From the Relativistic Mixture of Gases to the Relativistic Cucker–Smale Flocking

We present a relativistic model for a mixture of Euler gases with multiple temperatures. For the proposed relativistic model, we explicitly determine production terms resulting from the interchange of energy–momentum between the constituents via the entropy principle. We use the analogy with the homogeneous solutions of a mixture of gases and the thermomechanical Cucker–Smale (in short TCS) flocking model in a classical setting (Ha and Ruggeri in Arch Ration Mech Anal 223:1397–1425, 2017) to derive a relativistic counterpart of the TCS model. Moreover, we employ the theory of a principal subsystem to derive the relativistic Cucker–Smale (in short CS) model. For the derived relativistic CS model, we provide a sufficient framework leading to the exponential flocking in terms of communication weights and also show that the relativistic CS model reduces to the classical CS model, as the speed of light tends to infinity in any finite-time interval.