Thermodynamics and galactic clustering with a modified gravitational potential

Based on thermodynamics, we study the galactic clustering of an expanding Universe by considering the logarithmic and volume (quantum) corrections to Newton's law along with the repulsive effect of a harmonic force induced by the cosmological constant ($\Lambda$) in the formation of the large scale structure of the Universe. We derive the $N$-body partition function for extended-mass galaxies (galaxies with halos) analytically. For this partition function, we compute the exact equations of states, which exhibit the logarithmic, volume and cosmological constant corrections. In this setting, a modified correlation (clustering) parameter (due to these corrections) emerges naturally from the exact equations of state. We compute a corrected grand canonical distribution function for this system. Furthermore, we obtain a deviation in differential forms of the two-point correlation functions for both the point-mass and extended-mass cases. The consequences of these deviations on the correlation function's power law are also discussed.