Brownian chain

CHAIN WITH INDEPENDENT LINKS AND THE BROWNIAN CHAIN 

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Brownian Chain Tension in a Melt and the Tube Potential.213... 

The study of the Brownian chain, the simplest model, will enable us to introduce and to illustrate various basic notions which can easily be generalized to more realistic cases. 

We may now go to the continuous limit, which is called a Brownian chain. This limit is obtained by reducing the size of the links and by increasing their number so as to keep %,r2y fixed. 

The area s can thus be used as a curvilinear coordinate to locate a point on the continuous curve. Therefore, the configuration of a Brownian chain is defined by the vectorial function r(s) (where r(s) is the continuous limit of r ). 

Let us note that the length of a Brownian chain is infinite. In fact, let L be the length of a chain with independent links and /0 the common length of the links. We can define L by the equality... 

This is the reason why the curvilinear coordinates which is used to define the abscissa of a point on a Brownian chain is not a length but an area. This is connected with the fact that the HausdorfT dimensionality for a Brownian chain is not equal to one, as it is for a rectifiable curve, but is equal to two thus in a sense, the Brownian chain has the characteristics of a surface (see Section 2.3). 

Any portion of a Brownian chain, is in average and to a dilatation, absolutely identical to the whole chain. One says that there is internal similarity. The scale of a Brownian chain is determined only by one variable, namely S, which defines the mean square distance between the end points. 

The general probability law associated with a Brownian chain... 

I. CHAIN WITH INDEPENDENT LINKS AND THE BROWNIAN CHAIN 49... 

It is now possible to define infinite Brownian chains in the limit S - oo. The position of a point of the chain is given by the vector r(s) with — oo < s < + oo, and we can assume that the chain goes through the origin of coordinates. Thus, we postulate... 

This, we see that an infinite Brownian chain does not depend on any parameter. Moreover, we note that "W r(s) is invariant with respect to any transformation of the form s - X2s, r(s) - /r(s). In other words, an infinite Brownian chain is statistically scale-invariailt. 

This scale-invariance is a characteristic property of critical phenomena. Thus, an infinite Brownian chain is a critical object. 

An infinite Brownian chain is a unique object, since such a chain does not depend on any parameter. This example shows that the appearance of scale-invariance, which goes along with the elimination of the microstructure details, is characterized by a drastic reduction of the parameters of the system. Thus, the final properties of a critical system present special features of universality and simplicity which the scientists consider to be highly interesting. 

Thus, just as the chain with independent links has a continuous limit, which is the Brownian chain, the chain with excluded volume also has a continuous limit, which we call the Kuhnian chain. Like the Brownian chain, the Kuhnian chain has an infinite length. In fact, the length L of a chain with excluded volume is (to a proportionality factor) equal to... 

Only one length is associated with a Kuhnian chain and it defines its mean end-to-end distance. In fact, the size of a Kuhnian chain is defined by the course S which has neither the dimension of a length (as for ordinary rectifiable curves), nor the dimension of an area (as for a Brownian chain), but an intermediate dimension, as can be seen from (2.2.7). This arises from the fact that the exponent v of a Kuhnian chain is intermediate between 1/2 and 1, whereas the exponent of a normal curve is v = 1 and the exponent of a Brownian curve is v = 1/2. 

In particular, a Brownian chain must be considered as an object of dimension two. 

One-dimensional chains with repulsive interactions have properties which can be studied rigorously, and we shall describe them below. In d-dimensional space, the effect of the excluded volume on the behaviour of the chains diminishes when d increases and simple arguments show that beyond d = 4, the chains with excluded volume behave like chains with independent links (quasi-Brownian chains). 

For a Brownian chain v = 1/2 independently of the dimension of the space in which the chain is embedded consequently for d > 4, the preceding condition is not realized. 

Polymers appear as chains with excluded volume, but the mathematical difficulties which are encountered when one tries to study these chains have induced many author to use simpler models. These simple chains are either chains with independent links or other asymptotically Brownian chains, and we shall now review these types of chain. 

Asymptotically Brownian chain with independent links... 

Finally, we note that, for a Brownian chain, N and / are not independent. Setting S = Nl2, we can express all the results in terms of S alone. 

Non-asymptotically Brownian chains with independent links chains with intermittence (Levy flights)... 

A Brownian chain is characterized by the fact that the characteristic function associated with a link u is of the form... 

We can generalize this definition. A v-Brownian chain (a Levy flight) is a chain with independent links, such, as... 

Of course, such chains, if they exist, are homogeneous, like the Brownian chain, and the function PN (r) must be of the form... 

In order to prove the existence of these chains, we have only to verify that the function PN(r) is really a probability distribution, i.e. that f(x), the Fourier transform of exp ( — k1/v) is positive. It is clean that this condition can be realized, and the simplest example of a v-Brownian chain corresponds to the values v = 1, d = 1, in which case... 

However, as we shall see, v-Brownian chains exist only for v > 1/2. In fact, the size of a v-Brownian chain which is proportional to N cannot be smaller than the size of a Brownian chain which is proportional to N112. 

These v-Brownian chains which are also called Levy flights exhibit the intermittence property. This means that the global size of the chain does not result from an accumulation of links hut from a very small number of very large links. When the number of links becomes larger, the probability that the chain contains a larger link increases simultaneously, and this is why the size of an intermittent chain is a rapidly growing function of the number of links... 

Let us note that the projection of a v-Brownian chain embedded in a d-dimensional space, on a if-dimensional subspace is also a v-Brownian chain, as can easily be seen by considering the definition (3.4.17). 

Thus, the characteristic function of a v-Brownian chain in three-dimensional space is... 

To remedy this deficiency, one must consider locally rigid chains. The simplest example of such chains is the Kratky-Porod chain (wormlike chain),16 which is a continuous chain, and which, unlike the Brownian chain, has a finite length and, at each point, a well-defined tangent. 

Thus, we verify that the critical exponent y has the value y = 1 which is typical for a Brownian chain. In the same way, for large values of N, we have ( j remaining small)... 

On the contrary, the quasi-Brownian chains (with v = 1/2) correspond to the case where the Bjt decrease rapidly when j — / increases. A Gaussian chain is quasi-Brownian if... 

Nevertheless, these quasi-Brownian chains can be rather rigid and the coefficients Ay can be chosen so as to take stretching and bending elasticities simultaneously into account (Papadopoulos and Thomchick 1977)21. 

For Brownian chains X = 1 as we have seen in Chapter 3, Section 3.1. 

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