Statistical Physics of the Freely Jointed Chain

PHYSICAL REVIEW E VOLUME 53, NUMBER 6 JUNE 1996

Statistical physics of the freely jointed chain

Martial Mazars Laboratoire de Physique Theórique et Hautes Energies, Universite´ de Paris XI, Baˆtiment 211, 91405 Orsay Cedex, France ͑Received 31 July 1995͒ Using the method of constraints proposed by S. F. Edwards and A. G. Goodyear ͓J. Phys. A 5, 965 ͑1972͒; 5, 1188 ͑1972͔͒, we do a complete calculation of the canonical partition function of a freely jointed chain ͑FJC͒ from its classical Hamiltonian. We show how the constraints reduce the phase space of an ideal gas of monomers to the phase space of a FJC, and how they permit one to find the canonical partition function. By using this function, it is possible to study thermodynamical properties of FJC’s and to build other thermodynamical ensembles via Laplace transforms. Thus we define a grand canonical ensemble where the monomer number of the FJC can fluctuate; in this ensemble, the FJC of infinite length is the asymptotic state at low and high temperatures. The critical exponents ␥ and ␯ for FJC’s are calculated and found to be equal to the Gaussian polymer exponents. Connections between the properties of FJC’s and random walks on regular lattices are also discussed. ͓S1063-651X͑96͒07106-1͔

PACS number͑s͒: 36.20.Ϫr, 05.40.ϩj

I. INTRODUCTION simplest possible geometrical constraints, namely, bonds with constant lengths. Thus a FJC is made up of ͑Nϩ1͒ In the statistical theory of polymers, the first system massive points, linearly and freely jointed by N links of con- which has been studied is the ideal polymer. When the de- stant length ͑Fig. 1͒. This freely jointed chain was first stud- gree of polymerization of the polymer is large, the polymer’s ied by Kramers in 1946 ͓7͔, and since then by many others end to end distance has a Gaussian distribution. This fact can ͓8–12͔. If we remove the constraints, we will have an ideal be very well understood with the help of the central limit theorem ͓1͔. The Gaussian model of a polymer is very useful because it retains important characteristics of the polymers, and allows one to interpret and calculate some of the critical exponents ͓2,3͔. The values of these exponents can serve as a reference for more realistic models of polymers. This ideal model is thus used as a reference model for the study of macromolecules and polymers, in the same way as the ideal gas is used as a reference model for the study of real gases and liquids ͓2–6͔. However, the description of an ideal polymer, based on a Gaussian distribution, or a random walk on a regular lattice, does not provide a direct relationship to the fundamental laws of statistical physics, especially the relationship between the Hamiltonian of the polymer and the canonical partition function. The present work aims at filling this gap. To achieve this aim, we will use the simplest possible polymer model, the freely jointed chain ͑FJC͒, which we will also dub the ideal ghost polymer ͑IGP͒. This latter name seems to be a pleonasm in consideration of the customary use of the term ‘‘ideal’’ in polymer physics. In fact, in models of macromolecules and polymers, the interactions between monomers are taken to be ͑1͒ chemical and physical interactions ͑such as excluded volume effects, electrostatic interactions, etc.͒ and ͑2͒ geometrical constraints which define the structure and the geometry of the molecule. The freely jointed chain has two major characteristics; it is a ghost polymer and an ideal polymer. By ‘‘ghost,’’ we then mean that there is interaction neither between the bonds nor between the monomers of the polymer. In this sense, the FJC is different from a polymer at its ␪ point ͓6͔, but some of the physical properties, independent of the ghost characteris- FIG. 1. Conformations of a freely jointed chain in three dimen- tic, will be common to both systems. For ideal we take the sions: ͑a͒ Nϭ20; ͑b͒ Nϭ500.

1063-651X/96/53͑6͒/6297͑23͒/$10.0053 6297 © 1996 The American Physical Society 6298 MARTIAL MAZARS 53 gas of ͑Nϩ1͒ distinguishable monomers. The geometrical instance, in binary mixtures ͑for a review see ͓17͔͒ or liquid constraints transform the system of ͑Nϩ1͒ distinguishable crystals ͓18,19͔. Finally, in order to compare the FJC model monomers into a macromolecule by changing the topology with other ideal polymer models, we compute the critical of the phase space associated with the ͑Nϩ1͒ monomers. exponents ␥ and ␯ associated with the FJC. They are found The study of the thermodynamical and statistical properties to be equal to the critical exponents of the Gaussian polymer. of the macromolecule is equivalent to the study of an ideal In this work, d is the dimension of the space containing gas in this modified phase space; this is the point of view the polymers. It is taken to be a real number greater than or adopted in the late 1960s by the Soviet school ͓6,13͔.Atthe equal to 2. Comparison with random walks on a regular lat- beginning of the 1970s, Edwards and Goodyear put forward tice does not lead to any serious problems of interpretation as a study of the dynamics of a polymer using the microcanoni- long as d has an integer value. When d takes noninteger values the random walks may be ill defined 20 . Recent cal ensemble and expressing the geometrical constraints with ͓ ͔ work by Bender and co-workers ͓20,21͔ permits one, subject the help of Dirac distributions ͓9͔. The main purpose of the to topological regularity 22 , to extend the model to nonin- present work is to compute the canonical partition function ͓ ͔ teger dimensions of space. of the FJC and to study its equilibrium thermodynamical properties, especially the equilibrium states of a FJC which can exchange monomers with an infinite ideal gas of mono- II. REDUCTION OF THE PHASE SPACE mers. BY CONSTRAINTS The present work is organized as follows. In Sec. II, we Following Edwards and Goodyear ͓9͔, we express the study the reduction of phase space of a system of ͑Nϩ1͒ geometrical constraints with Dirac distributions. These dis- ideal and indistinguishable monomers under the action of the tributions reduce the phase space ␸ of an ideal gas of ͑Nϩ1͒ geometrical constraints expressed via Dirac distributions. monomers to the restricted phase space ␸rest of the FJC. First, The Dirac distributions allow us to express the partition the monomers within the polymer have to become distin- function as an integral over the full phase space rather than guishable. Nevertheless, in the polymer an indistinguishabil- over the reduced phase space which is topologically more ity remains due to the fact that one can label the monomers complicated. In this way we will derive two nonintegrated in one direction or the other. A memory effect is induced by analytical expressions for the partition function. In Sec. III, the geometrical constraints imposed on the system. This ef- the temperature dependence of the partition function is ex- fect may be interpreted as follows. At any point of the phase tracted. We show that the partition function is the product of space ␸rest , the monomer numbered i remembers that its left a function depending on the temperature and the degree of and right neighbors are, e.g., the monomers ͑iϪ1͒ and ͑iϩ1͒, polymerization, for which a closed analytical expression is respectively. It is possible to reverse the labeling, so that the obtained, and a function depending only on the degree of number of allowed permutations for the monomers in ␸rest is polymerization. The latter is expressed as an integral involv- only 2!, whereas the number of allowed permutations in ␸ is ing Bessel functions. Sections IV and V are devoted to the ͑Nϩ1͒! The memory effect results in a factor ͑Nϩ1͒!/2! in study of this integral. In Sec. IV, we present an analytical the partition function. Thus it becomes impossible to distin- study by series expansions of the exponentials appearing in guish the beginning of the polymer from its end. We will see the integral. These series expansions permit us to explore the that the Feynman rules associated with the polymer initiator analogy between quantum mechanics and polymer physics and polymer terminator are the same. via Feynman formulation rather than via the Fokker-Planck To use Dirac distributions in the partition function with- equation ͓1,6,9͔. Then, by using a method close to conven- out any dimensionality problem, we have to pay attention to tional renormalization theory and similar to the decimation their actions and meaning. The Dirac distributions are de- method proposed by de Gennes ͓14,3,6͔, we give analytical fined in such a way that for a physical observable A, there is approximations, involving hypergeometric functions, for the the dimensional equation partition function of the FJC. In Sec. V, we perform a numerical study of the integral using a Monte Carlo algorithm together with an importance sampling method. Comparison dnA␦͑A͒ϭ1 ͓␦͑A͔͒ϭ͓A͔Ϫn. ͑1͒ ͵ ⇒ of the numerical results with the analytical results of Sec. IV is made and an interpretation is given for the odd-even os- So with the normalization ͑1͒ of the Dirac distribution, the cillation effect observed in polymers ͓15,16͔: The agreement probability distribution of the FJC in ␸ derived from that in between the numerical and analytical results highlights the the space ␸rest by expressing the geometrical constraint as a quality of the analytical approximations for the partition Dirac distribution must be multiplied by a constant having function of the FJC. dimension such that in ␸ the distribution will be dimension- From the known canonical partition function of the FJC, less ͓cf. Eq. ͑1͔͒. we can derive its thermodynamical and statistical properties. For the FJC, the constraints on the bonds are In Sec. VI, we begin the study of these properties in the 2 2 canonical ensemble for a system of FJC’s. Then, considering ␣␦„͑riϪ1Ϫri͒ Ϫa …, ͑2͒ the FJC as a gas of ͑Nϩ1͒ monomers, we can build the microcanonical and the grand canonical ensembles by where ␣ is the multiplicative constant. Bonds with a constant Laplace transforms. In the grand canonical ensemble the low length induce, from a dynamical point of view, a relation and high temperature states are FJC’s of infinite length. This between the conjugate momenta associated with the degrees reentrant behavior is similar to the reentrant behavior of the of freedom of the system ͓9͔. This classical mechanics effect less ordered phases observed in other systems, such as, for is one of the major difficulties encountered in simulations 53 STATISTICAL PHYSICS OF THE FREELY JOINTED CHAIN 6299 using the molecular dynamics method for the study of com- N N ͑d͒ 1 1 ␤ 2 plex molecules when some degrees of freedom are frozen QN ϭ ͑Nϩ1͒d ͟ drndpn exp Ϫ ͚ pn , 2 h ͵␸ nϭ0 ͩ 2m nϭ0 ͪ ͑see ͓23,24͔, and for a review ͓25͔͒. In the FJC, the distribu- rest tion ͑2͒ induces the dynamical constraint ͓9͔ ͑10͒ 1 where the factor 2 results from the indistinguishability effect ␥␦„͑piϪ1Ϫpi͒•͑riϪ1Ϫri͒…, ͑3͒ discussed above. The topology of ␸rest may be complicated; where ␥ plays the same role as ␣. in particular, it depends on the value of d. With the method The two constants ␥ and ␣ must be related to physical of constraints, the domain of integration simplifies at the parameters of the system. To do so, a convention concerning expense of a modification of the integrand. With the con- the canonical partition function must be chosen. A plausible stants ␣ and ␥ as determined above the partition function choice, in view of the symmetry properties of the system, written in ␸ is (d) appears to be to take the partition function Q छ (T,L)ofa 1 1 particle in a box of length L in a space of dimension d, equal ͑d͒ 2 N QN ϭ ͑Nϩ1͒d ͑ha ͒ dr0dp0 (dϩ1) 2 h ͵␸ to the partition function Q (T,L) of a particle moving on a hypersphere of radius L in a space of dimension ͑dϩ1͒. (d) N Q छ (T,L) is given by 2 2 ϫ dundpn␦͑unϪa ͒␦„͑pnϪpnϪ1͒•un… n͟ϭ1 1 ͑d͒ ␤ 2 Qछ ͑T,L͒ϭ dr dp exp Ϫ p N hd ͵छ͑d͒ ͩ 2m ͪ ␤ 2 ϫexp Ϫ ͚ pn , ͑11͒ ͩ 2m nϭ0 ͪ mL2 d/2 ϭ , ͑4͒ ͩ 2␲␤ប2ͪ where the bond vectors are defined by

(dЈ) unϭrnϪrnϪ1 for 1рnрN. ͑12͒ and Q (T,L) is given by In the next section, we factorize the temperature depen- 1 ␤ ͑dЈ͒ 2 dence in the partition function ͑11͒. Q ͑T,L͒ϭ dr dp exp Ϫ p . ͑5͒ hdЈ ͵͑dЈϪ1͒ ͩ 2m ͪ III. DEPENDENCE ON TEMPERATURE With use of the Dirac distribution the latter can be written OF THE PARTITION FUNCTION OF A FJC To extract the dependence on temperature in the partition d 1 Q͑ Ј͒͑T,L͒ϭ ␣␥ dr dp ␦͑r2ϪL2͒␦͑p r͒ function of the FJC, the integration over all momenta must dЈ ͵ dЈ dЈ • h R ϫR be done, and the variables chosen such that the nonintegrated part of the partition function depends only on N. As in Sec. ␤ ϫexp Ϫ p2 . ͑6͒ II, the dynamical constraints are expressed through their ͩ 2m ͪ Fourier transforms

Use of the Fourier transform of the dynamical constraint, 1 ϩϱ ␦ ͑p Ϫp ͒ u ϭ d⍀ „ n nϪ1 • n… 2␲ ͵ n 1 ϩϱ Ϫϱ ␦͑p•r͒ϭ d⍀ exp͑ j⍀p•r͒, ͑7͒ 2␲ ͵Ϫϱ ϫexp͓ j⍀nun•͑pnϪpnϪ1͔͒. ͑13͒ gives Using the property ␦ (␭x)ϭ͑1/͉␭͉͒␦ (x) of the Dirac distribution and replacing ͑13͒ in ͑11͒, we obtain 2 ͑dЈϪ1͒/2 ␣␥ mL N Q͑dЈ͒͑T,L͒ϭ . ͑8͒ 1 1 h N u2 hL2 ͩ2␲␤ប2ͪ ͑d͒ n QN ϭ ͑Nϩ1͒d dr0 ͟ dun␦ 2Ϫ1 h 2 ͩ 2␲ͪ ͵ nϭ1 ͩ a ͪ Thus, with dЈϭdϩ1, in order to have N N N ␤ ͑d͒ ͑dϩ1͒ ϫ d dp dp exp Ϫ p2 Qछ ͑T,L͒ϭQ ͑T,L͒, ͑9͒ ͟ ⍀n 0 ͟ n ͚ n ͵ nϭ1 ͵ nϭ1 ͩ 2m nϭ0 2 the equality ␣␥ϭhL is required. To satisfy the dimensional N equation ͑1͒, dimensional analysis gives ␣ϭL2 and ␥ϭh.Of ϩ j ͚ ⍀nun•͑pnϪpnϪ1͒ . ͑14͒ course, by choosing another convention for the partition nϭ1 ͪ functions, for instance, by taking a hypersurface with a different geometry or by taking a different radius for the hyper- The integral over pn is a Gaussian integral, sphere, other relations could be obtained for the constants ␣ A Ϫ1/2 and ␥. 1 1 dX exp͑Ϫ XtAXϩJtX͒ϭ det exp͑ JtAϪ1J͒, With the convention ͑9͒, the partition function of the FJC ͵ 2 ͫ ͩ 2␲ͪͬ 2 in the canonical ensemble is deﬁned by the integral ͑15͒ 6300 MARTIAL MAZARS 53 where N 2 N ma 2 ϫ ͟ d⍀n exp Ϫ ͚ ⍀n ͵ nϭ1 ͩ ␤ nϭ1 ͪ

