352)

where N is an appropriate normalization factor. The distribution is peaked around

L = 3

M

2R1+δ/3α−1/6 1

δ + 3. (15.353)

This shows the most important consequence of the excluded-volume effect: Theaverage value of R2 grows like

〈R2〉 ≈ α1/(D+2)

D + 2

3

2

ML

6/(D+2)

. (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 reachout further into space than a random chain [although less than a completely stiffchain, which is always reached by the solution (15.354) for D = 1].

The restriction D < 4 in (15.348) is quite important. The value

Duc = 4 (15.356)

is called the upper critical dimension. Above it, the set of all possible intersectionsof a random chain has the measure zero and any short-range repulsion becomesirrelevant. In fact, for D > Duc it is possible to show that the polymer behaves likea random chain without any excluded-volume effect satisfying 〈R2〉 ∝ L.

15.11 Flory’s Argument

Once we expect a power-like scaling law of the form

〈R2〉 ∝ L2ν , (15.357)

the critical exponent (15.355) can be derived from a very simple dimensional argu-ment due to Flory. We take the action

A =∫ L

0dτ

M

2x2 − vaD

2

∫ L

0dτ∫ L

0dτ ′ δ(D)(x(τ)− x(τ ′)), (15.358)

with M = D/a, and replace the two terms by their dimensional content, L for theτ -variable and R- for each x-component. Then

A ∼ M

2LR2

L2− vaD

2

L2

RD. (15.359)

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1082 15 Path Integrals in Polymer Physics

Extremizing this expression in R at fixed L gives

R

L∼ R−D−1L2, (15.360)

implying

R2 ∼ L6/(D+2), (15.361)

and thus the critical exponent (15.355).

15.12 Polymer Field Theory

There exists an alternative approach to finding the power laws caused by theexcluded-volume effects in polymers which is superior to the previous one. It isbased on an intimate relationship of polymer fluctuations with field fluctuations ina certain somewhat artificial and unphysical limit. This limit happens to be acces-sible to approximate analytic methods developed in recent years in quantum fieldtheory. According to Chapter 7, the statistical mechanics of a many-particle ensem-ble can be described by a single fluctuating field. Each particle in such an ensemblemoves through spacetime along a fluctuating orbit in the same way as a randomchain does in the approximation (15.80) to a polymer in Section 15.6. Thus we canimmediately conclude that ensembles of polymers may also be described by a singlefluctuating field. But how about a single polymer? Is it possible to project outa single polymer of the ensemble in the field-theoretic description? The answer ispositive. We start with the result of the last section. The end-to-end distributionof the polymer in the excluded-volume problem is rewritten as an integral over thefluctuating field ϕ(x):

PL(xb,xa) =∫

Dϕ e−A[ϕ]P ϕL (xb,xa), (15.362)

with an action for the auxiliary ϕ(x) field [see (15.312)]

A[ϕ] = −1

2

dDxdDx′ϕ(x)V −1(x,x′)ϕ(x′), (15.363)

and an end-to-end distribution [see (15.315)]

P ϕL (xb,xa) =

Dx exp{

−∫ L

0dτ[

M

2x2 + ϕ(x(τ))

]

}

, (15.364)

which satisfies the Schrodinger equation [see (15.316)]

[

∂L− 1

2M∇

2 + ϕ(x)

]

P ϕL (x,x

′) = δ(3)(x− x′)δ(L). (15.365)

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15.12 Polymer Field Theory 1083

Since PL and P ϕL vanish for L < 0, it is convenient to go over to the Laplace

transforms

Pm2(x,x′) =1

2M

∫ ∞

0dLe−Lm

2/2MPL(x,x′), (15.366)

P ϕm2(x,x

′) =1

2M

∫ ∞

0dLe−Lm

2/2MP ϕL (x,x

′). (15.367)

The latter satisfies the L-independent equation

[−∇2 +m2 + 2Mϕ(x)]P ϕ

m2(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 P ϕm2(x,x′) describe the probability of a polymer of any length run-

ning from x′ to x, with a Boltzmann-like factor e−Lm2/2M 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 cor-

relation function of an auxiliary fluctuating complex field ψ(x):

