Emergent behaviors of the discrete thermodynamic Cucker-Smale model on complete Riemannian manifolds

We propose an intrinsic discrete-time counterpart of the abstract thermomechanical Cucker-Smale (TCS) model on connected, complete, and smooth Riemannian manifolds and study its emergent dynamics. Our proposed discrete model is expressed in terms of exponential map on the tangent bundle endowed with the Sasaki metric. Compared to projection-based discrete models on the manifold, it is embedding free and enjoys the same structural properties as the corresponding continuous models. For the proposed model, we provide a sufficient framework leading to asymptotic velocity alignment in which all particles' velocity align when they lie in the same tangent plane via the parallel transport along the length-minimizing geodesic. For the unit-d sphere (Sd), we provide explicit representations of the Sasaki metric and the corresponding geodesics on TSd and show that the TCS model exhibits a dichotomy in asymptotic spatial patterns (either energy tends to zero or all particles move along a common geodesic on Sd, which is a great circle). We also provide several numerical examples and compare them with analytical results.