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A Polymer Chain Near the Single Obstacle

As was mentioned above, there are a lot of different ways of considering the Edwards-Frisch problem. However, from the methodological point of view and for the sake of a better clarification of non-euclidean geometry ideas for the description of topological constraints, we would like to present the method of conformal transformation. [Pg.5]

The main idea is as follows. Let us consider the plane in which our chain is placed as a complex one, z = x + iy. (z = z(x, )) and let us find the conformal transformation, z = z( ), of the plane z with the obstacle to the Riemann surface, = + b], which does not contain an obstacle (such a transformation means the transfer to the covering space). Due to the conformal invariance of Brownian motion1, in the covering space a random process will be obtained corresponding to the initial one on the plane z but without any topological constraints. [Pg.6]

Introducing the polar coordinates (p, 6) on the initial z-plane, we can rewrite Eq. (4) as follows  [Pg.6]

Taking into account that under the conformal transformation, the Laplace operator is transformed in the following way  [Pg.6]

1 Conformal invariance of random walk means that after the conformal transformation this process will be random again. [Pg.6]


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