Adiabatic elimination procedure

The major aim of the present volume is to use the same rigorous and generally valid theoretical structure to explain a variety of physical phenomena in a self-consistent way. A basically important part of this theory is the adiabatic elimination procedure (AEP) that can be derived from Eq. (2.18) of Chapter I. 

Assume that 0 is a slow relaxation variable compared with u. Using the adiabatic elimination procedure (AEP) describe in Chapter II, we obtain for the probability distribution of the variable 8, a 8 t) the following, generally valid, equation of motion ... 

In the presence of an orientation potential, one should also consider the effective frequencies corresponding to the harmonic approximation of this potential. The adiabatic elimination procedure (AEP), as developed in Chapter II, should allow us to take into account the inertial corrections to the standard adiabatic elimination, thereby making it possible to determine the influence of inertia on EPR spectra within the context of a contracted description that retains only the variable Q. This allows us to arrive at an equation of the form... 

Analytic solutions of Eq. (1.16) cannot be obtained, and thus one must make some approximations. In many physical systems, y and X are very large (y 00, X oo) therefore both v and f can be considered as fast relaxing variables and eliminated by an adiabatic elimination procedure (AEP). The AEP of Chapter II allows us to derive from Eq. (1.16) the equation of motion of a(x t) regarded as being the contracted distribution ... 

The problem of eliminating fast variables plays a decisive role in the subject of synergetics. The concept of order parameters and slowing in nonequilibrium systems has been proved by Haken to be a powerful tool for analyzing instabilities far from thermal equilibrium. Furthermore, Haken used this concept to elucidate important analogies among many-component systems from completely difierent disciplines. For these reasons the Haken school has also been motivated to devdop a systematic procedure of adiabatic elimination. ... 

It is of interest to note that V R) is obtained by a procedure equivalent to adiabatic elimination of all coordinates other than R of the system of interest. 

In the literature a different technique has been widely used to construct effective Hamiltonians, based on the partitioning technique combined with an approximation procedure known as adiabatic elimination for the time-dependent Schrodinger equation (see Ref. 39, p. 1165). In this section we show that the effective Hamiltonian constructed by adiabatic elimination can be recovered from the above construction by choosing the reference of the energy appropriately. Moreover, our stationary formulation allows us to estimate the order of the neglected terms and to improve the approximation to higher orders in a systematic way. 

This system in many cases can be simplified further. For example, if we have a broad spectral line excitation with a not very intense laser radiation, we have a situation for an open transition when 7 Ti, H. In practical cases this condition is often fulfilled at excitation with cw lasers operating in a multimode regime. If the homogeneous width of spectral transition usually is in the range of 10 MHz, then the laser radiation spectral width broader than 100 MHz usually can be considered as a broad line excitation. In this case we can use a procedure known as adiabatic elimination. It means that we are assuming that optical coherence pi2 decays much faster than the populations of the levels puJ = 1,2. Then we can find stationary solution for off-diagonal elements for the density matrix and afterwards find a rate equations for populations in this limit. For the two level system we will have... 

The procedure of adiabatic elimination in this case leads to rather complicated cejuations for diagonal elements of the density matrix, which in a general case can not be reduced to simple rate equations for populations. To examine the essential characteristics of the eciuations for populations we shall make a number of simplifying assumptions. First, let us consider an exact resonance... 

Following the concept of adiabatic elimination, the velocity gradients dzVx, dzVy can be expressed with the components of n from (1) and substituted into (12) [14, 9]. The procedure is straightforward, but expressions are complicated, so the equations obtained can be solved only numerically. 

In the last few years several modifications to the traditional mixed acid nitration procedure have been reported. An adiabatic nitration process was developed for the production of nitrobenzene (9). This method eliminated the need to remove the heat of reaction by excessive cooling. The excess heat can be used in the sulfuric acid reconcentration step. An additional advantage of this method is the reduction in reaction times to 0.5—7.5 minutes. 

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