Adiabatic elimination

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. 

BASIC DESCRIPTION OF RULES LEADING TO ADIABATIC ELIMINATION 33... 

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. ... 

BASIC DESCRIPTION OF RULES LEADING TO ADIABATIC ELIMINATION 35 Fig. 1. Its dynamics are described by Newton s deterministic law as follows ... 

BASIC DESCRIPTION OF RULES LEADING TO ADIABATIC ELIMINATION 45 which reduces to Eq. (2.14) when a = u. Recall that... 

BASIC DESCRIPTION OF RltLES LEADING TO ADIABATIC ELIMINATION 77 slowest eigenvalue of the matrix... 

When the different approaches to adiabatic elimination are compared, substantial general agreement is found. The approach of the Barcelona group based on a generalization of the cumulant method, the Chapman-... 

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 ... 

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. 

Assuming that some reaction processes are fast, they can be adiabatically eliminated. Also, most fast reversible reactions can be eliminated by assuming that they are already balanced. Then we need to discuss only the concentration (number) of molecular species, which changes relatively slowly. For example, by assuming that an enzyme is synthesized and decomposed fast, the concentrations can be eliminated to give catalytic reaction network dynamics consisting of the reactions with... 

---