Pure exponential decay

When plotted as a function of drop size (Fig. 9), the contact angle was found to decrease with increasing drop height. A different analysis of these data was performed in the original paper. In that case the maximum slope near the drop edge was used, as well as a direct inversion of the droplet shape. The data could be fit to an empirical 1/z function. In the present analysis we use the method of the effective contact angles defined earlier, together with Eq. (18). For the Flamaker constant A, we calculated a value of approximately -2 X 10 ° J. However, the best fit to Eq. (18) is for a pure exponential decay of the form ... 

Associated with the pole of the S-matrix is a Seigert state, I-Ves, which has purely outgoing boundary conditions and satisfies (with some caveats) the equation, // I res = z les,H being the system Hamiltonian.44 If a square integrable approximation to I res is constructed, then its time evolution, k . (/,), wiH exhibit pure exponential decay after a transient induction period. Of course any L2 state will show quadratic, and hence non-exponential, decay at short times since... 

The phasor method associates the decay dynamics with a vector in a so-called phasor space. In particular, purely exponential decay corresponds to a phasor with its end point on a semicircle of radius 1/2 and centered at (1/2, 0). Tuning of the decay time from zero to infinity results in a counterclockwise displacement of the end point from (1,0) to (0,0) along the semicircle. Multi-exponential decay is equivalent to a point inside the semicircle, but its dependence on the weight-averaged decay... 

The proper procedure should be quite the opposite. Rq has to be found by fitting the non-Markovian asymptote (3.59) to the experimental data, provided D has been measured or calculated from the Stokes equation. Only then can the true Markovian rate constant k be found as AkRqD. The pure exponential decay with this rate constant is shown by the dashed line in Figure 3.7 for comparison with the true nonstationary kinetics. 

Figure 3.7. The fit of the non-Markovian electron transfer kinetics (thick curve) to the experimental data obtained in propylene glycol in Ref. 97. The thin line represents the tangent to the kinetic curve at the largest time within the available interval (t = 4 ns). Its slope is the time-dependent rate of ionization cki(t), which differs essentially from that of the dashed straight line. The latter represents the purely exponential decay, with the stationary rate constant k — 4tiRqD. (From Ref. 18.)...

The first explicit example of the validity of Khalfin s prediction was given by Winter [63]. He studied the motion on the half-line x > 0, with a delta barrier at X = a and zero potential elsewhere. He took the initial wave function to be the ground state of an infinite well, T o(x,t = 0) = 2/asm nxia) a-x), and showed, by making a few analytical approximations, that the mean momentum and the current at long times deviated from those of pure exponential decay. 

This Hamiltonian may hold Nioc discrete, localized eigenstates of complex energy with purely exponential decay. The detector is placed at the barrier... 

In a source/filter vocal model such as LPC or parallel/cascade formant synthesis, periodic impulses are used to excite resonant filters to produce vowels. We could construct a simple alternative model using three, four, or more tables storing the impulse responses of the individual vowel formants. Note that it isn t necessary for the tables to contain pure exponentially decaying sinusoids. We could include aspects of the voice source, etc., as long as those effects are periodic. FOFs (originally introduced as Formant Onde Functions in French, translates to Formant Wave Functions in English) were created for... 

Hence, according to the approximation (43) (corresponding to a straight line for log Vj, which means a purely exponential decay of we should find that, for the transition case, the l<5g curves touch each other (i.e., thei... 

If several resonator modes within the bandwidth of the laser pulse are excited, beat signals are superimposed onto the exponential decay curve. These beats are due to interference between the different modes with differing frequencies. They depend on the relative phases between the excited resonator modes. Since these phase differences vary from pulse to pulse when the cavity is excited by a train of input pulses, averaging over many excitation pulses smears out the interference pattern, and again a pure exponential decay curve is obtained. 

Equation (6.61) anticipates a pure exponential decay. This is justified if a single upper level k) is selectively populated. If several levels are simultaneously excited the fluorescence power Rpi(t) represents a superposition of decay functions with different decay times x/t. In such cases the phase shifts < (f2) and the amplitudes a/(l + have to be measured for different modulation frequencies The... 

There has been, from time to time, considerable interest in the possibility of pure exponential decay of various observable quantities in a relaxing system the necessary and sufficient conditions for exponential decay have been formulated in the form of a set of sum rules [63.S2 64.A], constraints to which the elements of the relaxation matrix must conform, given the energy levels of the system and their degeneracies. If... 

For the times restricted by condition (188a), one therefore obtains the pure exponential decay law ... 

Here, is the relative permittivity of the solvent, o is the electric constant, R the universal gas constant, T the temperature, I the ionic strength, and zt and c , are the valency and molar concentration of the ionic species. For a purely exponential decay, the Debye length equals the distance from the Stem layer, at which the potential ij/ is decreased by the factor e from (i.e. 0.37 x Equation (3.14) has to be modified for extremely low electrolyte concentrations in the solvent in order to account for the counter ions that compensate for the surface charge (Thies-Weesie et al. 1995 Tellez and Biben 2000 Belloni 2005). 

This latter expression implies that the approach to equilibrium of MJt) will always be slower than iotf t). Although M t) is not a pure exponential decay function, the time-dependent portion resembles one, in which the initial value is 1 and the final value is pi (p+a) and the decay rate is j8. An important approximate relationship between the decay rate of the effective decay of the time-dependent part of M t) and the constants a and p was derived to be ... 

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