The Wiener Integral

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 

Since (3.4b) gives the probability of a particular chain configuration To = 0, Fj. r , it is clear that this chain configuration r can be taken to be the discrete representation of the continuous curve r(j). Therefore let 

We now take the limit - 0, n - oo, n As = L to obtain a representation of a continuous equivalent random flight chain. In this limit, denoted by lim, the expression in the exponential in (3.4b) is 

Equation (3.7) follows from the definition of derivatives and integrals in elementary calculus. In this limit, the probability E( rjfe ) becomes 

Equation (3.4d) has led mathematicians frequently to claim that the representation of the Wiener measure in (3.10) is undefined. Their complaint is reminiscent of the disrepute in which Dirac delta functions were held by mathematicians for a number of years. There are mathematically acceptable formulations, or notational transcriptions, of these functional integrals. These formulations may make for good mathematics, but they are physically unnecessary. When in doubt, we just remember that the functional integrals are defined in terms of the limit of an iterated integral. 

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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... 

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. 

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

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). 

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

Duduchava R., Wendland W. (1995) The Wiener-Hopf method for system of pseudodifferential equations with applications to crack problems. Integr. Eqs. and Oper. Theory 23, 294-335. 

From the Wiener-Kinchine theorem, the integrated PSD yields the r.m.s. roughness vs the lenglh/scale / ... 

We are permitted to specify the integrals for positive co only, because of the even property of the integrand. This simplication, in turn, stems from the real nature of all the x-space components of the integrand. Minimizing expression (9) is equivalent to asking that the physical solution conform to the Wiener inverse-filter estimate in the sense of minimum mean-square error after suitable weighting of the positive solution to ensure best conformance at frequencies of greatest certainty. 

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)... 

A similar procedure gives the corresponding result for the Wiener process the first term of which was obtained by Feynman and Hibbs4 using a path integral approach. 

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. 

The power sjjectrum is merely the Fourier transform of the VAF via the Wiener-Khintchine theorem. The integration is carried out as a discrete sum over the jjeriod of time in which the VAF decays to a zero value. This quantity gives the number of oscillators at a given frequency and is a very informative indicator of the transition from rigid, quasiperiodic motion to nonrigid, chaotic motion. Note that I(co = 0) is proportional to the diffusion constant. This quantity was calculated by Dickey and Paskin - in the study of phonon frequencies in solids and also by Kristensen et al. in simulations of cluster melting. 

Equation (41a) means that the function B( r) is equivalent to the volume integral of the density matrix y(ri, ri) under the condition of r = r - r, and Eq. (41b) means that B(r) is the autocorrelation function of the position wave function (r). The latter is an application of the Wiener-Khintchin theorem (Jennison, 1961 Bracewell, 1965 Champeney, 1973), which states that the Fourier transform of the power spectrum is equal to the autocorrelation function of a function. Equation (41c) implies not only that B(r) is simply the overlap integral of a wave function with itself separated by the distance r (Thulstrup, 1976 Weyrich et al., 1979), but also that the momentum density p(p) and the overlap integral S(r) are a pair of the Fourier transform. The one-dimensional distribution along the z axis, B(0, 0, z), for example, satisfies... 

The integral over the Gaussian white noise gives the Wiener process which stands for the trajectory of a Brownian particle. The integral during... 

Next, unlike in the case of the Riemann sums, it turns out that the result depends on where we place the points tk. Using the properties of the Wiener process we may easily show that the choice 4 = kSt (left endpoint) and the choice tk = (k+ l/2)St yield different formulas for the integral. The first choice is commonly referred to as ltd integration, and the second as Stratonovich integration. As an example, consider the case where g(t) = W(t). We suppose this to be approximated by a sum... 

Proposition 6.3 Let g be a smooth deterministic function and W(f) a Wiener process. The stochastic integral Y(t) = fQg(s)dW(s) is, for all times t > 0, a normally distributed random variable such that... 

The first part of this section presents the model that we are using in order to tackle the issue of risk management during a financial crisis. In the second part of this section we give an overview of the MaUiavin derivative in the Wiener space and of its adjoint, i.e. the Skorohod integral. We refer the reader to Nualart [13] and... 

Let S represent the Skorohod integral in Wiener space. One can observe that 6 is the adjoint ofD as showing in the next proposition, which is an extension of the ltd integral. 

The integral with respect to the Wiener process is taken using the Ito approach that has an important advantage over the Stratonovich approach, namely the integral in this case will be a martingale. Definitions and... 

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... 

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. 

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