Xϭ͑pn͒0рnрN , N 2 ϫ duˆn␦͑uˆnϪ1͒ ͵ n͟ϭ1 Jϭ j͓Ϫ⍀1u1 ,͕͑⍀nunϪ⍀nϩ1unϩ1͖͒,⍀NuN͔, ͑16͒ ma2 NϪ1 ˆ ˆ ϫexp ͚ ͑⍀nun͒•͑⍀nϩ1unϩ1͒ . ͫ ␤ ͩ nϭ1 ͪͬ Aϭ͑␤/m͒͑I͑Nϩ1͒ Id͒, ͑21͒

A Ϫ1/2 2␲m d͑Nϩ1͒/2 The last equation shows how the dynamical constraints in- det ϭ , ͑17͒ duce a coupling between the bond vectors. The integration ͫ ͩ 2␲ͪͬ ͩ ␤ ͪ over the set of variables ͕uˆ n͖ is not straightforward. Deﬁning and N ͑d͒ ˆ ˆ 2 U ϭ ͵ ͟ dun␦͑unϪ1͒ m N NϪ1 nϭ1 1 t Ϫ1 2 2 2 J A JϭϪ ͚ ⍀nunϪ ͚ ͑⍀nun͒•͑⍀nϩ1unϩ1͒ . 2 NϪ1 ␤ ͩnϭ1 nϭ1 ͪ ma ˆ ˆ ϫexp ͚ ͑⍀nun͒•͑⍀nϩ1unϩ1͒ ͑22͒ ͑18͒ ͫ ␤ ͩ nϭ1 ͪͬ

2 To integrate over r0 , the box which contains the polymer and xϭ(ma /␤)⍀NϪ1⍀N , the integral over uˆ N is is assumed to be larger than the typical size of the polymer. After integration over all the momenta and over r Eq. ͑14͒ 0 z͑d͒͑x͒ϭ duˆ ␦͑uˆ 2 Ϫ1͒exp͑xuˆ uˆ ͒, ͑23͒ becomes ͵ N N NϪ1• N

x ϩ jϱ N d͑Nϩ1͒/2 ͑d͒ 2 1 1 h 2␲m z ͑x͒ϭ duˆ N du exp͓xu͑1ϪuˆN͔͒ Q͑d͒ϭ V 2␲ j ͵ ͵Ϫ jϱ N 2 h͑Nϩ1͒d ͩ 2␲ͪ ͩ ␤ ͪ ϫexp͑xuˆ NϪ1•uˆ N͒, ͑24͒

N u2 N x ␣ϩ jϱ n z͑d͒͑x͒ϭ dr exp͑xr͒ duˆ ϫ ͟ dun␦ 2Ϫ1 ͟ d⍀n ͵ ͵ N ͵ nϭ1 ͩ a ͪ ͵ nϭ1 2␲ j ␣Ϫ jϱ ˆ 2 ˆ ˆ ϫexp͓Ϫx͑ruNϪuNϪ1•uN͔͒. ͑25͒ m N 2 2 In Eqs. ͑23͒–͑25͒, the Dirac distribution has been expressed ϫexp Ϫ ͚ ⍀nun ͫ ␤ ͩ nϭ1 through its Fourier transform, the order of integrations has been permuted, and we have set rϭuϩ␣ with ␣ such that 2 ͑uϩ␣͒uˆ NϪuˆ NϪ1 uˆ NϾ0. NϪ1 • The integral over uˆ N is Gaussian, with the result Ϫ ͚ ͑⍀nun͒•͑⍀nϩ1unϩ1͒ͪ ͬ. ͑19͒ nϭ1 x ␣ϩ jϱ ␲ d/2 1 z͑d͒͑x͒ϭ dr exp x rϩ , ͑26͒ 2␲ j ͵␣Ϫ jϱ ͩxrͪ ͫ ͩ 4rͪͬ

1 x 2␲ d/2 2␣ϩ jϱ ͑d͒ Ϫd/2 The definition z ͑x͒ϭ dw͑w͒ 2 2␲ j ͩ x ͪ ͵2␣Ϫ jϱ x 1 1 ϫexp wϩ , ͑27͒ uˆ ϭ u ͑20͒ ͫ2 ͩ wͪͬ n a n 1 2␲ d/2 z͑d͒͑x͒ϭ x I ͑x͒. ͑28͒ allows us to work with dimensionless bond vectors. 2 ͩ x ͪ ͑d/2Ϫ1͒ The integrand in ͑19͒ is always positive; permutation of the order of integrations and use of the property The function I␮(x) is the modified Bessel function of the ͐ f (x)g(x)␦(x)dxϭf(0)͐g(x)␦(x)dx gives first kind. The integral z(d)(x) is independent of the direction of uˆ , but depends on ⍀ and ⍀ . The integrals over N d͑Nϩ1͒/2 NϪ1 NϪ1 N d 1 1 h 2␲m each bond are independent and the integral over the bond uˆ Q͑ ͒ϭ VaNd 1 N 2 h͑Nϩ1͒d ͩ 2␲ͪ ͩ ␤ ͪ produces a factor S ͑1͒, the surface of a sphere with a unit d 53 STATISTICAL PHYSICS OF THE FREELY JOINTED CHAIN 6301 radius in a space of d dimensions. The dependence on ⍀1 is Using this transformation in ͑32͒, the nonintegrated part be- included in the integral over the bond uˆ 2 . Equation ͑22͒ can comes be written in the form N N ma2 ͑NϪ1͒ J͑d͒ϭ d␻ ͉det͓Ja͑w͔͉͒ ͑␻ ͒͑1Ϫd/2͒ ͑d͒ N ͵ ͟ n ͟ n U ϭSd͑1͒ nϭ1 ͩnϭ2 ͩ 2␤ ͪ N nϪ1 ͑NϪ1͒ 2␲␤ d͑NϪ1͒/2 2 ͑Ϫ1͒p 2 ͑1Ϫd/2͒ ϫI͑d/2Ϫ1͒͑␻n͒ exp Ϫ ͚ ͟ ͑␻͑nϪp͒͒ ␻n ϫ 2 ͟ ͑⍀n⍀nϩ1͒ ͪ ͩ nϭ2 pϭ1 ͩ ma ͪ nϭ1

ma2 2 ϫI ⍀ ⍀ . ͑29͒ Ϫ␻1 . ͑35͒ ͑d/2Ϫ1͒ͩ ␤ n nϩ1ͪ ͪ

(d) A particular case of this result has been obtained by Stan- In order to identify the contribution to Q N from each bond, ley ͓26͔ for a chain of classical isotropic spins with nearest we set neighbor interactions. Stanley’s result is recovered if all ⍀n’s are taken to be equal. In this case U(d) is proportional to the NϪ1 ͑Ϫ1͒p partition function obtained by Stanley. If some other sets of ANϭ ͟ ͑␻͑NϪp͒͒ ͑36͒ ͯ pϭ1 ͯ ͕⍀n͖ values are chosen, partition functions of different spin systems can be obtained; for instance, if we set ⍀nϭ⍀ for and nÞi and ⍀iÞ⍀, the partition function of a classical isotropic spin chain with an impurity spin on the site i is obtained. In spin systems, the coupling between spins results from the ͑d͒ ͑1Ϫd/2͒ magnetic field induced by the spins. In the FJC, the coupling f N ϭAN ͵ d␻N␻N I͑d/2Ϫ1͒͑␻N͒ between bond vectors results from the dynamical constraints NϪ1 induced by the bonds. 2 ͑Ϫ1͒p 2 Use of the transformed variables ϫexp Ϫ ͟ ͑␻͑NϪp͒͒ ␻N . ͑37͒ ͩ pϭ1 ͪ ma2 1/2 ␥ ϭ ⍀ , ͑30͒ For the Bessel function, we use the representation n ͩ ␤ ͪ n ͑x/2͒␮ 1 leads to an integral which depends on N only. After straight- 2 ͑␮Ϫ1/2͒ xt I␮͑x͒ϭ ͑1Ϫt ͒ e dt. forward simplifications, the partition function ͑21͒ becomes ⌫͑␮ϩ1/2͒⌫͑1/2͒ ͵Ϫ1 ͑38͒ d/2 2/͑dϪ1͒ 2 N͑dϪ1͒/2 ͑d͒ m 1 ma ͑d͒ QN ϭSd͑1͒V 2 2 2 JN , ͩ 4␲ ␤ប ͪ ͫͩ 2ͪ ␤ប ͬ IV. ANALYTICAL STUDY OF J „d… ͑31͒ N In this section, devoted to the degree of polymerization, (d) where បϭh/2␲ and we try to compute the function J N for all values of N. The computation is made by extracting the contribution of each N ͑NϪ1͒ bond to J (d). ͑d͒ ͑1Ϫd/2͒ N JN ϭ ͟ d␥n ͟ ͑␥n␥nϩ1͒ By series expansion of the exponentials, each integral is ͵ nϭ1 ͩ nϭ1 transformed into a sum over the natural numbers. There are N ͑NϪ1͒ sums of that kind for a FJC with N bonds. These 2 (d) ϫI͑d/2Ϫ1͒͑␥n␥nϩ1͒ exp Ϫ ͚ ␥n . ͑32͒ sums show that J N can be expressed in terms of a special ͪ ͩ nϭ1 ͪ 1 function having the special value 4 for its arguments. If one were able to find an analytical form for this function, then All the temperature dependence is explicitly contained in one would have found an analytical form for the partition 31 , whereas the integral 32 contains all the nontrivial de- ͑ ͒ ͑ ͒ function of a FJC. Unfortunately, this function is directly pendence on N. The main difficulty in computing the latter related to the multiple hypergeometric functions, so that an integral comes from the coupling between and . ␥n ␥nϩ1 analytical form of the partition function of a FJC can easily For the transformation be obtained only for small values of N. On the other hand, the series expansion of J (d) is useful to exhibit the analogy ␻ ϭ␥ ␥ for 2рnрN, N n nϪ1 n between polymer physics and quantum mechanics via the ͑33͒ Feynman formulation. Usually, this analogy is demonstrated ␻1ϭ␥1 , by using the Fokker-Planck equation for the probability distribution function of the end to end distance ͓1,6,9,27͔;it the Jacobian matrix is triangular, giving the Jacobian may also be described via Green’s functions ͓6͔. This anal- N nϪ1 ogy enables the use of methods closely related to the con- ͑Ϫ1͒p ventional renormalization of the full propagator ͓28͔ to de- det͓Ja͑␻i͔͒ϭ͟ ͟ ͑␻͑nϪp͒͒ . ͑34͒ (d) nϭ2 pϭ1 rive various approximations for J N . 6302 MARTIAL MAZARS 53

To test the analytical approximations derived in this sec- which, after a straightforward integration, leads to tion a Monte Carlo algorithm described in the next section is (d) used to obtain numerical results for J N . The agreement be- d/2 ϱ ϩ n tween the approximate analytical results of this section and ͑d͒ 1 1 1 ⌫͑n 1/2͒ 1 2n 2n f N ϭ ͚ ANϪ1␻NϪ1 . the numerical results of the next section will show the reli- ͩ 2ͪͩ2ͪ nϭ0 n! ⌫͑nϩd/2͒ ͩ 4ͪ ability of the theoretical approach. ͑43͒ The present section is divided into five subsections. In (d) (d) Sec. IV A, we make the series expansion of J N . In Sec. Substituting ͑43͒ into ͑35͒, f NϪ1 is extracted as (d) IV B, we define the function related to J N . This function can be defined in several ways depending on the number of ϱ 1 1 d/2 1 1 ⌫͑nϩ1/2͒ 1 n variables the special value 4 is assigned to. The series expan- ͑d͒ 2n (d) f NϪ1ϭ ͚ ANϪ1ANϪ1 sion of J N is composed of ͑NϪ1͒ sums; thus the function ͩ 2ͪ 2 nϭ0 n! ⌫͑nϩd/2͒ ͩ 4ͪ (d) giving J N will be a scalar field on a Euclidean space of dimension ͑NϪ1͒. In Sec. IV C, we write the Feynman rules (d) ͑1/2͒͑d/2Ϫ1͒ 1 for the computation of J N . In Sec. IV D, we use these Feyn- 2 ͑dϪ3͒/2 ϫ dtNϪ1͑1ϪtNϪ1͒ man rules to develop the conventional renormalization and ⌫͓͑dϪ1͔/2͒⌫͑1/2͒ ͵Ϫ1 derive approximations. Finally, in Sec. IV E, we compute (d) J N exactly for small values of N. ϩϱ 2n 2 2 „d… ϫ d␻NϪ1␻NϪ1 exp͑ϪANϪ1␻NϪ1 A. Series expansion of J N ͵Ϫϱ (d) In the relation ͑37͒ defining f N , the Bessel functions are expressed with their representation ͑38͒; this gives ϩtNϪ1␻NϪ1), ͑44͒ ͑d/2Ϫ1͒ ϩϱ ͑d͒ ͑1/2͒ f N ϭ AN d␻N and becomes, after integration over ␻NϪ1, ⌫͓͑dϪ1͔/2͒⌫͑1/2͒ ͵Ϫϱ

