Stochastic Dynamics Of Active Brownian Particles

The stochastic dynamics of active Brownian particles is introduced by illuminating the biological active motion. And then, the theory of stochastic dynamics for quasi Hamiltonian systems is applied to the study of dynamics of active Brownian particles. A series of stationary analytical solutions are obtained for the first time for the dynamical system of Brownian particles with Rayleigh friction model and parabolic potential under Gaussian white noise exciatation. In the case of far from equilibrium, the stationary analytical solution in 4-d phase space is two diffused limit cycles which is verified by comparing with results from digital simulation. As for the other cases and system structures, such as elliptical and quartic potential, parametric noise exciatation, some satisfactory analytical results are also obtained by studying the particles system in detail. For studying the motion of a swarm of particles, some coupling mechanism is introduced between the particles. Our digital simulation verifies other scholar 's judgement that the mass center is at rest in stationary case. Based on this judgement, those analytical solutions obtained for single particle are applied to swarms. Particularly, the analytical solution of total energy is obtained and verified by digital simulation. Furthermore, the first-passage time of particles system with other structure is studied. These obtained results agree with those from digital simulation. The theory of stochastic dynamics for quasi Hamiltonian systems is the best theory for the investigation of the first-passage problem of Brownian particles with multi-well potential. Finally, a stochastic averaging method based on generalized harmonic functions has been introduced and applied to the study of dynamics of Brownian particles.