Allan Variance

The cornerstone of the recent optical frequency measurements in Paris is the LD/Rb standard laser [51,52,53]. Three identical systems have been built, two at the LPTF and a third in Laboratoire Kastler Brossel. As the two laboratories are linked by two 3 km long optical fibers, it is possible to compare the frequencies of the three systems. The frequency shift due to the fiber has been checked with the highly stabilized titanium-sapphire laser. After a round trip of 6 km through the fibers, a maximum frequency shift of 3 Hz is observed [54]. This shift is completely negligible for the optical frequency measurements. The main metrological features of the LD/Rb laser are a frequency stability (Allan variance) of about 4 X 10-13t-1/2 per laser over 1000 s and a day-to-day repeatability of 400 Hz. 

To characterise its frequency stability, we have measured the root Allan variance locked lasers [22]. A complicate experimental arrangement allowed us to use the same fiber amplifier and crystal for both sources. A stability value of 1.9 x 10-12 in 1 s (corresponding to 540 Hz at 1083 nm) is achieved, which improves to 4.1 x 10-13 at 300 s (see Fig. 7). In... 

Fig. 7. Allan variance

If we make the simplifying assumption that the time-domain Ramsey method is used to interrogate the clock transition, then the frequency stability (two-sample Allan variance- - ) is given as... 

Figure 6. The Allan variance of potential energy fluctuations, (a) The original BLN model and (b) the Go-like BLN model.

Figure 6 shows the so-called Allan variance [61-63] of time series of the potential energies V(t) for these models at several temperatures. The Allan variance c> (N) is defined by... 

Figure 7. The Allan variances for time series of a geometrical parameter of Ao for the cases of (a) the liquid-like state and (b) coexistence region where the frequent transitions between solid-like and liquid-like states are observed. The Allan variance for the solid-like state is qualitatively the same aside from the magnitude. [Reprinted with permission from C. Seko and K. Takatsuka, J. Chem. Phys. 104, 8613 (1996). Copyright 1996, American Institute of Physics.]...

Both models exhibit a 1 //-noise like spectral density that can be expressed as 1 // , where a = 1.5 for the BLN model and 1.6 for the Go-like model. It is noteworthy that in the case of the Go-like model this behavior persists over three decades of frequency extending into the low-frequency regions, while for the BLN model the 1 //-like spectrum extends for only two decades in frequency with the crossover to white noise occurring at higher frequencies. This supports the conclusions from the analysis of the Allan variance that the nonstationarity of the Go-like model is enhanced at the transition temperature. 

Figure 9 presents exponents a for both models as a function of temperature. The Go-like model exhibits a much sharper transition to a white noise spectrum than the BLN model as the temperature departs from the transition temperature. This finding also mirrors the results obtained from the Allan variance where the Go-model exhibits a sharp transition to nonstationarity around the transition temperature, while this transition is diffuse in the case of the BLN model. It was also shown [64,65] in a two-state-like helix-coil transition of a helical polypeptide that a 1//-noise structure of the potential energy fluctuations... 

The duality between the power spectrum and the Allan variance has been intriguing in the context of nonstationary chaos by Aizawa and co-workers [63,67,68,71]. Nonstationary chaos reveals a strong long-range order, so that the... 

Aizawa s Relation Between the Index a in the Power Spectrum and the Index y in the Allan Variance... 

Figure 10. The Allan variances of the individual energy fluctuations of (a) bond energy Vr, (b) bending energy V (c) torsional angle energy Vtn, and (d) nonbonded vdW interaction energy Vr at T = 0.2,0.4,0.72, and 2.0 for the original BLN model.

In the following, we will use solely the Allan variance to further inquire into the nonstationarity of the multibasin transition processes on many-dimensional protein landscapes. 

Bond Energy Fluctuation. Figures 12 and 13 show the Allan variances of each bond energy fluctuation Vn(t) and their superpositions Vr(t)(= (0) at... 

Below the transition temperature, the individual V .(f) exhibit somewhat different behavior in the Allan variances, implying that the energy landscapes in the torsional angle DOFs space look much more rugged compared with stretching and bending DOFs spaces for both the models [see also the average of the Allan variance of Ve.(f) and F (t) at T 0.4]. 

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