Geometry of phase transitions in systems of infinite degrees of freedom

Abstract Recent investigations have demonstrated that the origin of phase transitions may be explained by particular changes in the geometry of hypersurfaces, specifically Hamiltonian level sets, within phase space. This study offers a geometric explanation for phase transitions in systems of infinite degrees of freedom, demonstrating the versatility of this framework of phase transitions. An alternative microcanonical ensemble is proposed, whose partition function is well defined on infinite dimensional phase spaces, and whose phase transitions are associated with transitions in the canonical ensemble. The derivatives of the associated partition function and its entropy are shown to depend solely on the curvatures and covariant derivatives of curvatures of the potential energy level sets. Consequently, the occurrence of phase transitions in systems of infinite degrees of freedom may be explained by certain geometric behaviors of potential energy level sets, expanding and reinforcing the geometric explanation of the origin of phase transitions.