Wiener measurement

The Bloch equation is Eq. (6) as it stands the Schrodinger equation is Eq. (6) with 0 = i x time Wiener measure goes into Feynman measure under the latter transformation. We shall speak throughout in the language of the Bloch equation and Wiener measure. Our results are, at least formally, transformable into the Schrodinger-Feynman situation. 

The usual Wiener measure, Dw r, is concentrated on trajectories with fractal dimension df = 2. Instead of that, for description of an unknotted ring the measure 3sf rj with fractal dimension df = 3 should be used. 

In order to develop good sampling paths for the Gibbs-Boltzmann distribution, we need to introduce a method of adding random perturbations into a differential equation. Let us first consider the definition of the integral with respect to Wiener measure. 

In a dense melt, the excluded volume of the monomeric units is screened and chains adopt Gaussian conformations on large length scales. In the following, we shall describe the conformations of a polymer as space curves r(r), where the contoiu" parameter r runs from 0 to 1. The probability distribution P[r] of such a path r(r) is given by the Wiener measure... 

Fig. 2. A discontinuous polymer chain which is given zero weight by the Wiener measure (3.10). The arrows denote the arbitrarily chosen direction from chain beginning...

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. 

Before we consider the reduced endpoint distributions, it is convenient to note the analogy between (5.10) and (5.11) and situations involving Brownian motion. " - If we consider cases of Brownian motion for which the inertial term mf(t) (t is the time) in the equations of motion can be neglected, then the Wiener measure... 

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. 

Imagine that this integral defines a "Wiener measure of the set of functions x(t) ... 

In the standard model, the bonded interactions, which describe the Gaussian chain architecture, are given by the Wiener measure... 

Also, Ho [Rp] is the chain connectivity part, which comes from the fact that in the absence of interactions, the probability distribution function for the chains must be a Wiener measure. In the continuum representation [50], this term is written... 

The starting point is the probability of finding a conformation i (s) for the linear chain. This problem is discussed in detail in the chapter by Edwards and Muthukumar, and we give a brief description in Section 8.3.2 within this chapter, because of use in rubber theory, and we refer the mathematically interested reader to these parts of this series (Volume 2, Chapter 9). Assuming Gaussian conformations, it is given by the Wiener measure " ... 

The first method is the use of the chain variables R s) directly, and we remember equation (86) for the probability of finding a conformation. If a crosslink fixes segment s from chain i to segment 5 j on chain j we have the constraint Ri s ) = Rj s)) permanently. The probability distribution of the network is given by the constraint averaged over Gaussian conformations of chains. In terms of the Wiener measure this probability can be written as... 

Funaki, T. (2005). Concentration Property for Wiener Measure with Density Having Two Large Deviation Minimizers, preprint, available on the webpage of the author. 

---