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Employing profits data of Japanese firms in 2003--2005, we kinematicallyexhibit the static log-normal distribution in the middle scale region. In thederivation, a Non-Gibrat's law under the detailed balance is adopted togetherwith following two approximations. Firstly, the probability density function ofprofits growth rate is described as a tent-shaped exponential function.Secondly, the value of the origin of the growth rate distribution divided intobins is constant. The derivation is confirmed in the database consistently.  This static procedure is applied to a quasi-static system. We dynamicallydescribe a quasi-static log-normal distribution in the middle scale region. Inthe derivation, a Non-Gibrat's law under the detailed quasi-balance is adoptedtogether with two approximations confirmed in the static system. The resultantdistribution is power-law with varying Pareto index in the large scale regionand the quasi-static log-normal distribution in the middle scale region. In thedistribution, not only the change of Pareto index but also the change of thevariance of the log-normal distribution depends on the parameter of thedetailed quasi-balance. As a result, Pareto index and the variance of thelog-normal distribution are related to each other.

The log-normal distribution from Non-Gibrat's law in the middle scale region of profits