A theory of self-propelled particles is developed in two dimensions assuming that the particles can be deformed from a circular shape when the propagating velocity is increased. A coupled set of equations in terms of the velocity and a tensor variable to represent the deformation is introduced to show that there is a bifurcation from a straight motion to a circular motion of a single particle. Dynamics of assembly of the particles is studied numerically where there is a global interaction such that the particles tend to cause an orientational order.

We formulate the theory for a self-propelled domain in an excitable reaction-diffusion system in two dimensions where the domain deforms from a circular shape when the propagation velocity is increased. In the singular limit where the width of the domain boundary is infinitesimally thin, we derive a set of equations of motion for the center of gravity and two fundamental deformation modes. The deformed shapes of a steadily propagating domain are obtained. The set of timeevolution equations exhibits a bifurcation from a straight motion to a circular motion by changing the system parameters.

The fluctuation theorems obtained in a stochastic Markovian process are generalized to a non-Markovian system governed by the non-linear generalized Langevin equation with a Gaussian colored noise. We derive the non-Markovian version of the Crooks fluctuation theorem that relates the statistical averages of the two different dynamics characterized by the forward process and the reverse process. In contrast to the similar study by Zamponi et al, ours does not assume a stationary state asymptotically in time so that the present fluctuation theorem can deal explicitly with the dependence of the initial condition and the transient behavior. The Jarzynski equality for the non-equilibrium work relation and the representation of the linear response in the non-equilibrium steady state are also discussed. The conditions for the memory kernel that the fluctuation theorems hold are examined by analyzing a solvable model and are confirmed by a direct derivation of the fluctuation theorems for the cases of an exponential decay and a power law decay of the memory kernel.

We investigate the dynamical coupling between the motion and the deformation of a single selfpropelled domain based on two different model systems in two dimensions. One is represented by the set of ordinary differential equations for the center of gravity and two tensor variables characterizing deformations. The other is an active cell model which has an internal mechanism of motility and is represented by the partial differential equation for deformations. Numerical simulations show a rich variety of dynamics, some of which are common to the two model systems. The origin of the similarity and the difference is discussed.

No abstract