1 2 2 ϱ 2 ͑dϪ3͒/2 ϪA ␻ ϩ␻ t d/2 n1 ϫ ͑1Ϫt ͒ e N N N NϪ1dt . 1 1 1 ⌫͑n1ϩ1/2͒ 1 2n ͵ NϪ1 NϪ1 ͑d͒ 1 Ϫ1 f NϪ1ϭ ͚ ANϪ1 2 ͩ 2ͪ n ϭ0 n ! ⌫͑n ϩd/2͒ ͩ 4ͪ 1 1 1 ͑39͒

1 Since in ͑39͒ the integrand is always positive and the 2 ͑dϪ3͒/2 (d) ϫ dtNϪ1͑1ϪtNϪ1͒ value of f N ﬁnite, the integration orders can be permutated. ͵Ϫ1 The integral over ␻N is Gaussian; thus

d/2 2 1/2 1 2 2 2n1Ϫ1 ͑d͒ ͑ ͒ 2 dϪ3 /2 Ϫt /4A 1 d tNϪ1 f ϭ dt 1Ϫt ͑ ͒ e NϪ1 N. N NϪ1͑ NϪ1͒ ϫ tNϪ1 exp . ⌫͓͑dϪ1͔/2͒ ͵0 A2 dt2n1Ϫ1 ͫ ͩ 4A2 ͪͬ NϪ1 NϪ1 NϪ1 ͑40͒ ͑45͒

The factor AN deﬁned in ͑36͒ and stemming from the 2 Expanding the exponential in ͑45͒, taking the derivative, and Jacobian ͑34͒ is exactly cancelled by a factor ͱANϭ͉AN͉ integrating gives ϭAN coming from the Gaussian integration. Starting from (d) ͑40͒, we may either put ͑40͒ into ͑35͒ to extract f NϪ1 and make the Gaussian integration over ␻ , or make a series d/2 d/2 ϱ ϩ NϪ1 ͑d͒ 1 1 1 ⌫͑n1 1/2͒ expansion of the exponential of ͑40͒ and integrate over t . f NϪ1ϭ ͚ NϪ1 ͩ 2ͪ ͩ 2ͪ n ϭ0 n ! ⌫͑n ϩd/2͒ In this section the latter way is chosen, while the former will 1 1 1 be used in the next section. Since the integral is convergent ϱ both ways must lead to the same result. From the deﬁnition 1 n1 1 n1 1 1 of AN ͑36͒, the following relation holds: ϫ ͚ ͩ 4ͪ ͩ 4ͪ 2 n ϭ0 ͑n ϩn Ϫ1͒! 2 2 1

1 2 2 2 ϭANϪ1␻NϪ1. ͑41͒ A n2 2n2 N ͓2͑n2ϩn1͒Ϫ1͔! 1 ⌫͑n2ϩ1/2͒ 1 ϫ . ͑2n ͒! 4 ⌫͑n ϩd/2͒ A The series expansion of the exponential gives 2 ͩ ͪ 2 ͩ NϪ1 ͪ ͑46͒ ϱ 1/2͒d/2 1 1 n 1 n ͑d͒ ͑ From the relation f N ϭ ͚ 2 ⌫͓͑dϪ1͔/2͒ nϭ0 n! ͩ 4ͪ ͩ A ͪ N 1 22n 2 ͑dϪ3͒/2 2n 1 ϫ dtN͑1ϪtN͒ tN , ͑42͒ ͑2n͒!ϭ n!⌫͑nϩ 2 ͒, ͑47͒ ͵0 ͱ␲ 53 STATISTICAL PHYSICS OF THE FREELY JOINTED CHAIN 6303

ϱ it follows that d/2 ϩ n1 d/2 ͑d͒ 1 1 1 ⌫͑n1 1/2͒ 1 1 f NϪ1ϭ ͚ ͩ 2ͪ 2 n ϭ0 n ! ⌫͑n ϩd/2͒ ͩ 4ͪ ͩ 2ͪ 1 1 n1 1 ͓2͑n2ϩn1͒Ϫ1͔! 4 1 ⌫͑n2ϩn1ϩ1/2͒ ϭ ϱ 1 1 ⌫͑n ϩn ϩ1/2͒ 1 n2 ͑n2ϩn1Ϫ1͒!͑2n2͒! 2 ͑n2͒! ⌫͑n2ϩ1/2͒ 1 2 2n2 2n2 ϫ ͚ ANϪ2␻NϪ2. ͑48͒ ͩ 2ͪ n ϭ0 n ! ⌫͑n ϩd/2͒ ͩ 4ͪ 2 2 2 ͑49͒ and so with use of relation ͑41͒ the second sum is found to be The method is repeated until NϪpϭ2; thus

ϱ ϱ d͑NϪ1͒/2 ͑NϪ1͒ ϩ n1 ϩ ϩ n2 ͑d͒ 1 1 1 ⌫͑n1 1/2͒ 1 1 ⌫͑n1 n2 1/2͒ 1 f 2 ϭ ͚ ͚ ••• ͩ 2ͪ ͩ 2ͪ n ϭ0 n ! ⌫͑n ϩd/2͒ ͩ 4ͪ n ϭ0 n ! ⌫͑n ϩd/2͒ ͩ 4ͪ 1 1 1 2 2 2 ϱ ϱ ϱ np nNϪ2 1 ⌫͑npϪ1ϩnpϩ1/2͒ 1 1 ⌫͑nNϪ3ϩnNϪ2ϩ1/2͒ 1 1 ϫ ͚ ••• ͚ ͚ n ϭ0 n ! ⌫͑n ϩd/2͒ ͩ 4ͪ n ϭ0 n ! ⌫͑n ϩd/2͒ ͩ 4ͪ n ϭ0 n ! p p p NϪ2 NϪ2 NϪ2 NϪ1 NϪ1

nNϪ1 ⌫͑nNϪ2ϩnNϪ1ϩ1/2͒ 1 ϫ A2nNϪ1␻2nNϪ1. ͑50͒ ⌫͑n ϩd/2͒ ͩ 4ͪ 1 1 NϪ1

(d) (d) Finally, by deﬁnition of f 1 and J N ,

ϩϱ ͑d͒ ͑d͒ ͑d͒ 2 JN ϭ f 1 ϭ d␻1 f 2 exp͑Ϫ␻1͒, ͑51͒ ͵Ϫϱ

ϩϱ 2nNϪ1 2 1 (d) with ͐Ϫϱd␻1␻1 exp(Ϫ␻1)ϭ⌫(nNϪ1ϩ 2), which closes the expansion. Thus the series expansion of J N is

ϱ ϱ ϱ ͑NϪ1͒ d͑NϪ1͒/2 ϩ n1 ϩ ϩ n2 ϩ ϩ ͑d͒ 1 1 1 ⌫͑n1 1/2͒ 1 1 ⌫͑n1 n2 1/2͒ 1 1 ⌫͑npϪ1 np 1/2͒ JN ϭ ͚ ͚ ••• ͚ ͩ 2ͪ ͩ 2ͪ n ϭ0 n ! ⌫͑n ϩd/2͒ ͩ 4ͪ n ϭ0 n ! ⌫͑n ϩd/2͒ ͩ 4ͪ n ϭ0 n ! ⌫͑n ϩd/2͒ 1 1 1 2 2 2 p p p ϱ ϱ np nNϪ2 1 1 ⌫͑nNϪ3ϩnNϪ2ϩ1/2͒ 1 1 ⌫͑nNϪ2ϩnNϪ1ϩ1/2͒ ϫ ••• ͚ ͚ ͩ 4ͪ n ϭ0 n ! ⌫͑n ϩd/2͒ ͩ 4ͪ n ϭ0 n ! ⌫͑n ϩd/2͒ NϪ2 NϪ2 NϪ2 NϪ1 NϪ1 NϪ1

1 1 nNϪ1 ϫ⌫ n ϩ . ͑52͒ ͩ NϪ1 2ͪͩ4ͪ

͑This relation can be seen as a propagation over a chain of for the hypergeometric function and integer number.͒ The relation ͑52͒ is far from simple, but has (d) ϱ the advantage of showing that J N is given by the particular 1 ͑n,a͒ 1 (d) n value of a special function that we shall call g (x). The 1F1͑a;c;x͒ϭ x , ͑54͒ 4 N ͚ n! ͑n,c͒ series expansion of this function is obtained from ͑52͒ by nϭ0 1 (NϪ1) replacing 4 by x, imposing the constraint nϭ͚ iϭ1 ni on for the degenerate hypergeometric function; the ni , and extending the sum over n from 0 to ϱ. In each 1 sum, the factor 4 can be associated with an independent vari- ⌫͑nϩa͒ able xi , the ith component of a vector x in the Euclidean ͑n,a͒ϭ ͑55͒ (d) ⌫͑a͒ space of dimension ͑NϪ1͒. g N ͑x͒ is, with this definition, a (d) scalar field. We will use this latter definition of g N which is is the Pochhammer symbol ͓29͔, and easier to manipulate, rather than the functional definition (NϪ1) Ϫ Ϫ Ϫ where there is a constraint nϭ͚ iϭ1 ni . ͑N 1͒ d͑N 1͒/2 ͑N 1͒ ͑d͒ 1 1 1 ⌫͑1/2͒ gN ͑0͒ϭ ⌫ . B. The scalar field g „d… ͩ 2ͪ ͩ 2ͪ ͩ 2ͪͩ⌫͑d/2͒ͪ N ͑56͒ (d) The field g N can be expressed in terms of hypergeomet- 1 ni ric functions. The following definitions will be used ͑see In each sum over the integer ni , the factors ( 4) are ͓29͔͒: ni changed into xi where xi is the ith component of a vector x ϱ of a Euclidean space of dimension ͑NϪ1͒. With the previous 1 ͑n,a͒͑n,b͒ (d) n definitions, the series expansion of J N leads to the multiple 2F1͑a,b;c;x͒ϭ ͚ x , ͑53͒ (d) nϭ0 n! ͑n,c͒ series expansion of g N ͑x͒ according to 6304 MARTIAL MAZARS 53

ϱ ϱ ϱ 1 ͑n1,1/2͒ 1 ͑n1ϩn2,1/2͒ 1 ͑npϪ1ϩnp ,1/2͒ ͑d͒ ͑d͒ n1 n2 np gN ͑x͒ϭgN ͑0͒ ͚ ͑x1͒ ͚ ͑x2͒ ••• ͚ ͑x p͒ ••• n1ϭ0 n! ͑n1 ,d/2͒ n2ϭ0 n2! ͑n2 ,d/2͒ npϭ0 np! ͑n p ,d/2͒ ϱ ϱ 1 ͑nNϪ3ϩnNϪ2,1/2͒ 1 ͑nNϪ2ϩnNϪ1,1/2͒ 1 nNϪ2 nNϪ1 ϫ ͚ ͑x͑NϪ2͒͒ ͚ nNϪ1 , ͑x͑NϪ1͒͒ . n ϭ0 n ! ͑n ,d/2͒ n ϭ0 n ! ͑n ,d/2͒ ͩ 2ͪ NϪ2 NϪ2 NϪ2 NϪ1 NϪ1 NϪ1 ͑57͒

Derivatives of all orders at the origin of the scalar ﬁeld ͑NϪ1͒ sites; with each site a multiplicity ni and a weight ni are thus known. They can be expressed with the help of the (1/ni!)͓1/(ni ,d/2)͔(xi) are associated. On site 1, a propa- derivatives of the hypergeometric functions. For example, gation is initialized with a multiplicity n1 by a polymer ini- (d) 1 the gradiant of g N at the origin is tiator ͑n1 , 2͒. This perturbation of the integer chain propa- 1 gates itself to site 2 with a propagator ͑n1ϩn2 , 2͒, from site 2 xץ 1 1F1͑1/2;d/2;0͒ to site 3 in the same way, and so on until the perturbation Ӈ Ӈ reaches a polymer terminator on site ͑NϪ1͒ of the same ͑d͒ ͑d͒ 1 ͓ gN ͑x͔͒0ϭgN ͑0͒ , ͑58͒ form as the polymer initiator ͑n , ͒. The sum over all the x F ͑1/2;d/2;0͒ NϪ1 2ץ “ NϪ2 1 1 allowed propagations gives the behavior in N of the partition x 2F1͑1/2,1/2;d/2;0͒ͬץ ͫ NϪ1 function of the FJC. This is, of course, closely related to the decomposition with Green’s functions ͓6,30͔. The Feynman and derivatives of all orders in the direction of each axis of rules associated with this propagation are, for the sites, the Euclidean space at the origin are given by 1 1 ni ͑d͒ ͑d͒ ni p 1F1͑1/2;d/2;0͒ ᭺ pϵ ͑x͒ , ͑61͒ ץgN ͑x͉͒xϭ0ϭgN ͑0͒ ץ xi xi p! ͑p,d/2͒ for 1рiрNϪ2, ͑59͒ and for the propagators, nNϪ1 ͑d͒ ͑d͒ nNϪ1 2F1͑1/2,1/2;d/2;0͒ ץgN ͑x͉͒xϭ0ϭgN ͑0͒ ץ xNϪ1 xNϪ1

for NϪ1. 1 1 nu pϵ͑nϩp, 2 ͒ϭ͑pϩn, 2 ͒ ͑62͒ (d) One can note that the value of the ﬁeld g N along each axis of the Euclidean space is known, since, for ͉xi͉Ͻ1, ͑one can observe that the propagators are not oriented͒. For the polymer initiator and the polymer terminator the rules are g͑d͒͑x e ͒ϭg͑d͒͑0͒ F ͑1/2;d/2;x ͒ for 1рiрNϪ2, N i i N 1 1 i the same: ͑60͒ ͑d͒ ͑d͒ gN ͑xNϪ1eNϪ1͒ϭgN ͑0͒ 2F1͑1/2,1/2;d/2;xNϪ1͒ 1 for NϪ1. ᭹nϵ͑n, 2 ͒᭺n . ͑63͒

„d… C. Feynman rules for J N The rules ͑62͒ and ͑63͒ preserve the symmetry by inver- Equation ͑52͒ can be interpreted as a sum over all allowed sion of the labeling in the partition function. With the Feyn- propagations on an integer linear chain. The chain is made of man rules Eq. ͑57͒ becomes

ϱ ϱ ϱ g͑d͒͑x͒ϭg͑d͒͑0͒ ••• ••• ͑᭹ ᭺ ᭺ •••᭺ ᭺ •••᭺ ᭹ ͒. ͑64͒ N N ͚ ͚ ͚ n1u n2u n3 npϪ1u np nNϪ2u nNϪ1 n1 np nNϪ1

The memory effect appears explicitly in the last equation. If one considers that each site is associated with an event and the numbering of the sites with a discrete time, then the system will keep memory of the relative chronology of events but not of the direction of time.

