Gaussian fixed point

In homogeneous, stationary turbulence, the probability density of pi, must coincide at all times with that of the velocity recorded at any fixed point, pi. The probability densities of u iin + 1) At, x ) and (n At, x ) are thus identical. Under homogeneous, stationary conditions, pI is known to be Gaussian (see Section III). We indicate the distribution... 

Thus all seems perfect. We have constructed an RG mapping, wliich indeed shows a fixed point. However, the expression (8.32) for / is not satisfactory. It must be independent of A, otherwise dilatation by A2 does not lead to the same result as repeated dilatation by A. Now Eq. (8.32) is only approximate since in Eq. (8,31) we omitted terms O 0 2. This is justified only if 0 is small. We thus need a parameter which allows us to make. If arbitrarily small, irrespective of A. Only e — 4 — d can take this role. In all our results the dimension of the system occurs oidy in the form of explicit factors of d or It thus can be used formally as a continuous parameter. To make our expansion a consistent theory, we have to introduce the formal trick of expanding in powers of e — 4 — d. 3 vanishes for = 0, consistent with the observation (see Chap, fi) that the excluded volume is negligible above d = 4, not changing the Gaussian chain behavior qualitatively. For e > 0 Eq. (8.32) to first order in yields... 

This structure of the critical manifold is valid for d < 4. For d —> 4 the nontrivial fixed point merges into the Gaussian fixed point. For d > 4 only the Gaussian fixed point is physical, governing the system for all ft > 0. In RG language this is the reason for the triviality of the results for d > 4, pointed out in Chap. 6. 

Let us now turn to the nontrivial fixed point. For d —> 4 it merges into the Gaussian fixed point. Analyticity in the parameter d of the RG flow presupposed by the -expansion then guarantees that the scaling fields at u as well as their dimensions for d < 4 smoothly develop from their equivalent at the Gaussian fixed point, d = 4, Indeed it has recently been shown [KWP93] that... 

In the construction of the RGf dimension d = 4 plays a special role as upper critical dimension of the thebry. This for instance shows up in the estimate of the nonuniversal corrections to the theorem of renormalizability, or in the feature that the nontrivial fixed point u merges with the Gaussian fixed point for d — 4. It naturally leads to the e-expansion. However, the RG mapping constructed in minimal subtraction only trivially depends on e. Also results of renormalized perturbation theory do not necessarily ask for further expansion in e. Equation (12.25) gives an example. We should thus consider the practical implications of the -expansion in some more detail. 

This representation shows two fixed points. We first consider the trivial, or Gaussian , fixed point l/n — 0,/3e = 0. The associated linear representation is of the form (10.19) with resulting... 

Appendix 8.A — Tension in a Gaussian Chain Between Two Fixed Points... 

The critical exponent v at Gaussian fixed point reads z/ = 1/2, which corresponds to the fact that SAWs at dimensions above the upper critical one behave like RWs (simple random walks). There are essentially two ways to proceed in order to obtain the qualitative characteristics of the critical behavior of the model. The first is to substitute the loop integral in equations (94), (95) by its e-expansion ... 

We note that the collapse transition on fractal lattices corresponds to a new fixed point, intermediate between swollen (SAW) phase and the collapsed phase and cannot be viewed as a perturbation of the Gaussian fixed point describing random walks. 

In another fixed point (C) of a nondraining Gaussian chain, where = 8ir e/3,... 

Release Geometry. An ideal release for Gaussian dispersion models would be from a fixed point source. Real releases are more likely to occur as a line source (from an escaping jet of material), or as an area source (from a boiling pool of liquid). 

For 4fourth order terms are irrelevant in Eq. (78). An anisotropic (/i -renor-malized theory was constructed [85]. The Gaussian and Ising fixed point are unstable, as expected. A stable non-trivial fixed point exists and anisotropic critical exponents were calculated in 6-f dimensions. 

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