Canonical average

The additional factor of Qi(V, T) in Eq. (21) makes the leading term in the sum unity, as suggested by the usual expression for the cluster expansion in terms of the grand canonical partition function. Note that i in the summand of Eq. (20) is not explicitly written in Eq. (21). It has been absorbed in the n , but its presense is reflected in the fact that the population is enhanced by one in the partition function numerator that appears in the summand. Equation (21) adopts precisely the form of a grand canonical average if we discover a factor of (9(n, V, T) in the summand for the population weight. Thus... 

Finally, in terms of the canonical average of the probability exponent... 

It should be noticed that a((3) and a((3) satisfy the same algebraic relation as those given in Eq. (3), and also that a(/3) 0(/ )) = a(j3) 0(/ )) = 0. Then the thermal state 0(/3)) is a vacuum for a((3) and a(/3) (otherwise, 0,0) is the vacuum for the operators a and a). As a result, the thermal vacuum average of a non-thermal operator is equivalent to the Gibbs canonical average in statistical physics. As a consequence, the thermal problem can be treated by a Bogoliubov transformation, such that the thermal state describes a condensate with the mathematical characteristics of a pure state. 

By expressing the time derivatives in terms of the Poisson bracket and canonically averaging, we get for the second binary moment... 

This expression is squared and canonically averaged, to get the fourth binary moment [79],... 

Here A is the Helmholtz free energy, X is an arbitrary parameter of the Hamiltonian, and <> denotes a canonical average. For more information about computer calorimetry see refs. 17, 18, and 19. 

The interpretation of the above expressions is rather remarkable. The centroid constraints in the Boltzmann operator, which appear in the definition of the QDO from Eqs. (19) and (20), cause the canonical ensemble to become non-stationary. Equally important is the fact that the non-stationary QDO, when traced with the operator it (or P) as in Eq. (37), defines a dynamically evolving centroid trajectory. The average over the initial conditions of such trajectories according to the centroid distribution [ cf. Eq. (36) ] recovers the stationary canonical average of the operator (or fi). However, centroid trajectories for individual sets ofinitial conditions are in fact dynamical objects and, as will be shown in the next section, contain important information on the dynamics of the spontaneous fluctuations in the canonical ensemble. 

We have seen that free energies like Z rl (X) are not themselves naturally expressible as canonical averages, but their derivatives with respect to field-parameters are expressible this way. Specifically,... 

It would be more circumspect to say that the partition function cannot in general be naturally or usefully expressed as a canonical average since that average inevitably falls into the category for which canonical sampling is inadequate [13]. 

We see that the rate constant may be determined as the time integral of the canonical averaged flux autocorrelation function for the flux across the dividing surface between reactants and products. It is also clear that we only need to calculate the flux correlation function for trajectories starting on the dividing surface, for otherwise F(p(0), q(0)) = 0 and there will be no contributions to the product formation. 

Derive Eq. (3.24) using the expression for a grand 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. 

Over the last several years much effort has gone into the extension of MD to describe other than microcanonical systems (Nose and references therein). Nose, in particular, has provided a method that is capable in principle of producing canonical averages from time averages of static physical properties along continuous, deterministic trajectories. This method can also be used for the calculation of dynamic propierties that are inaccessible from MC simulations. However, the interpretation of time-dejjendent properties calculated from this and similar methods is not straightforward, particularly for very small systems. ... 

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. 

Formally, these equations are valid for an arbitrary choice of the weight function, w E), but a failure to sample all configurations that significantly contribute to canonical averages for a temperature in the interval T < T < T2... 

This implies that when the pure liquid solvent is chosen as the reference state, only terms involving canonical averages over the potential energy of interaction between the solute and the solvent (plus the changes in the internal free energies discussed in the previous section) contribute to the free energy of solvation at infinite dilution. At finite concentration, the solute-solute interaction terms have to be considered as well. 

The angular brackets ( ) denote a canonical average. For the case of short-range interactions the contribution from U can be further simplified by using... 

Here, rijm = r, r, + mLb, and the canonical average has not been denoted... 

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