菲波纳契猫映射的弛豫和遍历性特征

Relaxation and Ergodicity of the Fibonacci Cat Map

研究一个准周期猫映射的弛豫及遍历性特征,其中系统的一个参数随时间的变化由菲波那契序列确定。研究表明,在弛豫阶段,系统的粗粒熵随时间线性增长,增长率为系统的正李亚普诺夫指数。该结果与周期猫映射的弛豫特征一致。但当系统达到平衡态以后,其量子粗粒熵的时间平均值在半经典极限下正比于普朗克常数的平方,和在周期猫映射中观测到正比于普朗克常数的结果截然不同。文章进一步讨论这一结果的普遍性及鲁棒性。

We study relaxation and ergodicity properties of a quasiperiodic cat map where the system parameter is coded into a Fibonacci sequence. We find that during the relaxation stage, coarse--grained entropy increases linearly in time with the rate given by the positive Lyapunov exponent in both the classical and quantum cases, which is the same as in the periodic cat maps. However, in the equilibrium stage, the time averaged quantum coarse--grained entropy is proportional to the square of the Planck constant, in clear contrast to the linear dependence on the Planck constant as observed in the corresponding periodic cat map. The generality and robustness of this novel result is also discussed.