Applications of ensemble theory

The above expressions hold also for the quantum mechanical ensembles. The only difference is that the partition function is in this case defined as the trace of the density matrix for example, in the quantum canonical ensemble, the partition function becomes 

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As a last application of ensemble theory to the quantum mechanical ideal gas, we obtain the equation of state for fermions and bosons. To this end, the most convenient approach is to use the grand canonical partition function and the momentum representation, in which the matrix elements of the density are diagonal. This gives for the canonical partition function... 

The most characteristic type of primary activations are the electronic transitions of molecules which are much faster than other response of the irradiated medium. This enables one to consider separately the physical stage of radiolysis, at the end of which a certain ensemble of excited and ionized molecules is formed in the medium. Each of the activated molecules possesses a particular amount of energy available for subsequent processes. The initial distribution and yields of individual primary activations are dealt with by the theory of primary radiation chemical yield (PRCY). We have studied the application of this theory to the radiolysis of gases in detail during the last years (16, 17, 18, 19, 20). Thus, in the formal expression—see (5), for the yield G(X) = %ngncompetitive reaction ways and remain much more obscure at the present. 

Probability theory is the study of random events. The mathematical study of probability was begun by Pascal and Fermat. The principal applications of probability theory in physical chemistry are in the analysis of experimental errors and in quantum-mechanical theory. Probability theory begins with a large set of people, objects, or numbers, called a population. In the study of random experimental errors, the population is an imaginary set of infinitely many repetitions of a given measurement. In quantum mechanics, thepopulation is an imaginary set of infinitely many replicas of the physical system, called an ensemble. 

Knowing the functions (26) and(27) it is possible by means of the formalism of the theory of Markovian processes [53] to find any statistical characteristic in an ensemble of macromolecules with labeled units. A subsequent label erasing procedure is carried out by integration of the obtained expressions over time of the formation of monomeric units. Examples of the application of this algorithm are reported elsewhere [25]. 

In order to get significant results, the initial data must be formed by a set of clearly non-A -representable second-order matrices, which would generate upon contraction a closely ensemble A -representable 1-RDM. It therefore seemed reasonable to choose as initial data the approximate 2-RDMs built by application of the independent pair model within the framework of the spin-adapted reduced Hamiltonian (SRH) theory [37 5]. This choice is adequate because these matrices, which are positive semidefinite, Hermitian, and antisymmetric with respect to the permutation of two row/column indices, are not A -representable, since the 2-HRDMs derived from them are not positive semidefinite. Moreover, the 1-RDMs derived from these 2-RDMs, although positive semidefinite, are neither ensemble A -representable nor 5-representable. That is, the correction of the N- and 5-representability defects of these sets of matrices (approximated 2-RDM, 2-HRDM, and 1-RDM) is a suitable test for the two purification procedures. Attention has been focused only on correcting the N- and 5-representability of the a S-block of these matrices, since the I-MZ purification procedure deals with a different decomposition of this block. 

Abstract The theoretical basis for the quantum time evolution of path integral centroid variables is described, as weU as the motivation for using these variables to study condensed phase quantum dynamics. The equihbrium centroid distribution is shown to be a well-defined distribution function in the canonical ensemble. A quantum mechanical quasi-density operator (QDO) can then be associated with each value of the distribution so that, upon the application of rigorous quantum mechanics, it can be used to provide an exact definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables can thus be defined and explored. Importantly, this perspective shows that the centroid constraint on the imaginary time paths introduces a non-stationarity in the equihbrium ensemble. This, in turn, can be proven to yield information on the correlations of spontaneous dynamical fluctuations. This exact formalism also leads to a derivation of Centroid Molecular Dynamics, as well as the basis for systematic improvements of that theory. 

S. A. Rice I agree with Prof. Kohler that the use of a density matrix formalism by Wilson and co-workers generalizes the optimal control treatment based on wave functions so that it can be applied to, for example, a thermal ensemble of initial states. All of the applications of that formalism I have seen are based on perturbation theory, which is less general than the optimal control scheme that has been developed by Kosloff, Rice, et al. and by Rabitz et al. Incidentally, the use of perturbation theory is not to be despised. Brumer and Shapiro have shown that the perturbation theory results can be used up to 20% product yield. Moreover, from the point of view of generating an optimal control held, the perturbation theory result can be used as a first guess, for which purpose it is very good. 

While the NVE (microcanonical) ensemble theory is sound and useful, the NVT (canonical) ensemble (which fixes the number of particles, volume, and temperature while allowing the energy to vary) proves more convenient than the NVE for numerous applications. 

The application of this model in physics and chemistry has had a long history. We shall give some examples of the early works. The work of Einstein S2) on the theory of Brownian motion is based on a random walk process. Dirac S3) used the model to discuss the time behavior of a quantum mechanical ensemble under the influence of perturbations this development enables one to discuss the probability of transition of a system from one unperturbed stationary state to another. Pauli 34) [also see Tolman 35)], in his treatment of the quantum mechanical H-theorem, is concerned with the approach to equilibrium of an assembly of quantum states. His equations are identical with those of a general monomolecular... 

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