P ϕm2(x,x

′) = Gϕ0 (x,x

′) = 〈ψ∗(x)ψ(x′)〉ϕ≡

∫ Dψ∗(x)Dψ(x) ψ∗(x)ψ(x′) exp {−A[ψ∗, ψ, ϕ]}∫ Dψ∗(x)Dψ(x) exp {−A[ψ∗, ψ, ϕ]} , (15.370)

with a field action

A[ψ∗, ψ, ϕ]=∫

dDx{

∇ψ∗(x)∇ψ(x) +m2ψ∗(x)ψ(x) + 2Mϕ(x)ψ∗(x)ψ(x)}

.(15.371)

The second part of Eq. (15.370) defines the expectations 〈. . .〉ψ. In this way, weexpress the Laplace-transformed distribution Pm2(xb,xa) in (15.366) in the purelyfield-theoretic form

Pm2(x,x′) =∫

Dϕ exp {−A[ϕ]} 〈ψ∗(x)ψ(x′)〉ϕ

=∫

Dϕ exp{

1

2

dDydDy′ϕ(y)V −1(y,y′)ϕ(y′)}

×∫ Dψ∗Dψ ψ∗(x)ψ(x′) exp {−A[ψ∗, ψ, ϕ]}

∫ Dψ∗∫ Dψ exp {−A[ψ∗, ψ, ϕ]} . (15.372)

It involves only a fluctuating field which contains all information on the path fluc-tuations. The field ψ(x) is, of course, the analog of the second-quantized field inChapter 7.

Consider now the probability distribution of a single monomer in a closed poly-mer chain. Inserting the polymer density function

ρ(R) ≡∫ L

0dτδ(D)(R− x(τ)) (15.373)

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1084 15 Path Integrals in Polymer Physics

into the original path integral for a closed polymer

PL(R) =∫ L

0dτ

DDx∫

Dϕ exp {−AL −A[ϕ]} δ(D)(R− x(τ)), (15.374)

the δ-function splits the path integral into two parts

PL(R) =∫

Dϕ exp {−A[ϕ]}∫ L

0dτP ϕ

L−τ (0,R)P ϕτ (R, 0). (15.375)

When going to the Laplace transform, the convolution integral factorizes, yielding

Pm2(R) =∫

Dϕ(x) exp {−A[ϕ]}P ϕm2(0,R)P ϕ

m2(R, 0). (15.376)

With the help of the field-theoretic expression for Pm2(R) in Eq. (15.370), the prod-uct of the correlation functions can be rewritten as

P ϕm2(0,R)P ϕ

m2(0,R) = 〈ψ∗(R)ψ(0)〉ϕ〈ψ∗(0)ψ(R)〉ϕ. (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 termin the Wick expansion (recall Section 3.10) of the four-field correlation function

〈ψ∗(R)ψ(R)ψ∗(0)ψ(0)〉ϕ. (15.378)

This would be equal to the sum of pair contractions

〈ψ∗(R)ψ(R)〉ϕ〈ψ∗(0)ψ(0)〉ϕ + 〈ψ∗(R)ψ(0)〉ϕ〈ψ∗(0)ψ(R)〉ϕ. (15.379)

There are no contributions containing expectations of two ψ or two ψ∗ fields whichcould, 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):

P ϕm2(0,R)P ϕ

m2(0,R) = 〈ψ∗(R)ψ(R)ψ∗(0)ψ(0)〉ϕ − 〈ψ∗(R)ψ(R)〉ϕ〈ψ∗(0)ψ(0)〉ϕ.(15.380)

The right-hand side only contains correlation functions of a collective field, thedensity field [8]

ρ(R) = ψ∗(R)ψ(R), (15.381)

in terms of which

P ϕm2(0,R)P ϕ

m2(0,R) = 〈ρ(R)ρ(0)〉ϕ − 〈ρ(R)〉ϕ〈ρ(0)〉ϕ. (15.382)

