Exponential approximation

Pack R. T. Relation between some exponential approximations in rotationally inelastic molecular collisions, Chem. Phys. I ett. 14, 393-5 (1972). 

In this section we will analyze the validity of exponential approximation of observables in wide range of noise intensity [88,89,91,92]. 

Figure 10. Evolution of the survival probability for the potential Tf.v) — ax2 - for3 for different values of noise intensity the dashed curve denoted as MFPT (mean first passage time) represents exponential approximation with MFPT substituted into the factor of exponent.

Finally, we have considered an example of metastable state without potential barrier decay time (6.5)] gives an adequate description of the probability evolution and that this approximation works better for larger noise intensity. 

Let us consider now some examples of symmetric potentials and check the applicability of exponential approximation ... 

The second considered example is described by the monostable potential of the fourth order (x) = ax4/4. In this nonlinear case the applicability of exponential approximation significantly depends on the location of initial distribution and the noise intensity. Nevertheless, the exponential approximation of time evolution of the mean gives qualitatively correct results and may be used as first estimation in wide range of noise intensity (see Fig. 14, a = 1). Moreover, if we will increase noise intensity further, we will see that the error of our approximation decreases and for kT = 50 we obtain that the exponential approximation and the results of computer simulation coincide (see Fig. 15, plotted in the logarithmic scale, a = 1, xo = 3). From this plot we can conclude that the nonlinear system is linearized by a strong noise, an effect which is qualitatively obvious but which should be investigated further by the analysis of variance and higher cumulants. 

Temporal behavior of the correlation function was studied in Ref. 91 using a particular example of the correlation function of sin x(f) in a periodic potential with periodic boundary conditions. In that case the use of single exponential approximation had also given a rather adequate description. The considered... 

The single exponential approximation works especially well for observables that are less sensitive to the location of initial distribution, such as transition probabilities and correlation functions. 

In all other cases it is usually enough to apply a double exponential approximation to obtain the required observable with a good precision,... 

The exponential approximation may lead to a significant error in the case when the noise intensity is small, the potential is tilted, and the barrier is absent (purely dynamical motion slightly modulated by noise perturbations). But, to the contrary, as it has been observed for all considered examples, the single exponential approximation is more adequate for a noise-assisted process either (a) a noise-induced escape over a barrier or (b) motion under intensive fluctuations. 

The d, b n, and cmn coefficients are given numerically as polynomials of S0 and their expansion coefficients are tabulated in Table 1 of Voids for the Maier-Saupe potential. In the single exponential approximation for gmn(t),119 the spectral densities simplify to... 

Numerous approaches have been published to improve the accuracy of IE data. However, the uncertainty of electron energy remains, causing the ionization efficiency curves not to directly approach zero at IE. Instead of being linear, they bend close to the ionization threshold and exponentially approximate zero. Even though the electron energy scale of the instrument has been properly calibrated against lEs of established standards such as noble gases or solvents, IE data obtained from direct readout of the curve have accuracies of 0.3 eV (Fig. 2.19a). 

Selected entries from Methods in Enzymology [vol, page(s)] Analysis of GTP-binding/GTPase cycle of G protein, 237, 411-412 applications, 240, 216-217, 247 246, 301-302 [diffusion rates, 246, 303 distance of closest approach, 246, 303 DNA (Holliday junctions, 246, 325-326 hybridization, 246, 324 structure, 246, 322-324) dye development, 246, 303, 328 reaction kinetics, 246, 18, 302-303, 322] computer programs for testing, 240, 243-247 conformational distribution determination, 240, 247-253 decay evaluation [donor fluorescence decay, 240, 230-234, 249-250, 252 exponential approximation of exact theoretical decay, 240, 222-229 linked systems, 240, 234-237, 249-253 randomly distributed fluorophores, 240, 237-243] diffusion coefficient determination, 240, 248, 250-251 diffusion-enhanced FRET, 246, 326-328 distance measurement [accuracy, 246, 330 effect of dye orientation, 246, 305, 312-313 limitations, 246,... 

Selected entries from Methods in Enzymology [vol, page(s)[ Analysis, 240, 290-310 anisotropic, 240, 301-310 effect of material diffusion, 240, 219, 221 evaluation of donor fluorescence decay, 240, 230-234 optimal length of time step, 240, 224-229 exponential approximation of exact theoretical decay, 240, 222-229 linked systems, 240, 234-237 measurement techniques... 

