Rarefied Particle Motions On Hillslopes - Part 3: Entropy

Theoretical and experimental work (Furbish et al., 2021a, b) indicates that the travel distances of rarefied particle motions on rough hillslope surfaces are described by a generalized Pareto distribution. The form of this distribution varies with the balance between gravitational heating, due to conversion of potential to kinetic energy, and frictional cooling, due to particle-surface collisions; it varies from a bounded form associated with rapid thermal collapse to an exponential form representing isothermal conditions to a heavy-tailed form associated with net heating of particles. The generalized Pareto distribution in this problem is a maximum entropy distribution constrained by a fixed energetic "cost" - the total cumulative energy extracted by collisional friction per unit kinetic energy available during particle motions. That is, among all possible accessible microstates - the many different ways to arrange a great number of particles into distance states where each arrangement satisfies the same fixed total energetic cost - the generalized Pareto distribution represents the most probable arrangement. Because this idea applies equally to the accessible microstates associated with net cooling, isothermal conditions and net heating, the fixed energetic cost provides a unifying interpretation for these distinctive behaviors, including the abrupt transition in the form of the generalized Pareto distribution in crossing isothermal conditions. The analysis therefore represents a novel generalization of an energy-based constraint in using the maximum entropy method to infer non-exponential distributions of particle motions. Moreover, the energetic costs of individual particle motions follow an extreme-value distribution that is heavy-tailed for net cooling and light-tailed for net heating. The relative contribution of different travel distances to the total energetic cost is reflected by the product of the travel distance distribution and the cost of individual particle motions - effectively a frequency-magnitude product.