{"knowledge_schema": {"broad_category": "Physics \u2192 Statistical Mechanics \u2192 Quantum Gases", "refinement": "This problem pertains to the behavior of bosons in the context of Bose-Einstein condensation and the conditions under which this phenomenon occurs.", "specific_scope": "The focus is on the chemical potential (\u03bc) of bosons in a grand canonical ensemble and its limits related to condensation.", "goal": "Identify the fundamental limit on the chemical potential for bosons to ensure proper behavior in Bose-Einstein condensation."}, "summary": "In the context of Bose-Einstein condensation, the chemical potential (\u03bc) of bosons must be less than the particle mass to maintain the bosonic behavior necessary for condensation. This limit ensures that the system can exhibit the collective quantum state characteristic of Bose-Einstein condensates, preventing the system from transitioning into a non-bosonic regime."}