Ensemble equivalence

Thus, given an ensemble generated by a particular (non-ideal) simulation, possibly consisting of a great many "snapshots," the key conceptual question is To how many i.i.d. configurations is the ensemble equivalent in statistical quality The answer is the effective sample size [1,9,10] which will quantify the statistical uncertainty in every slow observable of interest — and many "fast" observables also, due to coupling, as described earlier. 

From these results we conclude that the observed 2 Si nmr data are well accounted for by assuming pathway I and postulating essentially complete P2 ordering in the 6R ensemble, equivalent to assuming distributions calculated for the ordered sodalite cages in Figure 1. Table III summarizes the results of this calculation for the compositions R = 13/11, 14/10, 15/9, 16/8, 17/7, and 18/8. The first line (a) for each composition in Table III is the distribution calculated for the "pure" compositions, with an infinitely narrow distribution in %]. ... 

The microcanonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same energy U. In such an ensemble of isolated systems, any allowed quantum state is equally probable. In classical thennodynamics at equilibrium at constant n (or equivalently, N), V, and U, it is the entropy S that is a maximum. For the microcanonical ensemble, the entropy is directly related to the number of allowed quantum states C1(N,V,U) ... 

The statement of the mixing condition is equivalent to the followhig if Q and R are arbitrary regions in. S, and an ensemble is initially distributed imifomily over Q, then the fraction of members of the ensemble with phase points in R at time t will approach a limit as t —> co, and this limit equals the fraction of area of. S occupied by... 

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. 

There are 2 temis in the sum since each site has two configurations with spin eitlier up or down. Since the number of sites N is fmite, the PF is analytic and the critical exponents are classical, unless the themiodynamic limit N oo) is considered. This allows for the possibility of non-classical exponents and ensures that the results for different ensembles are equivalent. The characteristic themiodynamic equation for the variables N, H and T is... 

Here the ijk coordinate system represents the laboratory reference frame the primed coordinate system i j k corresponds to coordinates in the molecular system. The quantities Tj, are the matrices describing the coordinate transfomiation between the molecular and laboratory systems. In this relationship, we have neglected local-field effects and expressed the in a fomi equivalent to simnning the molecular response over all the molecules in a unit surface area (with surface density N. (For simplicity, we have omitted any contribution to not attributable to the dipolar response of the molecules. In many cases, however, it is important to measure and account for the background nonlinear response not arising from the dipolar contributions from the molecules of interest.) In equation B 1.5.44, we allow for a distribution of molecular orientations and have denoted by () the corresponding ensemble average ... 

Another way of looking at it is that Shannon information is a formal equivalent of thermodynamic entroi)y, or the degree of disorder in a physical system. As such it essentially measures how much information is missing about the individual constituents of a system. In contrast, a measure of complexity ought to (1) refer to individual states and not ensembles, and (2) reflect how mnc h is known about a system vice what is not. One approach that satisfies both of these requirements is algorithmic complexity theory. 

Now let us use the set, <0> to form a matrix representation of some operator Q at time hi assuming that Q is not explicitly a function of time. The expectation value of Q in the various states, changes in time only by virtue of the time-dependence of the state vectors used in the representation. However, because this dependence is equivalent to a unitary transformation, the matrix at time t is derived from the matrix at time t0 by such a unitary transformation, and we know that this cannot change the trace of the matrix. Thus if Q — WXR our result entails that it is not possible to change the ensemble average of R, which is just the trace of Q. 

The Bragg scattering of X-rays by a periodic lattice in contrast to a Mossbauer transition is a collective event which is short in time as compared to the typical lattice vibration frequencies. Therefore, the mean-square displacement (x ) in the Debye-Waller factor is obtained from the average over the ensemble, whereas (r4) in the Lamb-Mossbauer factor describes a time average. The results are equivalent. 

Fet us now consider the 3D equivalent of the aforementioned example an ensemble of uncorrelated homogeneous spheres - with polydispersity, meaning that the observed CLD... 

This equation forms the fundamental connection between thermodynamics and statistical mechanics in the canonical ensemble, from which it follows that calculating A is equivalent to estimating the value of Q. In general, evaluating Q is a very difficult undertaking. In both experiments and calculations, however, we are interested in free energy differences, AA, between two systems or states of a system, say 0 and 1, described by the partition functions Qo and (), respectively - the arguments N, V., T have been dropped to simplify the notation ... 

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