On the Stochastic Flocking Dynamics of the Random Discrete Thermodynamic Cucker–Smale Model

We study the emergent dynamics of the discrete thermodynamic Cucker-Smale (in short, DTCS) model with randomly switching network topologies. The thermodynamic Cucker-Smale (TCS) model corresponds to a generalized CS model with extra internal observables called “temperature”. For isothermal case in which all temperatures are the same constant, it exactly coincides with the CS model for flocking. Motivated by the work of discrete CS model in [12], we use a change of variables and reformulate the nonlinear coupling part of the DTCS model to rewrite the transformed system into a matrix form, and then we use an ergodicity coefficient to derive the exponential decay of state diameter functionals. At random switching instants, we choose one of network topologies randomly from the finite universe of network topologies whose union graph contains a spanning tree. Under some sufficient framework formulated in terms of system parameters and coupling weight functions, we show that the DTCS model exhibits a stochastic flocking with a probability one.