Probability distribution functions Gaussian chain

The probability distribution function of the tube length L for a chain with N monomers is approximately Gaussian, with mean-square fluctuation of the order of the mean-square end-to-end distance of the chain. The tube length fluctuates in time, leading to stronger molar mass dependences of relaxation time, viscosity, and diffusion coefficient resembling experi-mental observations over some range of molar masses ... 

Thus, as given by Eq. (1.42), the probability distribution function for the end-to-end vector R is Gaussian. The distribution has the unrealistic feature that R can be greater than the maximum extended length Nb of the chain. Although Eq. (1.42) is derived on the freely jointed chain model, it is actually valid for a long chain, where the central limit theorem is applicable, except for the highly extended states. 

Consider a strand of polymer chain between two cross-links. The vector R between the positions of the two cross-links changes with deformation. Any molecular theory on rubber elasticity is based on the probability distribution function for R. As seen in Chapter 1, if the number of segments N on the strand is large, the probability distribution (R, N) of the end-to-end vector R is a Gaussian function... 

Since this is a Gaussian distribution, a chain with this probability distribution function is called a Gaussian chain. The mean square end-to-end distance of a Gaussian chain is given by... 

The Gaussian scission probability distribution function, with a preference for mid-chain scission, is frequently encountered in shear-induced mechanochemical degradation. The parabolic distribution, on the contrary, indicates a preference for chain-end degradation. This phenomenon has been reported in the hydrolysis of dextran, as a result of chain branching. 

Random polymer coils are often called Gaussian chains because the probability distribution function (for finding a segment in a given volume element) has a Gaussian shape. For linear Gaussian chains, the r.m.s. radius of gyration is... 

In addition to the above result for the size exponent, several quantities such as the probability distribution function for finding a particular end-to-end distance can be derived for the Kuhn model. In particular, the results become simple if the end-to-end distance is smaller than the chain contour length Nl.ln this limit, for example, the probability distribution function for the end-to-end distance is a Gaussian function. In view of this, a Kuhn model chain with large enough N is called a Gaussian chain. The major properties of a Gaussian chain are now summarized. 

Figure 2.10 (a) The normalized end-to-end distance probability distribution function for a Gaussian chain. The radius of gyration Rg and the hydrodynamic radius R are shown in units of the root mean square end-to-end distance R. (b) Sketch of the long-ranged segment density profile falling inversely with the radial distance from the center of the chain. 

Applying the TABS model to the stress distribution function f(x), the probability of bond scission was calculated as a function of position along the chain, giving a Gaussian-like distribution function with a standard deviation a 6% for a perfectly extended chain. From the parabolic distribution of stress (Eq. 83), it was inferred that fH < fB near the chain extremities, and therefore, the polymer should remain coiled at its ends. When this fact is included into the calculations of f( [/) (Eq. 70), it was found that a is an increasing function of temperature whereas e( increases with chain flexibility [100],... 

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

This equation shows that W(x, y, z) is a Gaussian distribution function that has a maximum value at r = 0, as shown in Fig. A2.2(a) for a large chain of carbon-carbon links. The value of W(x, y, z) corresponds to an end-to-end distance r in a particular direction specified by the set of coordinates (x,y,z). However, there may be many such coordinates each of which gives rise to the same end-to-end distance r but in a different direction. Thus, a more important probability is that... 

In particular, it appears that the probability distribution of the vector joining the chain ends, for a chain with excluded volume, strongly differs from a Gaussian function, and this fact greatly diminishes the value of the Gaussian chains. 

The Gaussian chain has an important property the probability distribution of the vector R — R between any two beads n and m is a Gaussian function, i.e. 

A quenched random medium, such as a rough surface or a frozen gel network, is a complex structure that can in principle be modelled by a complicated potential function y(R). However, we will not be interested in the physical properties of a polymer chain immersed in a specific environment, but rather in an ensemble of similar environments. Hence, we will have to specify instead the probability distribution of the random potential y(R). Here, we will consider random potentials that are taken fi om a Gaussian distribution defined by... 

The next step in the description of the chain statistics is an effort to find a functional expression for the distribution of the density of matter within a random cod of a single macromolecule. Assuming that the distribution of the end-to-end vectors of a macromolecule is Gaussian, one can establish a full distribution function, as shown in Fig. 1.30. The function W( r ) represents the fractional probability to find a given end-to-end vector r defined by its length and direction. The general Gaussian curve has only two constants, a and b ... 

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. 

Another interesting quantity is the probability that a chain has a given end-to-end distance (77-81). This is called the Gaussian distribution function... 

The Gaussian distribution (eq. (12)) was obtained in the aforementioned treatment with the assumption that chains are far from their fiill extension. Moreover, the Gaussian distribution function predicts zero probability only for r = oo instead of for all r in excess of that for full chain extension, and does not adequately... 

Figure 2.41. Probability distribution for the distance r of other segments from a given segment in a Gaussian chain. The segment density autocorrelation function p(r)p(0))/p multi-phed by is plotted as a function of r/Rg. Short-distance and long-distance asymptotes are indicated.

We want the probability, P(m,N) that the chain takes m steps in the +x direction out of N total steps, giving N -m steps in the -x direction. Just like the probability of getting m heads out of N coin flips, this probability is given approximately by the Gaussian distribution function. Equation (4.34),... 

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