D. Calculations with Feynman rules From the deﬁnitions ͑53͒ and ͑54͒, it is straightforward that the relations

ϱ ͑᭺ ᭺ ᭺ ͒ϭ᭺ ᭺ , ͑65͒ ͚ n1u n2u n3 n1v n3 n2ϭ0 53 STATISTICAL PHYSICS OF THE FREELY JOINTED CHAIN 6305

ϱ ͑᭹ ᭺ ᭺ ͒ϭ᭹ ᭺ ͑66͒ ͚ n1u n2u n3 n1v n3 n2ϭ0 hold, where the double propagator is defined as 1 1 d ϵ F ϩn , ϩn ; ;x . ͑67͒ nϪ1vniϩ1 2 1ͩ 2 iϪ1 2 iϩ1 2 i ͪ As in particle physics, the last equation is the full propagator of a scalar field to first order in ␭, when the interaction is a 1 2 composite operator like 2␭⌽ ͓28͔. If N is odd, taking (NϪ1)ϭ2P, Eq. ͑64͒ becomes ϱ g͑d͒ ͑x͒ϭg͑d͒ ͑0͒ ͑᭹ ᭺ ᭺ •••᭺ ᭺ ᭹ ͒. ͑68͒ 2Pϩ1 2Pϩ1 ͚ n1v n3v n5 n͑2PϪ3͒v n͑2PϪ1͒u n2P n1 ,...,n͑2PϪ1͒ ,n2Pϭ0 On the contrary, if N is even, taking NϪ1ϭ2PϪ1, Eq. ͑64͒ becomes ϱ g͑d͒͑x͒ϭg͑d͒͑0͒ ͑᭹ ᭺ ᭺ •••᭺ ᭺ ᭹ ͒. ͑69͒ 2P 2P ͚ n1v n3v n5 n͑2PϪ5͒v n͑2PϪ3͒v n͑2PϪ1͒ n1 ,...,n2PϪ1ϭ0

Equations ͑68͒ and ͑69͒ show that there is a nontrivial with the rules ͑65͒ and ͑66͒ generates a kind of self-similar parity effect in the partition function of the FJC; the single fractal sequence. In Fig. 2, the first three generations of this propagator in ͑68͒ may be placed anywhere in the integer fractalization are shown for a fractal initiator given by Nϭ3. chain. This odd-even effect manifests itself by oscillations in There are holes in the prefractal sequence relative to the some functions derived from the partition function. It has (d) sequence g 2P ͑x͒. If the fractal initiator is made of P single been observed by Fisher and Hiley ͓15͔ in a counting of propagators, then the nth prefractal generation will have 2nP self-avoiding walks ͑SAW’s͒ and recently by Grassberger single propagators; it will correspond to the partition func- and Hegger ͓16͔ in simulations of ⌰ polymers in two and tion of a FJC made of Nϭ2(1ϩ2(nϪ1)P) bonds. This frac- three dimensions. talization looks like the reverse of the decimation method It is difficult to contract the double propagators ͑67͒ as is ͓14,6,3͔. done for the single propagators, because of other odd-even 1 effects of higher order that appear and because of the need To contract the propagators further we take xiϭ 4 in ͑67͒. (R) for multiple hypergeometric functions. On the other hand, Thus the function g N ͑xЈ͒ is defined as a scalar field on a the double propagators can be developed in single propaga- Euclidean space of a dimension smaller than the Euclidean tors ͑stage b͒, and the single propagators transformed into space defined in Sec. IV B. This space has a dimension ͑N double propagators ͑stage a͒. This manipulation repeated Ϫ1͒/2 if N is odd, and ͑N/2Ϫ1͒ if N is even.

FIG. 2. Schematic representation of the ﬁrst three stages associated with the building of the prefractal sequence for the butane skeleton as a fractal initiator. This looks like the reverse of the decimation method. 6306 MARTIAL MAZARS 53

Relations between the hypergeometric functions give ͑R͒ g2Pϩ1͑x2Peˆ2P͒ 1 1 d 1 ͑PϪ1͒ 1 1 1 d d 1 ϭ ͑d͒ F , , ; , ;x , g2Pϩ1͑0͒ 2F1 , ; ; 2ͩ 2 2 2 2 2 1 4ͪ ͫ ͩ 2 2 2 4ͪͬ ϱ 1 1 d 1 ͑n1,1/2͒ 1 1 d 1 ϫ F n ϩ , ; ; x ͑75͒ ϭ F n ϩ , ; ; xn1 ͑70͒ 2 1ͩ 1 2 2 2 2P ͪ ͚ 2 1 1 1 n ϭ0 n ! ͑n ,d/2͒ ͩ 2 2 2 4ͪ 1 1 1 for 3рiр2PϪ3 ͑i odd͒ and and ͑PϪ3͒ ͑R͒ ͑d͒ 1 1 d 1 1 1 1 1 d d d 1 1 g2Pϩ1͑xieˆi͒ϭg2Pϩ1͑0͒ 2F1 , ; ; F , , , ; , , ; ,x , ͫ ͩ 2 2 2 4ͪͬ Kͩ 2 2 2 2 2 2 2 4 1 4ͪ 1 1 1 1 d d d 1 1 ϱ ϫF , , , ; , , ; ,x , 1 1 d 1 1 d Kͩ 2 2 2 2 2 2 2 4 i 4ͪ ϭ ͚ 2F1 ,n1ϩ ; ; n1 , n ϭ0 ͩ 2 2 2 4ͪ n ! ͩ 2ͪ 1 1 ͑76͒ 1 1 d 1 ϫ F n ϩ , ; ; xn1, ͑71͒ for iϭ1or2PϪ1. For N even 2 1ͩ 1 2 2 2 4ͪ 1 1 1 d 1 ͑PϪ2͒ where F2 is the second Appell double hypergeometric func- g͑R͒͑x eˆ ͒ϭg͑d͒͑0͒ F , ; ; 2P i i 2P ͫ 2 1ͩ 2 2 2 4ͪͬ tion, and FK the triple hypergeometric function of Lauricella-Saran ͑see ͓29͔͒. 1 1 1 d d 1 From Eqs. ͑68͒ and ͑69͒, we obtain derivatives of all or- ϫF , , ; , ;x , ͑77͒ (R) 2ͩ 2 2 2 2 2 i 4ͪ ders of g N at the origin. If N is odd and similar relations on the other axis. n2P ͑R͒ g2Pϩ1͑xЈ͉͒0 ץ x2P The Feynman rules of the FJC may be used for calculations involving hypergeometric functions. ͑PϪ1͒ ͑d͒ 1 1 d 1 ϭg2Pϩ1͑0͒ 2F1 , ; ; ͫ ͩ 2 2 2 4ͪͬ d „ … E. J N for small N 1 1 d (d) n2Pϩ1 In this subsection, we calculate g N ͑x͒ for a few small 2F1 n1ϩ , ; ;x2P ͉x ϭ0 ͑72͒ ץϫ x2Pϩ1 ͩ 2 2 2 ͪ 2P values of the degree of polymerization.

(1؍for 3рiр2PϪ3 ͑i odd͒ and 1. Ethane skeleton (N For this molecule, since NϪ1ϭ0, the previous calcula- n (d) i ͑R͒ tions do not apply. However, J 1 can be computed from g2Pϩ1͑xЈ͉͒0 ץ xi ͑32͒: 1 1 d 1 ͑PϪ3͒ ϭg͑d͒ ͑0͒ F , ; ; ϩϱ 1 2Pϩ1 ͫ 2 1ͩ 2 2 2 4ͪͬ ͑d͒ 2 J1 ϭ d␥1 exp͑Ϫ␥1͒ϭ⌫ . ͑78͒ ͵Ϫϱ ͩ 2ͪ 1 1 1 1 d d d 1 1 ni , FK , , , ; , , ; ,xi ץϫ xi ͩ 2 2 2 2 2 2 2 4 4ͪ ͯ (2؍xiϭ0 2. Propane skeleton (N ͑73͒ The Feynman diagram associated with this molecule is 1 (n1 , 2)᭹n . So we have for iϭ1or2PϪ1. If N is even 1 ϱ n R ͑d͒ ͑d͒ 1 i ͑ ͒ g ͑x͒ϭg ͑0͒ ͓͑n1 , 2 ͒᭹n ͔ g2P ͑xЈ͉͒0 2 2 ͚ 1 ץ xi n1ϭ0

͑PϪ2͒ ͑d͒ 1 1 d 1 ϭg ͑0͒ 2F1͑1/2,1/2;d/2;x1͒, ͑79͒ ϭg͑d͒͑0͒ F , ; ; 2 2P ͫ 2 1ͩ 2 2 2 4ͪͬ and consequently 1 1 1 d d 1 ni F2 , , ; , ;xi , ͑74͒ ץϫ xi ͩ 2 2 2 2 2 4ͪͯ 1 1 1 d/2 1 xiϭ0 J͑d͒ϭg͑d͒ ϭ ⌫ 2 2 ͩ 4ͪ ͩ 2ͪͩ2ͪ ͩ 2ͪ and relations similar to Eqs. ͑72͒ and ͑73͒.Asfor͑60͒, the restriction of g (R) to each axis of the space for x Ͻ1is ⌫͑1/2͒ 1 1 d 1 2Pϩ1 ͉ i͉ ϫ F , ; ; . ͑80͒ given by ͩ ⌫͑d/2͒ͪ 2 1ͩ 2 2 2 4ͪ 53 STATISTICAL PHYSICS OF THE FREELY JOINTED CHAIN 6307

1 -For x1ϭ•••ϭxiϭ•••ϭxNϪ1ϭ 4 , all the nontrivial depen (3؍Butane skeleton (N .3 dence on the degree of polymerization of the partition func- The Feynman diagram associated with this molecule is (N) ᭹ ᭹ , giving tion is included in F G . This function is far from simple; n1u n2 however, with Eq. ͑60͒, its restriction to each axis of the

ϱ Euclidean space is known. With the method of the previous ͑d͒ ͑d͒ subsections, we can build more and more accurate approxi- g ͑x͒ϭg ͑0͒ ͑᭹ ᭹ ͒ (N) 3 3 ͚ n1u n2 mations of F ͓31͔. n1 ,n2ϭ0 G

͑d͒ „d… ϭg3 ͑0͒F2͑1/2,1/2,1/2;d/2,d/2;x1 ,x2͒, V. NUMERICAL STUDY OF J N ͑81͒ To test the precision of the analytical method of the pre- and thus vious section, the results will be compared with a numerical calculation. A feature of the method used in Sec. IV is that it 2 d uses the dimension of the physical space in which the poly- ͑d͒ ͑d͒ 1 1 1 1 1 J3 ϭg3 , ϭ ⌫ mer is contained as a real parameter and not as the number of ͩ 4 4ͪ ͩ 2ͪ ͩ 2ͪ ͩ 2ͪ independent components of the bond vectors. To obtain another nonintegrated form of the partition function with d as a ⌫͑1/2͒ 2 1 1 1 d d 1 1 real parameter, we must not make the integration over the set ϫ F2 , , ; , ; , . ͩ ⌫͑d/2͒ͪ ͩ 2 2 2 2 2 4 4ͪ of variables ͕⍀n͖ before the integration over the variables ˆ ͑82͒ ͕un͖ as is done in ͓12͔, because the values allowed for d in that way would be the integer values, but the method must be ,generalized to noninteger values of d too. For this purpose (4؍Pentane skeleton (N .4 the integration over ͕uˆ ͖ is still made before the integration For this molecule the associated Feynman diagram is n over ͕⍀ ͖, then the Bessel functions are expressed with their ᭹ ᭺ ᭹ . So we have n n1u n2u n3 representation ͑38͒, and thus the space dimension d becomes a parameter of which the partition function is a function. ϱ Then we make the complete integration over the variables ͑d͒ ͑d͒ g4 ͑x͒ϭg4 ͑0͒ ͑᭹n ᭺n ᭹n ͒ ͕␻ ͖; this allows us to recover the metric determinant of the n ,n͚,n ϭ0 1u 2u 3 n 1 2 3 freely jointed chain ͓8,10,12͔, which in this way is independent of the space dimension. Since we do not know the ana- 1 1 1 1 d d ͑d͒ lytical form of the integral containing this determinant, we ϭg ͑0͒F , , , ; , ;x ,x ,x . 4 Kͩ 2 2 2 2 2 2 1 2 3 ͪ estimate it with a Monte Carlo algorithm using importance ͑83͒ sampling. The numerical results obtained with this algorithm permit a comparison with the approximate analytical results With the contractions ͑65͒ and ͑66͒, the Feynman diagram is obtained in Sec. IV. also This section is divided into three subsections. In Sec. V A, the analytical form of J (d) is established with the met- ϱ N ric determinant of the FJC for any dimension of space. In ᭹n ᭺n ᭹n ϭ᭹n ᭹n . ͑84͒ n͚ϭ0 1u 2u 3 1v 3 Sec. V B, the integral is computed with the Monte Carlo 2 algorithm. In Sec. V C, the analytical results of Sec. IV are We have compared with the numerical results of Sec. V B, and an accurate approximation for the partition function of the FJC 1 1 1 made of N bonds is given. J͑d͒ϭg͑d͒ , , 4 4 ͩ 4 4 4ͪ „d… A. Another integral equation for J N 1 3 1 3d/2 1 ⌫͑1/2͒ 3 To obtain the FJC metric determinant, a recurrence based ϭ ⌫ on the computational rules for the determinants is used. The ͩ 2ͪ ͩ 2ͪ ͩ 2ͪͩ⌫͑d/2͒ͪ (d) (d) ﬁrst stage of the recurrence is to determine f NϪ1 from f N . (d) With the relation ͑41͒, f NϪ1 is deﬁned by 1 1 1 1 d d d 1 1 1 ϫF , , , ; , , ; , , . Kͩ 2 2 2 2 2 2 2 4 4 4ͪ ϩϱ ͑d͒ ͑1Ϫd/2͒ ͑85͒ f NϪ1ϭANϪ1 d␻NϪ1␻NϪ1 I͑d/2Ϫ1͒͑␻NϪ1͒ ͵Ϫϱ

