Search for: All records

AbstractHard disks systems are often considered as prototypes for simple fluids. In a statistical mechanics context, the hard disk configuration space is generally quotiented by the action of various symmetry groups. The changes in the topological and geometric properties of the configuration spaces effected by such quotient maps are studied for small numbers of disks on a square and hexagonal torus. A metric is defined on the configuration space and the various quotient spaces that respects the desired symmetries. This is used to construct explicit triangulations of the configuration spaces as$$\alpha$$ $\alpha$-complexes. Critical points of the hard disk potential on a configuration space are associated with changes in the topology of the accessible part of the configuration space as a function of disk radius, are conjectured to be related to the configurational entropy of glassy systems, and could reveal the origins of phase transitions in other systems. The number of critical points and their topological and geometric properties are found to depend on the symmetries by which the configuration space is quotiented. Graphic abstract 

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 epsilon-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 epsilon-mixing time coincides with the solid-fluid phase transition in the thermodynamic limit. 

Abstract 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. 

Recently, machine learning potentials have been advanced as candidates to combine the high-accuracy of electronic structure methods with the speed of classical interatomic potentials. A crucial component of a machine learning potential is the description of local atomic environments by some set of descriptors. These should ideally be invariant to the symmetries of the physical system, twice-differentiable with respect to atomic positions (including when an atom leaves the environment), and complete to allow the atomic environment to be reconstructed up to symmetry. The stronger condition of optimal completeness requires that the condition for completeness be satisfied with the minimum possible number of descriptors. Evidence is provided that an updated version of the recently proposed Spherical Bessel (SB) descriptors satisfies the first two properties and a necessary condition for optimal completeness. The Smooth Overlap of Atomic Position (SOAP) descriptors and the Zernike descriptors are natural counterparts of the SB descriptors and are included for comparison. The standard construction of the SOAP descriptors is shown to not satisfy the condition for optimal completeness and, moreover, is found to be an order of magnitude slower to compute than that of the SB descriptors.