Leading logarithmic approximation

Finally, we have to comment on the approximations used in this calculation. The first approximation was to neglect the forward scattering term g2, the other was to consider the leading and next to leading logarithmic corrections only, i. e. a second-order renormalization was performed and the higher order corrections were neglected. 

In a one—dimensional metal the formation of Cooper pairs is accompanied by the Peierls dielectric pairing [f]. This leads to a summation of "parquet" diagrammes instead of a "ladder" occuring in the three-dimensional case. The "parq.uet" corresponds to a logarithmic approximation and the calculation of the next approximations encounters considerable difficulties.. In the logarithmic approximation one obtains that in case of attraction of electrons both types of pairing appear simultaneously. It is not clear however whether this result is reliable and cannot it happen really that the dielectric coupling prevents the formation of Cooper pairs... 

These solutions matches logarithmically in the intermediate region and a uniform leading order approximation to the neck shape is unif( f ) = o( ) +, (r) — aln(2r/a) in the vicinity of the (0,0) site. 

The number of steps is always much larger than the displacement x, since there is a good deal of back-and-forth cancellation. Hence the ratio x/nl is less than unity and the logarithms may be approximated by the leading terms of a series expansion... 

Next we use the Flory-Huggins theory to evalute AG by Eq. (8.44). As noted above, the volume fraction occupied by polymer segments within the coi domain is small, so the logarithms in Eq. (8.44) can be approximated by the leading terms of a series expansion. Within the coil N2 = 1 and Nj = (1 - 0 VuNa/Vi, where is the volume of the coil domain. When all of these considertions are taken into account, Eq. (8.108) becomes... 

Experimental conditions and initial rates of oxidation are summarized in Table V. For comparison, initial rates of dry oxidation at the same temperature and pressure of oxygen predicted by Equation 9 are included in parentheses. The predicted dry rate, measured dry rate, and measured wet rates are compared in Figure 2. The logarithms of the initial rates of heat production during wet oxidation increase approximately linearly (correlation coefficient = 0.92) with the logarithm of the partial pressure of oxygen and lead to values of In k = 2.5 and r = 0.9, as compared with values of In k = 4.8 and r = 0.6 for dry oxidation at this temperature. 

In Bethe theory the shell correction ALsheii is conveniently defined as the difference between the stopping number LBom in the Born approximation and the Bethe logarithm LBethe —in (2mv /I). Fano [12] wrote the leading correction in the form... 

A similar procedure for the hard sphere RDFs leads to the usual PY approximation expressions for gff(R) and gfC(R). However, for gcc(R) it is the logarithm of gCc(R) that is equal to the usual PY expression. Thus, the contact values are... 

These equations result from the intimate mixing of electron-electron and electron-hole channels (the Parquet summation). This is of crucial importance in one dimension. The f-matrix or random-phase approximations are incapable of doing this and are fundamentally wrong in one dimension. Notice also that g4 is absent because it does not alone contribute any logarithmic term. It leads only to charge and spin velocity corrections. It is normally neglected in the RG treatments (see Refs. 15 and 39 for a discussion of this). It will only be taken into account for the uniform susceptibility in part d. 

Before concluding this discussion of cell walls, we note that the case of elasticity or reversible deformability is only one extreme of stress-strain behavior. At the opposite extreme is plastic (irreversible) extension. If the amount of strain is directly proportional to the time that a certain stress is applied, and if the strain persists when the stress is removed, we have viscous flow. The cell wall exhibits intermediate properties and is said to be viscoelastic. When a stress is applied to a viscoelastic material, the resulting strain is approximately proportional to the logarithm of time. Such extension is partly elastic (reversible) and partly plastic (irreversible). Underlying the viscoelastic behavior of the cell wall are the crosslinks between the various polymers. For example, if a bond from one cellulose microfibril to another is broken while the cell wall is under tension, a new bond may form in a less strained configuration, leading to an irreversible or plastic extension of the cell wall. The quantity responsible for the tension in the cell wall — which in turn leads to such viscoelastic extension — is the hydrostatic pressure within the cell. 

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