Reminder of Jaynes’ views on two key issues of Gibbsian statistical mechanics

Abstract Gibbs’ approach to statistical mechanics incorporates key foundational concepts and is at the core of modern statistical thermodynamics. However, two aspects of Gibbs’ approach still cause problems for students and teachers alike. The first is the conservation of phase space density in a thermally isolated system (the Liouville theorem) which means that the entropy of the system, according to Gibbs’ definition, is necessarily constant. This then leads to measures such as coarse-graining to enable an approach to statistical equilibrium. The second aspect is the Gibbs’ paradox, concerning the extensivity of the entropy of ideal gas mixtures in classical statistical mechanics. However, Edwin Jaynes addressed both of these issues in a logically consistent manner many decades ago, and provided interesting and insightful alternative ways to view them. Surprisingly Jaynes’ treatment of both these issues seems to be largely forgotten. In the present work Jaynes’ treatment is recalled and linked back to Gibbs’ original work. Several important issues merit being highlighted once again for a new generation of teachers and students, including the details of Gibbs’ statistical mechanical definition of entropy, the exact definition of entropy in classical thermodynamics, as well as Jaynes’ insights on the distinction between physical and thermodynamical systems, the relationship to the anthropomorphic nature of entropy, and the information known about a system.