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

Abstract

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.

Emergence of coherent motion in aggregates of motile coupled maps

“…In the context of animal societies, sharp transitions of collective behavior have been reported to occur at specific values of the population density (see for instance [18,30,1,13,15,29,28,22]). As it has been already mentioned in the introduction, models belonging to the same class as Vicsek's exhibit an order-disordered phase transition at a critical value of the particle density.…”

“…From a purely deterministic perspective, such systems can be represented as high dimensional dynamical systems, which can be either discrete or continuous in their time evolution. This approach has been mainly spearheaded by Smale and collaborators [26,27,18,21]. On the other hand, providing understanding of the connection between macroscopic collective features and microscopic scale interactions in multi-particle systems is well within the principal aims of statistical physics.…”

“…The range of applications of such studies is indeed extensive [1,2,3,4,5,6,7]. As a matter of fact, coherent patterns of collective motion found in distinct families of biological species such as fish schools, flocks of birds, swarms of insects and even colonies of bacteria [6,5,8,1,4,9,10,11] have also been detected in granular matter systems, self-propelled particles with inelastic collisions and active Brownian particles in autonomous-motor groups [12,13,14,15,16,17,18,19,20,21,22]. Early in the study of such collectives, modeling and simulation have been recognized as playing a crucial role in gaining insight of the mechanisms underlying such an emergence of global features from a set of simple rules [23,24,25].…”

Nonlinear Brownian motion - mean square displacement

“…Let us replace γ 0 by a function γ(v) and the constant D 0 also by a function D(v). Then we speak about "nonlinear Brownian motion" [1,2], while in other connection we speak about "active Brownian movement" [3][4][5][6]. The corresponding Langevin equation of motion reads:…”

Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space

“…In this context, the concept of canonical-dissipative systems has been developed to address self-propagating systems, self-excited systems, and in particular limitcycle oscillators that are supplied by energy sources [1][2][3][4][5]. Several studies have demonstrated that the canonical-dissipative approach can yield helpful insights into this kind of open systems [6][7][8][9][10][11][12][13][14][15][16][17][18].…”