Time ordered product

The stochastic evolution operator Y(t 110) might be written as a time-ordered product as in (3.1) but there is no need for an explicit form. From (4.1) one finds, on the one hand,... 

This example, within 1/(1) electrodynamics can then be seen in the light of non-Abelian electrodynamics. This may simply be seen by the replacement where f is a structure constant that obeys [f, tb] = 2eabcf. The time ordered product is written as... 

The counterterm is computed by performing a perturbation expansion of Green s function, or propagator for the free electron. The entire process is represented by r " , which in general is determined by a time ordered product of fields... 

Figure 6. Blob diagram for a time ordered product of fields.

Due to the fact that the Lagrangian incorporates the creation and destruction of field quanta, not even the time-development of a single particle is a simple matter. The time development can be expressed in terms of the electron (fermion) and photon propagators, which are defined as the vacuum expectation values of the time-ordered product of field operators. For the fermions one has... 

Here /L( ) is the Liouvillian corresponding to the fully time-dependent dynamics of Eqs. [91], and the subscript R denotes a right-time-ordered product, with later times appearing to the right. ... 

If the Hamilton operator depends on time in a harmonic fashion, the time dependence can be eliminated by transformation into a rotating reference frame in analogy to the transformation of the Bloch equations. A representation in the rotating frame is also called interaction representation in quantum mechanics. If the time dependence is more general, the Schrodinger equation is solved for small enough time increments, during which H is approximately constant. For each of the n time increments At a solution of the form (2.2.46) applies. The complete evolution operator is the time ordered product of the incremental evolution operators. This operator is written in short hand as... 

This Green function is analytic in the complex energy plane except for the bound-state poles at En, with branch points at = 1 and cuts along the real axis for E > 1. Bound states occur only at energies E > 0. The firee-photon propagator appears as a time-ordered product of firee-photon field operators (in Feynman gauge)... 

The electric conductance of a molecular junction is calculated by recasting the Keldysh formalism in Liouville space. Dyson equations for non-equilibrium many-body Green functions (NEGF) are derived directly in real (physical) time. The various NEGFs appear naturally in the theory as time-ordered products of superoperators, while the Keldysh forward/backward time loop is avoided. 

However, for Fermi superoperators life is more complicated. The anticommutator corresponding to only the left or the right Fermi superoperators are numbers but that for the left and right superoperators, in general, is not a number. Thus, the Fermi superoperators are not Gaussian. However, since the left and right superoperators always commute, the following Wick s theorem [49] can be applied to the time-ordered product... 

For free fields the vacuum e3q>ectation value of the time ordered product of two bosonic fields is... 

Here indicates an infinitesimal positive time-shift of t, i.e. = lime o(t + e ). The KS and the full many-body Green s function are defined by the ground-state expectation values of the time-ordered product of the corresponding field operators, and... 

There exists a hierarchy of Green s functions (GF). These functions are defined as the //-electron ground-state expectation value of a time-ordered product of annihilation and creation operators. The simplest and probably most relevant member of the hierarchy is the one-particle GF... 

Let the time-ordering operator 16 S, when applied to the product... 

By introducing the time-ordering chronological operator8 P which rewrites a product of time labeled operators as the product of these same operators ordered according to their time label, the one with the earliest time label standing farthest to the right, we can rewrite the expansion (10-151) in the form... 

To determine the rate constant one uses the same methods as mentioned under the first-order reaction, i.e. a plot of log[A] versus time. The product k[B]o assumes the role of a pseudo-first order rate constant, from which the true second-order rate constant is easily obtained. 

In 1958, Forrester started studies on an effect which is nowadays often referred to as the bullwhip effect. The bullwhip effect describes the amplification of temporal variations of the orders in a supply chain the more one moves away from the retail customer. Forrester showed that small changes in consumer demand result in large variations of orders placed upstream [4, 5]. It is interesting that this effect occurs even if the demand of final products is almost stable. For his studies, he assumed that some time delay exists between placing an order and the realization of this order (production). Furthermore, he assumed that each part of the supply chain plans its production and places its orders upstream taking into account only the information about the demands of its direct customer. 

Besides all these quant-related constraints there are numerous others such as down times for production equipment when maintenance or rebuilding activities have to be carried out or fixed production orders. Finally, there are constraints such as varying shift models, and so forth. 

Zeolites are formed by crystallization at temperatures between 80 and 200 °C from aqueous alkaline solutions of silica and alumina gels in a process referred to as hydrothermal synthesis.15,19 A considerable amount is known about the mechanism of the crystallization process, however, no rational procedure, similar to organic synthetic procedures, to make a specifically designed zeolite topology is available. The products obtained are sensitive functions of the reaction conditions (composition of gel, reaction time, order of mixing, gel aging, etc.) and are kinetically controlled. Nevertheless, reproducible procedures have been devised to make bulk quantities of zeolites. Procedures for post-synthetic modifications have also been described.20 22... 

The theory of statistical mechanics provides the formalism to obtain observables as ensemble averages from the microscopic configurations generated by such a simulation. From both the MC and MD trajectories, ensemble averages can be formed as simple averages of the properties over the set of configurations. From the time-ordered properties of the MD trajectory, additional dynamic information can be calculated via the time correlation function formalism. An autocorrelation function Caa( = (a(r) a(t + r)) is the ensemble average of the product of some function a at time r and at a later time t + r. 

Figure 1. Pseudo-first-order kinetic plot of - n(A -Ao)/ A -Ag) versus time for production of p-nitrophenolate from reaction of sodium perborate at various concentrations with EPMP at 27.5 oc, pH = 8.

---