Probability distribution functions jointed chain

Demonstrate that the probability distribution function of the end-to-end distance i of a freely jointed chain can be expressed in terms of the inverse Langevin function C (x ) of the ratio x = RjR y of the end-to-end distance R to its maximum value i max =... 

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. 

To determine the probability distribution function for the chain end-to-end distance, we first consider a freely jointed, one-dimensional chain having links of length = //VI, which are aU constrained to lie along the x axis. What is the probabihty that the end-to-end distance of this one-dimensional chain is mLl The... 

To find the anisotropy of the whole molecule in molecular coordinates h, the value of cos d should be averaged over all chain elements and all its conformations corresponding to a given value of h. To solve this problem Kuhn and Griin ) have considered the statistics of the intramolecular orientational distribution of segments with respect to h in a freely jointed chain and have shown that for the most probable distribution the value of cos d averaged over all chain segments can be expressed as a unique function of the h/L ratio... 

At equilibrium, the distribution of conformations in a solvent at the theta temperature (see Section 2.3.1.2), or in a concentrated solution, is given by a set of random walks or, equivalently, by the conformations of a. freely jointed chain (see Section 2.2.3.2). If one end of the freely jointed chain with links, each of length bjc, lies at the origin, then-the probability, jrodR, that the other end lies at a position between R and R + dR is approximately a Gaussian function (Flory 1969 Larson 1988) ... 

The freely jointed chain model has an exact analytical solution for the distribution function of the end-to-end vector. The probability that the chain of n bonds has the end-to-end vector r is... 

The problem is characterized by the set of joint probability density - discrete distribution functions qjizi, Z2, i) of the response state variables - the displacement Zi(t) and the velocity Z2(0, and the m states S t) of a pertinent Markov chain, defined as... 

To establish a useful equation of state for the mechanical behavior of a rubber network, it is necessary to predict the most probable overall dimensions of the molecules under the influence of various externally applied forces. An interesting approach to rubber elasticity consists of simulating network chain configurations (and thus the distribution of end-to-end distances) by the rotational isomeric state technique cited above. Based on the actual chemical structure of the chains, it enables one to circumvent the limitations of the Gaussian distribution function in the high deformation range. Nonetheless, the Gaussian distribution function of the end-to-end distance is very useful. It is obtained from a simple hypothetical model, the so-called freely jointed chain, which can be treated either exactly or at various levels of approximation. 

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