Observable canonical average

The x-ray results presented here show both consistencies and discrepancies with NMR observations. The most serious discrepancy is the implied coexistence of static and mobile C nuclei well below our Tc, deduced from the NMR observation of superposed motionally narrowed and powder pattern signals at temperatures as low as 140 K. On the other hand, a minimum in 7 i at 233 K is observed in one NMR experiment. In fact, the two techniques probe different aspects of the structure. NMR experiments to date cannot distinguish between free rotation and jump rotational diffusion between symmetry-equivalent orientations. X-ray diffraction is sensitive to orientational order (as a canonical average of snapshots) even in the presence of substantial thermal disorder, as long as one set of orientations is statistically preferred and the orientational order is long range. Indeed, our measurements indicate that much of the sc order is reduced by orientational fluctuations at Tc. 

To observe the relaxation process, we use the magnetization M(t). Note that observing M(t) corresponds to observing 2K t) / N by using Eq. (3), and 2K(t) /TV is the time series of the temperature, since the canonical average of 2K/N coincides with the canonical temperature. 

In the limit where the number of configurations M generated tends to infinity, the distribution of states X obtained by this procedure is proportional to the equilibrium distribution Pe (X), provided there is no problem with the ergodicity of the algorithm (this point will be discussed later). Then, the canonical average of any observable Z(X) is approximated by a simple arithmetic average. 

If it is possible to generate a Markov chain of transitions X —> X —> X one can show that, in the limit when the number M of configurations generated is very large, i.e., Ai oo, the canonical average of some observable A X) ... 

To calculate canonical averages of an observable f x,p) using the entropy distribution (Eq. [47]), we can implement umbrella sampling methods... 

The partition function Z is given in the large-P limit, Z = limp co Zp, and expectation values of an observable are given as averages of corresponding estimators with the canonical measure in Eq. (19). The variables and R ( ) can be used as classical variables and classical Monte Carlo simulation techniques can be applied for the computation of averages. Note that if we formally put P = 1 in Eq. (19) we recover classical statistical mechanics, of course. 

In view of the ergodic hypothesis the average value of an observable property may be regarded as the quantity measured under specified conditions. In this way the internal energy of a system corresponds to the average energy of the canonical distribution ... 

According to (39) the average of an observable over the grand canonical ensemble is... 

In order to describe the signal observed, it is necessary to average over the canonic ensemble of N identical particles j, or, in other words, to pass over to the density matrix Jmm [73]. The oscillatory term does not disappear after such a procedure, because at pulsed excitation all excitation moments are synchronized. 

Let us consider a system in equilibrium, described in the absence of external perturbations by a time-independent Hamiltonian Ho. We will be concerned with equilibrium average values which we will denote as (...), where the symbol (...) stands for Trp0... with p0 = e H"/ Vre the canonical density operator. Since we intend to discuss linear response functions and symmetrized equilibrium correlation functions genetically denoted as Xba(, 0 and CBA t,t ), we shall assume that the observables of interest A and B do not commute with Ho (were it the case, the response function %BA(t, t ) would indeed be zero). This hypothesis implies in particular that A and B are centered A) =0,... 

It is important to understand the conceptual difference between the quantities E and S in Eqs (1.161) and (1.15 8), and the corresponding quantities in Eq. (1.149). In the microcanonical case E, S, and the other derived quantities (P, T, /.i are unique numbers. In the canonical case these, except for T which is defined by the external bath, are ensemble averages. Even T as defined by Eq. (1.151) is not the same as T in the canonical ensemble. Equation (1.151) defines a temperature for a closed equilibrium system of a given total energy while as just said, in the canonical ensemble T is determined by the external bath. For macroscopic observations we often disregard the difference between average quantities that characterize a system open to its environment and the deterministic values of these parameters in the equivalent closed system. Note however that fluctuations from the average are themselves often related to physical observables and should be discussed within their proper ensemble. 

Recall that for canonical sampling, the (stationary or invariant) average of an observable (j> q,p) is given as... 

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