Now, the right-hand side is the connected correlation function of the density fieldρ(R):

〈ρ(R)ρ(0)〉ϕ,c ≡ 〈ρ(R)ρ(0)〉ϕ − 〈ρ(R)〉ϕ〈ρ(0)〉ϕ. (15.383)

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15.12 Polymer Field Theory 1085

In Section 3.10 we have shown how to generate all connected correlation func-tions: The action A[ψ, ψ∗, ϕ] is extended by a source term in the density field ρ(x)

Asource[ψ∗, ψ,K] = −

dDxK(x)ρ(x) =∫

dDxK(x)ψ∗(x)ψ(x)), (15.384)

and one considers the partition function

Z[K,ϕ] ≡∫

DψDψ∗ exp {−A [ψ∗, ψ, ϕ]−Asource[ψ∗, ψ,K]} . (15.385)

This is the generating functional of all correlation functions of the density fieldρ(R) = ψ∗(R)ψ(R) at a fixed ϕ(x). They are obtained from the functional deriva-tives

〈ρ(x1) · · · ρ(xn)〉ϕ = Z[K,ϕ]−1 δ

δK(x1)· · · δ

δK(xn)Z[K,ϕ]

K=0.

(15.386)

Recalling Eq. (3.559), the connected correlation functions of ρ(x) are obtained sim-ilarly from the logarithm of Z[K,ϕ]:

〈ρ(x1) · · ·ρ(xn)〉ϕ,c =δ

δK(x1)· · · δ

δK(xn)logZ[K,ϕ]

K=0.

(15.387)

For n = 2, the connectedness is seen directly by performing the differentiationsaccording to the chain rule:

〈ρ(R)ρ(0)〉ϕ,c =δ

δK(R)

δ

δK(0)logZ[K,ϕ]

K=0

δK(R)Z−1[K,ϕ]

δ

δK(0)Z[K,ϕ]

K=0

= 〈ρ(R)ρ(0)〉ϕ − 〈ρ(R)〉ϕ〈ρ(0)〉ϕ. (15.388)

This agrees indeed with (15.383). We can therefore rewrite the product of Laplace-transformed distributions (15.382) at a fixed ϕ(x) as

P ϕm2(0,R)P ϕ

m2(0,R) =δ

δK(R)

δ

δK(0)logZ[K,ϕ]

K=0. (15.389)

The Laplace-transformed monomer distribution (15.376) is then obtained by aver-aging over ϕ(x), i.e., by the path integral

Pm2(R) =δ

δK(R)

δ

δK(0)

Dϕ(x) exp {−A[ϕ]} logZ[K,ϕ]∣

K=0. (15.390)

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1086 15 Path Integrals in Polymer Physics

Were it not for the logarithm in front of Z, this would be a standard calculation ofcorrelation functions within the combined ψ, ϕ field theory whose action is

Atot[ψ∗, ψ, ϕ] = A[ψ∗, ψ, ϕ] +A[ϕ]

=∫

dDx(

∇ψ∗∇ψ +m2ψ∗ψ + 2Mϕψ∗ψ

)

− 1

2

dDxdDx′ϕ(x)V −1(x,x′)ϕ(x′). (15.391)

To account for the logarithm we introduce a simple mathematical device called thereplica trick [9]. We consider logZ[K,ϕ] in (15.388)–(15.390) as the limit

logZ = limn→0

1

n(Zn − 1) , (15.392)

and observe that the nth power of the generating functional, Zn, can be thoughtof as arising from a field theory in which every field ψ occurs n times, i.e., with nidentical replica. Thus we add an extra internal symmetry label α = 1, . . . , n to thefields ψ(x) and calculate Zn formally as

Zn[K,ϕ] =∫

Dψ∗αDψα exp {−A[ψ∗

α, ψα, ϕ]−A[ϕ]−Asource[ψ∗α, ψα, K]} ,

(15.393)

with the replica field action

A[ψα, ψ∗α, ϕ] =

dDx(

∇ψ∗α∇ψα +m2ψ∗

αψα + 2Mϕψ∗αψα

)

