The formation of a droplet of kaon condensate is connected with the production of strangeness. In this case the nucleation should involve weak interaction processes. For kaon condensation in the npep matter, whose equilibrium properties were described in § 7.4, the strange kaon-phase has to nucleate in the non-strange medium. In this respect kaon condensation is different from pion condensation or (B-Q) phase transition from baryon matter to quark matter, where the first step is a strong-interaction process. For example, the first step in the B-Q phase transition is the formation of a droplet of two flavor, non-strange (ud) quark matter in the non-strange baryon matter.
The surface tension at the interface between the normal and condensed phases was calculated by Christiansen et al. (2000) using the non-uniform rela-tivistic mean-field model. In the approximation in which only linear terms in the curvature of the droplet surface are kept, the surface contribution to the thermodynamic potential of a spherical droplet is a = as + 2ac/R (see § 3.4.2), where as is the surface tension, ac is the curvature coefficient, and R is the droplet radius. Christiansen et al. (2000) obtain as = 30 MeV for a small admixture of kaon-condensed droplets in a nucleon matter. Unfortunately, their curvature coefficient is negative, probably because of several oversimplified assumptions made by the authors. Let us remind that the calculation of ac is ambiguous even for a basic problem of nuclear surface; ac depends on the assumed position of the phase interface (see, e.g., Douchin et al., 2000, and references therein).
The nucleation of kaon condensate in a non-strange npe matter was studied by Norsen (2002). He considered only the thermal regime and used the nucleation theory of Langer (1969). He analyzed local density fluctuations at a fixed volume and a fixed nucleon number of a matter element. In addition, he assumed that the matter element is electrically neutral and contains one density and charge fluctuation (in the Wigner-Seitz cell approximation). The central quantities of this study are the free energy excess F* implied by the fluctuation, and the number of nucleons Acrit in the critical droplet. It was supposed that the critical drop has enough time to acquire strangeness corresponding to the kaon-condensed state. Following Christiansen et al. (2000), Norsen (2002) used as = (20-30) MeV fm"2 and neglected the curvature term in the surface contribution to F* .
It turns out that weak interaction processes producing strangeness via e + N — ve + K" + N (where an additional nucleon is needed for momentum conservation) and n — p + K" are too slow to create a critical droplet of kaon condensate from a density fluctuation during the fluctuation lifetime. Norsen (2002) suggested that strangeness can be produced at a reasonable rate from thermal kaon-antikaon (K"K+) pairs, but this mechanism can operate only at extremely high temperatures typical for protoneutron stars (§ 1.4.2). For kBT < 10 MeV it could operate only at a very high overcompression, where Acrit and therefore F* are sufficiently small. However, as Norsen (2002) argues, a protoneutron star cools so rapidly that kaon condensate has actually no time to nucleate.
The conclusion of the above discussion is unfavorable for kaon condensation in neutron stars. Thus, a neutron star, whose central pressure at birth is too low for the nucleation, but exceeds P0 later owing to accretion in a binary system, may remain in a metastable non-kaon condensed state forever. However, as soon as ¡ie > _ (where _ is the minimum energy of a single zero momentum kaon in dense matter), a spontaneous formation of kaons is possible. This occurs at pc = pcrit, Fig. 7.6, and triggers the kaon condensation. Kinetics of kaon condensation at P > Pcrit was studied in detail by Muto et al. (1997), Muto et al. (2000a), and Muto et al. (2000b). The timescale for relaxation of dense matter to the equilibrium kaon-condensed state at p ~ 4p0 and T = 1011 K is only 10-4 s, but grows to ~ 10 s at 1010 K (Muto et al., 2000b).
The "minimal model" of the npe/ matter may be an oversimplification. The matter at densities relevant for kaon condensation can contain some fraction of hyperons. As hyperons are strongly interacting carriers of strangeness, their presence seems favorable for kaon condensation. A hyperon density fluctuation with sufficiently large strangeness could be a seed to nucleate a droplet of kaon-condensed matter without any strangeness production via weak interactions, for instance, via n + A ^ p + n + KSuch a strong interaction process is allowed provided the in-medium energy of K- satisfies uK- < /e. However, the presence of hyperons in thermodynamic equilibrium reduces the electron number density and chemical potential. Consequently, it pushes up the equilibrium threshold pressure for kaon condensation or even blocks kaon condensation at any density. Thus, the presence of hyperons alleviates the problem of strangeness condensation but actually hinders kaon condensation in neutron star cores (see Ramos et al., 2001, for a more detailed discussion).
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