By S. Albeverio, L.S. Ferreira, L. Streit
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Extra info for Resonances - Models and Phenomena
The spatial form of a chain in a Gaussian coil is described by the transfer operator with continuous spectrum;for a continuous spectrum, correlations along the chain extend over its entire length. In the two previous subsections, we considered the case of the discrete spectrum of the transfer operator ~. The results obtained will be used repeatedly below. An example of the opposite situation of a continuous spectrum is considered now. Let us return to the problem of the spatial form of the free ideal coil for the standard Gaussian model of a polymer chain.
Actually, the function ~b(x) varies slowly far from the potential well, where q~(x) ~0, and the integral operator ~ in Eq. 8) so that Eq. 4} in this region is reduced to (a2/6) A~= (A-- 1)~(x). 29) The solution of this equation takes the form const [ x . 30) in exact correspondence with Eq. 24). 12: long Gaussian loops fluctuate around a localization region of the field q)(x). According to Eq. 30), the characteristic loop size is approximately a(A-- 1) -~/2, that is, because the chains are Gaussian, their characteristic length is m~ (A--1)-~, and the number of loops in the chain is approximately N/m ~N(A--1).
3), as where the integral is treated as continual, that is, as an integral over all continuous paths x(~-). ~i~di: also well known from the theory of diffusion, describing the probability of finding a diffusing particle at the point x(t) at the time moment t provided that it was located at the initial moment 0 at the point x. It should be recalled that Eq. 26) does not describe the random walk of one Brownian particle (or a fixed number of particles) but rather the diffusion of a cloud of such particles randomly appearing or being absorbed in proportion to qg(x).