Microscopic explanation for black hole phase transitions via Ruppeiner geometry: two competing factors-the temperature and repulsive interaction among BH molecules

Charged dilatonic black hole (BH) has rather rich phase diagrams which may contain zeroth-order, first-order as well as reentrant phase transitions (RPTs) depending on the value of the coupling constant $\alpha$ between the electromagnetic field and the dilaton. We try to give a microscopic explanation for these phase transitions by adopting Ruppeiner's approach. By studying the behaviors of the Ruppeiner invariant $R$ along the co-existing lines, we find that the various phase transitions may be qualitatively well explained as a result of two competing factors: the first one is the low-temperature effect which tends to shrink the BH and the second one is the repulsive interaction between the BH molecules which, on the contrary, tends to expand the BH. In the standard phase transition without RPT, as temperature is lowered, the first kind of factor dominates over the second one, so that large black hole (LBH) tends to shrink and thus transits to small black hole (SBH); While in the RPT, after the LBH-SBH transition, as temperature is further decreased, the strength of the second factor increases quickly and finally becomes strong enough to dominate over the first factor, so that SBH tends to expand to release the high repulsion and thus transits back to LBH. Moreover, by comparing the behavior of $R$ versus the temperature $T$ with fixed pressure to that of ordinary two-dimensional thermodynamical systems but with fixed specific volume, it is interesting to see that SBH behaves like a Fermionic gas system in cases with RPT, while it behaves oppositely to an anyon system in cases without RPT. And in all cases, LBH behaves like a nearly ideal gas system.