O(N) Generalized nonlinear sigma model and its applications
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ELEMENTARY PARTICLES AND FIELDS Theory
O(N ) Generalized Nonlinear Sigma Model and Its Applications* F. Ferrari1)** , J. Paturej1) , and T. A. Vilgis2) Received April 17, 2009
Abstract—In this work the results of an interdisciplinary research between field theory and statistical mechanics will be presented. It is shown that the dynamics of an inextensible chain is described by a slight generalization of the O(d) nonlinear sigma model. It is checked, that in the large time limit the correct equilibrium distribution is reached. Our approach is based on path integrals, but it may be also connected to the usual description of the dynamics of a chain as a diffusion stochastic process. Some applications of our results will be discussed, like for instance the calculation of the average distance between two polymer segments. DOI: 10.1134/S1063778810020158
1. INTRODUCTION In this report the authors briefly review the results of their studies on the dynamics of a chain fluctuating in a solution held at constant temperature. In the language of polymer physics, the chain can be described as the continuous limit of the so-called freely jointed chain. This means that one starts from a discrete system of small massive particles connected together by segments of constant length and performs the limit in which the length of the segments approaches zero, while their number grows to infinity. In doing this limit, the total length and mass of the chain are preserved. The theoretical treatment of a system of this type requires the imposition of rigid constraints, which assure the inextensibility of the chain. To deal with these constraints is cumbersome. In the statistical mechanics of polymers, which can also be considered as long, flexible chains, most often the gaussian chain model is applied, a variant of the freely jointed chain in which the constraints are relaxed. One can show that, if the number of segments is large, the gaussian chain model becomes equivalent to the freely jointed chain model [1, 2]. In the case of dynamics, the situation is more complicated. The formulation of the dynamics of the continuous limit of the freely jointed chain using the stochastic equation of Langevin has been discussed in a series of seminal works by Edwards and Goodyear [3–5]. ∗
The text was submitted by the authors in English. Institute of Physics and CASA*, University of Szczecin, Poland. 2) Max Planck Institute for Polymer Research, Mainz, Germany. ** E-mail: [email protected] 1)
The reader who is interested in possible applications to polymer physics of the freely jointed model may consult these references. Our aim is rather to provide a path integral formulation of an inextensible chain and to investigate its consequences. The ideas presented here have been developed in [6–9]. To avoid the complications related to the presence of the constraints, the fluctuations of the chain from a given initial configuration to a final configuration3) are regarded as the motion of a string which spans a given surface in a d-dimensional space.
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