Probability distribution function ideal chain

The probability distribution function for the end-to-end vector R of an ideal linear chain of N monomers is the product of the three independent distribution functions [Eq. (2.81)] ... 

The entropy of an ideal chain with JVmonomers and end-to-end vector R is thus related to the probability distribution function ... 

The probability distribution of the end-to-end vector of an ideal chain is well described by the Gaussian function ... 

We know that an ideal polymer chain is a random walk. Consider a distribution function q r, r, t), which tells us the probability that a chain that is t steps long has started at a position r and finished at a position r. We know that a random walker in free space has a distribution function that obeys a diffusion equation, so for an ideal, isolated chain (with no excluded volume interaction) we can write... 

The simplest model to describe the structure of a linear chain made of N units of length I each is the random walk. This is an ideal chain where no interactions are present between monomers. The distribution function P(r,N), which is the probability that a chain made of N steps starts at the origin and ends at point r, is a Gaussian. In three-dimensional space. 

Fig. 2 The distribution function of probability density P r) that the ends of an ideal chain are separated by distance r...

We have seen that at a distance r from a monomer the monomers within a range n ia/r) of m along the chain make the major contribution to On occasion, especially for photophysical processes where two monomers must come to proximity, one needs to know the behavior of a more detailed correlation function, p(r,/i), the probability that two monomer units n monomers apart on the chain are separated in space by a vector r. For an ideal chain this is the well-known Gaussian distribution... 

The examples of self-similar functions considered above fall into an especially simple category all are functions of the generalized time or group parameter and functionals (either linear or nonlinear) of the initial condition. However, there are many self-similar physical properties and mathematical objects that depend on additional, unsealed variables. For example, the probability density for the distribution of end-to-end distances of a linear, ideal (phantom) polymer chain is given by the expression... 

Figure 15 Composition profiles shown as a function of reduced monomer number i /Vfor 512-unit MIST-tuned copolymer sequences having 1 1 AB composition. The present definition assumes that the A-type monomers are coded by symbol +1, whereas symbol -1 is assigned to the B-type monomers. For an ideal RCP in which chemically different units follow each other in a statistically random fashion, the probability Pa that monomer A is iocated at the th position in the chain is % for any . (a) Optimization of the transition temperature V at different Ti leads to sequences that have a gradient (or S-iike) distribution of A and B monomers, as schematically depicted In the right panel, (b) Optimization of the characteristic length scale r at different leads to sequences that have a tapered (or bolas-like ) distribution of A and B monomers, as schematically depicted in the right panel.

The conformational distribution (defining the probability of each particular conformation) can he expressed as the proh-ability density function of all sets of hond orientations (nii, m2, m3,. ..). The distribution depends on the mechanism of the chain flexibility and on the interactions between nonbonded chemical units. Below we consider the simplest case of an ideal polymer chain with no interactions between its units apart from interactions between dose enough neighbors along the chain. 

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