Vanishing fluctuations

Spectral fluctuations are a distinctive property of atomic spectra, but have rarely been studied in their own right. One example involving fluctuations is the procedure of Gailitis averaging near a series limit (see chapter 3), This involves taking the mean of the maximum and minimum excursions as the series limit is approached, and emphasises the fairly obvious result that spectral fluctuations tend to zero as the limit of a Rydberg series is approached. 

Another situation where experimental fluctuations in the cross section tend to zero is when a vanishing width occurs and the instrumental resolution is finite. The fluctuations then pass through zero at an energy which 

In nuclear physics, fluctuations are studied in their own right through the variance  

Spectral fluctuations are an observable, and therefore exhibit the same properties of continuity across thresholds as other obervables in atomic physics. Although the variance is strictly of fourth order in elements of the S-matrix, a useful theorem, due to Moldauer [433] states that the fluctuation in the total cross section for a given entrance channel a is of second order in S-matrix elements  

var 7a behaves like a cross section. For example, it is continuous across thresholds [374] and may possess interesting variations and zeros, just as one finds when considering interferences between Rydberg series of resonances and broad intruder states. 

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The non-vanishing of the subsystem average of the commutator implies a fluctuation in the value of the observable G over the subsystem as measured by the flux of its vector current density through the surface of the subsystem. Thus one anticipates and finds non-vanishing fluctuations in subsystem expectation values for observables which do not commute with H. 

Fig. 8.23. Vanishing fluctuations in a doubly-excited series of Ba. Note the existence of a point (marked X in the figure) at which spectral structure disappears, and which is not a series limit. Note also the q reversal about a window resonance (after J.-P. Connerade and S.M. Farooqi [442]).

We infer that a vanishing width in the case of a single open channel becomes a vanishing fluctuation in the presence of several open channels The data also demonstrate that the influence of a single perturber is enough for a disappearance of fluctuations to occur, as witnessed by the simultaneous occurrence of a single q reversal. 

One should also examine the conditions under which the standard expressions of Lu and Fano [440] remain applicable when all the lines are autoionising resonances, and the extent to which the usual parametri-sation must be reinterpreted or may exhibit differences, bearing in mind potential complications such as symmetry reversals and vanishing fluctuations. Fortunately, even in such cases, the two-dimensional representation remains a valid map. 

K becomes independent of the sum over poles for series 1. Near this energy, the spectral fluctuations disappear, but no change is induced in the appearance of the two-dimensional graph. The existence of vanishing fluctuations in atomic spectra at energies distinct from series limits was suggested by Lane [379] and occurrences have been confirmed experimentally in the spectra of Ba [442] and Sr [443] (see section 8.33). 

E = Eb, there is a pole. For very weak coupling, the point of vanishing fluctuations coincides with this pole, which may, however, be detuned from the maximum in the broad resonance. 

In the presence of continua into which series 1 and 2 do not autoionise, Evf is unchanged, but vanishing fluctuations tend to disappear when a background continuum 6 into which series can autoionise becomes conspicuous, as one sees from equation (8.109). 

One of the most important effects of direct inter series coupling is the existence of vanishing fluctuations for an appropriate combination of par rameters, as discussed in the previous section. Fig. 8.33 shows such a case, and demonstrates that the presence of vanishing fluctuations does not lead to any change in structure of the two-dimensional graph. In-... 

If this condition is not satisfied, there is no unique way of calculating the observed value of ff, and the validity of the statistical mechanics should be questioned. In all physical examples, the mean square fluctuations are of the order of 1/Wand vanish in the thennodynamic limit. 

This behaviour is characteristic of thennodynamic fluctuations. This behaviour also implies the equivalence of various ensembles in the thermodynamic limit. Specifically, as A —> oo tire energy fluctuations vanish, the partition of energy between the system and the reservoir becomes uniquely defined and the thennodynamic properties m microcanonical and canonical ensembles become identical. 

This result is identical to that obtained from a canonical ensemble approach in the thennodynamic limit, where the fluctuations in N vanish and (N) = N. The single-particle expression for the canonical partition fiinction = (-" can be evaluated using ih r rV i f<,2M) or a particle in a cubical box of volume V. 

Therefore, the locus of the values ( ) with a vanishing second derivative of A delimits the region of the miscibility gap in which spinodal decomposition occurs. This locus is referred to as the spinodal (figure C2.1.10 (bl). The length scale of the concentration fluctuations at the beginning of the separation process is controlled by... 

