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Comparison between Borel resummation and renormalization group descriptions of polymer expansion
1.H. Yamakawa, Modern Theory of Polymer Solutions (Harper and Row, New York, 1971);
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2.M. Muthukumar and B. G. Nickel, J. Chem. Phys. 80, 5839 (1984).
3.M. Muthukumar and B. G. Nickel, J. Chem. Phys. 86, 460 (1987).
4.The calculations by J. des Cloizeaux, R. Conte, and G. Jannink, J. Phys. Lett. 46, 595 (1985) sum the sixth‐order perturbation series of Muthukumar and Nickel (Ref. 2). While des Cloizeaux et al. use techniques derived from RG methods, their work should be viewed as an alternative resummation to that of MN (Ref. 3).
5.K. F. Freed, Renormalization Group Theory of Macromolecules (Wiley‐Interscience, New York, 1987).
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8.J. des Cloizeaux, J. Phys. (Paris) 42, 635 (1981);
8.J. des Cloizeaux, 43, 1743 (1982) has independently presented a chain space direct RG method at about the same time as the developments in Ref. 7. Both approaches can be shown to produce identical results for dimensionless ratios when it is recognized that des Cloizeaux uses the dimensionless variable g which is proportional to where is the osmotic second viral coefficient and M is the molecular weight. (To order ε this g is equivalent to the parameter u of the methods of Refs. 5–7.) Some differences emerge for dimensional properties because Ref. 7 uses Eq. (1.3), while des Cloizeaux uses Eq. (1.2) and these differences are removed in this note. However, other differences remain because des Cloizeaux does not base his crossover analysis on the Gell‐Mann‐Low style RG equation; instead he introduces Padé‐type methods building upon assumptions concerning the analytic structure of these dimensional properties, some of which are based on properties proven on the basis of the renormalization group equation. His method provides complimentary insight into the RG resummation of the bare perturbation series., J. Phys. (Paris)
9.This interpretation neglects contributions from residual interactions at the theta point, but it is quite rigorous within the traditional two‐parameter model (Ref. 1). The work of J. F. Douglas, B. J. Cherayil, and K. F. Freed, Macromolecules 18, 2455 (1985)
9.and B. J. Cherayil, J. F. Douglas, and K. F. Freed, J. Chem. Phys. 83, 5293 (1985);
9.B. J. Cherayil, J. F. Douglas, and K. F. Freed, 87, 3089 (1987). Indicates that this neglect may not introduce serious problems, provided the binary interaction and unperturbed dimensions are interpreted as effective quantities containing the influence of the residual interactions. Use of the Edwards Hamiltonian to evaluate the unperturbed dimensions with RG calculations yields the value thereby giving to the additional interpretation as the renormalized size of the unperturbed polymer. Hence, may be interpreted as the renormalized chain length in the ideal chain or noninteracting state., J. Chem. Phys.
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12.When and are evaluated to a given order in ε, the presence of the power of in Eq. (2.5) implies that the factor contributes at one higher order in e than the order to which and have been determined. Some caution must be exercised to only retain contributions from the factor to the order to which has been determined. If this procedure is not followed, it is found that the highest order in ε contibutions to exponents in the dependence of ζ on are altered in the next order of ε expansion and inconsistent relations are obtained (Refs. 5–7). For instance, the spurious first‐order equation is thereby generated. Fortunately, the spurious factor of does not appear in the final results of previous works (Refs. 5–7) because of the subsequent use of the inversion transformation which “kills off” the spurious terms. In this paper, however, the matter is of some importance as it is necessary to define ζ consistently to give orders of ε.
13.C. Domb and A. J. Barrett, Polymer 17, 179 (1976).
14.J. F. Douglas and K. F. Freed, J. Chem. Phys. 86, 4280 (1987), Appendix A.
15.This structure emerges directly from the alternative trajectory formulation of the solution to the RG equation (Refs. 5 and 6).
16.The form (3.3a) is required because the order terms in the exponent in Eq. (3.3a) are found below to contribute to the good solvent limit in the form of contributions of order to ν, while the second‐order evaluation of Eq. (2.6) also produces second‐order terms in the exponential involving and However, the coefficients of these terms have important additional contributions emerging in order calculations, so these terms are dropped. These consistency considerations are of the same kind as those dictating the forms (2.8) and (2.9) used in calculating to order ε and respectively (see also Ref. 20).
17.J. F. Douglas, S. Q. Wang, and K. F. Freed, Macromolecules 19, 2207 (1986).
18.S. Q. Wang and K. F. Freed, J. Phys. A 19, L637 (1986).
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20.When the exponential factor from is expanded in u to the order of ε of the calculation, it generates contributions to the prefactor portion. The influence of these terms can be understood by noticing how the coefficient of changes between the first‐ and second‐order RG calculations. The first‐order treatment yields a coefficient of the leading two terms in the ε expansion of the exact coefficient In second order, becomes modified, and another term in the ε expansion of the exact coefficient is retained. This alone would produce the coefficient which upon ε expansion yields The “extraneous” term in (11/16), emerging from ε expansion of is precisely the one that is removed by the linear term in the expansion of exponential in Eq. (3.3), and this is why the (11/16) is formally taken as of order ε. This cancellation shows how the RG crossover maintains the proper coefficients of the leading terms in the expansion in The analysis also indicates the importance of requiring the crossover description to be consistent with the original perturbation expansion. It is also interesting to note that the transformation (2.1) is found, after evaluating to give a change in variables reminescent of the closely related Euler transform (Ref. 21) appearing in the Borel summation methods (Ref. 19). Perhaps, further analysis might uncover additional correspondences.
21.G. H. Hardy, Divergent Series (Clarendon, Oxford, 1949).
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