5. General form for any N NϪ2 (N) p We can deﬁne a multiple hypergeometric function F G 2 ͑Ϫ1͒ 2 ͑d͒ ϫexp Ϫ ͟ ͑␻͑NϪ1Ϫp͒͒ ␻NϪ1 f N , such that ͩ pϭ1 ͪ ͑87͒ 1 1 d d g͑d͒͑x͒ϭg͑d͒͑0͒F͑N͒ ,..., ; ,..., ;x ,...,x . N N G ͩ 2 2 2 2 1 NϪ1ͪ which, using the integral representation of the Bessel func- ͑86͒ tions ͑38͒, can be cast in the form 6308 MARTIAL MAZARS 53

͑1/2͒͑d/2Ϫ1͒ ͑1/2͒d/2 ϩϱ d/2 ͑pϩ1͒ ͑d͒ ͑d͒ 1 1 f NϪ1ϭANϪ1 d␻NϪ1 f NϪpϭ ⌫͓͑dϪ1͔/2͒⌫͑1/2͒ ⌫͓͑dϪ1͔/2͒ ͵Ϫϱ ͫͩ 2ͪ ⌫͓͑dϪ1͔/2͒ͬ p 1 1 1/2 1 2 ͑dϪ3͒/2 2 ͑dϪ3͒/2 ͑␻ t ͒ ϫ ͟ dt͑NϪn͒͑1Ϫt͑NϪn͒͒ ͑pϩ1͒ ϫ dt ͑1Ϫt ͒ e NϪ1 NϪ1 ͵0 nϭ0 ͩ det M ͪ NϪ1 NϪ1 ͵Ϫ1 2 ͑p͒ tNϪp det M ϫexp A2 ␻2 . ͑92͒ 1 ͩ 4 det M ͑pϩ1͒ NϪpϪ1 NϪpϪ1 ͪ 2 ͑dϪ3͒/2 2 2 2 ϫ dtN͑1ϪtN͒ e͓ϪANϪ1͑1ϪtN/4͒␻NϪ1͔. ͵0 ͑88͒ Then we have

͑d/2Ϫ1͒ The integral over ␻ is Gaussian; thus 1 1 NϪ1 f ͑d͒ ϭA NϪpϪ1 ͑NϪpϪ1͒ͫͩ 2ͪ ⌫͓͑dϪ1͔/2͒⌫͑1/2͒ͬ d/2 2 1 ͑d͒ 1 1 1 f NϪ1ϭ dt͑NϪ1͒ 2 ͑dϪ3͒/2 ͫͩ 2ͪ ⌫͓͑dϪ1͔/2͒ͬ ͵0 ϫ dt͑NϪpϪ1͒͑1Ϫt͑NϪpϪ1͒͒ ͵Ϫ1

ϩϱ 1 2 ͑dϪ3͒/2 2 ͑dϪ3͒/2 ϫ d␻NϪpϪ1 exp͑␻͑NϪpϪ1͒t͑NϪpϪ1͒͒ ϫ dtN͑1Ϫt͑NϪ1͒͒ ͑1ϪtN͒ ͵Ϫϱ ͵0 2 2 ͑d͒ ϫexp͑ϪANϪpϪ1␻NϪpϪ1͒f NϪp . ͑93͒ 1/2 t2 1 NϪ1 2 2 ϫ 2 exp 2 ANϪ2␻NϪ2 . The integral over is Gaussian with cf. Eq. 15 ͩ ͑1Ϫt /4͒ͪ ͩ 4͑1Ϫt /4͒ ͪ ␻NϪpϪ1 ͓ ͑ ͔͒ N N ͑89͒ Xϭ␻NϪpϪ1 , The relation ͑41͒ permits one to ﬁnd the dependence of (d) f NϪ1 on ␻NϪ2 and thus to build a recurrence relation. Let JϭtNϪpϪ1 , ͑94͒ M (p) be a sequence of square matrices pϫp, such that they 2 ͑p͒ verify the recurrence relation tNϪp det M Aϭ2 1Ϫ A2 ͩ 4 det M ͑pϩ1͒ ͪ NϪpϪ1 det M ͑p͒ϭdet M ͑pϪ1͒Ϫc2 det M ͑pϪ2͒, ͑90͒ pϪ1 ͑pϩ2͒ det M 2 ϭ2 ͑pϩ1͒ ANϪpϪ1, where ͕c p͖ is a sequence of real numbers, and det M

͑1͒ 1 det M ϭ1, and c pϭ 2 tNϪp . Using the relation ͑91͒ ͑2͒ 2 1 2 2 det M ϭ1Ϫc1. 2 ϭANϪpϪ2␻NϪpϪ2, ͑95͒ ANϪpϪ1 (d) With these deﬁnitions, assume that f NϪp can be written as ͑93͒ becomes

pϩ1 d/2 ͑pϩ2͒ 1 ͑d͒ 1 1 f NϪpϪ1ϭ ͟ dt͑NϪn͒ ͫͩ 2ͪ ⌫͓͑dϪ1͔/2͒ͬ ͵0 nϭ0

1/2 2 ͑pϩ1͒ 1 tNϪpϪ1 det M ϫ͑1Ϫt2 ͒͑dϪ3͒/2 exp A2 ␻2 . ͑96͒ ͑NϪn͒ ͩ det M ͑pϩ2͒ ͪ ͩ 4 det M ͑pϩ2͒ NϪpϪ2 NϪpϪ2 ͪ

(d) (d) For pϭ0, the relation between f NϪ1 ͓Eq. ͑89͔͒ and f N is in agreement with ͑92͒ on account of the deﬁnitions ͑90͒ and ͑91͒. Equation ͑96͒ shows the stability of Eq. ͑92͒ under the recurrence, so this form is right for all 0рpрNϪ2. In particular, for pϭNϪ2 one has

NϪ2 d/2 NϪ1 1 ͑d͒ 1 1 f 2 ϭ ͟ dt͑NϪn͒ ͫͩ 2ͪ ⌫͓͑dϪ1͔/2͒ͬ ͵0 nϭ0

1/2 2 ͑NϪ2͒ 1 t2 det M ϫ͑1Ϫt2 ͒͑dϪ3͒/2 exp ␻2 , ͑97͒ ͑NϪn͒ ͩ det M ͑NϪ1͒ ͪ ͩ 4 det M ͑NϪ1͒ 1 ͪ 53 STATISTICAL PHYSICS OF THE FREELY JOINTED CHAIN 6309

(d) (d) which by deﬁnition of f 1 and J N gives a Gaussian integration over ␻1 and with x pϭtNϪp NϪ1 d/2 ͑NϪ1͒ 1 d 1 1 JNϭͱ␲ ͟ dxn ͫͩ 2ͪ ⌫͓͑dϪ1͔/2͒ͬ ͵0 nϭ1 1 1/2 ϫ͑1Ϫx2͒͑dϪ3͒/2 . ͑98͒ n ͩ det M ͑N͒ ͪ If the matrix M (p) is taken as

1 0 1 2 x1 0 ••• 0 0 1 1 0 0 0 2 x1 1 2 x2 ••• 1 ••• 0 0 0 0 2 x2 1 Ӈ Ӈ Ӈ Ӈ Ӈ Ӈ Ӈ ͑p͒ 0 0 0 M ϭ 0 0 0 ••• , ͑99͒ Ӈ Ӈ Ӈ Ӈ Ӈ Ӈ Ӈ 1 1 0 0 0 0 ••• 2 x͑pϪ2͒ 1 1 0 0 0 ••• 2 x͑pϪ2͒ 1 2 x͑pϪ1͒ 0 0 ••• 0 1 x 1 ΂ 0 2 ͑pϪ1͒ ΃

NϪ1 then it veriﬁes the conditions ͑90͒ and ͑91͒. This result, when 1 ͑d͒ ͑NϪ1͒ 2 ͑dϪ3͒/2 replaced in ͑31͒, gives the partition function of a FJC as ͑No ͒ ϭ dxn͑1Ϫxn͒ ͵0 n͟ϭ1 ͑d/2Ϫ1͒ d/2 1 1 ⌫͑1/2͒ m ͑NϪ1͒ Q͑d͒ϭ S ͑1͒ V 1 ⌫͑1/2͒⌫͓͑dϪ1͔/2͒ N 2 2 ⌫͓͑dϪ1͔/2͒ d 4␲2␤ប2 ϭ . ͑103͒ ͫͩ ͪ ͬ ͩ ͪ ͩ 2 ⌫͑d/2͒ ͪ 1 ͑dϩ2͒/͑dϪ1͒ ϫ ͫͩ 2ͪ We have applied the Monte Carlo integration method to FJC’s for which the number of bonds takes all values be- 1 2/͑dϪ1͒ ma2 N͑dϪ1͒/2 ͑d͒ tween Nϭ2 and 500 and considered 28 values of d between ϫ 2 IN , ͑100͒ ͩ ⌫͓͑dϪ1͔/2͒ͪ ␤ប ͬ 2 and 8 ͑including noninteger dimensions͒. For each value of d, 499 partition functions of the FJC have been obtained. where The Monte Carlo results are shown in Fig. 3. The data may NϪ1 be very well represented by a straight line ﬁt 1 1 1/2 ͑d͒ 2 ͑dϪ3͒/2 IN ϭ ͟ dxn͑1Ϫxn͒ ͑N͒ . ͑101͒ ͵0 nϭ1 ͩ det M ͪ

(d) It is the value of I N that we estimate by a Monte Carlo algorithm. In this relation d is only an integration parameter, which would not be the case if we had made the Gaussian integration over the set of variables ͕⍀n͖ in Eq. ͑19͒.

B. Monte Carlo integrations (d) From Eq. ͑101͒, I N may be considered as the average value of ͑1/det M (N)͒1/2 over the distribution NϪ1 2 (dϪ3)/2 ␳N͑x͒ϭ⌸ nϭ1 dxn(1Ϫx n) , i.e., 1/2 ͑d͒ 1 IN ϭ dx ␳N͑x͒ ͑N͒ . ͩ ͵͓0,1͔͑NϪ1͒ ͪͳ ͩ det M ͪ ʹ ␳N͑x͒ ͑102͒

Importance sampling is done over the distribution function ␳ ͑x͒ where d appears as a parameter. Results for non- FIG. 3. Results obtained for ln (1/det M(N))1/2 with a N ͗ ͘␳N(x) integer values of d can thus be obtained as well. The nor- Monte Carlo algorithm using importance sampling. The straight malization of the distribution function ␳N͑x͒ is easily line ﬁt is not represented on the curves. The symbols are placed calculated to be every 15 points. 6310 MARTIAL MAZARS 53

C. Analytical form for the partition function of the FJC From the results of the previous subsection, the accuracy of the analytical expressions presented in Sec. IV can be tested on the coefﬁcients of the ﬁt. The partition function of the FJC is given by

T N͑dϪ1͒/2 Q͑d͒͑V,T͒ϭq͑d͒Q͑d͒͑V,T͒ , ͑107͒ N o छ ͩ T ͑d͒ͪ o with

␲ 1/2 B͑d͒ q͑d͒ϭ , o ͩ 2 ͪ ⌫͓͑dϪ1͔/2͒2

mkT 1/2 Q͑d͒͑V,T͒ϭ V, ͑108͒ छ ͩ 4␲2ប2ͪ

1 1 ͑dϩ4͒/͑dϪ1͒ ␤ ͑d͒ϭ ϭ o kT ͑d͒ ͩ2ͪ o

⌫͑1/2͒ 2/͑dϪ1͒ ma2 ϫ A͑d͒ . ͩ ⌫͑d/2͒ͪ ប2

In Fig. 5, A(d) and B(d) are shown as functions of d. From Sec. IV B, the scalar ﬁeld h1͑x͒ deﬁned by

Ϫ FIG. 4. Accuracy of the straight line fit for data from Fig. 3 1 1 d ͑N 2͒ 1 d versus number of Monte Carlo iterations. On both graphs the first ͑d͒ h1͑x͒ϭgN ͑0͒2F1 , ; ;xNϪ1 ͟ 1F1 ; ;xi curve from the top corresponds to dϭ2.0, and the following curves ͩ 2 2 2 ͪ iϭ1 ͩ 2 2 ͪ are obtained by increasing d by 0.5; thus on both graphs the lowest ͑109͒ ϭ curve corresponds to d 4.5. (d) coincides with the field g N on each axis. Furthermore, from Sec. IV D, the scalar field h (R) x defined by 1 1/2 1 ͑ Ј͒ ln ϭln B͑d͒ϩN ln A͑d͒. ͑104͒ ͳ ͩ det M ͑N͒ ͪ ʹ 1 1 1 d d 1 ␳ ͑x͒ N h͑R͒͑xЈ͒ϭg͑d͒ ͑0͒F , , ; , ;x , 1 2Pϩ1 2ͩ 2 2 2 2 2 1 4ͪ After only a few thousand Monte Carlo iterations, almost 2PϪ3 all the points of Fig. 3 are in agreement with the straight line 1 1 1 1 d d d 1 1 ϭ fit to within 1–3 %, except for the point with N 3. How- ͟ FK , , , ; , , ; ,xi , (d) iϭ3,odd ͩ 2 2 2 2 2 2 2 4 4ͪ ever, for Nϭ3, the analytical function J 3 is given by Eq. ͑85͒. If correlations between two random variables (X,Y) are 1 1 1 d d 1 ϫF , , ; , ; ,x exactly according to a straight line fit, one has, with 2ͩ 2 2 2 2 2 4 2PϪ1 ͪ 2 ␴ uvϭ͗uv͘Ϫ͗u͗͘v͘ and YϭaXϩb, 1 1 d ␴2Ϫa2␴2ϭ0, ϫ F , ; ;x ͑110͒ Y X 2 1ͩ 2 2 2 2Pϩ1 ͪ ͑105͒ 2 2 ␴ Ϫa␴ ϭ0. (R) XY X coincides with the field g 2Pϩ1 on each axis. Of course, h1͑x͒ (R) (d) 2 2 2 2 2 and h 1 ͑xЈ͒ are not exactly equal to the fields g N and In Fig. 4, log10(␴ YϪa ␴ X) and log10(␴ XYϪa␴ X) are (R) g 2Pϩ1 away from the axis, but they may serve as approxi- shown as functions of the number of Monte Carlo integra- mations of these fields. The difference between Secs. IV and tions. The accuracy of the fit is so good than after only a few 2 2 2 V resides only in the integration order in which the integrals thousand iterations, the differences between ␴ Y and a ␴ X, 2 2 have been computed. As the integrals are convergent, both and ␴ XY and a␴ X, are less than 0.01. Thus the approxima- results must be equal. Thus, by fixing the value of the fields tion (R) 1 h1͑x͒ and h 1 ͑xЈ͒ at x1ϭ•••ϭxiϭ•••ϭxNϪ1ϭ 4 , after sim- ͑d͒ plification, one has the following. d B d ͑ ͒ ͑ ͒ ͑d͒ N For h1 IN ϭ ͑d͒ ͑No A ͒ ͑106͒ No ͑d͒ ͑d͒ N ͑NϪ2͒ B ͑A ͒ ϭ2F1͑ 1F1͒ , ͑111͒ appears to be extremely good. The results for A(d) and B(d) for several values of d are given in Table I. giving 53 STATISTICAL PHYSICS OF THE FREELY JOINTED CHAIN 6311

(d) (d) (d) TABLE I. The numerical values of the coefficients A and B from the straight line fit of the Monte Carlo estimates of I N and the analytical approximations of both coefficients ͓we use the reduced notations (1F1)ϭ1F1͑1/2,d/2,1/4͒ and (2F1)ϭ2F1͑1/2,1/2,d/2,1/4͔͒.

