Random lifetime assumption

In the end, the random-lifetime assumption was confirmed for a large number of the hypothetical models, but not all. The fact that the randomlifetime assumption did not apply universally could have been a problem for the RRKM theory, but Bunker was able to devise a theoretical model that could predict whether a particular model would or would not have a random lifetime. This had never been done before, as Bunker pointed out, because a variety of models had been considered to be plausible, and until the Monte Carlo results were available, a choice could not be made. As it turned out, nonrandom lifetimes occurred only under special experimental conditions, which had never been carried out, so in addition to being able to endorse the RRKM theory for reactions it had already been applied to, Bunker was able to suggest experiments with conditions under which the RRKM theory should fail. ... 

The random lifetime assumption is perhaps most easily tested by classical trajectory calculations (Bunker, 1962 1964 Bunker and Hase, 1973). Initial momenta and coordinates for the Hamiltonian of an excited molecule can be selected randomly, so that a microcanonical ensemble of states is selected. Solving Hamilton s equations of motion, Eq. (2.9), for an initial condition gives the time required for the system to reach the transition state. If the unimolecular dynamics of the molecule are in accord with RRKM theory, the decomposition probability of the molecule versus time, determined on the basis of many initial conditions, will be exponential with the RRKM rate constant. That is, the decay is proportional to exp[-k( )t]. The observation of such an exponential distribution of lifetimes has been identified as intrinsic RRKM behavior. If a microcanonical ensemble is not maintained during the unimolecular decomposition (i.e., IVR is slower than decomposition), the decomposition probability will be nonexponential, or exponential with a rate constant that differs from that predicted by RRKM theory. The implication of such trajectory studies to experiments and their relationship to quantum dynamics is discussed in detail in chapter 8. 

Rigorous tests of the assumptions are not easily performed. The first assumption of RRKM theory can be tested only by single quantum state selection of the reactants. Because state selection is possible in only very small molecules, most efforts have been directed at energy selection which produces a distribution of initial states. While not ideal, this does permit testing one of the consequences of the random lifetime assumptions which is that k E) increases monotonically with increasing E. 

Selective excitation experiments on unimolecular reactions have two main aims. First, to test the proposition, inherent in the statistical theories of unimolecular reactions, that intramolecular energy transfer is extremely rapid and therefore the random lifetime assumption [see equation (1.52)] is valid. Secondly, to measure specific rate constants, k e and compare them with theoretical predictions. In the rest of this section, some of the experimental studies which have had greatest success in fulfilling these objectives are reviewed. 

In principle, photoactivation allows the selection of a well-defined reactant state. In practice, the quasicontinuous absorption of polyatomic molecules, arising from the high internal state density and the thermal distribution of absorbers over closely packed rovibrational levels, usually frustrate this desirable objective. In single-photon excitation relying on fast internal conversion from an electronically excited state, the distribution is unlikely to be completely random, or to be entirely independent of excitation wavelength, but it is not possible to control the initial nonrandomness. The results of such experiments (see Section 1.4.3) are consistent with rapid energy randomization but no direct tests of the random lifetime assumption have been made. 

Figure A3,12.2(a) illnstrates the lifetime distribution of RRKM theory and shows random transitions among all states at some energy high enongh for eventual reaction (toward the right). In reality, transitions between quantum states (though coupled) are not equally probable some are more likely than others. Therefore, transitions between states mnst be snfficiently rapid and disorderly for the RRKM assumption to be mimicked, as qualitatively depicted in figure A3.12.2(b). The situation depicted in these figures, where a microcanonical ensemble exists at t = 0 and rapid IVR maintains its existence during the decomposition, is called intrinsic RRKM behaviour [9].

The z(C) dependence has been investigated with the help of microscopic foam bilayers of both ionics and nonionics [419,420]. Due to the fluctuation character of the film rupture, the film lifetime is a random parameter. Experimentally, the film mean lifetime r has been determined by averaging from a great number of measurements. Because of the assumption that the monomer and the total surfactant concentrations are practically equal, in all t(C) dependences given below, C refers to the total concentration. Using Eq. (3.120) to analyse the experimentally obtained time dependence of the probability P(t) of film rupture it was found... 

As regards the orientation factor k, it is usually taken as 2/3, which is the isotropic dynamic average, i.e. under the assumptions that both donor and acceptor transient moments randomize rapidly during the donor lifetime and sample all orientations. However, these conditions may not be met owing to the constraints of the microenvironment of the donor and acceptor. A variation of from 2/3 to 4 results in only a 35% error in r because of the sixth root of... 

