Saddle-point entropy states of equilibrated self-gravitating systems

In this Letter, we investigate the stability of the statistical equilibrium of spherically symmetric collisionless self-gravitating systems. By calculating the second variation of the entropy, we find that perturbations of the relevant physical quantities should be classified as long- and short-range perturbations, which correspond to the long- and short-range relaxation mechanisms, respectively. We show that the statistical equilibrium states of self-gravitating systems are neither maximum nor minimum, but complex saddle-point entropy states, and hence differ greatly from the case of ideal gas. Violent relaxation should be divided into two phases. The first phase is the entropy-production phase, while the second phase is the entropy-decreasing phase. We speculate that the second-phase violent relaxation may just be the long-wave Landau damping, which would work together with short-range relaxations to keep the system equilibrated around the saddle-point entropy states.Comment: 5 pages, 1 figure, MNRAS Letter, in the pres

Twist-3 contributions to γγ→π+π−,K+K−\gamma\gamma\rightarrow\pi^+\pi^-,K^+K^- processes in perturbative QCD approach

As one of the simplest hadronic processes, $\gamma\gamma\rightarrow M^{+}M^{-}$ ($M=\pi,K$) could be a good testing ground for our understanding of the perturbative and nonperturbative structure of QCD, and will be studied with high precision at BELLE-\RNum{2} in the near future. In this paper, we revisit these processes with twist-3 corrections in the perturbative QCD approach based on the $k_{T}$ factorization theorem, in which transverse degrees of freedom as well as resummation effects are taken into account. The influence of the distribution amplitudes on the cross sections are discussed in detail. Our work shows that not only the transverse momentum effects but also the twist-3 corrections play a significant role in the processes $\gamma\gamma\rightarrow M^{+}M^{-}$ in the intermediate energy region. Especially in the few GeV region, the twist-3 contributions become dominant in the cross sections. And it is noteworthy that both the twist-3 result of the $\pi^{+}\pi^{-}$ cross section and that of the $K^{+}K^{-}$ cross section agree well with the BELLE and ALEPH measurements. For the pion and kaon angular distributions, there still exist discrepancies between our results and the experimental measurements. Possible reasons for these discrepancies are discussed briefly.Comment: 19 pages, 7 figures and 2 tables. Contents improved and more discussions adde