(d) (d) (1F1) Ϫ1 Ϫ1 dAB͑1F1͒͑2F1͒ 2 ͑2F1͒͑1F1͒ (2F1) 2.0 1.092͑5͒ 0.894͑0͒ 1.137 58 1.073 18 0.829 297 0.931 81 0.879 06 2.2 1.077͑9͒ 0.912͑7͒ 1.124 50 1.066 00 0.843 021 0.938 09 0.889 28 2.4 1.069͑0͒ 0.921͑8͒ 1.113 65 1.060 07 0.854 745 0.943 33 0.897 95 2.5 1.065͑3͒ 0.928͑3͒ 1.108 90 1.057 48 0.859 978 0.945 64 0.901 79 2.6 1.061͑9͒ 0.932͑4͒ 1.104 51 1.055 09 0.864 872 0.947 79 0.905 38 2.8 1.056͑1͒ 0.938͑0͒ 1.096 71 1.050 85 0.873 691 0.951 61 0.911 82 3.0 1.051͑3͒ 0.944͑9͒ 1.089 97 1.047 20 0.881 453 0.954 93 0.917 46 3.2 1.047͑2͒ 0.949͑3͒ 1.084 10 1.044 02 0.888 320 0.957 84 0.922 42 3.4 1.043͑8͒ 0.952͑4͒ 1.078 94 1.041 24 0.894 454 0.960 39 0.926 84 3.5 1.042͑2͒ 0.955͑3͒ 1.076 58 1.039 97 0.897 279 0.961 57 0.928 87 3.6 1.040͑7͒ 0.956͑5͒ 1.074 36 1.038 78 0.899 960 0.962 67 0.930 79 3.8 1.038͑1͒ 0.959͑0͒ 1.070 28 1.036 59 0.904 928 0.964 70 0.934 33 4.0 1.035͑8͒ 0.962͑0͒ 1.066 62 1.034 63 0.909 422 0.966 53 0.937 54 4.2 1.033͑2͒ 0.964͑5͒ 1.062 54 1.032 45 0.914 486 0.968 57 0.941 14 4.5 1.031͑0͒ 0.967͑7͒ 1.058 92 1.030 53 0.919 042 0.970 37 0.944 36 5.0 1.027͑3͒ 0.971͑4͒ 1.052 81 1.027 28 0.926 810 0.973 44 0.949 84 6.0 1.022͑1͒ 0.972͑7͒ 1.043 70 1.010 87 0.927 988 0.989 25 0.958 13 7.0 1.018͑5͒ 0.981͑1͒ 1.037 26 1.019 09 0.947 194 0.981 27 0.964 08 8.0 1.015͑8͒ 0.983͑5͒ 1.032 46 1.016 59 0.953 671 0.983 68 0.968 56

͑d͒ ͑d͒ 1/2 A ϭ͑ 1F1͒, A ϭ͑FK͒ , ͑112͒ ͑114͒ ͑d͒ 2 3/2 ͑d͒ 2 B ϭ͑F2͒ ͑ 2F1͒/͑FK͒ , B ϭ͑ 2F1͒/͑ 1F1͒ . and for h (R), when Nϭ2P, (R) 1 For h 1 , when Nϭ2Pϩ1, ͑d͒ ͑d͒ 2P 2 ͑PϪ1͒ B ͑A ͒ ϭ͑F2͒ ͑FK͒ , ͑115͒ ͑d͒ ͑d͒ ͑2Pϩ1͒ 2 ͑PϪ1͒ B ͑A ͒ ϭ͑F2͒ ͑2F1͒͑FK͒ , ͑113͒ leading to giving ͑d͒ 1/2 A ϭ͑FK͒ , ͑116͒ ͑d͒ 2 B ϭ͑F2͒ /͑FK͒,

where the arguments of the hypergeometric functions have been omitted. In Eqs. ͑114͒ and ͑116͒ the odd-even effect appears explicitly in the value of B(d). In both equations A(d) takes identical values. Thus with the use of the approximation given by Eqs. ͑107͒ and ͑108͒ the odd-even effect must de- crease when the degree of polymerization increases. This is in agreement with the results of Fisher and Hiley ͓15͔ and those of Grassberger and Hegger ͓16͔. With the relations

1 1 1 d d 1 1 1 d 1 F , , ; , ;0, ϭ F , ; ; ͑117͒ 2ͩ 2 2 2 2 2 4ͪ 2 1ͩ2 2 2 4ͪ

and FIG. 5. Comparison between the theoretical values for A(d) and (d) (d) 1 1 1 1 d d d 1 1 B obtained with an approximation of the function g N and the FK , , , ; , , ; ,0, values of A(d) and B(d) obtained from the straight line ﬁt of the ͩ 2 2 2 2 2 2 2 4 4ͪ Monte Carlo data. The numerical values are given in Table I. We 1 1 d 1 1 1 d 1 use the reduced notations (1F1)ϭ1F1(1/2;d/2;1/4) and ϭ F , ; ; F , ; ; ͑118͒ ( F )ϭ F (1/2,1/2;d/2;1/4). 2 1ͩ 2 2 2 4ͪ 2 1ͩ 2 2 2 4ͪ 2 1 2 1 6312 MARTIAL MAZARS 53 the odd-even effect is canceled and another approximation is ␰͑d͒͑V,T͒ϭ␰͑d͒͑V,T͒ϪkT ln͓Q͑d͒͑V,T͔͒ϪkT ln͑q͑d͒͒, (d) N ex o o possible. To zeroth order, with the scalar ﬁeld g N , it gives ͑123͒ ͑d͒ ͑d͒ ͑d͒ Ϫ1 ␰ex ͑V,T͒ϭ͑N/2͒͑dϪ1͒kT ln͑To͑d͒/T͒. A Ϸ͑ 1F1͒ and B Ϸ͑ 1F1͒ , ͑119͒

(R) The average energy of the gas is given by and to first order, with the scalar field g N , it gives ͑d͒ lnZN d N ץ A͑d͒Ϸ͑ F ͒ and B͑d͒Ϸ͑ F ͒Ϫ1. ͑120͒ ¯E ϭϪ ϭMe¯ϭMkT ϩ ͑dϪ1͒ . ͪ c ͩ2 2 ͯ ␤ץ c 1 2 1 2 M,V In Table I, the values of Eqs. ͑119͒ and ͑120͒ are reported, ͑124͒ as well as those of Eq. ͑112͒. In Fig. 5, these analytical approximations are shown as functions of d. The good agree- Equation ͑124͒ shows that equipartition is verified, as was ment between these results confirms the validity of the already obvious from Eq. ͑31͒ of Sec. III. The entropy is given by method described in Sec. IV and the approximation ͑107͒ made for the partition function of the FJC. 1 Finally, the asymptotic behavior for d ϱ gives S͑d͒͑M,V,T͒ϭ ͓¯E ϪF͑d͒͑M,V,T͔͒, ͑125͒ → N T c N 1F1͑1/2;d/2;1/4͒ 1 and 2F1͑1/2,1/2;d/2;1/4͒ 1. In this limit, the only nonzero→ contribution to I (d) is given→ by xϭ0; N ͑d͒ ϭ ͑d͒ ϩ ͑d͒ then SN ͑M,V,T͒ Sid ͑M,V,T͒ MsN ͑T͒, ͑126͒ where 1 1/2 1, ͳ ͩ detM ͑N͒ ͪ ʹ → d ␳N͑x͒ S͑d͒͑M,V,T͒ϭk MϩM ln͓Q͑d͒͑V,T͔͒Ϫln͑M!͒ , id ͩ 2 छ ͪ and A(d) 1 and B(d) 1. With the approximation ͑107͒ for → → the partition function the study of the thermodynamical and N s͑d͒͑T͒ϭk ͑dϪ1͒ϩln͑q͑d͒͒Ϫs͑d͒͑T͒ , ͑127͒ statistical properties is straightforward. N ͩ 2 o ex ͪ

VI. THERMODYNAMICAL AND STATISTICAL N To͑d͒ s͑d͒͑T͒ϭ ͑dϪ1͒ln . PROPERTIES OF FJC’s ex 2 ͫ ͩ T ͪͬ The two preceeding sections give the canonical partition The relations ͑127͒ are the Sakur-Tetrode relation for the function of the FJC. Then it is possible to study the equilib- (d) (d) rium thermodynamical properties of a gas made of M FJC’s. FJC. In Fig. 6, sex (T) and ␰ex (V,T) are shown as functions As explained in the Introduction and explicitly used in Eqs. of T*ϭT/To(d) for a few values of the degree of polymer- ͑10͒ and ͑11͒, the FJC can be considered as an ideal gas with ization and dϭ3. As long as the size of the box is greater than the typical size of the polymer ͓6͔, the equation of state phase space ␸rest . It is straightforward to build the microcanonical and grand canonical ensembles relative to this ideal of the gas of FJC’s is the same as that of an ideal monatomic gas via Laplace transform. The meaning of the partition gas, PVϭMkT. The chemical potential is given by function permits one to give the connection with random ͑d͒ϭ ͑d͒ ϩ Ϫ ͑d͒ walks and to compute the critical exponents ␥ and ␯. ␮N FN ͑M 1,V,T͒ FN ͑M,V,T͒ ͑d͒ ϭkT ln͑Mϩ1͒Ϫ␰N ͑V,T͒. ͑128͒ A. Gas of FJC’s in the canonical ensemble The canonical partition function ͑107͒ of the FJC permits This chemical potential may serve as a reference value for one to express the canonical partition function of a gas of M the computation of chemical potentials in realistic models, FJC’s as with the conﬁgurational bias method ͓32–34͔. Considering that the FJC is an ideal gas with phase space 1 ␸rest , the degree of polymerization can be associated with a Z͑d͒͑M,V,T͒ϭ ͓Q͑d͒͑V,T͔͒M N M! N second chemical potential ␹ called the bonding chemical potential. In the canonical ensemble, ␹ is given by 1 ͑d͒ M ͑d͒ M ϭ ͑qo ͒ ͓Qछ ͑V,T͔͒ 1 To͑d͒ M! ␹ϭ␰͑d͒ ͑V,T͒Ϫ␰͑d͒͑V,T͒ϭ ͑dϪ1͒kT ln . Nϩ1 N 2 ͩ T ͪ T ͑MN/2͒͑dϪ1͒ ͑129͒ ϫ . ͑121͒ T ͑d͒ ͩ o ͪ With N and ␹, thermodynamical ensembles of the same kind The free energy of the gas of FJC’s is given by as the grand canonical ensemble may be associated ͓41͔. This second chemical potential has been used previously by F͑d͒͑M,V,T͒ϭϪkT ln͓Z͑d͒͑M,V,T͔͒ de Gennes ͓3,35͔ and Redner and Reynolds ͓36͔ in a study of N N SAW’s, Daoud and Family ͓37͔ in a study of polydispersity, ͑d͒ ϭkT ln͑M!͒ϪM␰N ͑V,T͒, ͑122͒ Grassberger and Hegger ͓38͔ in simulations of a polymer, and by many others ͑for a review, see ͓4͔͒. In a recent work, (d) where ␰ N (V,T) is the free energy of one FJC, Gujrati ͓40͔ has used several chemical potentials of this kind 53 STATISTICAL PHYSICS OF THE FREELY JOINTED CHAIN 6313

