Relaxation exponential

For this reaction alone, one would thus obtain a simple exponential relaxation witii relaxation time... 

A single calculation of the discrete path integral with a fixed length of time t can be employed to compute the state conditional probability at many other times. It is possible to use segments of the path of time length At, 2At,..., NAt sampled in trajectories of total length of NAt and to compute the corresponding state conditional probabilities. The result of the calculations will make it possible to explore the exponential relaxation of P Ao B,t) for times between 0 and t. 

Equation (2.41) describes either damped oscillations (at tls < 2do) or exponential relaxation OItls > 2do). Since tls grows with increasing temperature, there may be a cross-over between these two regimes at such that 2h QoJ Ao)coth P hAo) = 2Aq. If the friction coefficient... 

Coupling to these low-frequency modes (at n < 1) results in localization of the particle in one of the wells (symmetry breaking) at T = 0. This case, requiring special care, is of little importance for chemical systems. In the superohmic case at T = 0 the system reveals weakly damped coherent oscillations characterised by the damping coefficient tls (2-42) but with Aq replaced by A ft-If 1 < n < 2, then there is a cross-over from oscillations to exponential decay, in accordance with our weak-coupling predictions. In the subohmic case the system is completely localized in one of the wells at T = 0 and it exhibits exponential relaxation with the rate In k oc - hcoJksTY ". 

The ohmic case is the most complex. A particular result is that the system is localised in one of the wells at T = 0, for sufficiently strong friction, viz. rj > nhjlQo. At higher temperatures there is an exponential relaxation with the rate Ink oc (4riQllnh — l)ln T. Of special interest is the special case t] = nhl4Ql. It turns out that the system exhibits exponential decay with a rate constant which does not depend at all on temperature, and equals k = nAl/2co. Comparing this with (2.37), one sees that the collision frequency turns out to be precisely equal to the cutoff vibration frequency Vo = cojln. 

Table (a) shows experimental data [24] for the initial charge density exiting a fuel filter Qq plus the charge density Q remaining 30 s downstream. At low conductivity the charge decays much faster than predicted by an exponential relaxation law [Eq. (2-3.7)] and instead follows a hyperbolic relaxation law [24] given by... 

Up to now it has been tacitly assumed that each molecular motion can be described by a single correlation time. On the other hand, it is well-known, e.g., from dielectric and mechanical relaxation studies as well as from photon correlation spectroscopy and NMR relaxation times that in polymers one often deals with a distribution of correlation times60 65), in particular in glassy systems. Although the phenomenon as such is well established, little is known about the nature of this distribution. In particular, most techniques employed in this area do not allow a distinction of a heterogeneous distribution, where spatially separed groups move with different time constants and a homogeneous distribution, where each monomer unit shows essentially the same non-exponential relaxation. Even worse, relaxation... 

Deviations from the impact (exponential) relaxation of angular momentum found in (1.84a) are rather insignificant. Taking account of finite... 

Of course, knowledge of the entire spectrum does provide more information. If the shape of the wings of G (co) is established correctly, then not only the value of tj but also angular momentum correlation function Kj(t) may be determined. Thus, in order to obtain full information from the optical spectra of liquids, it is necessary to use their periphery as well as the central Lorentzian part of the spectrum. In terms of correlation functions this means that the initial non-exponential relaxation, which characterizes the system s behaviour during free rotation, is of no less importance than its long-time exponential behaviour. Therefore, we pay special attention to how dynamic effects may be taken into account in the theory of orientational relaxation. 

Though Kj(t) decays from 1 to 0 it is in general non-exponential relaxation. Its conventually defined correlation time... 

Figure 6. Shown is the correlation between the liquid s fragility and the exponent p of the stretched exponential relaxations, as predicted by the RFOT theory, superimposed on the measured values in many liquids taken from the compilation of Bohmer et al. [50]. The dashed line assumed a simple gaussian distribution with the width mentioned in the text. The solid line takes into account the existence of the highest barrier by replacing the barrier distribution to the right of the most probable value by a narrow peak of the same area the peak is located at that most probable value. Taken from Ref. [45] with permission.

Fig. 5. The characteristic frequencies QR and time exponents (3 in the stretched exponential relaxation function obtained for the randomly labelled PDMS melt at 100 °C. (Reprinted with permission from [44]. Copyright 1989 Steinkopff Verlag, Darmstadt)...

The rate constant for the exponential relaxation of the latter system to the starting system was calculated to be 1.4 x 10 s . From this value, an approximate second order rate constant of 1.0 x 10 L mol" -s"l was calculated for the reaction between IV and CO. Given the above determination of the limiting rate constant for CO dissociation... 

