Temporal Averaging

To illustrate the general idea, let us consider a diagonal two-qubit density matrix with real diagonal elements a, b, c and, d, corresponding to the populations of the states 00), 01), 10), 11) (note that such matrix can be seen as a general representation of an equilibrium density matrix). 

Lets then apply the unitary transformation Uq, Ui, and U2 given by (4.3.4) to pm  

Taking the average over po. Pi and pi we obtain the effective state. 

As a specific example of generating pseudo-pure states in NMR spin systems, lets consider a case of two /-coupled spin 1/2 (see Section 4.1). The equilibrium density matrix of this system can be written as (4.3.3). Remember that the pulse sequence for the CNOT gate, the operations Uq, Ui, and U2 can be written as. 

Notice that the z rotations in the CNOT gates are not necessary for producing U, and U2. This is a typical example of the pulse simplification discussed in the last section. 

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Although, in general, there may be many distinct invariant measures, we can single out one particular equilibrium measure by demanding that the spatial average over the distribution for (almost) all initial points xq be equal to the temporal average over the trajectory, [xq, x, X2,. ... ... 

If Pm x) is independent of xq then the system is said to be ergodic. In this case, various temporal averages over some function h x) may be conveniently rewritten as spatial averages over Pm x) ... 

Modeling biogeochemical cycles normally involves estimating the spatial and temporal averages for concentrations and fluxes in and out of reservoirs (i.e., reservoir modeling). The... 

SL thermometry) exhibiting a strong dependence on thermal conductivity, the more spatially and temporally averaged temperature (from the MRR method), which is more representative for sonochemical processes, does not show such dependence. It is probable that it is actually the water vapour content within the bubble not the gas itself that dictates the bubble temperature. 

The spatio-temporal averaged transport measurements of [3H]PVP and [14C]sorbitol have been performed in a vertical test tube whose contents were fractionated at various times (Fig. 10). The distribution of both PVP and sorbitol appears to be sigmoidal, with gradual spreading with time. We have previously claimed 46) that... 

As in the Danckwerts equation, the frequency s does not depend on the diffusion coefficient. Unlike the Danckwerts equation, however, it results from both a spatial and temporal averaging. 

When K is in equilibrium, 0 must be constant in time, and this will be die case if il is a function of llie (time-indepeiidem) Hamiltonian H of K. Ensemble averages are now assumed to coincide with temporal averages. When, in particular. K is in diathermic equilibrium with its surroundings one can show that

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For low values of the inlet gas temperatures (30°-80°C) oscillatory states within the particle have been observed. The periodic activity of the pellet has been observed both for the freely suspended and imbedded particles. While the temporal average temperature was 170°C, the temperature oscillations amounts to 60°C. The frequence of oscillations is very low the period is almost 1 hr. The temperature of the gas and solid oscillated in phase with the reaction rate. 

The results discussed in this chapter demonstrate that 2H NMR is a powerful technique for investigating microscopic properties in rubber networks. Most of the experiments described here are easy to handle on standard NMR equipment. Due to the absence of interactions between 2H nuclei, spectra and line shapes are easy to interpret and give quite direct information, at least in the first step of analysis, which is that generally required to correlate microscopic to macroscopic properties in these systems. Additionally, in contrast to optical techniques (as birefringence, infrared dichroism, fluorescence polarisation) the information which is obtained is very specific, because spatial and temporal averaging processes are clearly distinguishable in NMR. 

The key challenge for the successful use of NMR velocity-imaging techniques to characterize fluid flow properties is the interpretation of the measured parameters. Different experimental strategies provide information about flow processes at different spatial and dynamic scales in porous media. In principle, the flow velocity can be probed either as a local quantity with an image resolution below the pore level,2425 or as a macroscopic flow property corresponding to local volume and temporal averages of fluid molecular displacements.26 One must develop a suitable methodology to correctly determine the parameters that best describe the properties of interest. 

For continuity with flow within the canopy, the velocity is taken as a spatial and temporal average, (n), although above the canopy this spatial average is not needed. The effective momentum boundary is the vertical position within the canopy below which momentum from the overflow does penetrate. That is, for flow above the canopy, z = zm is an effective lower boundary and (H - zm) is the effective depth of the overflow. Friction velocity for the overflow may then be estimated as... 

The mechanism responsible for the creation of x grating in glass fibers was proposed originally by Baranova and Zeldovich [165] in terms of a polychromatic interference of input fields at co and 2co frequencies, leading to a non-zero temporal average poling field ... 

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