FIG. 7. Schematic representation of all isochoric thermodynamical ensembles of the FJC in equilibrium with their reservoirs. Ar- rows with T( xX) symbolize Laplace transforms according to ͑3͒ ␤ FIG. 6. ͑a͒ Excess free energy ␰ex (V,T) for a FJC of a few ϩϱ f (x)ϭ␤͐ 0 dX exp(␤xX)F(X). different degrees of polymerization as a function of T*ϭT/To(d). ͑3͒ ͑b͒ Excess entropy s ex (T) for FJC’s as in ͑a͒. 1. The microcanonical ensemble (d) From the canonical partition function Q N (V,T), given to give a geometrical description of the phase transitions and by Eq. ͑107͒, the microcanonical partition function is ob- has applied this formalism to branched polymers. tained by the Laplace transform In most of the previous works, the authors perform calcu- ϩϱ lations on lattices using a grand canonical ensemble in which ͑d͒ Ϫ␤E ͑d͒ QN ͑V,T͒ϭ␤ dE e ⍀N ͑V,E͒. ͑130͒ the number of bonds in the polymer can fluctuate. In the ͵0 present work, ␹ is the continuous version of these chemical potentials. The definition of ␹ is neither a constraint to the A straightforward calculation gives geometry of a regular lattice nor a particular dimension of N͑dϪ1͒/2 the physical space. ͑d͒ ͑d͒ ͑d͒ ͓␤o͑d͒E͔ ⍀N ͑V,E͒ϭ␻o ⍀छ ͑V,E͒ , ⌫„͓N͑dϪ1͒ϩd͔/2ϩ1… ͑131͒ B. Some thermodynamical ensembles for the FJC where In this subsection, the FJC is considered as an ideal gas with phase space ␸ . From the canonical partition function ͑d͒ ͑d͒ rest ␻o ϭqo , of this gas, seven other thermodynamical ensembles may be ͑132͒ built using Laplace transforms ͓41͔. mE d/2 The extensive variables are (N,V,S) and the associated ⍀͑d͒͑V,E͒ϭ V. छ ͩ 2␲ប2ͪ intensive variables are (␹,ϪP,T). The canonical partition function Q (d)(V,T) is obtained with an approximation on (d) N ⍀ N (V,E) is the number of microstates of a FJC made of the size of the box containing the polymer. The dependence N bonds isolated in a volume V with an energy E.Inthe (d) (d) on V in Q N (V,T) is only linear; thus the Laplace trans- work of Edwards and Goodyear the contribution ⍀ छ (V,E) forms for the conjugated variables V and ϪP are straightfor- due to the motion of the first monomer is absent, the calcu- ward and do not give any extra information. lations having been done in the frame where this monomer is In Fig. 7, all the isochoric ensembles are shown in equi- at rest. If we omit this contribution we recover, for dϭ3, the librium with their reservoirs. We study two of these en- same behavior with E as found by these authors, which is a sembles, the microcanonical ensemble partly studied to some consequence of the equipartition. extent by Edwards and Goodyear ͓9͔ and the grand canonical The behavior with N is not trivial because of the presence ensemble in which the conjugated variables are ͑N,␹͒. of the gamma function. 6314 MARTIAL MAZARS 53

The entropy is given by

͑d͒ ͑d͒ sN ϭk ln⍀N ͑N,E͒ N ϭs͑d͒ϩ ͑dϪ1͒ln͓␤ ͑d͒E͔ id 2 o N͑dϪ1͒ϩd ϩk ln ␻͑d͒Ϫk ln⌫ ϩ1 , ͑133͒ o ͩ 2 ͪ

(d) where sid is the entropy of an ideal monomer. In the microcanonical ensemble the temperature is deﬁned by (d) -E)͉N,V which leads to ͑124͒ and again to equiץ/ s Nץ)Tϭ/1 partition.

2. Grand canonical ensemble with respect to variables (␹,N) This ensemble is called the grand canonical ensemble by FIG. 8. Diagram of the different domains of the FJC in the plane * * * * some authors ͓3,4͔, and the equilibrium ensemble by others ͑␹ , T ͒, with ␹ ϭ␹/kTo͑3͒ and T ϭT/To(3). The domains are (M) the monatomic domain, (P) the polyatomic domain, and (F) ͓39͔ to distinguish it from the truly grand canonical ensemble the forbidden zone. The point C is like a double critical point whose where the number of polymers can ﬂuctuate. In this en- coordinates are T*ϭ(1/2e)2/(3Ϫ1) and ␹*ϭ͓(3Ϫ1)/2͔T* . semble the partition function is given by c c c

ϱ ͑d͒ ͑d͒ J ϭϪkT lnQ0 ͑V,T͒ϩkT ͑d͒ ,V,T ϭ Q͑d͒ V,T e␤␹N, 134 ⌶ ͑␹ ͒ ͚ N ͑ ͒ ͑ ͒ dϪ1 /2 Nϭ0 ␹ T ͑ ͒ ϫln 1Ϫexp . ͑138͒ (d) (d) (d) ͫ ͩ kTͪͩT ͑d͒ͪ ͬ o where, for Nϭ0, Q 0 (V,T)ϭq o Q छ (V,T). In ͓6͔, the function ⌶(d)(␹,V,T) is called the generating The average number of bonds in the FJC is function. In Eq. ͑134͒ the first contribution to the sum cannot be given by a single monomer, because the first monomer ͑d͒ ͑dϪ1͒/2 ln ⌶ exp͑␹/kT͒͑T/To͑d͒͒ ץ 1 must have the ability to be a polymer initiator for the poly- N¯͑d͒ϭ ϭ . ␹ ͯ 1Ϫexp ␹/kT͒ T/T d͒͒͑dϪ1͒/2ץ ␤ merization ͑see Sec. IV C͒. This is the reason why there is a V,T ͑ ͑ o͑ (d) (d) (d) factor q o in the definition of Q 0 (V,T). This factor q o is ͑139͒ the continuous version of the activity for creating a chain end ␹͒ to ͑139͒ once moreץ/ץ͑͒␤/Applying the operator ͑1 .6,42͔͓ ¯(d) ¯(d) With the use of Eqs. ͑108͒, the partition function is given gives the fluctuations ⌬N for the average number N of by the bonds in the FJC. An elementary computation gives

ϱ ␤ ͑d͒ ͑dϪ1͒/2 N ¯͑d͒ ͑d͒ ͑d͒ o ⌬N ͑d͒ ⌶ ͑␹,V,T͒ϭQ0 ͑V,T͒ ͚ exp͑␤␹͒ . ϭexp͕␤͓␹Ϫ␹max͑T͔͖͒Ͻ1. ͑140͒ Nϭ0 ͫ ͩ ␤ ͪ ͬ N¯͑d͒ ͑135͒ In the plane ͑␹,T͒ three domains are easily identified. If The series ͑135͒ are geometrics, as for Bose-Einstein statis- ¯(d) (d) (d) (d) N Ͻ1, then on average there is only one monomer in the tics ͓43͔. ⌶ is defined only if ␹Ͻ␹ (T) where ␹ (T)is (d) max max polymer; this leads to the definition of ␹ (T)as given by mono

1 ͑dϪ1͒ To͑d͒ ͑dϪ1͒ To͑d͒ ͑d͒ ␹͑d͒ ͑T͒ϭkT ln . ͑136͒ ␹mono͑T͒ϭkT ln ϩkT ln . ͑141͒ max 2 ͩ T ͪ ͩ 2ͪ 2 ͩ T ͪ (d) The domain where ␹Ͻ␹(d) (T) is the monatomic domain. ␹max(T) is the continuous analog of the critical value of the mon (d) (d) activity appearing in the study of SAW’s in the equilibrium If ␹mono(T)Ͻ␹Ͻ␹max(T), then the FJC is made of several ensemble ͓36,38,39͔. The case of the subcritical phase is the bonds; this domain is the polyatomic domain. When only case studied in this work. When ␹ approaches ␹(d) (T)a ␹ ␹(d) (T) then N¯(d) ϱ; a FJC of infinite length is found max → max → careful study is needed for two reasons, which are ͑a͒ the as in ͓36͔. This phenomenon is similar to the Bose conden- (d) limitation of the box size ͓36,45͔ and ͑b͒ exchange between sation since ␹max(T) is defined with the same criterion ͓43͔. (d) the system and its reservoir. The domain ␹Ͼ␹max(T) is forbidden since the series ͑134͒ (d) For ␹Ͻ␹max(T), the grand partition function is are divergents. The forbidden domain is an artifact of the limiting case studied here where the size of the box is con- Q͑d͒͑V,T͒ ͑d͒ 0 sidered to be larger than the typical size of the FJC. If ones ⌶ ͑␹,V,T͒ϭ ͑dϪ1͒/2 , ͑137͒ 1Ϫexp͑␹/kT͒͑T/To͑d͒͒ take into account the finite size of the box then the forbidden domain disappears and the series ͑134͒ is convergent for all and the grand potential values of ␹ ͓45͔. These three domains are shown in Fig. 8. In 53 STATISTICAL PHYSICS OF THE FREELY JOINTED CHAIN 6315

A is a constant, z is the coordination number of the lattice on which the random walk is performed, and ␥ the critical exponent. Thus, for the FJC, the same value for the exponent ␥ is found as for the Gaussian polymer model. The exponent ␯ is related to the size of the polymer. The distance between two distinct monomers of the same FJC is given by ͉R(n,p)͉ where

R͑n,p͒ϭa uˆ k ͑nϽp͒. ͑145͒ k͚ϭn

R(n,p) is the bond vector between monomer ͑nϪ1͒ and p. The square distance is

p 2 2 ˆ 2 ˆ ˆ ͑R͑n,p͒͒ ϭa ͚ uk ϩ2 ͚ ui•uj ͑146͒ FIG. 9. Average number of links N¯͑3͒ in a FJC studied in the ͩ kϭn nрiϽ jрp ͪ internal grand canonical ensemble as a function of T*ϭT/To(3) for different values of ␹*ϭ␹/kTo͑3͒. For ␹*ϭ0.15, polyatomic be- and its average value havior is reentrant while for ␹*Ͻ0 it is not. For ␹*ϭ0.25 monatomic behavior is absent. 2 R 2 ϭa2 pϪn 1ϩ uˆ uˆ . 147 (d) ͗͑ ͑n,p͒͒ ͘ ͑ ͒ ͚ ͗ i• j͘ ͑ ͒ Fig. 9 N¯ is shown as a function of T*ϭT/T (d) for a few ͩ pϪn nрiϽ jрp ͪ o values of ␹*ϭ␹/kTo(d). The coordinates of the point marked C in Fig. 8 are given by With the results of Sec. III, the correlations between the bonds i and j are given by 1 2/͑dϪ1͒ Tc*ϭ , ͩ 2eͪ N Q͑␤͒ uˆ uˆ ϭ d⍀ ͗ i• j͘ ͑d͒ ͵ ͟ n ͑dϪ1͒ QN nϭ1 ␹c*ϭ Tc* . ͑142͒ 2 N ma2 N 2 ˆ ˆ2 ˆ ˆ ϫexp Ϫ ͚ ⍀n ͟ dun␦͑unϪ1͒ui•uj In Fig. 8, the polyatomic domain is reentrant; the C point ͩ ␤ nϭ1 ͪ͵ nϭ1 is similar to the double critical point found in some binary NϪ1 mixtures ͓17͔. The polyatomic domain appears strange. In ma2 ˆ ˆ (d) Ͻ Ͻ (d) ϫexp ͚ ͑⍀nun͒•͑⍀nϩ1unϩ1͒ , ͑148͒ this domain for ␹mono(T) ␹ ␹max(T), Eq. ͑140͒ gives ͫ ␤ ͩ nϭ1 ͪͬ 1/2Ͻ⌬N¯(d)/N¯(d)Ͻ1, and thus predicts large fluctuations in this domain, of the size of the whole system. The results for leading, after some algebra, to ⌬N¯(d)/N¯(d) indicate important polydispersity in this domain. These large values of ⌬N¯(d)/N¯(d) may be an artifact of the ͑d͒ N approximation on the size of the box. Gij 1 uˆ uˆ ϭ ϭ d␥ ␥ ␥ ͑1Ϫd/2͒ ͗ i• j͘ ͑d͒ ͑d͒ ͵͟ n͟ ͑ n nϩ1͒ Jn Jn nϭ1 n෈I C. Statistical properties and critical exponents The exponent ␥ is the first exponent that we compute. ͑1Ϫd/2͒ ϫI͑d/2Ϫ1͒͑␥n␥nϩ1͒ ͟ ͑␥k␥kϩ1͒ This exponent is related to the total number of conformations k෈J of the FJC ͓44͔. A FJC can be related to a random walk with N the partition function interpreted as the total number of con- 2 ϫId/2͑␥k␥kϩ1͒exp Ϫ ͚ ␥n , ͑149͒ formations of the FJC. The total number of random walks ͩ nϭ1 ͪ (d) made of N steps RN is identified with Q N ; thus with where Iϭ͕1,iϪ1͖ഫ͕jϩ1,N͖ and Jϭ͕i, j͖. The transforma- R Q͑d͒ӍAzNN␥Ϫ1 ͑143͒ N→ N tion ͑33͒ gives one has, using Eq. ͑108͒, N ͑d͒ ͑d͒ϭ ͑1Ϫd/2͒ AϭqoQछ ͑V,T͒, Gij ͟ d␻n͉det͓Ja͑␻͔͉͒͟ ͑␻n͒ I͑d/2Ϫ1͒͑␻n͒ ͵nϭ1 n෈I T ͑dϪ1͒/2 N zϭ , ͑1Ϫd/2͒ 2 2 2 ͩ T ͑d͒ͪ ϫ ͑␻k͒ I͑d/2͒͑␻k͒exp Ϫ An␻nϪ␻1 . o ͟ ͚ k෈J ͩ nϭ2 ͪ ␥ϭ1. ͑144͒ ͑150͒ 6316 MARTIAL MAZARS 53

