Large k Limit

Numerical simulations of the k = oo case reveal a sharp phase transition at Ac = 0.27 [wootters]. Simulations also suggest that the spread in values of entropy decreases with increasing k, and that the width of the transition region probably goes as k f [woot90]. 

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It turns out, in fact, that the large k limit sought leads to an approximation in which the kinetic energy is suppressed and thus the leading contribution to the ground-state energy is determined by the minimum of the effective potential V(u, v). The stationary points of K = V(u, v) are evidently roots of the algebraic equations dV/du = 0 = 6K/0t . Explicitly, one then finds that [25]... 

The second term in the sqnare bracket is negligible compared with N in both the small k and large k limits. Therefore,... 

For large k, the ratio of expansion eoeffieients reaehes the limit, whieh... 

For large K (K = 1000 in Fig. 17) an upper limit is reached, where the interfacial kinetics are sufficiently fast - on the time scale of the SECM measurement - such that the concentrations of Red in the two phases, adjacent to the interface, are always in equilibrium even though Red is generally depleted. The tip current response is then dependent... 

The normalized steady-state current vs. tip-interface distance characteristics (Fig. 18) can be explained by a similar rationale. For large K, the steady-state current is controlled by diffusion of the solute in the two phases, and for the specific and y values considered is thus independent of the separation between the tip and the interface. For K = 0, the current-time relationship is identical to that predicted for the approach to an inert substrate. Within these two limits, the steady-state current increases as K increases, and is therefore diagnostic of the interfacial kinetics. 

Let us consider the large volume limit L —> oo. In this limit we have a continuous spectrum for the field modes k, and we replace summations over modes by integrations... 

The photoelectron wave-vector k is evaluated using = 2m(E — E ) where E is the energy of the X-ray photon, , a reference energy and m, the mass of the electron. x(k) is multiplied by k"(n = 2 or 3 usually) to magnify the faint EXAFS at large k (Lytle et al, 1975) /c"x(k) is Fourier transformed to yield the RSF, < (R). In the model compound, the first peak at a distance Rj represents the distance to the nearest-neighbour shell and may be compared to R[, the known distance. We can then define a as (R — Rj), which represents the experimentally determined phase correction. In principle, 2a should be equal to the theoretically estimated k-dependent part of /k), viz. if the identity of the scatterer environment has been correctly assumed. It must be emphasized that wherever scatterer identities are obscure (e.g. in several covalently bonded and disordered systems) use of a (and not j) is advisable. Further, the k-dependence of < /k) introduces an intrinsic limitation to its quantitative accuracy. 

This code is typically called by a MATLAB command such as f luidbed(4000,. 4,. 9,1,1, 45/99,10 s8,10 sll,.4,.6,18,27,1,0,10), which uses a large time limit of 4000 dimensionless time units so that the individual solution curves of the IVPs have time to run to the global steady state at (0.17619, 0.72225, 0.86446). We do this with the original parameter data of p. 183, a controller gain of K = 10, and several different initial values. Since the value of K is greater than 4, the middle steady state with maximal x i> yield is both unique and (statically) stable. 

S.T. Rittenhouse, C.H. Greene, The degenerate Fermi gas with density-dependent interactions in the large-N limit under the K-harmonic approximation, J. Phys. B At. Mol. Opt. Phys. 41 (2008) 205302. 

Here, we report on high-field ESR properties of the 5=1 Heisenberg chain system Ni(C2H8N2)2Ni(CN)4 (NENC) [16,31], This material is characterized by strong in-plane anisotropy D/k= 6 K, and so far can be considered as the best known candidate for a quantum 5=1 chain in the large-D limit (D/J=1.5) [32],... 

The special case of Eq. 6.27 that obtains where m = n, that is, k°n= 2K , is trivially scale invariant, so this property ought to be implicit in the corresponding solution of the von Smoluchowski rate law, given in Eq. 6.18. That this is the case can be seen by noting the large-time limit of p,(t),... 

The mere observation of a burst implies that a step after chemistry is at least partially rate limiting for steady-state turnover. In addition, if no observable burst occurs, then the data imply that chemistry is largely rate limiting. When a burst can be observed, quantitative fitting of the amplitude and rate of the burst phase relative to the steady-state phase affords estimates for three rate constants k2, k-2 and ks in the pathway shown above (Scheme 2). 

Given that if is often not large, a limiting rate (as expected for Eq. (12)) is often not seen, and Eq. (14) operates. Separation of ku and K is then not accessible from the kinetics. Rarely, the pseudo first-order rate constants have been reported to be a relatively complicated function of [M ]. This is the case for [Co(NH3)5l] + and [RWNHslsI] with Ag+ (32, 174), in which the complications are reported to result from the existence of species such as [((NH3)5MI)2Ag] + in addition to [(NHslsMIAg] . In the presence of excess chloride ion, Hg -promoted aquation can be complicated by separate pathways involving Hg , ... 

Potential of I Approaches at small o> or large k in limit of large o>, falls... 

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