Wiener integral

Wiener integrals in general, often useful in statistical mechanics1,4, can be expressed in terms of the propagator in Eq. (4). In particular, putting 0 = 1 /kT, p = ha/kT, we have for the partition function of the particle obeying Eq. (6)... 

We now give a brief description of a new expansion which is more accurate for Z(p). It is based on a suggestion of Feynman and Hibbs4 who calculated the Wiener integral corresponding to the first term in this expansion. Our method of calculation uses probability techniques and treats both the Wiener and Uhlenbeck-Ornstein processes. In the latter case we first rewrite Eq. (20) in the form... 

It must be noted that the expression (5.14) was obtained in Ref.36) not as a result of the smoothing procedure, but with the help of some other more formal consideration. The main idea of this consideration can be outlined as follows. It is well known24 that the smoothing procedure for the model of beads leads to the operator of the type g a= 1 + a2 A, and in the same approximation the partition function reduces to the Wiener integral33 . In Ref.36) it was shown that the role of the operator (5.14) for the integration over the space of smooth curves is the same as the role of the Laplace operator for the usual Wiener integral... 

Here, e (t, X) is given to the computation with the relation (4.93). We obtain formula (4.96) where we can observe that Ex = Exo is a Wiener integral. 

If we relax the unit vector condition of u, we can obtain an analytical solution to the differential equation. The relaxation of this condition allows the use of the Wiener integral in the case. 

Although the Wiener integral formulation for the distribution functions of flexible polymer chains rests upon general considerations of random walks and Brownian motion, it is easily introduced, heuristically, through the concept of an equivalent chain. In this section, only those flexible polymer chains are considered which are composed of equivalent gaussian links. Here L is the maximum contour length of the real chain at full extension, and (R ) for the equivalent chain is taken to be that for the real chain. Thus we have... 

G is then the Green s function for the diffusion equation with the diffusion constant D = //6 [see (3.23)]. The diffusing particle is initially at R at time I, = 0. [Wiener integrals were initially used to describe... 

But (4.7b) is also a diffusion equation which admits the Wiener integral representation... 

Equations (5.15) and (5.21) are familiar Wiener integrals which are easily evaluated in closed form, provided (5.2a) is not invoked. Before considering this case, it is instructive to recover the results of STY. 

Equation (5.25) is a simple Wiener integral which is subject to (5.2a). Therefore, from (5.25) with (3.20)-(3.22), satisfies the diffusion equation... 

Therefore, in an attempt to obtain simple analytic expressions for the distribution functions of stiff polymer chains, condition (5.2a) is relaxed. The relaxation of this condition is in the original spirit of the use of Wiener integrals. If this condition were imposed for flexible polymer chains, the Wiener measure would be 2[t s)] exp (—3L/2/) and would give equal weight (measure) to all continuous configurations of the polymer. Thus the use of (5.2a) would not yield the correct gaussian distribution for flexible chains. 

As noted previously (5.21) is a simple Wiener integral which can be... 

But jP(R L) is still gaussian as in (5.45) with the rodlike mean square (R ) of (5.46d). The limit (5.46a) implies that the average bond length l—ylL and that /Se oo. This is of course consistent with rodlike behavior. However, in this limit there is probably no real need for the use of Wiener integral formulations in the first place. 

The relationship between the Wiener integral (3.20) and the simple diffusion equation (3.21) suggests that it might be instructive to convert (6.12) to a differential equation. As also noted by Whittington, for the case of a discrete chain (6.12) can be expressed only in terms of the solution of a hierarchy of integro-differential equations. The derivation in the continuous case is presented in Appendix B for convenience, although the result is quoted here. Define the three-point Green s function as... 

In (6.38) we could try first to perform (approximately) the [r(s)] integration to yield G(RO LO[]) and then (approximately) the remaining S integration. From (6.25) it is clear that the Wiener integral is not in general exactly evaluable since < (R) is a random variable. One possible simplification arises from the fact that a solution is desired in the asymptotic limit L 00. Proceeding by analogy with the semiclassical approach... 

The formulation of the foundations of the statistical mechanics of polymers in bulk is just in its infancy. A number of questions of principle still require careful attention. This review can then only be of a state-of-the-art nature, and only a few very simple applications can be considered. In the next section, a simple model is considered which serves as a zeroth-order model for a system of polymers in bulk. This model, which employs the Wiener integral formulation for flexible chains, has great pedagogical value since it introduces some of the formidable problems to be encountered in any statistical mechanical description of polymers in bulk. This model then naturally leads to a discussion of the nature of statistical mechanics for systems with internal constraints (Section IX). 

If N is finite and x(t) is constant within each interval, then the associated Wiener integral of some functional ajx(t)j = a(x, .., Xj ) looks like an ordinary multiple integral, and in fact this expression can be evaluated exactly whenever a is the exponential of some linear or bilinear function of the x s. For the more general case of infinite N, Kac proved that if F(x) is a real, positive, continuous function of x, then... 

The connection between these meanderings and electrons in simple liquids is that if one can somehow calculate the "conditional Wiener integral (Simon, 1979, 1985 Kac and Luttinger, 1974)... 

A different kind of mathematics then applies via the Wiener integral where the probability over large distances of finding a curve R(s) is... 

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