Statistical thermodynamics is valuable as a conceptual structure that shapes our thinking about equilibrium thermodynamic states. A cloud of unresolved questions surrounding the foundations of the theory could lead an impartial observer to conclude that statistical thermodynamics is in a state of crisis though. Indeed, the discussion about the microscopic origins of irreversibility has continued in the scientific community for more than a hundred years. This paper considers these questions while beginning to develop a statistical thermodynamics 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.

As phenomena that necessarily emerge from the collective behavior of interacting particles, phase transitions continue to be difficult to predict using statistical thermodynamics. A recent proposal called the topological hypothesis suggests that the existence of a phase transition could perhaps be inferred from changes to the topology of the accessible part of the configuration space. This paper instead suggests that such a topological change is often associated with a dramatic change 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 by evaluating the diffusion diameter and -mixing time of the configuration spaces of hard disk and hard sphere systems of increasing size. Explicit geometries are constructed for the configuration spaces of these systems, and numerical evidence suggests that a discontinuity in the -mixing time coincides with the solid-fluid phase transition in the thermodynamic limit.