  Employing profits data of Japanese firms in 2003--2005, we kinematically
exhibit the static log-normal distribution in the middle scale region. In the
derivation, a Non-Gibrat's law under the detailed balance is adopted together
with following two approximations. Firstly, the probability density function of
profits growth rate is described as a tent-shaped exponential function.
Secondly, the value of the origin of the growth rate distribution divided into
bins is constant. The derivation is confirmed in the database consistently.
  This static procedure is applied to a quasi-static system. We dynamically
describe a quasi-static log-normal distribution in the middle scale region. In
the derivation, a Non-Gibrat's law under the detailed quasi-balance is adopted
together with two approximations confirmed in the static system. The resultant
distribution is power-law with varying Pareto index in the large scale region
and the quasi-static log-normal distribution in the middle scale region. In the
distribution, not only the change of Pareto index but also the change of the
variance of the log-normal distribution depends on the parameter of the
detailed quasi-balance. As a result, Pareto index and the variance of the
log-normal distribution are related to each other.
