We study the effect of the running gravitational constant on a system of particles (galaxies) interact via gravitational potential. The gravitational potential is derived based on the entropic modified Newtonian force by logarithmic and power correction terms. We derive the partition function for many-body system and the exact equations of state including the thermodynamic properties. A modified parameter B emerged from the thermodynamics derivation. Moreover, an extension of the study is done to determine the distribution function for point mass and extended mass structure galaxies. In particular, the rotation velocity of some galaxies is studied, and the results confronted the observed data which fit well only for short distance with considering the logarithmic correction effect. This leads us to conclude that the logarithmic correction effect explains the rotation velocity data for short and long distances at least qualitatively.
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