Active sorting of particles as an illustration of the Gibbs mixing paradox

The Gibbs Mixing Paradox is a conceptual touchstone for understanding mixtures in statistical mechanics. While debates over the theoretical subtleties of particle distinguishability continue to this day, we seek to extend the discussion in another direction by considering devices which can only distinguish particles with limited accuracy. We introduce two illustrative models of sorting devices which are designed to separate a binary mixture, but which sometimes make mistakes. In the first model, discrimination between particle types is passive and sorting is driven, while the second model is based on an active proofreading network, where both discrimination and sorting have a tunable active component. We show that the performance of these devices may be enhanced out of equilibrium, and we further probe how the quality of particle sorting is maintained by trade-offs between the time taken and the energy dissipated. Considering these examples, we demonstrate how increasing the similarity between particles gradually increases the work required to sort them, eliminating the paradox, while preserving the limits imposed by standard equilibrium statistical mechanics.