By Alexei R. Khokhlov, Alexander Yu Grosberg, Vijay S. Pande
Marketplace: experts and graduate scholars in polymer physics, statistical physics, actual chemistry of polymers, fabrics technology, molecular biophysics, and chemical engineering. This introductory quantity offers in-depth descriptions of basic innovations in addition to key commercial purposes in polymer actual chemistry and molecular biophysics. subject matters contain statistical theories of polymer ideas, melts, polymer liquid crystals, polymer networks, and polyelectrolytes; data of perfect chains; the viscoelastic habit of polymer platforms; and diverse good points of biopolymers, DNA, and proteins.
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Extra resources for Statistical Physics of Macromolecules
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).