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First‐Order Phase Transitions in Quantum‐Coulomb Plasmas
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31.Notice that this separation is done without redundant variables and subsidiary conditions; Eqs. (4) hold as identities. We here use the notations
32.At there always remains, of course, a (usually very small) finite probability for the excitation of higher collective levels. We shall indeed take it into account in our calculations [Eq. (20)], but in the region of interest it will give a negligible contribution. Physically, the point is that this probability is much smaller than it would be if the collective levels made a continuum.
33.The destruction of a mode means that no longer obeys a self‐determining equation and one has to go back to the full (exact) equation . The second, “thermal” term, which couples to all particle degrees of freedom, shows that this destruction first occurs at highest k’s.
34.It may be assumed that the situation shown in Fig. 2 has been arrived at as the limiting case of a finite‐width pipeline with a finite‐width particle in it, in such a way that, in the limit, the particle had no “elbowroom” to exercise any motion at all.
35.To simplify the notation, we henceforth write and drop the prime.
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37.Here one has to imagine as expressed in terms of its Fourier components and the integration as carried out.
38.Although (19) is usually justified (see Ref. 36) by the RPA, it is also closely connected with the assumption of a single homogeneous phase. This can be shown by expanding , where Since Z is, after the change of variables to an integral over all values of this would have been, under a change of variables to the function space equivalent to an integration over all possible functions The functions are either of the type say, for which in which case one may neglect third‐ and higher‐order terms in the above series, or of the type for which this is not so. If the are neglected, one easily obtains (19), using Paraseval’s identity. There will be some range of T and n in which the contributions of the functions to Zind will be negligible. Because of we shall call this range the range of existence of a single phase. In this range the use of (19) is justified. In the range of which the contribution of the is not negligible, the use of (19) is not justified. If one nevertheless uses (19) there, the mistake made thereby may show up by giving an with a concave portion. It has not been proved that this will always show up this way. However, it seems reasonable to conjecture that whenever a concave part of has shown up, it is because the were not negligible.
39.In the RPA, one can in principle define functions such that . This is because the are linear in the and contain phase factors which the are (in the RPA) uncorrelated to the
40.To avoid confusion, we denote the average density in this calculation by Note again the connection (11) between s and and the obvious notation
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