{"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, particularly within the framework of the grand canonical ensemble.", "specific_scope": "The focus is on the conditions that govern the chemical potential (\u03bc) of bosons as they transition into a Bose-Einstein condensate, especially at low temperatures.", "goal": "Identify the correct fundamental limit on the chemical potential for bosons to achieve Bose-Einstein condensation."}, "summary": "In the context of Bose-Einstein condensation, the chemical potential (\u03bc) of bosons must be less than or equal to the ground state energy of the system to allow for the formation of a condensate. This is crucial because if \u03bc were to exceed the ground state energy, it would lead to an unphysical situation where the number of particles in the ground state would become negative. Therefore, the correct answer is that \u03bc must be less than or equal to the ground state energy, which aligns with the fundamental limit on the chemical potential for bosons in this phase transition."}