Thermodynamics and phase transitions in galaxy clustering

The phenomenon of galaxy clustering was studied from the perspective of the gravitational phase transition, which is somewhat different from a phase transition in materials science. There is evidence that the phase transition describing galaxy clustering in an expanding universe is a first-order phase transition exhibiting a mixed phase. As such, the Clausius-Clapeyron equation is relevant for studying such a system. In this work, we derive a general analog of the Clausius-Clapeyron equation that applies not only toward the coexistence curve in pressure-temperature space, but to a more general parameter space. One key finding is that a cusp exists at the critical point in this mixed phase when viewed in this more general parameter space. We extend this formalism to derive an equation for the curvature of the phase coexistence curve in pressure-temperature space and a more general parameter space. We also verify previous findings of hysteresis in the system via an independent free energy analysis.