Statistical Mechanical Properties of Polymer Configurations which Enclose a Constant Area
TL;DR: In this article, the statistical properties of plane polymer loops enclosing a constant area were investigated, using a continuous model from the start, and an analytic expression for the generating functional was obtained, which in turn was used to derive the distribution function for the enclosed area, the average squared distance of a given repeating unit from the origin, and the entropic force on a repeating unit.
Abstract: The statistical mechanical properties of plane polymer loops enclosing a constant area are investigated, using a continuous model from the start. For this purpose an analytic expression for the generating functional is obtained, which in turn is used to derive (1) the distribution function for the enclosed area, (2) the average squared distance of a given repeating unit from the origin, and (3) the entropic force on a repeating unit.
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31 Oct 2012
TL;DR: In this paper, the probability density for the area A enclosed by a polymer loop in crossed electric-magnetic fields is evaluated using the Hida-Streit formulation, where the many possible conformations of the polymer, x(v) and y(v), are represented by paths and parametrized in terms Brownian motion.
Abstract: The probability density for the area A enclosed by a polymer loop in crossed electric-magnetic fields is evaluated using the Hida-Streit formulation. In this approach, the many possible conformations of the polymer, x(v) and y(v), are represented by paths and are parametrized in terms Brownian motion. When the magnetic field is switched off, results agree with the works of Khandekar and Wiegel5
10 Jun 2015
TL;DR: In this paper, the probability distribution for the area enclosed by a polymer loop in a gel and under different external fields using white noise analysis is derived, where the external fields considered are electric field, and crossed electric-magnetic fields.
Abstract: This paper presents derivation of the probability distribution for the area enclosed by a polymer loop in a gel and under different external fields using white noise analysis. In this context, the polymer loop is represented by Brownian paths and its immersion in a gel constraints it to occupy a constant area[1]. The external fields considered are electric field, and crossed electric-magnetic fields.
TL;DR: In this article, the statistical properties of plane polymer configurations which enclose a fixed area and are subject to an external electric field are investigated, and an exact expression for the generating functional is obtained and subsequently used to derive: (a) the distribution function for the enclosed area; (b) the mean square distance of a given repeating unit from the origin; (c) the entropic force on a repeating unit.
Abstract: The statistical mechanical properties of plane polymer configurations which enclose a fixed area and are subject to an external electric field are investigated. For this purpose an exact expression for the generating functional is obtained and subsequently used to derive: (a) the distribution function for the enclosed area; (b) the mean square distance of a given repeating unit from the origin; (c) the entropic force on a repeating unit.
References
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Book•
01 Jan 1986
TL;DR: In this article, the viscoelasticity of polymeric liquids was studied in the context of rigid rod-like polymers and concentrated solutions of rigid rods like polymers.
Abstract: Introduction Static properties of polymers Brownian motion Dynamics of flexible polymers in dilute solution Many chain systems Dynamics of a polymer in a fixed network Molecular theory for the viscoelasticity of polymeric liquids Dilute solutions of rigid rodlike polymers Semidilute solutions of rigid rodlike polymers Concentrated solutions of rigid rodlike polymers Index.
10,225 citations
"Statistical Mechanical Properties o..." refers background or methods in this paper
...Recently, Brereton and Butler (5) suggested a new way to do this: (1) represent a polymer by a random walk of N steps in a plane; each step has length I; (2)approximate the steric and topological effects of the other polymers in the system by the constraint that this walk encloses a fixed (algebraic) area....
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...For this purpose an analytic expression for the generating functional is obtained, which in turn is used to derive (1) the distribution function for the enclosed area, (2) the average squared distance of a given repeating unit from the origin, and (3) the entropic force on a repeating unit....
[...]
Book•
01 Jan 1986
TL;DR: In this article, the authors present an elementary account on the Wiener path integral as applied to Brownian motion, and the author progresses on to the statistics of polymers and polymer entanglements.
Abstract: This monograph distills material prepared by the author for class lectures, conferences and research seminars. It fills in a much-felt gap between the older and original work by Feynman and Hibbs and the more recent and advanced volume by Schulman.After presenting an elementary account on the Wiener path integral as applied to Brownian motion, the author progresses on to the statistics of polymers and polymer entanglements. The next three chapters provide an introduction to quantum statistical physics with emphasis on the conceptual understanding of many-variable systems. A chapter on the renormalization group provides material for starting on research work. The final chapter contains an over view of the role of path integrals in recent developments in physics. A good bibliography is provided for each chapter.
241 citations
"Statistical Mechanical Properties o..." refers background or methods in this paper
...Recently, Brereton and Butler (5) suggested a new way to do this: (1) represent a polymer by a random walk of N steps in a plane; each step has length I; (2)approximate the steric and topological effects of the other polymers in the system by the constraint that this walk encloses a fixed (algebraic) area....
[...]
...(2) A new way to work directly in covering space forms the subject of ref....
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...For this purpose an analytic expression for the generating functional is obtained, which in turn is used to derive (1) the distribution function for the enclosed area, (2) the average squared distance of a given repeating unit from the origin, and (3) the entropic force on a repeating unit....
[...]
138 citations
TL;DR: The status of exactly solvable problems within the path integral formulation of non-relativistic quantum mechanics is reviewed in this paper, where some applications of these exact results are presented.
131 citations
TL;DR: In this article, a random walk in a plane, constrained to enclose a given area, can be used to approximately represent the properties of an entangled polymer molecule, and the statistical mechanical properties of the loop are calculated exactly and the distribution function for the enclosed areas is found.
Abstract: The authors show how a random walk in a plane, constrained to enclose a given area, can be used to approximately represent the properties of an entangled polymer molecule. The statistical mechanical properties of the loop are calculated exactly and the distribution function for the enclosed areas is found. For the case of a random walk with free ends joined by a straight line segment, the distribution function is given by the Cauchy distribution. This implies that the area has statistical fractal properties but does not have a mean. For a genuinely closed random walk, a mean exists but the distribution of areas is not fractal. The spatial and mechanical properties of the constrained configurations have also been calculated analytically. If the unrestricted coil can be regarded as an entropic spring of zero natural length, then the area-constrained configurations behave qualitatively like springs with a finite natural length. The deformation behaviour also shows both softening and hardening dependent on the area imposed.
34 citations