Recurrence time

My impression is that most in vitro experimentally observed periodic reactions have recurrence times of the order of minutes, seldom more than an hour. This may be an artifact of choice by the experimenter to choose conditions to have this convenient range I do not know. Those of us here who have just arrived from California are suffering from our built-in 24 hour clock. There is at least one well-known perodicity of four weeks affecting a large fraction of humans. Probably the periodicity of one year observable in most temperate-zone organisms is a reaction to external stimuli such as temperature or relative day-night lengths, but I... 

Find an expression for the mean recurrence time at the site n = 0. That is, the system starts at n = 0 and one wants to know the average of the time at which the first transition from n = 1 into n = 0 occurs. Evaluate it for the above example and note that it is the reciprocal of ps0. 

H. Rabitz The information in the recurrence time alone is minimal. However, the temporal structure of the recurrence signal contains detailed information on the surface explored by the wandering scout wavepacket during its excursion. Further experiments may be necessary to follow (i.e., track) the wavepacket through its excursion over the potential surface. Such pump-probe experiments go beyond conventional spectroscopy. 

Note that eq. (2-11) does not satisfy the limiting condition (2-6). Provided that the density of states in the energy spectrum is sufficiently large, we can define a recurrence time, tr, for the evolution of the metastable state such that the metastable state will decay on the time scale t r,/2. On the other hand, for time t tr 2, the amplitude of TjfXj, X2 t) will increase again toward its initial value. To demonstrate this point we notice... 

It is meaningful to consider a decay process on this limited time scale, which in turn is determined by the properties of the system under consideration. The introduction of the recurrence time defines, appropriately, the notion of irreversibility for the decay process. In particular, we see that if the recurrence time is sufficiently long (p sufficiently large) it is meaningful to consider a relaxation process even in a system with only a quasicontinuum of levels. 

Hence, the nonradiative lifetime can be displayed as the recurrence time diluted by the factor n. 

Exponential decay of this particular form (shortened radiative lifetime) is not expected on a time scale longer than the recurrence time. See (e) below. 

This second point is quite an interesting one, for there is a theorem known as the Poincare recurrence theorem which states that an isolated system (like our molecule left to itself) will in the course of time return to any of its previous states (e.g. the initial state), no matter how improbable that state may be. This recurrence can be observed with very small molecules but not with polyatomic molecules, because in the latter there are far too many levels of the final state the recurrence time is then far longer than any practicable observation time. 

The origin of this resonance was identified by extending Reinhardt s21 wave packet notion. Realizing that the wave packet evolves along the classical trajectories of the electron, Holle et al. searched for classical trajectories in which the electron leaving the origin returned in a time of 9.5 ps, 1/2tt (0.64[Pg.153]

It is clear that it is only possible to observe these resonances if their recurrence times are shorter than the coherence time of the exciting laser. With this point in mind, Main et al.23 made a five fold improvement in their spectral resolution and were able to see new resonances in the Fourier transform spectrum with longer recurrence times, as shown in Fig. 9.8. Tc is the recurrence time for a cyclotron orbit. The spectrum vs tuning energy is not shown since it is composed of... 

If <<(>11i (t)> and < 21 2(t)> peak at different times, the product <<(> <(>(t)> may peak at some intermediate time. The compromise recurrence time tm is not just the average of ti and t2. The MIME frequency may be smaller than any of the individual frequencies. It is usually between the highest and lowest frequencies. It cannot be larger than the highest frequency. 

A much simpler picture emerges in the time domain. The corresponding autocorrelation function, depicted in Figure 8.4, exhibits three well resolved recurrences with very small amplitudes. The recurrence times 2"i, T2, and T3 are incommensurable which indicates that they reflect different types of molecular motion. Since the recurrences are well separated we can write S(t) as a sum S(t) = Si(t), i = 0,..., 3, with So representing the main peak at t = 0. The Fourier transformation is linear so that the absorption cross section also splits into four individual terms,... 

Because of the superposition of three distinct types of periodic motion with different periods the wavepacket by itself does not reveal a clear picture in the present case, i.e., the classical skeleton is hardly visible through the quantum mechanical flesh . The perfect agreement between the recurrence times of the quantum mechanical wavepacket and the periods of the classical periodic orbits, however, provides convincing evidence that the structures in the absorption spectrum are ultimately the consequence of the three generic unstable periodic orbits. This correlation is... 

Le Quere and Leforestier (1990, 1991) calculated the autocorrelation function directly using the same PES and found fair agreement in regard of the recurrence times while the amplitudes were in remarkable disagreement. This may be due to either deficiencies of the calculated PES or the neglect of nonzero total angular momentum states in the theory. If the recurrences in the autocorrelation function are rescaled, the quantum mechanically calculated spectrum agrees well with experiment. 

Trec scaling as 1/Afinite system the recurrence time is finite, but for an infinite number of degrees of freedom, corresponding to a heat bath, the solution yields a continuum of eigenvalues, Aco = 0 and rrec = oo. 

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