基于Langevin问题探讨广义M-J集的物理意义

Study on the physical meaning for generalized Mandelbrot-Julia sets based on the Langevin problem

基于对一典型Langevin问题———在双势井和变化的磁场中并受一恒冲量不断作用的运动带电粒子的动力学分析 ,利用频闪采样法 ,给出了描述粒子速度变化规律的复差分方程 .选取适当的磁场强度和时间间隔 (采样周期 ) ,将这一差分方程简化为用来构造广义M_J(Mandelbrot_Julia)集的复映射 ,并基于粒子的动力学特征探讨了广义M_J集的物理意义 .结果发现 :1)广义M_J集的分形结构特征可形象地反映出粒子速度的变化规律 ;2 )选取的时间间隔有、无意义 ,决定了广义M_J集的分形结构是否具有连续性 ;3)广义M_J集的演化 ,即粒子速度的变化规律依赖于相角主值范围的不同选取 ;4 )若改变磁场强度和时间间隔的选取 ,如选取一随机波动的磁场 ,则此时广义J集可能会出现内部被填充的结构特征 ,即在速度空间中粒子的不稳定周期轨道的闭包出现“爆炸”

Based on a dynamics research of the typical Langevin problem, i.e., a moving charged particle under the continuous influence of a constant impulse in a double-well potential and a time-dependent magnetic field, using the stroboscopic sampling, we propose complex difference equations which can describe the change rule of particle's velocity. By selecting appropriate magnetic intensity and time intervals (sampling period), we reduce the difference equations to complex mapping which is used to construct the generalized M-J sets. Based on the particle's dynamics characteristic, we discussed the physical meaning of the generalized M-J sets. The authors found that: (1) The fractal structure of the generalized M-J sets may visually reflect the change rule of particle's velocity. (2) Whether the selected time intervals is significative determines whether the fractal structure of the generalized M-J sets has the continuity. (3) The evolution of the generalized M-J sets, i.e., the change rule of particle's velocity, depends on the different choices of the principal range of the phase angle. (4) If we change the choices of the magnetic intensity and time intervals, for example, choose a random fluctuant magnetic field, the generalized J sets may present the interior-filling structure feature, i.e., “explosion' phenomena appear in the closure of the particle's instable periodic orbits in the velocity space.