Statistical Topology of the Configuration Spaces of Hard Disks and Spheres
As phenomena that necessarily emerge from the collective behavior of interacting particles, phase transitions continue to be difficult to predict using statistical thermodynamics. A phase is generally defined as a part of a thermodynamic system with uniform physical properties. For example, the liquid and vapor forms of water are distinct phases that can be mechanically separated. One question to be asked is whether the transition between them always involves a discontinuous change in properties though. The short answer to this question is that it depends on the path that the system takes. It is known that there is a first-order transition if the path crosses the liquid-vapor line in phase diagram of water, but there is no transition if the path goes around the critical point. One can then ask whether there is an entirely unambiguous definition of a phase, or only of a phase transition. Further, setting aside the question of whether a phase can be precisely defined, can a phase transition be predicted on the basis of statistical thermodynamics?
A phase transition is usually identified by an observed discontinuity in one of the derivatives of a thermodynamic potential. There have been several proposals concerning the origin of these discontinuities. For example, Landau theory associates first-order phase transitions with spontaneous symmetry breaking as quantified by an appropriately-constructed order parameter. This work instead suggests that phase transitions are often associated with dramatic changes in the configuration space geometry, and that the geometric change is the actual driver of the phase transition. More precisely, a geometric change that brings about a discontinuity in the mixing time required for an initial probability distribution on the configuration space to reach steady-state is conjectured to be related to the onset of a phase transition in the thermodynamic limit.
This conjecture is tested in the configuration spaces of hard disk and hard sphere systems of increasing size. These systems are often used to model simple fluids as they are one of the simplest systems that can be computationally studied and yet are complicated enough to undergo phase transitions. Initially, a framework to geometrically construct the symmetry invariant configuration spaces of hard disks and spheres is developed and applied to systems of a few hard disks and spheres. Particularly, a metric is defined on the base configuration space and various quotient spaces that respects the desired symmetries. This is used to construct explicit triangulations of the configuration spaces as alpha-complexes. The critical configurations are found to be associated with geometric changes to the configuration space that connect previously distant regions and reduce the configuration space diameter. The number of such critical configurations around the packing fraction of the solid-fluid phase transition increases exponentially with the number of disks and spheres, suggesting that the onset of the first-order phase transition in the thermodynamic limit is associated with a discontinuity in the configuration space diameter.
The implication is that a phase transition in the thermodynamic limit is associated with a discontinuous change in the probability distribution on the phase space, and this concept is initially explored for finite non-equilibrium systems. Definitions are proposed for all of the extrinsic variables of the fundamental thermodynamic relation that are consistent with existing results in the equilibrium thermodynamic limit. The probability density function on the phase space is interpreted as a subjective uncertainty about the microstate, and the Gibbs entropy formula is modified to allow for entropy creation without introducing additional physics or modifying the phase space dynamics. Resolutions are proposed to the mixing paradox, Gibbs' paradox, Loschmidt's paradox, and Maxwell's demon thought experiment. Finally, the extrinsic variables of the fundamental thermodynamic relation are evaluated as functions of time and space for a diffusing ideal gas, and the initial and final values are shown to coincide with the expected equilibrium values when interpreted in a classical context.