Statistical mechanics of canonical-dissipative systems and applications to swarm dynamics.

We develop the theory of canonical-dissipative systems, based on the assumption that both the conservative and the dissipative elements of the dynamics are determined by invariants of motion. In this case, known solutions for conservative systems can be used for an extension of the dynamics, which also includes elements such as the takeup/dissipation of energy. This way, a rather complex dynamics can be mapped to an analytically tractable model, while still covering important features of nonequilibrium systems. In our paper, this approach is used to derive a rather general swarm model that considers (a) the energetic conditions of swarming, i.e., for active motion, and (b) interactions between the particles based on global couplings. We derive analytical expressions for the nonequilibrium velocity distribution and the mean squared displacement of the swarm. Further, we investigate the influence of different global couplings on the overall behavior of the swarm by means of particle-based computer simulations and compare them with the analytical estimations.

Active Brownian particles with energy depots modeling animal mobility.

In the model of active motion studied here, Brownian particles have the ability to take up energy from the environment to store it in an internal depot and to convert internal energy into kinetic energy. Considering also internal dissipation, we derive a simplified model of active biological motion. For the take-up of energy two different examples are discussed: (i) a spatially homogeneous supply of energy, and (ii) the supply of energy at spatially localized sources (food centers). The motion of the particles is described by a Langevin equation which includes an acceleration term resulting from the conversion of energy. Dependent on the energy sources, we found different forms of periodic motion (limit cycles), i.e. periodic motion between 'nest' and 'food'. An analytic approximation allows the description of the stationary motion and the calculation of critical parameters for the take-up of energy. Finally, we derive an analytic expression for the efficiency ratio of energy conversion, which considers the take-up of energy, compared to (internal and external) dissipation.