, (15.394)

and the source term

Asource[ψ∗α, ψα, K] = −

dDxψ∗α(x)ψα(x)K(x). (15.395)

A sum is implied over repeated indices α. By construction, the action is symmetricunder 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

Zn[K,ϕ] =∫

Dψ∗αDψα exp {−An[ψ∗

α, ψα]−Asource[ψ∗α, ψα, K]} , (15.396)

with the action

An[ψ∗α, ψα] =

dDx(

∇ψ∗α∇ψα +m2ψ∗

αψα)

+1

2(2M)2

dDxdDx′ψα∗(x)ψα(x)V (x,x

′)ψβ∗(x′)ψβ(x

′). (15.397)

It describes a self-interacting field theory with an additional U(n) symmetry.

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15.12 Polymer Field Theory 1087

In the special case of a local repulsive potential V (x,x′) of Eq. (15.319), thesecond term becomes simply

Aint[ψ∗α, ψα] =

1

2(2M)2vaD

dDx [ψα∗(x)ψα(x)]

2 . (15.398)

Using this action, we can find logZ[K,ϕ] via (15.392) from the functional integral

logZ[K,ϕ] ≡ limn→0

1

n

(∫

Dψ∗αDψα exp {−An[ψ∗

α, ψα]−Asource[ψ∗α, 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 de-scription with complex replica fields it was assumed that the two orientations canbe distinguished. If they are indistinguishable, the polymer fields Ψα(x) have to betaken as real.

Such a field-theoretic description of a fluctuating polymer has an importantadvantage over the initial path integral formulation based on the analogy with aparticle orbit. It allows us to establish contact with the well-developed theory ofcritical phenomena in field theory. The end-to-end distribution of long polymersat large L is determined by the small-E regime in Eqs. (15.330)–(15.349), whichcorresponds to the small-m2 limit of the system here [see (15.369)]. This is preciselythe regime studied in the quantum field-theoretic approach to critical phenomenain many-body systems [10, 11]. It can be shown that for D larger than the uppercritical dimension Duc = 4, the behavior for m2 → 0 of all Green functions coincideswith the free-field behavior. For D = Duc, this behavior can be deduced from scaleinvariance arguments of the action, using naive dimensional counting arguments.The fluctuations turn out to cause only logarithmic corrections to the scale-invariantpower laws. One of the main developments in quantum field theory in recent yearswas the discovery that the scaling powers for D < Duc can be calculated via anexpansion of all quantities in powers of

ǫ = Duc −D, (15.400)

the so-called ǫ-expansion. The ǫ-expansion for the critical exponent ν which rulesthe relation between R2 and the length of a polymer L, 〈R2〉 ∝ L2ν , can be derivedfrom a real φ4-theory with n replica as follows [12]:

ν−1 = 2 + (n+2)ǫn+8

{

−1 − ǫ2(n+8)2 (13n + 44)

+ ǫ2

8(n+8)4[3n3 − 452n2 − 2672n − 5312

+ ζ(3)(n + 8) · 96(5n + 22)]

+ ǫ3

32(n+8)6[3n5 + 398n4 − 12900n3 − 81552n2 − 219968n − 357120

+ ζ(3)(n + 8) · 16(3n4 − 194n3 + 148n2 + 9472n + 19488)

+ ζ(4)(n + 8)3 · 288(5n + 22)

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1088 15 Path Integrals in Polymer Physics

− ζ(5)(n + 8)2 · 1280(2n2 + 55n + 186)]

+ ǫ4

128(n+8)8 [3n7 − 1198n6 − 27484n5 − 1055344n4

−5242112n3 − 5256704n2 + 6999040n − 626688

− ζ(3)(n + 8) · 16(13n6 − 310n5 + 19004n4 + 102400n3

− 381536n2 − 2792576n − 4240640)

