Correction for detailed balance

Here J = j +1 is the total angular momentum of the system which is conserved during collisions. 

The latter relation results from energy conservation and forbids rotational transitions when translational energy is deficient. The back processes with transfer of the rotational energy to translational energy are unrestricted. As a consequence, the lower limit of integration in Eq. (5.18) equals f — ej at j j and otherwise it is equal to 0. It is this very difference that leads to an exact relation between off-diagonal elements of the impact operator 

Being applied for the relaxation of populations (k = 0), this equality expresses the demands of the detailed balance principle. This is simply a generalization of Eq. (4.25), which establishes the well-known relation between rates of excitation and deactivation for the rotational spectrum. It is much more important that equality (5.21) holds not only for k = 0 but also for k = 1 when it deals with relaxation of angular momentum J and the elements should not be attributed any obvious physical sense. The non-triviality of this generalization is emphasized by the fact that it is impossible to extend it to the elements of the four-index 

It follows from the definition of the impact operator and the S-matrices unitarity that f(0) obeys not only relation (4.65) but also Eq. (4.66), instead of Eq. (5.14) of EFA. Consequently we obtain an equilibrium (not equiprobable) distribution of populations. The property (5.9) as well as (5.16) are not confirmed. They are peculiar only to EFA and cannot 

This relaxation proceeds without energy exchange between rotational and translational degrees of freedom and is supposed to be the same in EFA as in exact theory f = With this assumption we obtain a result identical to the ELIOS approximation [190]  

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According to eq. (8.11) the gas atoms move in an average potential (a mean field potential) of the phonons. However, in order to obtain a temperature-dependent description we need to add some corrections to the mean field theory [106]. These corrections have to do with the correction for detailed balance and are well known from gas-phase molecular dynamics [109], These corrections will be described in further detail below. 

The second method can be applied to mixtures as well as pure components. In this method the procedure is to find the final temperature by trial, assuming a final temperature and checking by entropy balance (correct when ASp t, = 0). As reduced conditions are required for reading the tables or charts of generalized thermodynamic properties, the pseudo critical temperature and pressure are used for the mixture. Entropy is computed by the relation. See reference 61 for details. ... 

As an example, Fig. 6.20 below compares the Ai AL = 0001 and 2023 line profiles at 195 K which were computed with and without (solid and dashed curves, respectively) accounting for the vibrational dependences of the interaction potential. The correct profiles (solid curves) are more intense in the blue wing, and less intense in the red wing by up to 25% relative to the approximation (dashed), over the range of frequencies shown. Whereas the dashed profiles satisfy the detailed balance relation, Eq. 6.59, if a> is taken to be the frequency shift relative to the line center, the exact profiles deviate by up to a factor of 2 from that equation over the range of frequencies shown. In a comparison of theory and measurement the different symmetries are quite striking use of the correct symmetry clearly improves the quality of the fits attainable. 

A modified Thiele-Geddes method, programmed for an IBM 370-155, was used to perform the calculations needed to size each required column. Experimental activity coefficient data were used to allow for nonideal liquid phase behavior while energy balances, using estimated enthalpy data, were used to correct for non-constant molal overflow. The Theta Method was used for convergence, and all plate efficiencies were assumed to be 100%. (See Reference 7 for additional calculational details and a program listing.)... 

A valid Monte Carlo scheme must not only obey detailed balance, but must also be ergodic. This means that there is a path in phase space from every state to every other state, via a succession of trial moves. Clearly, if the trial states are chosen in such a manner that certain states can never be reached, then the estimator for a thermodynamic observable can differ severely from the correct expectation value. 

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