A bracket formulation of the Toner-Tu equations

The Toner-Tu equations and their variants form an important set of equations useful to the study of phenomena such as flocking, herding, and spatiotemporal chaos observed in collections of self-propelled organisms. While the Toner-Tu equations share certain properties with the Navier-Stokes equations for equilibrium fluids, they model systems that are active and intrinsically out of thermodynamic equilibrium. One bracket formulation for the Navier-Stokes equations comprises of four building blocks, namely, the total energy and the corresponding skew-symmetric linear operator, and the entropy and the corresponding symmetric linear operator. The significance of bracket formulations lies in their immediately highlighting the time evolution of important physical quantities, serving as a source for the development of new dynamic equations, and their use in establishing structure-preserving numerical schemes. Here, we propose a bracket formulation for the Toner-Tu equations and present a related geometric integrator.