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A polymer is a macromolecule, composed of many similar or identical repeated subunits. Polymers are common, but not limited to organic media. They range from familiar synthetic plastics to natural biopolymers such as DNA and proteins. Their unique elongated molecular structure produces unique physical properties, including toughness, viscoelasticity, and a tendency to form glasses and semicrystalline structures. The modern concept of polymers as covalently bonded macromolecular structures was proposed in 1920 by Hermann Staudinger.〔H.R Allcock; F.W. Lampe; J.E Mark, ''Contemporary Polymer Chemistry (3 ed.)''. (Pearson Education 2003). p. 21. ISBN 0-13-065056-0.〕 One sub-field in the study of polymers is polymer physics. As a part of soft matter studies, Polymer physics concerns itself with the study of mechanical properties〔P. Flory, ''Principles of Polymer Chemistry'', Cornell University Press, 1953. ISBN 0-8014-0134-8.〕 and focuses on the perspective of condensed matter physics. Because polymers are such large molecules, bordering on the macroscopic scale, their physical properties are usually too complicated for solving using deterministic methods. Therefore, statistical approaches are often implemented to yield pertinent results. The main reason for this relative success is that polymers constructed from a large number of monomers are efficiently described in the thermodynamic limit of infinitely many monomers, although in actuality they are obviously finite in size. Thermal fluctuations continuously affect the shape of polymers in liquid solutions, and modeling their effect requires using principles from statistical mechanics and dynamics. The path integral approach falls in line with this basic premise and its afforded results are unvaryingly statistical averages. The path integral, when applied to the study of polymers, is essentially a mathematical mechanism to describe, count and statistically weigh all possible spacial configuration a polymer can conform to under well defined potential and temperature circumstances. Employing path integrals, problems hitherto unsolved were successfully worked out: Excluded volume, entaglment, links and knots to name a few.〔F.W. Wiegel, ''Introduction to Path-Integral Methods in Physics and Polymer science'' (World Scientific, Philadelphia, 1986).〕 Prominent contributes to the development of the theory include Nobel laureate P.G. de Gennes, M.Doi,〔M. Doi and S.F. Edwards, ''Dynamics of concentrated polymer systems''. Parts 1-4. (Journal of the Chemical Society, Faraday Transactions 2 74: 1978-9)〕 F.W. Wiegel〔 and H. Kleinert.〔H. Kleinert, ''PATH INTEGRALS in Quantum mechanics, Statistics, Polymer Physics, and Financial Markets'' (World Scientific, 2009).〕 == Path Integral Formulation == Early attempts at path integrals can be traced back to 1918.〔P.J. Daniell, ''Ann, Math.'' 19 (1918) 279.〕 A sound mathematical formalism wasn't established until 1921.〔N. Wiener, ''Proc. Nat. Acad. Sci.'' USA 7 (1921) 253.〕 This eventually lead Richard Feynman to construct a formulation for quantum mechanics,〔R.P. Feynman, "The Principle of Least Action in quantum Mechanics," Pd.d Thesis, Princeton University (1942), unpublished.〕 now commonly known as Feynman Integrals. In the core of Path integrals lies the concept of Functional integration. Regular integrals consist of a limiting process where a sum of functions is taken over a space of the function's variables. In functional integration the sum of functionals is taken over a space of functions. For each function the functional returns a value to add up. Path integrals should not be confused with line integrals which are regular integrals with the integration evaluated along a curve in the variable's space. Not very surprisingly functional integrals often diverge, therefore to obtain physically meaningful results a quotient of path integrals is taken. This article will use the notation adopted by Feynman and Hibbs,〔R.P. Feynman and A.R. Hibbs, ''Quantum Mechanics and Path Integrals'' (McGraw-Hill, New York, 1965).〕 denoting a path integral as: : with as the functional and the functional differential. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Path integrals in polymer science」の詳細全文を読む スポンサード リンク
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