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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 processes in perturbative QCD approach
As one of the simplest hadronic processes, () 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 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 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 cross
section and that of the 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
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