An alternative way to describe the phenomenon is to consider that the ground state of a chain is already divided into domains at any temperatures. In order for the system to follow a small variation of the magnetic field some domains have to reverse their spin orientation. This occurs through a random walk of the DWs, that is, equal probability for the DW to move backward or forward, which implies that the DW needs a time proportional to d2 to reach the other end of a domain of length d. Given that d scales as the two spins correlation length, ., which, for the Ising model, is proportional to exp(2///rB7 ), for unitary spins, the same exponential relaxation is found... 

It can be observed that these thermal conductances G(7) are typical of phonon conduction between two solids at very low temperature, as already reported [45], The value of the heat capacity was calculated from equation C = r G, where the thermal time constant r is obtained from the fit to the exponential relaxation of the wafer temperature. 

The H20 exchange mechanism was studied by Hunt et al. (32) who reported that exchange between aqueous solvent and Fein(TPPS)(H20)2 occurs with a first-order rate constant (kex = 1.4xl07s-1 in water at 298 K) far exceeding the k0 s values determined at any [NO]. If the steady state approximation was applied with regard to the intermediate Fem(Por)(H20), the kohs for the exponential relaxation of the non-equilibrium mixture generated by the flash photolysis experiment would be,... 

Ad. 4. Alteration of anatomical features. Finally, several studies have attempted to manipulate the macroscopic features of muscle/meat tissue in order to verify or disprove the intra-/extra-cellular model. Efforts have been made to disrupt cell membranes by glycerination and DMSO treatment in order to make assessment of the potential role of membranes possible.30,40,41 All these studies showed unaltered relaxation behaviour upon membrane disintegration, which suggests that intact cell membranes in themselves are not necessary for a non-mono-exponential relaxation. Irrelevance of membranes... 

A particular characteristic feature of dynamic processes in the vicinity of the glass transition is the ubiquity of the Kohlrausch-Williams-Watts (KWW) stretched exponential relaxation 1,7-9... 

Note that similar curves are obtained for the eight glitches of the Vela pulsar. As we see from Fig. 1, the behavior of AQ + AQ.S. is quite different in different parts of the relaxation region. The first part is the region of exponential relaxation with constants t and r2 and is located within the shell 9.533 < r < 9.61 km. The second part is the region of the exponential relaxation with t3 and linear relaxation, located within the shell 9.36 < r 5 9.533... 

The ideal stress relaxation experiment is one in which the stress is instantaneously applied. We have seen in Section 4.4.2 the exponential relaxation that characterises the response of a Maxwell model. We can consider this experiment in detail as an example of the application of the Boltzmann Superposition Principle. The practical application of an instantaneous strain is very difficult to achieve. In a laboratory experi-... 

Most methods assume an exponential decay for the resonances in the time domain giving rise to Lorentzian lineshapes in the frequency domain. This assumption is only valid for ideal experimental conditions. Under real experimental circumstances multi-exponential relaxation, imperfect shimming, susceptibility variations and residual eddy current usually lead to non-ideal... 

These various processes have several important implications for plant tissue relaxometry. First, the fact that plant cells are compartmentalized means that, in general, the transverse and longitudinal relaxation will be multiple exponential when measured with the CPMG sequence and enough data points to give well characterized decay to the baseline. Such multiple exponential relaxation is observed with apple. 

The singular character of the diffusive modes allows their exponential relaxation at the rate given by the dispersion relation of diffusion. Their explicit construction can be used to perform an ab initio derivation of entropy production directly from the microscopic Hamiltonian dynamics [8, 9]. 

Using a simple kinetic model, Solomon demonstrated that the spin-lattice relaxation of the I and S spins was described by a system of coupled differential equations, with bi-exponential functions as general solutions. A single exponential relaxation for the I spin, corresponding to a well-defined Tu, could only be obtained in certain limiting situations, e.g., if the other spin, S, was different from I and had an independent and highly efficient relaxation pathway. This limit is normally fulfilled if S represents an electron spin. The spin-lattice relaxation rate, for the nuclear spin, I, is in such a situation given by ... 

In the case of hi- or multi-exponential relaxation curves the treatment involved can be rather complex (119-123). It becomes even more problematic. Needles to say, the same is true for systems with suspected continuous distributions of relaxation rates, whose evaluation by numerical analysis of the decay curves (124-128) represents one of the most arduous mathematical problems (124-128). In general, evaluation tasks of this kind need to be treated off-line, using specific programs and algorithms. 

To retain the analogy with a simple exponential function, it is considered in the cases described by Equations (1.8) and (1.9) that there is a distribution of barrier heights, g(G), each height corresponding to an exponential relaxation (Austin et al. 1975 Nagy et al. 2005). The concentration profile is in this case described by... 

Aging. If we assume independent exponential relaxations for the CRRs, we obtain the following expression for the two-times correlation function ... 

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