One can immediately see that c can vary from 0 if all angles are perpendicular to 4 if all angles are parallel. If the orientation of the dipoles is random, because they are moving rapidly (within the donor lifetime) then = 2/3. This assumption is often made, even if not strictly true, and accounts for much of the uncertainty in FRET measurements. If just the donor or just the acceptor is randomized, then 1/3 < < 4/3. In this... 

Recalling the spin-sorting nature of the radical pair mechanism, we can anticipate that in the absence of nuclear spin relaxation, random recombination will eventually lead to exact cancellation of the + polarization when the + polarization in A is transferred to A (making the usual assumption that chemical reaction preserves nuclear spin orientation). In such a situation, polarization in A could only be observed in a time-resolved experiment before all the radicals had recombined. Relaxation of the nuclei A , however, allows some of the escape polarization to "leak away preventing complete cancellation (II). Thus, unless the radical lifetime is very much smaller than the nuclear... 

Most probabilistic models of complex systems assiune fiiU iadependence of their components. In particular, it means that the component lifetimes and repair times are independent random variables. Obviously, this assumption is made for the sake of computational simplicity. In reality, however, such independence rarely occurs. In order to make the considered network model more true-to-life, it is assumed that the functioning of Ci depends on the states of components located between Co andci in the following way if all these components are operable, then ei s time-to-failure is distributed according to Fi, otherwise (one of the components is failed) it is distributed according to Gi. It is also assumed that Fj > Gi, which conveys the idea that the components being under load are more failure prone, as in many real-life systems. Note that Gi = 0 if Ci disconnected from eo cannot fail. It is also assumed that ei functions independently of all components not located between ej and cq, and Cj s repair time is independent on the states of all other components. 

Initially, PET in DNA was studied using the electron donors and acceptors intercalated in DNA in a random manner. In 1992, Harriman and Brun reported fluorescence lifetimes of calf thymus DNA using ethidium bromide (EB+) and V,V -dimethyl-2,7-diazapyrenium dichloride (DAP +) as intercalators. They found the short fluorescence lifetime components to be due to PET between EB+ and DAP +. On the basis of the assumption that PET occurred between EB+ and DAP + separated by 3-5 base pairs, PET rates were estimated. Since the electron transfer rate ( et) can be expressed as a function of the donor-acceptor distance (r) in (7), the value was evaluated to be 0.88 a value close to that estimated for proteins. 

In ZF-/tSR the longitudinal muon spin relaxation function G t) is directly deduced from the time-differential measurement of the forward/backward muon decay asymmetry, without any disturbance of the spin-glass system by an external field. (No depolarization of the muon spin means G = l, complete depolarization G =0.) The observed time evolution G (t) of muon-spin polarization reflects amplitudes, randomness, and fluctuations of local magnetic fields at muon sites in the specimen. There appear two essential problems in analyzing pSR experiments on spin glasses (i) One has to make model assumptions about the shape of G (t) (ii) Any relaxation slower than 10 s appears as a static component in pSR (lifetime of the muon is = 2.2 x 10 s). 

The second approximation is to release the radical purpose entrance assumption but maintain the constant initiation rate, which is reasonable considering the short period of stage I and the initiator s long half-lifetime. Radicals have nonpreference of nricelles over polymer parride but enter the organic phases randomly according to eqn [199]. With the assistance of Am-l-Ap =NAvas([S] -[S]cMc) NAvas[S] , the total parride surface area is easily fotmd from eqn [203] ... 

The dynamic terms in eqn (1) depend upon the assumptions used to describe the motion. For the intermolecular motion a diffusive process is assumed (rotation through a sequence of small angular steps). In that case intermolecular reorientation can be characterized by two rotational correlation times, and Tru. The correlation time for reorientation of the symmetry axis of a molecular diffusion tensor is Tr, while Tru refers to rotation about the axis. For the intramolecular motion a random jump process is assumed. Thus, isomerization occurs through jumps between different conformations with an average lifetime Tj. 

Trajectory calculations have also been employed to examine the proposition, fundamental to statistical theories of unimolecular processes, that isolated molecules behave as if< ° all accessible states at the same energy are in rapid communication. This implies that the lifetime of a species of defined energy and angular momentum is independent of its distinctive mode of formation, and leads to the assumption that the lifetimes with respect to chemical rearrangement (t) are distributed randomly according to the equation... 

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