Emergent behaviors of thermodynamic Cucker–Smale ensemble with a unit-speed constraint

We propose the thermodynamic Cucker–Smale system with unit-speed constraints (TCSUS) and study its emergent dynamics rigorous proofs. For this, three dissipative differential inequalities are derived in terms of $ L^\infty $-diameters position–velocity–temperature to verify mono-cluster flocking TCSUS model under suitable frameworks using two approaches. Motivated by first approach, we provide a sufficient framework for bi-cluster model. Unlike behavior TCSUS, decay rate velocity–temperature each cluster group is not exponential dependent on corresponding communication weight, respectively.