− ζ2(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)

+ 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-polymerproperties must emerge in the limit n→ 0. There, ν−1 has the ǫ-expansion

ν−1 = 2− ǫ

4− 11

128ǫ2 + 0.114 425 ǫ3 − 0.287 512 ǫ4 + 0.956 133 ǫ5. (15.402)

This is to be compared with the much simpler result of the last section

ν−1 =D + 2

3= 2− ǫ

3. (15.403)

A term-by-term comparison is meaningless since the field-theoretic ǫ-expansion hasa grave problem: The coefficients of the ǫn-terms grow, for large n, like n!, so thatthe series does not converge anywhere! Fortunately, the signs are alternating andthe series can be resummed [13]. A simple first approximation used in ǫ-expansionsis to re-express the series (15.402) as a ratio of two polynomials of roughly equaldegree

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 Pade approximation. Its Taylor coefficients up to ǫ5 coincide with those ofthe initial series (15.402). It can be shown that this approximation would convergewith increasing orders in ǫ towards the exact function represented by the divergentseries. In Fig. 15.7, we plot the three functions (15.403), (15.402), and (15.404), thelast one giving the most reliable approximation

ν−1 ≈ 0.585. (15.405)

Note that the simple Flory curve lies very close to the Pade curve whose calculationrequires a great amount of work.

There exists a general scaling relation between the exponent ν and another ex-ponent appearing in the total number of polymer configurations of length L whichbehaves like

S = Lα−2. (15.406)

H. Kleinert, PATH INTEGRALS

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15.13 Fermi Fields for Self-Avoiding Lines 1089

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 Pade 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

α ∼ 1

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 treat-

ment, they are due to the use of the saddle point approximation and the fact thatthe δ-function does not completely rule out the crossing of the lines, as required bythe true self-avoidance of the polymer. The field theoretic ǫ-expansion, on the otherhand, which in principle can give arbitrarily accurate results, has the problem ofbeing divergent for any ǫ. Resummation procedures are needed and the order of theexpansion 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 fluctuatingcomplex fields Ψα and a U(n)-symmetric fourth-order self-interaction as in the action

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1090 15 Path Integrals in Polymer Physics

(15.397), the symmetric incorporation of a set ofm anticommuting Grassmann fieldsremoves the effect of m of the Bose fields. For free fields this observation is trivialsince 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 theform (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). Thisexponent is purely quadratic in the replica field, and each path integral over a Fermifield cancels a functional determinant coming from the Bose field.

This boson-destructive effect of fermions allows us to study theories with a neg-ative number of replica. We simply have to use more Fermi than Bose fields. More-over, we may conclude that a theory with n = −2 has necessarily trivial criticalexponents. From the above arguments it is equivalent to a single complex Fermifield theory with fourth-order self-interaction. However, for anticommuting Grass-mann fields, such an interaction vanishes:

(θ†θ)2 = [(θ1 − iθ2)(θ1 + iθ2)]2 = [2iθ1θ2]

2 = 0. (15.410)

Looking back at the ǫ-expansion for the critical exponent ν in Eq. (15.401) we canverify that up to the power ǫ5 all powers in ǫ do indeed vanish and ν takes themean-field value 1/2.

Appendix 15A Basic Integrals

∆(0, 0) = ∆(L,L) = L/3, (15A.1)

I1 =

∫ L

0

ds∆(s, s) = L2/6, (15A.2)

I2 =

∫ L

0

ds ∆ 2(s, s) = L/12, (15A.3)

I3 =

∫ L

0

ds∆2(s, s) = L3/30, (15A.4)

I4 =

∫ L

0

ds∆(s, s) ∆ 2(s, s) = 7L2/360, (15A.5)

I5 =

∫ L

0

ds

∫ L

0

ds′ ∆2(s, s′) = L4/90, (15A.6)

∆(0, L) = ∆(L, 0) = −L/6, (15A.7)

I6 =

∫ L

0

ds[

∆2(s, 0) + ∆2(s, L)]

