Initial relaxation-rate definition

By definition irreversible electronic relaxation processes cannot occur in isolated small and too-many level small (intermediate) case molecules because of the insufficient density of final levels. For long times the molecule senses the presence of a finite number of possible final levels instead of the effective continuum that is required to drive irreversible electron relaxation. When collisional processes are appended, it is clear that the continuous density of states of the colliding pair can provide the necessary driving force for irreversible relaxation. The observed magnitudes of electronic relaxation rates as well as dependencies on the initial state, perturbing molecules, temperature, and so on, are the aspects of the processes that are of central interest. 

The lifetime of fluorescence (or phosphorescence), r0, is defined using the rate constant kf as r0 = 1 /k. By this definition, the residual intensity according to equation (12.1) is only 36.7% of the initial intensity at r0 63.3% of the initial species have relaxed to a non-emissive state. Because fluorescence lifetimes are only a few nanoseconds, fluorimeters require measurements to be made at the same time as... 

Since the total concentration a + r + s follows the time evolution d(a + r + s)/ dt = F - k(a + r + s), it approaches the steady state value F/k with a relaxation time 1 /k. This is a consequence of unbiased outflow (Eq. 47) of all reactants with the same rate k. Consequently, even though we are dealing with an open system under a flow, the analysis is similar to the closed system by replacing the total concentration c with the steady state value F/k. Instead of recycling, therefore, constant supply of the substrate allows the system to reach a certain fixed point with a definite value of the order parameter 0i, independent of the initial condition. 

The competition between intramolecular vibrational relaxation and chemical reaction has been discussed in terms of the applicability of transition state theory to the kinetic analysis [6], If the environment functions mainly as a heat bath to ensure thermalization among the vibrational modes in the excited complex, then transition state theory is a good approximation. On the other hand, when the reaction is too fast for thermalization to occur the rate can depend upon the initial vibronic state. Prompt reaction and prompt intersystem crossing are, by definition, examples of the latter limit. 

We discuss now how the synchronization transition occurs, taking the applause in an audience as an example (experimental study of synchronous clapping is reported in [35]). Initially, each person claps with an individual frequency, and the sound they all produce is noisy.As long as this sound is weak, and contains no characteristic frequency, it does not essentially affect the ensemble. Each oscillator has its own frequency oJk, each person applauds and each firefly flashes with its individual rate, but there always exists some value of it that is preferred by the majority. Definitely, some elements behave in a very individualistic manner, but the main part of the population tends to be like the neighbor . So, the frequencies u>k are distributed over some range, and this distribution has a maximum around the most probable frequency. Therefore, there are always at least two oscillators that have very close frequencies and, hence, easily synchronize. As a result, the contribution to the mean field at the frequency of these synchronous oscillations increases. This increased component of the driving force naturally entrains other elements that have close frequencies, this leads to the growth of the synchronized cluster and to a further increase of the component of the mean field at a certain frequency. This process develops (quickly for relaxation oscillators, relatively slow for quasilinear ones), and eventually almost all elements join the majority and oscillate in synchrony, and their common output - the mean field - is not noisy any more, but rhythmic. 

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