Emergent flocking dynamics of the discrete thermodynamic cucker-smale model

We present two sufficient frameworks for the emergent dynamics to the discrete thermodynamic Cucker-Smale (TCS) model. Our proposed frameworks are formulated in terms of the initial data and system parameters. The TCS model was first introduced to incorporate the effect of a temperature field in the dynamics of the Cucker-Smale model, and it has been systematically derived from the hydrodynamic model for gas mixture under the spatial homogeneity assumption. The particle model by Cucker and Smale describes the temporal evolution of mechanical observables such as position and velocity, whereas our TCS model governs the dynamics of position, velocity, and temperature of thermodynamic C-S particles. The TCS model conserves the mass, momentum, and energy, and the total entropy is monotonically increasing so that it is consistent with the principle of thermodynamics.