= 2L3/45, (15A.8)

I7 =

∫ L

0

ds

∫ L

0

ds′ ∆(s, s) ∆ 2(s, s′) = L3/45, (15A.9)

I8 =

∫ L

0

ds

∫ L

0

ds′ ∆ (s, s)∆(s, s′) ∆ (s, s′) = L3/180, (15A.10)

I9 =

∫ L

0

ds∆(s, s)[

∆ 2(s, 0) + ∆ 2(s, L)]

= 11L2/90, (15A.11)

I10 =

∫ L

0

ds ∆ (s, s) [∆(s, 0) ∆ (s, 0) + ∆(s, L) ∆ (s, L)] = 17L2/360, (15A.12)

H. Kleinert, PATH INTEGRALS

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Appendix 15B Loop Integrals 1091

∆ (0, 0) = − ∆ (L,L) = −1/2, (15A.13)

I11 =

∫ L

0

ds

∫ L

0

ds′ ∆(s, s) ∆ (s, s′) ∆ (s′, s′) = L3/360, (15A.14)

I12 =

∫ L

0

ds

∫ L

0

ds′ ∆(s, s′) ∆ 2(s, s′) = L3/90. (15A.15)

Appendix 15B Loop Integrals

We list here the Feynman integrals evaluated with dimensional regularization rules whenever nec-essary. 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 dependingon ρn(s) ≡ δn + [δ(s) + δ(s− L)] /2 with δn = δ(0) − σn/L or ρk(s) = δk + [δ(s) + δ(s− L)] /2with δk = δ(0) − ik/L:

Hn(k)1 =

∫ L

0

ds

∫ L

0

ds′ρn(k)(s)ρn(k)(s′)∆2(s, s′),

= δ2n(k) I5 + δn(k) I6 +1

2

[

∆2(0, 0) + ∆2(0, L)]

, (15B.1)

Hn(k)2 =

∫ L

0

ds

∫ L

0

ds′ρn(k)(s′)∆(s, s) ∆ 2(s, s′) = δn(k) I7 +

I92, (15B.2)

Hn(k)3 =

∫ L

0

ds

∫ L

0

ds′ρn(k)(s′) ∆ d(s, s)∆2(s, s′) =

[

δn(k) I5 +I62

]

δ(0), (15B.3)

Hn(k)4 =

∫ L

0

ds

∫ L

0

ds′ρn(k)(s′) ∆ (s, s) ∆ (s, s′)∆(s, s′) = δn(k) I8 +

I102

. (15B.4)

When calculating 〈Rn 〉, we need to insert here δn = δ(0) − σn/L, thus obtaining

Hn1 =

L4

90δ2(0)+

L3

45(3−D−n) δ(0)+

L2

360

[

(45−24D+4D2)−4n(6−2D−n)]

, (15B.5)

Hn2 =

L3

45δ(0) +

L2

180(15−4D−4n), (15B.6)

Hn3 =

L4

90δ2(0)+

L3

90(3−D−n)δ(0), (15B.7)

Hn4 =

L3

180δ(0)+

L2

720(21−4D−4n) , (15B.8)

where the values for n = 0 correspond to the partition function Z = 〈R0 〉. The substitutionδk = δ(0) − ik/L required for the calculation of P (k;L) yields

Hk1 =

L4

90δ2(0) +

L3

45[2 + (−ik)] δ(0) +

L2

90(−ik) [4 + (−ik)] +

5L2

72, (15B.9)

Hk2 =

L3

45δ(0) +

L2

180[11 + 4(−ik)] , (15B.10)

Hk3 =

L4

90δ2(0) +

L3

90[2 + (−ik)] δ(0), (15B.11)

Hk4 =

L3

180δ(0) +

L2

720[17 + 4(−ik)] . (15B.12)

The other loop integrals are

H5 =

∫ L

0

ds

∫ L

0

ds′∆(s, s) ∆ 2(s, s′) ∆ d(s′, s′) = δ(0)I7 =L3

45δ(0), (15B.13)

