Perturbation Calculus

To find the time required until the flow-equilibrium in the transport zone is achieved, the eigenvalue problem of the system of Eqs. (14) with v = 1 must be solved. This was done in Ref. 4) the results show that this time is essentially longer than the time needed for conformational changes in the macromolecules transferred between sol and gel (e.g. = 10 ps for the P = 1082-mer at 15 °C). 

The phenomenological concept described above allows to find the partition function Q(P) = (cg/cs)flow of the flow-equilibrium by means of a perturbation calculus applied to Eq. (3 b) the reversible partition function K(P) = cjcs in Eq. (3 b) is replaced by Q(P) Q(P) is set equal to K(P) multiplied by an exponential factor containing the free enthalpy of deformation of the coils transported from the sol into the gel through the gel front, where a strong and steep velocity gradient of the column liquid deforms the coil chain with this a new non-linear integrated transport equation 

2 This was the idea behind concept (14b) in Ref.4). The corresponding formulation of Eq. (15i) of Ref.4), however, was unhappily chosen Eq. (15f)ofthis paper should have been used. If the kinetics of separation were explicitly introduced into the transport Equation of PDC (instead of the implicit concept (I7a-c) of the flow-equilibrium), an integrodifferential equation more complicated than (41 a-b) would be obtained, which could hardly be solved analytically 

It can be shown that the phenomenological function a(P T) is closely related to the retardation time of the P-mer in the gel 

Beyond the dynamic region of the PDC-column (strictly on the theta-point), AGdef in Eqs. (17c) tends to zero, giving the limits Q(P) - K(P), 5Q(P)/K(P) - 0, a(P) - 1, k - kg, and xflow(P) - 0 for all degrees of polymerization P Eq. (19) is then identical with Eq. (5 a) in agreement with Fig. 8, and Eq. (23) gives the chain part in Eq. (13). Only a slow resolution of the PDC column in observed in this reversible-thermodynamical region it vanishes at the theta point, as was shown above. 

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I now consider statement 3 How should an extension of dynamics be understood In the MPC theory the problem does not exist For the intrinsically stochastic systems there is no need for modifying the laws of dynamics. As for the LPS theory, one notes the presence of two essentially new concepts. The introduction of non-Hilbert functional spaces only concerns the definition of the states of the dynamical system, and not at all the law governing their evolution. It is an important precision introduced in statistical mechanics. The extension of dynamics thus only appears in the operation of regularization of the resonances. This step is also the one that is most difficult to justify rigorously it is related to the (practical) necessity to use perturbation calculus (see Appendix). 

It is well known that a flow-equilibrium must be treated by the methods of irreversible thermodynamics. In the case of the PDC-column, principally three flows have to be considered within the transport zone (1) the mass flow of the transported P-mer from the sol into the gel (2) the mass flow of this P-mer from the gel into the sol and (3) the flow of free energy from the column liquid into the gel layer required for the maintenance of the flow-equilibrium. If these flows and the corresponding potentials could be expressed analytically by means of molecular parameters, the flow-equilibrium 18) could be calculated by the usual methods 19). However, such a direct way would doubtless be very cumbersome because the system is very complicated (cf. above). These difficulties can be avoided in a purely phenomenological theory, based on perturbation calculus applied to the integrated transport Eq. (3 b) of the PDC-column in a reversible-thermodynamic equilibrium. 

The concentration c depends on the shape of the concentration profile in the transport zone of Fig. 13. If the perturbation calculus (see Sect. 3.2.2) is applied to the reversible-thermodynamic equilibrium in the zone, and if further the spreading of the zone remains small, then c = vcs with v 1 is approximately valid. 

Now, within the limits of the perturbation calculus (js/cs < 1 and jjcg < 1), the equations of system (15b) can be combined with the definitions of the excess concentrations, giving... 

The elution and migration effects are also found in PDC, but not the compression effect, since no precipitant and temperature gradients are applied. Unlike BWF, no simple transport Equation is applied in PDC because the replacement of the linear Eq. (3b) by the non-linear (17c) would lead to an integrodifferential equation similar to (41 a-b), but more complicated, if some explicit formulation were used instead of the implicit one, based on a flow-equilibrium and on a perturbation calculus, applied to an integrated transport Equation... 

Statistical Perturbation Calculus.—Statistical Mean Value. Let Q denote some physical quantity (electric polarization, magnetic polarization, exs,t%y), in general a function of the continuously varying canonic variables F (generalized momeita, generalized co-orclassical statistical mechanics, the mean value of Q is defined as ... 

The expansion of equation (100) is characteristic for classical statistical perturbation calculus. 

The two structure formulae, (48 3), (48 4), are at variance with chemical usage. The perturbation calculus of Heitler and London for this case shows, however, that the actual stationary state of the group CNH corresponds to a mixture of the three pure valence states, (48 2), (48 3), (48 4), in this sense that the quantum-mechanical wave function of the binding electrons is a linear superposition of the wave functions associated with each of the above pictures. The justification of the formula (48 2) lies in the fact that in the wave function associated with a stable molecule the part which it contributes predominates over the parts arising from the other idiotic structure formulae. 

Craubner claims - that the perturbation calculus which he has devised is the most successful means of dealing with such phase phenomena thus far. Multiphase Separation. Separation of polymer solutions into more than two liquid phases which coexist in equililnium is known and was predicted by Tompa on the basis of Flory-Huggins theory (see next section). It is one of the success stories in solution thermodynamics that a 3-phase region, so small as to be difficult to find experimentally by chance, was predicted for a polyethylene/ diphenylether system and subsequently confirmed experimentally. ... 

One of the interesting issues is the problem of variationality of the perturbation calculus built upon the variational Hamiltonian. This is, however, a sophisticated problem for advanced readers and will not be discussed here, it should be added, in summary, that the manner of... 

According to the perturbation calculus, the wave function corrected in the first order can be written as... 

Additionally, one needs to remember that for a powerful tool such as pertnrbation theory, there is no obstacle to applying the multi-determinant reference function as the unperturbed function in perturbation calculus. Thus, similar to the SCF-HF and MP2 approaches, CASPT2 would be the second-order perturbation theory complete active space method - the perturbationaUy corrected CASSCF. 

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