At temperatures above there is no instanton, and escape out of the initial well is accounted for by the static solution Q = Q with the action S ff = PVo (where Vq is the adiabatic barrier height here) which does not depend on friction. This follows from the fact that the zero Fourier component of K x) equals zero and hence the dissipative term in (5.38) vanishes if Q = constant. The dissipative effects come about only through the prefactor which arises from small fluctuations around the static solution. Decomposing the trajectory into Fourier series. 

Furthermore, one can infer quantitatively from the data in Fig. 13 that the quantum system cannot reach the maximum herringbone ordering even at extremely low temperatures the quantum hbrations depress the saturation value by 10%. In Fig. 13, the order parameter and total energy as obtained from the full quantum simulation are compared with standard approximate theories valid for low and high temperatures. One can clearly see how the quasi classical Feynman-Hibbs curve matches the exact quantum data above 30 K. However, just below the phase transition, this second-order approximation in the quantum fluctuations fails and yields uncontrolled estimates just below the point of failure it gives classical values for the order parameter and the herringbone ordering even vanishes below... 

The uniform phase becomes unstable if there exist fluctuations for which 0,2, given by Eq. (56), vanishes. Since a2 > 2 for all temperatures (see (48) and (49)), the instability of the uniform phase is related to fluctuations of the fields (f) and j li. O2 can be written in a diagonal form... 

In the case of strong surfactants the tcp corresponds to the lowest value of Ps for which CI4 vanishes at the bifurcation line for some fluctuation 0(k), with k = k. The value of H4 depends on the form of 0(k). Here we find the approximate position of the tcp, considering a selection of structures... 

Development in recent years of fast-response instruments able to measure rapid fluctuations of the wind velocity (V ) and of fhe tracer concentration (c ), has made it possible to calculate the turbulent flux directly from the correlation expression in Equation (41), without having to resort to uncertain assumptions about eddy diffusivities. For example, Grelle and Lindroth (1996) used this eddy-correlation technique to calculate the vertical flux of CO2 above a foresf canopy in Sweden. Since the mean vertical velocity w) has to vanish above such a flat surface, the only contribution to the vertical flux of CO2 comes from the eddy-correlation term c w ). In order to capture the contributions from all important eddies, both the anemometer and the CO2 instrument must be able to resolve fluctuations on time scales down to about 0.1 s. 

In fact, the clearly posed problem of the final state of an unstable laminar flame is a limiting case of turbulent flame for vanishing initial turbulence of the oncoming flow, but the general case, for any initial velocity fluctuations, is clearly of great interest in practical devices such as spark-ignited engines, turbojet, or gas turbine combustion chambers. 

The average of the quantity (f Ar) has been equated to zero in obtaining Equation (24) in as much as this quantity is equally likely to be positive or negative because of the fluctuating term Ar, and the average overall possible occurrences vanishes. Equation (24) is valid in both the deformed and undeformed states. It may be written in terms of Xx, Xy, and Xz as... 

From (2.70), it follows that the free energy cannot be divided simply into two terms, associated with the interactions of type a and type b. There are also coupling terms, which would vanish only if fluctuations in AUa and AUb were uncorrelated. One might expect that such a decoupling could be accomplished by carrying out the transformations that involve interactions of type a and type 6 separately. In Sect. 2,8.4, we have already discussed such a case for electrostatic and van der Waals interactions in the context of single-topology alchemical transformations. Even then, however, correlations between these two types of interactions are not... 

Therefore, Flory s theory concludes that as the functionality of a network increases, the constraint contribution, fc, should decrease and eventually vanish. Furthermore, in the extreme limit in which junction fluctuations are totally suppressed, the Flory theory reduces to the affine network model ... 

Now we consider thermodynamic properties of the system described by the Hamiltonian (2.4.5) it is a generalized Hamiltonian of the isotropic Ashkin-Teller model100,101 expressed in terms of interactions between pairs of spins lattice site nm of a square lattice. Hamiltonian (2.4.5) differs from the known one in that it includes not only the contribution from the four-spin interaction (the term with the coefficient J3), but also the anisotropic contribution (the term with the coefficient J2) which accounts for cross interactions of spins a m and s m between neighboring lattice sites. This term is so structured that it vanishes if there are no fluctuation interactions between cr- and s-subsystems. As a result, with sufficiently small coefficients J2, we arrive at a typical phase diagram of the isotropic Ashkin-Teller model,101 102 limited by the plausible values of coefficients in Eq. (2.4.6). At J, > J3, the phase transition line... 

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