All the permutations of the variables ͕␻j͖ are not allowed whose integration over ␻⌬nϩ1 gives, due to the fact that the because the memory effect in the FJC is included in the integrand is odd, 2 factor A n. However, taking E(A) as the number of elements ϱ k k in the set A, the relation ͑41͒ allows the ␴ permutations such 1 1 1 ͑2kϩ1͒! 2 W ͚ 2 that C ͑␴*[J]͒ϭC [J] and if (n1 ,n2)෈(␴*[J]) all natural kϭn k! ͩ 4ͪ ͩ A ͪ ͓2͑kϪn⌬nϩ1͒ϩ1͔! ⌬nϩ1 ⌬nϩ1 numbers between n and n are in the image ensemble ␴*[J] 1 2 1 of J with the permutation ␴. This limitation on the permuta- 2 ͑dϪ1͒/2 ͓2͑kϪn⌬nϩ1͒ϩ1͔ ϫ ͑1Ϫt⌬nϩ1͒ t⌬nϩ1 dt⌬nϩ1ϭ0. tions allows the correlations between the monomers i and j ͵Ϫ1 to propagate on the polymer, and this property must be the ͑156͒ same anywhere in the polymer. If a permutation of this type (d) is done in ͑150͒, G ij is invariant because ͉det͑␴͉͒ϭ1. More- Thus over, since it is right for any permutation which has these ͑d͒ properties, it is right for the permutation which puts i on 1 G⌬n ϭ0 ͑157͒ and j on C (J). Thus the invariance by propagation on the and polymer imposes the relation 2 2 ͗͑R͑n,p͒͒ ͘ϭa ͑pϪn͒. ͑158͒ ͑d͒ ͑d͒ ͑d͒ Gij ϭG͉iϪj͉ϭG⌬n ͑151͒ 1 The last equation gives ␯ϭ 2 which is the same value as that for the Gaussian polymer. For the same reasons as given and Eq. ͑147͒ becomes in Sec. IV, a simple argument could not give this result for any value of d because the analytical continuation to nonin- 2 2 2 ͑d͒ teger dimension is not always straightforward; the important ͗͑R͑n,p͒͒ ͘ϭa ͑pϪn͒ 1ϩ ͑d͒ ͚ Gij , ͫ ͑pϪn͒J nрiϽ jрp ͬ point is to preserve something similar to the Euclidean scalar N ͑152͒ product. These results are given for any dimension d, but not for any space of dimension d because a regularity on the 2 ͑pϪn͒ topological structure of the space is needed ͓20–22͔. 2 2 ͑d͒ ͗͑R͑n,p͒͒ ͘ϭa ͑pϪn͒ 1ϩ ͑d͒ ͚ G⌬n ͫ J ⌬nϭ1 N VII. CONCLUSION ͑pϪn͒ 2 ⌬n ͑d͒ We know that the Gaussian distribution for the end to end Ϫ ͑d͒ ͚ G⌬n . distribution of an ideal polymer is a consequence of the cen- J ⌬nϭ1 pϪn ͬ N tral limit theorem ͓1,6͔, and so this distribution is valid for Section IV gives N ϱ; for finite values of N there are corrections in 1/N ͓6͔. In→ the case of the FJC, the values of the critical exponents of ⌬n ⌬nϩ1 ⌬nϩ1 ideal polymers have been recovered and, moreover, a very ͑d͒ ͑1Ϫd/2͒ good approximation is given for its canonical partition func- G⌬n ϭ ͟ d␻k ͟ Ak ͟ ͑␻k͒ Id/2͑␻k͒ ͵ kϭ1 kϭ2 kϭ2 tion for almost all degrees of polymerization and for all ⌬nϩ1 physical dimensions of space dу2, subject to topological 2 2 2 ͑d͒ regularity of the spaces with noninteger dimensions. ϫexp Ϫ ͚ An␻nϪ␻1 f ͑⌬nϩ2͒ ͑153͒ ͩ nϭ2 ͪ Thus, in principle, this ideal model of a polymer is com- pletely solved. The analytical form of the partition function and we can perform the same series expansion as in Sec. IV ͑107͒ is approximate for two reasons: ͑1͒ The multiplicative on the function defined by coefficients A(d) and B(d), coming from the metric determinant, are only approximated by hypergeometric functions; ͑2͒ the size of the box containing the polymer has been con- h͑d͒ ϭA d␻ ␻ ͒͑1Ϫd/2͒I ␻ ͒ ͑⌬nϩ1͒ ͑⌬nϩ1͒ ͵ ⌬nϩ1͑ ⌬nϩ1 d/2͑ ⌬nϩ1 sidered greater than the typical size of the polymer. For the latter approximation, an analytical study is difficult to make, 2 2 ͑d͒ (d) (d) ϫexp͑ϪA⌬nϩ1␻⌬nϩ1͒f ͑⌬nϩ2͒ ͑154͒ while for the former, the coefficients A and B are computed from a straight line fit to the Monte Carlo data, and (d) to obtain G ⌬n . compared with the analytical approximation made with hy- (d) The series expansion of f (⌬nϩ2) gives for the last term of pergeometric functions. The good agreement between the (d) h (⌬nϩ1) analytical and numerical results validates the analytical method used. For each dimension studied by means of the 1 d/2 1 Monte Carlo integration, 498 points out of 499 are in agree- ͩ2ͪ ⌫͓͑dϩ1͔/2͒⌫͑1/2͒ ment, with an accuracy of 1–3%, with the straight line fit. The Monte Carlo method has been applied for 28 values of 1 the dimension of space between 2 and 8 including noninte- 2 ͑dϪ1͒/2 2nNϪ⌬nϪ2ϩ1 ͑ ϫ ͑1Ϫt⌬nϩ1͒ dt⌬nϩ1A⌬nϩ1 ͵Ϫ1 ger dimensions͒. Thus for the 13 972 partition functions computed, 13 944 are in agreement with the partition func- ϩϱ tion given by ͑107͒ with an accuracy of 1–3%. For each 2nNϪ⌬nϪ2ϩ1 2 2 ϫ d␻⌬nϩ1␻⌬nϩ1 exp͑ϪA⌬nϩ1␻⌬nϩ1 ͵Ϫϱ dimension studied, the only exception is for Nϭ3. However, in that case the analytical form of the partition function is ϩ␻⌬nϩ1t⌬nϩ1͒, ͑155͒ known exactly. In view of these results, Eq. ͑107͒ can be 53 STATISTICAL PHYSICS OF THE FREELY JOINTED CHAIN 6317 considered as nearly exact. The values of A(d) and B(d) given system, whereas these fluctuations are of the same size as the by the fit are all close to 1. A(d) is slightly bigger than 1 and system in the polyatomic domain. This domain is very B(d) is slightly smaller than 1 ͑see Table I͒. Both coefficients strange and requires a careful study. First, this domain has a arise from the metric determinant of the FJC; thus we can reentrant behavior with temperature ͑see Fig. 8͒ similar to conclude that the metric determinant produces only a weak the one occurring for the disordered phases in other systems perturbation of the value of the partition function ͓10͔. This ͓17–19͔; this would lead one to define a double critical point fact is known from numerical simulations where often the and to consider that the second chemical potential is similar contribution from the frozen vibrational degree of freedom to a hydrogen bond strength ͓17͔. The big fluctuations of the polyatomic molecules can be neglected ͓23,47͔. The present in the domain do not allow one to consider it as a analytical method used to obtain the approximations allows well defined phase. Because the fluctuations are of the same order of magnitude as the whole system, the polyatomic do- one to describe the analogy between quantum mechanics and main cannot be considered as a well defined phase. Never- polymer physics via the Feynman formulation ͓46͔.Inthe theless, the fluctuations always remain smaller than the sys- quantum theory of fields, the path formulation leads to the tem; thus the FJC is really polyatomic in this domain, but definition of the generating functional whose logarithmic de- with a very high polydispersity. rivatives with respect to the currents give the connected In the grand canonical ensemble, there is a limiting value Green’s functions ͓28͔. In statistical physics, the partition depending on the temperature for the second chemical poten- function is similar to the generating functional. So, in poly- tial, at which the system becomes infinite. This FJC of infi- mer physics, it seems natural to use the Feynman formula- nite length appears at high as well as low temperature for a tion to compute the partition function rather than the Fokker- positive second chemical potential. This phenomenon looks Planck equation. The use of Feynman rules allows us like the Bose condensation because the limit on the chemical naturally to use conventional renormalization methods and potential is defined with the same criterion on geometrical so to find naturally a method similar to the decimation series. It may be surprising to find that the reentrant infinite method ͓14,6͔. The first stage of the conventional renormal- FJC has the same behavior at high and low temperatures the ization, applied to the FJC, gives an odd-even effect in the ͑ critical exponent ␯ is independent of the temperature͒. This FJC. This effect has been observed by Fisher and Hiley ͓15͔ is, of course, a consequence of the fact that our system is and by Grassberger and Hegger ͓16͔. Because of the fracta- classical. In recent work by Golubovic´ and Xie 48 ,ithas lization process explained in Sec. IV D and shown for one ͓ ͔ been shown that if one takes quantum fluctuations into ac- case in Fig. 2, one can suppose that FJC’s whose degrees of count, one finds, for low temperatures and for a non-self- polymerization belong to the same prefractal sequence have avoiding chain, a crumpled ground state having the appear- some common properties; this also leads one to suppose that ance of a highly collapsed polymer with radius of gyration other odd-even effects of higher order exist. The fractaliza- growing very slowly with N as R ln(N) 1/2 48 .Ifwe tion appears as the reverse of the decimation method; this ϳ͓ ͔ ͓ ͔ apply the FJC model to polyethylene with the same geo- might be easily tested because all analytical terms have finite metrical factors as in Ref. 48 , we find using Eqs. 108 and values. The numerical estimate of the odd-even effect on the ͓ ͔ ͑ ͒ 142 for dϭ3, T 3 1.516 K, and value of the coefficient B(d) in Eqs. ͑114͒ and ͑116͒ is given ͑ ͒ o͑ ͒Ϸ to first order in Sec. V C. In view of this effect the strange To T ϭ Ӎ0.279 K, behavior of the FJC for Nϭ3 is interpreted as a reminiscence c 2e of the odd-even effect. This effect appears to be smaller for ͑159͒ the FJC than for realistic polymers models 16 and SAW’s Ϫ23 Ϫ5 ͓ ͔ ␹ Ӎ0.385ϫ10 JӍ2.403ϫ10 eV. ͓15͔; this certainly means that interaction between monomers c amplifies this effect. With the knowledge of the partition Tc is not far from the Ginzburg temperature defined in function of the FJC made of N bonds, we can study all the ͓48͔. The ‘‘infinite real’’ polymer in the low temperature equilibrium properties of the FJC. Thus we have recovered limit would be in the crumpled ground state by a pure quan- equipartition and found the Sakur-Tetrode relation for the tum effect, whereas our classical infinite FJC has the same FJC. behavior at low and high temperatures. Of course, in the case Considering the FJC as an ideal monatomic gas with the of these very low temperatures, our system ͑as well as the reduced phase space ␸rest it is possible to build the partition quantum system͒ is certainly meaningless because in the real function of the FJC via Laplace transform in other thermo- world various things may happen, such as, for instance, crys- dynamical ensembles. Thus, we have given the microcanoni- tallization, before reaching temperatures of the order of a few cal partition function and the partition function of the grand degrees kelvin. Even if one were able to experimentally cool canonical ensemble where the monomer number can fluctu- an ideal polymer down to a few kelvins, in the crumpled ate. In this grand canonical ensemble, a second chemical ground state excluded volume effects cannot be neglected. potential is associated with the number of bonds in the poly- Neither the FJC nor the ideal gas exists in the real world. mer. In the plane ͑␹,T͒ of this ensemble, we can define es- They are very crude models of real systems. Moreover, it sentially two domains in which the system has very different must be pointed out that the system studied in ͓48͔ is slightly behaviors: a monatomic domain where the average number different from a FJC since a harmonic potential between the of monomers in the FJC is less than 2, and in which the bonds and a bending energy are used. Constant length for the fluctuations of this number are quite small, and a polyatomic bonds is not compatible with the uncertainty principle, unless domain where the FJC is made of several monomers. In the the metric of the space is modified. monatomic domain the fluctuations of the average number of Throughout this work d is the dimension of the physical monomers in the FJC are small compared to the size of the space and is taken as a real parameter greater than or equal to 6318 MARTIAL MAZARS 53

2. The use of noninteger dimensions should be done with lattices ͑i.e., random walks͒. In particular, Eqs. ͑144͒ give caution and if the results of the FJC are given for any dimen- the coordination number of the equivalent lattice as a func- sion of space, they are not necessarily right for any space of tion of the temperature. Thus the FJC appears as a generali- dimension d. For example, the critical exponent ␯ must be zation of random walks on a lattice. Choosing a particular defined in the fractal metric, otherwise an anomalous diffu- geometry for a regular lattice on which a random walk is sion exponent ␪ appears that one may relate to the fractal performed is equivalent to choosing a temperature for the dimension d f by ␪ϭ2d f Ϫ2 ͓22͔; thus in the fractal space one FJC model. To compare a random walk on a regular lattice could recover ␯ϭ1/2. But, even if we define ␯ with caution, with a given symmetry with another on a lattice of different internal to the fractal space, it may happen that the relation- symmetry but with the same lattice spacing and the same ship between the fractal dimension and the anomalous diffu- number of steps is equivalent to studying a FJC made of a sion exponent is not valid ͑for ramification orders greater number of bonds equal to the number of steps in the random than 1͒. The properties of the FJC in space with noninteger walk and with a bond length equal to the lattice spacing at dimension depend on the topology of the space. Neverthe- two different temperatures. For instance, in a two dimen- less, in a recent study by Bender and co-workers ͓20,21͔, sional space, transferring a random walk from a triangular ‘‘random walks in noninteger dimension’’ make sense, if we lattice ztriϭ6 to a square lattice zsquaϭ4 is equivalent to cool- define rigorously the probabilities of walking outward or in- ing a FJC according to the rule ward in some regions of the space ͓20͔; otherwise, we may obtain unacceptable formulas for some probabilities ͑greater ͑2Ϫ1͒/2 Ttri ztri 3 than 1 or less than 0͒. With their spherically symmetric ran- ϭ ϭ . ͑160͒ dom walks, they recover also the right Hausdorff dimension ͩ Tsquaͪ zsqua 2 of a random walk ͓21͔. Their construction is equivalent to a generalization of the Buffon needle construction ͑see note 7 ACKNOWLEDGMENTS of ͓20͔͒. The FJC may be viewed as made of N Buffon needles, and one may use results of the FJC in noninteger I wish to thank J. J. Weis and J. M. Caillol for useful dimension if we have shown previously that the topology of discussions. I am very grateful to D. Levesque for a lot of the noninteger space is correct for the FJC. The results of the helpful discussions on the Monte Carlo methods. The Lab- FJC are not true for any fractal space. A criterion of validity oratoire de Physique Theórique et Hautes Energies is a lab- might be the value of the critical exponent ␯. Finally, Eqs. oratoire associe´ au Centre National de la Recherche Scienti- ͑144͒ give the connection with ideal polymers on regular fique ͑CURA No. Q63͒.

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