  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.
