Ergodic correlation function

The corresponding correlation functions are sketched for characteristic temperatures in Fig. 9b. The critical temperature Tc marks the crossover from an ergodic state (liquid) to a nonergodic state (glass). In the liquid state, a two-step process describes the decay of the correlation function, where/is the fraction relaxed by the slow process (a-process) and 1 / is the part decaying due to the... 

In the limit of long times and if the ergodic assumption holds, then we have 7 = (/), where (I) is the ensemble average. As usual, we may generate many intensity trajectories one at a time, to obtain the ensemble-averaged correlation function... 

Flere (1 - f) and (1 - S) are the losses brought about by the corresponding process and gfastj/3 (t -> oo) = 0 (see Fig. 6). The factor / can be regarded as a generalized non-ergodicity parameter and, hence, it is expected to show a similar anomaly as the Debye-Waller factor /g (see Fig. 5). Such decomposition of the correlation function is useful in spin-lattice relaxation studies, as will be discussed in Section 3.2.4. 

Whereas the correlation function S(Q, t) decays to zero in the liquid state, this is no longer the case below Tc, where the system becomes non-ergodic. Such scenario results from a generalized oscillator equation with non-linear damping. The slowing down of the molecular dynamics creates an enhanced damping, which in turn slows down the correlation function etc. The density is taken as a control parameter. In Fig. 38a, /(the Q dependence is omitted in the most simple approach) is displayed... 

If a system is uniformly hyperbolic, every point in phase space has both stable and unstable directions, and the maximum Lyapunov exponent with respect the maximum entropy measure is positive. The system has the mixing property and is therefore ergodic. The correlation function of observables also shows exponential decay. Uniformly hyperbolicity, which is sometimes rephrased as strong chaos in physical literature, is a well-established class of systems and is controllable by means of many mathematical tools [15]. In hyperbolic systems, there are no sources to make the relaxation process slow. 

It is instructive to examine properties of spatial correlation functions for specific cases. The stadium-billiard problem, discussed in previous sections in the context of classical-quantum correspondence, is a useful example. There is, in this case, no potential within the boundary, that is, the boundary alone induces ergodicity. Therefore, the q integration is readily done, yielding,... 

Figure 16 provides67 a comparison between the classical [Eq. (4.6)], and quantum [Eq. (4.1)] spatial correlation functions for four chaotic stadium eigenfunctions. The classical and quantum correlation functions are seen to agree very well for all distances other than those comparable to the size of the stadium s linear dimension. Thus, the rapid decay of the quantum spatial autocorrelation is a measure relating directly to classical ergodic behavior. 

The classical-quantum correspondence also results in useful scaling laws for chaotic states. For the stadium problem the classical correlation functions scale as (2m ) l/z, a feature which is solely a consequence of classical ergodicity. As shown in Fig. 17, quantum states labeled chaotic (according to the aperiodic nature of their correlation function) do obey this scaling relation. Specifically, Fig. 17 displays the correlation lengths (A,/2), defined as... 

Cross-Correlation Function If n (t) and ng t) are two ergodic random processes it is possible to define the cross-correlation function by... 

Here, the correlation function of the partial heterodyne approach, g, is that for the ergodic fluctuation part expressed as... 

A unique feature of SM-FRET experiments is their ability to provide information on the correlations between structure and dynamics. One way of extracting this information is by monitoring the distribution of (R)tw as a function of Tw [30] Correlations between conformational structure and the time scale on which it moves can be obtained from the different values of Tw at which different subsets of the ensemble which are characterized by different values of R reach the ergodic limit. 

There is one paradoxical subtlety here, insofar as the dimensionality one computes will usually be based on some finite-time sampling. Consequently if a system is truly ergodic but requires a very long time interval to exhibit its ergodicity, the dimensionality one obtains can be no better than a lower bound. We shall return to this topic later, when we examine local properties of several-body systems. Meanwhile, we just quote the results of computations of the dimensionality of the phase space for Ar3, as a function of energy [4]. The calculations were carried out by the method introduced by Grassberger and Procaccia [6,7]. Values of the dimension, specifically the correlation... 

Figure 17. The correlation length as a function of energy for the classical ergodic distributions (dashed) and ergodic quantum states (diamonds). States marked as ( and +) are regular by the adiabatic criterion.66 (From Ref. 67.)...

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