哈密顿力学的时间对称破缺
Time-Symmetry Breaking in Hamiltonian Mechanics
摘要：
哈密顿轨道是严格时间可逆的。任何满足哈密顿运动方程的哈密顿坐标{q}的时间序列在“倒放”时也同样满足它们，相应的动量改变符号：{+p} --> {-p}。在这里，我们采用Levesque和Verlet的精确位可逆运动算法，以确保轨迹可逆性是准确的，具有相同的前向和后向坐标集。然而，相关的瞬时李雅普诺夫不稳定性，或“敏感依赖于初始条件”的“混沌”（或“李雅普诺夫不稳定”）位可逆的坐标轨迹仍然可以表现出指数增长的时间对称性破坏不可逆性。令人惊讶的是，正和负指数，以及向前和向后的李雅普诺夫谱，通常是不密切相关的，所以给四个不同的拓扑措施的“本地”混沌。我们已经证明了这种对称性破缺的流体冲击波，自由膨胀，和混乱的分子碰撞。在这里，我们举例说明并讨论了三个物理力学系统的这种时间对称性破缺，[1]一个最小的（但仍然是混沌的）具有四维相空间的单体“细胞模型”; [2]相对较小的碰撞微晶，可以获得整个李雅普诺夫谱; [3]两个较大的400粒子球的近连续非弹性碰撞。在最后两个教学问题中，两个相互碰撞的物体结合在一起。最容易发生李雅普诺夫不稳定性的粒子在两个时间方向上有很大的不同。因此，这种基于李雅普诺夫的对称性破缺是一个有趣的时间之箭。
英文摘要：
Hamiltonian trajectories are strictly time-reversible. Any time series of Hamiltonian coordinates {q} satisfying Hamilton's motion equations will likewise satisfy them when played "backwards", with the corresponding momenta changing signs : {+p} --> {-p}. Here we adopt Levesque and Verlet's precisely bit-reversible motion algorithm to ensure that the trajectory reversibility is exact, with the forward and backward sets of coordinates identical. Nevertheless, the associated instantaneous Lyapunov instability, or "sensitive dependence on initial conditions" of "chaotic" (or "Lyapunov unstable") bit-reversible coordinate trajectories can still exhibit an exponentially growing time-symmetry-breaking irreversibility. Surprisingly, the positive and negative exponents, as well as the forward and backward Lyapunov spectra, are usually not closely related, and so give four differing topological measures of "local" chaos. We have demonstrated this symmetry breaking for fluid shockwaves, for free expansions, and for chaotic molecular collisions. Here we illustrate and discuss this time-symmetry breaking for three statistical-mechanical systems, [1] a minimal (but still chaotic) one-body "cell model" with a four-dimensional phase space; [2] relatively small colliding crystallites, for which the whole Lyapunov spectrum is accessible; [3] a near-continuum inelastic collision of two larger 400-particle balls. In the last two of these pedagogical problems the two colliding bodies coalesce. The particles most prone to Lyapunov instability are dramatically different in the two time directions. Thus this Lyapunov-based symmetry breaking furnishes an interesting Arrow of Time.
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