The importance of the carboxylate donors is underlined by a study of the lanthanide coordination chemistry of the similar terdentate ligand 2,6 -bis( 1 -pyrazol-3 -yl)pyridine, L24 (63). The complex structure of [Tb(L24)3][PF6]3, shown in Fig. 11, appears to be fairly robust in methanolic solution, with Horrocks analysis (q = 0.6) suggesting the 9-coordinate structure is retained the small quenching effect of outer sphere coordination explains the q-value. However, in aqueous solution, the lability of the ligands dramatically changes the luminescence. Whilst the emission decays are not exactly single exponential, approximate lifetimes in H20 and DoO suggest a solvation value of 4-5. 

We may represent clearly our required conditions for the exponential approximation in terms of the original quantities, in particular the temperature of the mixture and the ambient temperature. From the definitions for y and 0, the inequality y0 1 becomes T — ra ra, i.e. the temperature rise must be small compared with the absolute ambient temperature. For a system with Ta = 400 K in which there is 12 K self-heating, y6 = 0.03, so we would expect the exponential approximation to hold quite well. 

In the next few sections we will concentrate on the form of the governing equations (4.24) and (4.25) with the exponential approximation to f(0) as given by (4.27). We will determine the stationary-state solution and its dependence on the parameters fi and k, the changes which occur in the local stability, and the conditions for Hopf bifurcation. Then we shall go on and use the full power of the Hopf analysis, to which we alluded in the previous chapter, to obtain expressions for the growth in amplitude and period of the emerging oscillatory solutions. 

Exponential approximation stationary states and local stability... 

FIG. 4.2. Dependence of dimensionless pseudo-steady-state intermediate concentration and temperature excess on reactant concentration for the model with the exponential approximation (a) three-dimensional representation of universal locus (b) projection showing dependence of a5S on n (c) projection showing... 

Fig. 4.3. The ft-K parameter plane showing loci of changes in local stability or character for the model with the exponential approximation (a) full plane (b) enlargement of region near origin showing, in particular, the locus of Hopf bifurcation (change from stable to unstable focus) and the locus corresponding to the maximum in the ass(/r) curves (broken line).

Conditions for Hopf bifurcation with exponential approximation... 

We have already determined the following information about the behaviour of the pool chemical model with the exponential approximation. There is a unique stationary-state solution for ass, the concentration of the intermediate A, and 0SS, the temperature rise, for any given combination of the experimental conditions /r and k. If the dimensionless reaction rate constant k is larger than the value e-2, then the stationary state is always stable. If heat transfer is more efficient, so that k Hopf bifurcation points along the stationary-state locus as /r varies (Fig. 4.4). If these bifurcation points are /r and /z (with the stationary state... 

Example applications of the Hopf formulae for pool chemical model with exponential approximation. The two sets of data for each k correspond to lower and upper Hopf bifurcation points respectively... 

For small y, corresponding loosely to large values for the activation energy, the a5S locus has both the maximum which was displayed with the exponential approximation-and now a minimum which occurs at relatively distant values of h/k. This qualitative change in the locus means that the stationary-state concentration of A increases with high-enough reactant concentra-... 

The lower root for small y lies at fi/K slightly greater than unity and ass slightly greater than e tending to these values as y tends to zero. The upper root corresponds to the minimum in the curve and for small y clearly lies at large h/k with ass exponentially small. In the limit of the exponential approximation, y - 0, the minimum goes off to infinity and ass = 0. 

We now relax our implicit approximation in which the consumption of the reactant has been ignored. The full time-dependent behaviour of the dimensionless equations will be considered. The situation is not greatly affected by use of either the exponential approximation or the full Arrhenius form, so we return to the former for simplicity. We will take the example data from Table 4.1 k = 0.05, n0 = 0.5, and e = 10 2. As we are employing the exponential approximation the value of y is not important. 

In this chapter we give an introduction and recipe for the full Hopf bifurcation analysis for chemical systems. Rather than work in completely general and abstract terms, we will illustrate the various stages by using the thermokinetic model of the previous chapter, with the exponential approximation for simplicity. We can draw many quantitative conclusions about the oscillatory solutions in that model. In particular we will be able to show (i)that the parameter values given by eqns (4.49) and (4.50) for tr(J) = 0 satisfy all the requirements of the. Hopf theorem (ii)that oscillatory behaviour is completely confined to the conditions for which the stationary state is... 

We have already discussed the expressions resulting from a full Hopf bifurcation analysis of the thermokinetic model with the exponential approximation (y = 0). We may do the same for the exact. Arrhenius temperature dependence (y 0). Although the algebra is somewhat more onerous, we still arrive at analytical, expressions for the stability of the emerging or vanishing limit cycle and the rate of growth of the amplitude and period at... 

---