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1092 15 Path Integrals in Polymer Physics

H6 =

∫ L

0

ds

∫ L

0

ds′ ∆ d(s, s)∆2(s, s′) ∆ d(s′, s′) = δ2(0)I5 =L4

90δ2(0), (15B.14)

H7 =

∫ L

0

ds

∫ L

0

ds′ ∆ (s, s) ∆ (s, s′)∆(s, s′) ∆ d(s′, s′) = δ(0)I8 =L3

180δ(0), (15B.15)

H8 =

∫ L

0

ds

∫ L

0

ds′ ∆ (s, s) ∆ (s, s′) ∆ (s, s′) ∆ (s′, s′) = − L2

720, (15B.16)

H9 =

∫ L

0

ds

∫ L

0

ds′ ∆ (s, s)∆(s, s′) ∆ d(s, s′) ∆ (s′, s′) =I102

− I8L

−H8 =7L2

360, (15B.17)

H10 =

∫ L

0

ds

∫ L

0

ds′∆(s, s) ∆ (s, s′) ∆ d(s, s′) ∆ (s′, s′) =I94

− I72L

=7L2

360, (15B.18)

H11 =

∫ L

0

ds

∫ L

0

ds′∆(s, s) ∆ d2(s, s′)∆(s′, s′) = δ(0)I3 + (I1L

)2 − 2(I3 − I11)

L− 2I4,

+ 2[

∆2(L,L) ∆ (L,L) − ∆2(0, 0) ∆ (0, 0)]

− 2H10 =L3

30δ(0) +

L2

9, (15B.19)

H12 =

∫ L

0

ds

∫ L

0

ds′ ∆ 2(s, s′) ∆ 2(s, s′) =L2

90, (15B.20)

H13 =

∫ L

0

ds

∫ L

0

ds′∆(s, s′) ∆ (s, s′) ∆ (s, s′) ∆ d(s′, s) =I42

− I122L

− H12

2= − L2

720, (15B.21)

H14 =

∫ L

0

ds

∫ L

0

ds′∆2(s, s′) ∆ d2(s, s′) = δ(0)I3 −2(I3 − I12)

L− 2I4 +

I5L2

,

+ 2[

∆2(L,L) ∆ (L,L) − ∆2(0, 0) ∆ (0, 0)]

− 2H13 =L3

30δ(0) +

11L2

72. (15B.22)

Appendix 15C Integrals Involving Modified Green Func-tion

To demonstrate the translational invariance of results showed in the main text we use the slightlymodified Green function

∆(s, s′) =L

3a− | s− s′ |

2− (s + s′)

2+

(s2 + s′2)

2L, (15C.1)

containing an arbitrary constant a. The following combination yields the standard Feynman prop-agator for the infinite interval [compare (3.249)]

∆F(s, s′) = ∆F(s−s′) = ∆(s, s′)− 1

2∆(s, s)− 1

2∆(s′, s′) = −1

2(s−s′). (15C.2)

Other useful relations fulfilled by the Green function (15C.1) are, assuming s ≥ s′,

D1(s, s′) = ∆2(s, s′)−∆(s, s)∆(s′, s′) = (s−s′)

[

s(L−s)

L+

(s−s′)(s+s′)2

4L2−La

3

]

,

(15C.3)

D2(s, s′) = ∆(s, s)−∆(s′, s′) =

(s−s′)(s+s′−L)

L. (15C.4)

The following integrals are needed:

J1(s, s′) =

∫ L

0

dt∆(t, t) ∆ (t, s) ∆ (t, s′) =(s4+s′4)

4L2− (2s3+s′3)

3L,

+((a+3)s2+as′2)

6− sa

3L+

(20a−9)

180L2, (15C.5)

H. Kleinert, PATH INTEGRALS

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Notes and References 1093

J2(s, s′) =

∫ L

0

dt ∆ (t, t)∆(t, s) ∆ (t, s′) =(−2s4+6s2s′2+3s′4)

