Statistic operator matrix

Here pq is the diagonal matrix element of the equilibrium statistical operator of... 

We have so far limited ourselves to a classical description, the natural requirement for which is the condition /, /" —> oo. In order that the description is valid for any angular momentum value, it is necessary to employ the quantum mechanical approach. We presume that the reader is acquainted with the density matrix (or the statistic operator) introduced into quantum mechanics for finding the mean values of the observables averaged over the particle ensemble. Under the conditions and symmetry of excitation considered here one must simply pass from the prob-... 

At this stage we have to define a convenient form of projection operator. We denote this projection operator with the symbol II rather than P to stress the fact that the choice we adopt is inspired to that made by Kenkre [22] to study, as we do here, a Gibbs system. We have to stress, in fact, that in the present case the statistical density matrix under study is... 

This is identical to the projection operator adopted by Zwanzig [16]. The second possibility rests on the adoption of the ordinary statistical density matrix p(f). In this case, to obtain the same result we must use a different projection operator, denoted by the symbol II and defined by... 

Let us consider the kinetics of the excitonic excitation in the retinal using the method of nonequilibrium statistical operator [281]. We denote the statistical operator of the system by p(t) and shall study the evolution of its following matrix elements ... 

In all of these considerations cany the Ml conelations internal to their individual subsystems, A, B, respectively, with no relation to the other subsystem, B, A, respectively. As the operators internal to the statistical trace matrix elements of Gf in Eq.(5) involve the charge density operators of subsystem B,, it is... 

Following von Neumann, the "thermodynamic state" is expressed in terms of a statistical matrix or statistical operator that includes combinations of two kinds of probabilities, those that describe the uncertainties characterizing the knowledge of the... 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

The modified statistical weight matrix denoted by U. -is obtained by zeroing all columns of U,- except the column that indexes the desired state, 17. This operation has the effect of ignoring the statistical weights of all conformations of the chain where bond i is not in the desired state, while keeping intact the statistical weights of aU chain... 

In the next step all the resource limitations are introduced. Starting from an already statistical operation distribution to clock cycles we will calculate a statistical distribution of operations to components. This distribution is expressed by the matrix multiplication given in (2). 

Altematively, in the case of incoherent (e.g. statistical) initial conditions, the density matrix operator P(t) I 1>(0) (v(01 at time t can be obtained as the solution of the Liouville-von Neumann equation ... 

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

Entropy and Equilibrium Ensembles.—If one can form an algebraic function of a linear operator L by means of a series of powers of L, then the eigenvalues of the operator so formed are the same algebraic function of the eigenvalues of L. Thus let us consider the operator IP, i.e., the statistical matrix, whose eigenvalues axe w ... 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

This is where we see the convergence of Statistics and Chemometrics. The cross-product matrix, which appears so often in Chemometric calculations and is so casually used in Chemometrics, thus has a very close and fundamental connection to what is one of the most basic operations of Statistics, much though some Chemometricians try to deny any connection. That relationship is that the sums of squares and cross-products in the (as per the Chemometric development of equation 70-10) cross-product matrix equals the sum of squares of the original data (as per the Statistics of equation 70-20). These relationships are not approximations, and not within statistical variation , but, as we have shown, are mathematically (algebraically) exact quantities. 

Another development in the quantum chaos where finite-temperature effects are important is the Quantum field theory. As it is shown by recent studies on the Quantum Chromodynamics (QCD) Dirac operator level statistics (Bittner et.al., 1999), nearest level spacing distribution of this operator is governed by random matrix theory both in confinement and deconfinement phases. In the presence of in-medium effects... 

The operators Fk(t) defined in Eq.(49) are taken as fluctuations based on the idea that at t=0 the initial values of the bath operators are uncertain. Ensemble averages over initial conditions allow for a definite specification of statistical properties. The statistical average of the stochastic forces Fk(t) is calculated over the solvent effective ensemble by taking the trace of the operator product pmFk (this is equivalent to sum over the diagonal matrix elements of this product), so that = Trace(pmFk) is identically zero (Fjk(t)=Fk(t) in this particular case). The non-zero correlation functions of the fluctuations are solvent statistical averages over products of operator forces,... 

The density matrix method is useful in treating relaxation processes, linear and non-linear laser spectroscopies and non-equilibrium statistical mechanics. In this chapter, the definition of density matrix and the equation of motion (EOM) it follows are introduced. The projection operator technique, which makes the density matrix method a very powerful tool in non-equilibrium statistical mechanics, is presented. 

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