Formula, ensemble

It is important to note that this formula gives only an average value of the crystallite size if, for example, we have a polydispersed ensemble of spherical particles with density probability of the diameter sizes N D), the average... 

Upon expressing from the equilibrium condition the complex concentration M12 through the concentrations of monomers, and substituting the expression found into relationship (21) we obtain, invoking the formalism of the Markov chains, final formulas enabling us to calculate instantaneous statistical characteristics of the ensemble of macromolecules with colored units. A subsequent color erasing procedure is carried out in the manner described above. For example, when calculating instantaneous copolymer composition, this procedure corresponds to the summation of the appropriate components of the stationary vector jt of the extended Markov chain ... 

Fig. 3. Functions in the integrand of the partition function formula Eq. (6). The lower solid curve labeled Pq AU/kT) is the probability distribution of solute-solvent interaction energies sampled from the uncoupled ensemble of solvent configurations. The dashed curve is the product of this distribution with the exponential Boltzmann factor, e AJJ/kT r the upper solid curve. See Eqs. (5) and (6).

Here, ( )0 denotes an ensemble average over configurations sampled from the reference state. This is the fundamental FEP formula, which is the basis for all further developments in this chapter. It states that AA can be estimated by sampling only equilibrium configurations of the reference state. 

Using a similar approach we can derive a formula for the statistical average of any mechanical property, J (x. p., j in the target system in terms of statistical averages over conformations representative of the reference ensemble... 

Occasionally alternative expressions of the PDT (9.5) have been proposed [23-25], These alternatives arise from consideration of statistical thermodynamic manipulations associated with a particular ensemble, and the distinguishing features of those alternative formulae are relics of the particular ensemble considered. On the other hand, relics specific to an ensemble are not evident in the PDT formula (9.5). These alternative formulae should give the same result in the thermodynamic limit. 

As an example [24, 25] consider calculation of ((W pAUa))0/ ((v))o inan isothermal-isobaric ensemble, for which the volume V fluctuates. In this formula V is a relic of the ensemble considered. A calculation using the rule of averages (9.14) then leads to [10]... 

When a chemist writes down a molecular formula, it normally represents an ensemble of many molecules. The new challenge is to handle single molecules or defined aggregates of a few molecules. One would then want to visualize these nanoobjects, to move them around in real space, to record physical functions such as light emission and their response to a stimulus, and even to perform chemical reactions on them. 

It is clear from the integration limits in Eq. (A.4) that the sequence of interactions between particles 1 and 2 must precede the interactions of the other types. Moreover, the expression (A.4) corresponds to the ensemble of contributions where the interactions between particles 1 and 3 are arbitrarily ordered with respect to interactions between particles 2 and 4 and it follows from the fact that the group (1,3) ignores the group (2,4) during the interval that extends from the initial instant to the instant rn. One can easily verify that this chronological ordering of the events is already contained in formula (A.3) (see the definition (61) of (2)2>(34))-... 

An empirical formula indicates a set A of n atoms which is contained in a given molecule, or ensemble of molecules (EM). Thus an EM(A)... 

An FIEM(A), the family of isomeric ensembles of molecules of A, is the set of all EM(A), Thus, an FIEM is described simply by a gross, empirical formula an EM is described by a list of molecular species in terms of constitutional formulas. The constitutional chemistry of a set of atoms A can be described by the structure of its FIEM(A). 

In the constitutional model of Ugi, rather than molecules, "ensemble of molecules (EM) are used in which the molecules can be either chemically different or identical. Like molecules, an EM has an empirical formula, which is the sum of the empirical formulae of the constiment molecules and describes the collection A of atoms within the EM under consideration. All the EM s which can be formed from A have the same empirical formula . Therefore, an EM(A) consists of one or more molecules which can be obtained from A using each atom which belongs to A only once. Moreover, a FIEM(A) or a family of isomeric EM, is the collection of all EM(A) and it is determined by the empirical formula . On the other hand, a chemical reaction, or a sequence of chemical reactions, is the conversion of an EM into an isomeric EM, and therefore a FIEM contains all EMs which are chemically interconvertible, as far as stoichiometry is concerned. In summary, a FIEM(A) contains, at least in principle, the whole chemistry of the collection A of atoms and since any collection of atoms may be chosen here, Ugi concludes that a theory of FIEM is, in fact, a theory of all chemistry. 

The JE (Eq. (40)) indicates a way to recover free energy differences by measuring the work along all possible paths that start from an equihbrium state. Its mathematical form reminds one of the partition function in the canonical ensemble used to compute free energies in statistical mechanics. The formulas for the two cases are... 

After keeping on a water bath for x/7 hr, filter through a tared sintered glass crucible. Transfer the ppt (BaCOj) quantitatively into the crucible and wash it with w. Dry the ensemble to const wt at 100° and calc % total carbonates as Na2COj from the formula ... 

If a medium is considered as an ensemble of individual molecules, the average energy losses of a charged particle on electron retardation per unit path length can be calculated according to the formula... 

A and B (and thus of the controlling Hamiltonian ) at a point in w -space, averaged over that space. In the numerator this switch is from S h to A and the average is with respect to the canonical distribution in ensemble B in the denominator the roles of A and B are reversed. This is Bennett s acceptance ratio formula [61]. 

Equation (55) shows that for this strategy to work, one needs the two ensembles to overlap in the sense [somewhat less restrictive than in the case of the Zwanzig formula, Eq. (54)] that the two single-ensemble PDFs are measurable at some common value of Jt, the most obvious candidate being the Ji 0 region intermediate between the values typical of the two ensembles. Equation (56) shows that this requirement is effectively equivalent to the condition that the probabilities of acceptance of a Hamiltonian switch can be measured, in both directions. 

The last formula obtained by using the hybrid-model version refers to the quantity, slightly differing from this number, namely, to the number of reorientation cycles, averaged over the total ensemble. 

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