24L2,

+(3s3−3s2s′−6ss′2−s′3)

12L− (5s2−12ss′−4as′2)

24− s′a

6L+

(20a−3)

720L2,

(15C.6)

J3(s, s′) =

∫ L

0

dt∆(t, s)∆(t, s′) =−(s4+6s2s′2+s′4)

60L+

(s2+3s′2)s

6,

− (s2+s′2)

6L+

(5a2−10a+6)

45L3. (15C.7)

These are the building blocks for other relations:

〈f2(q(s), q(s′))〉 =(d−1)

2

[

D1(s, s′)− (d+1)

4D2

2(s, s′)

]

=−(d−1)(s−s′)

4

×[

2La

3+

(d−3)s−(d+1)s′

2− (d−1)s2−(d+1)s′2

L+d(s−s′)(s+s′)2

2L2

]

, (15C.8)

and

K1(s, s′) = J1(s, s

′)− 1

2J1(s, s)−

1

2J1(s

′, s′),

= − (s−s′)

[

La

6− (s+s′)

4+

(s2+ss′+s′2)

6L

]

, (15C.9)

K2(s, s′) = J2(s, s

′)+J2(s′, s)−J2(s, s)−J2(s

′, s′),

= − (s−s′)2[

1

4− (5s+7s′)

12L+

(s+s′)2

4L2

]

, (15C.10)

K3(s, s′) = J3(s, s

′)− 1

2J3(s, s)−

1

2J3(s

′, s′) =−(s−s′)2[

(s+2s′)

6− (s+s′)2

8L

]

, (15C.11)

K4(s, s′) = ∆(0, s)∆(0, s′)− 1

2∆2(0, s)− 1

2∆2(0, s′) =− (s−s′)2(s+s′−2L)2

8L2, (15C.12)

K5(s, s′) = ∆(L, s)∆(L, s′)− 1

2∆2(L, s)− 1

2∆2(L, s′) =− (s2−s′2)2

8L2. (15C.13)

Notes and References

Random chains were first considered byK. 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 byL.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 byS.F. Edwards, Proc. Phys. Soc. Lond. 85, 613 (1965); 88, 265 (1966); 91, 513 (1967); 92, 9 (1967).See alsoH.P. Gilles, K.F. Freed, J. Chem. Phys. 63, 852 (1975)and the comprehensive studies byK.F. Freed, Adv. Chem. Phys. 22,1 (1972);

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1094 15 Path Integrals in Polymer Physics

K. Ito and H.P. McKean, Diffusion Processes and Their Simple Paths, Springer, Berlin, 1965.

An alternative model to stimulate stiffness was formulated byA. 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, thedistribution functions agree quite well with those of the Kratky-Porod chain. The advantage ofthis 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 byR.J. Rubin, J. Chem. Phys. 21, 2073 (1953).The simple scaling law 〈R2〉 ∝ L6/(D+2), D < 4 was first found byP.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 polymerconfigurations byC. 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, seeP.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 inH. 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 byO. Kratky and G. Porod, Rec. Trav. Chim. 68, 1106 (1949).The lowest moments were calculated byJ.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 byR. Koyama, J. Phys. Soc. Japan, 34, 1029 (1973).Still higher moments are found numerically byH. Yamakawa and M. Fujii, J. Chem. Phys. 50, 6641 (1974),and in a large-stiffness expansion inT. 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 obtained from ibid.http/b5/pgm15. The program needsless 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

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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 seeH. 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 〈R2〉 and other critical expo-nents were given byP.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, Singa-pore, 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 orderǫ5, seeH. Kleinert, J. Neu, V. Schulte-Frohlinde, K.G. Chetyrkin, and S.A. Larin, Phys. Lett. B272, 39 (1991) (hep-th/9503230).

[13] For a comprehensive list of the many references on the resummation of divergent ǫ-expansions obtained in the field-theoretic approach to critical phenomena, see the textbook[11].

[14] A.J. McKane, Phys. Lett. A 76, 22 (1980).