翻倒的铅笔--微观涨落的宏观随机性
Toppling pencils -- Macroscopic Randomness from Microscopic Fluctuations
摘要：
我们构建了一个微观模型来研究耦合到多自由度环境中的离散随机性。一个四阶双势阱与有限个$N$谐振子双线性耦合。数值求解时间反演不变的哈密顿运动方程，我们发现，对于$N = 1$，系统表现出一个过渡，随着耦合强度的增加，从可积到混沌运动，以下的KAM方案。将$N$提高到10或更高的值，动力学将过渡到准松弛，接近两个最小值处的稳定平衡。我们证实了这种弛豫的不可逆性的系统的其他特征时间尺度上记录的时间依赖性的自相关性，偏熵，以及作为函数的$N$和其他参数的井之间的跳跃的频率。准备中央系统的不稳定平衡在顶部的障碍和浴在一个随机的初始状态从高斯分布，对称下的空间反射，我们证明了决定是否放松到左或右井确定可重复的剩余不对称的初始位置和动量的浴振荡器。这一结果将渐近态的随机性和自发对称性破缺与正则变换下的熵守恒以及系统的势和初始条件的明显对称性协调起来。
英文摘要：
We construct a microscopic model to study discrete randomness in bistable systems coupled to an environment comprising many degrees of freedom. A quartic double well is bilinearly coupled to a finite number $N$ of harmonic oscillators. Solving the time-reversal invariant Hamiltonian equations of motion numerically, we show that for $N = 1$, the system exhibits a transition with increasing coupling strength from integrable to chaotic motion, following the KAM scenario. Raising $N$ to values of the order of 10 and higher, the dynamics crosses over to a quasi-relaxation, approaching either one of the stable equilibria at the two minima of the potential. We corroborate the irreversibility of this relaxation on other characteristic timescales of the system by recording the time dependences of autocorrelation, partial entropy, and the frequency of jumps between the wells as functions of $N$ and other parameters. Preparing the central system in the unstable equilibrium at the top of the barrier and the bath in a random initial state drawn from a Gaussian distribution, symmetric under spatial reflection, we demonstrate that the decision whether to relax into the left or the right well is determined reproducibly by residual asymmetries in the initial positions and momenta of the bath oscillators. This result reconciles the randomness and spontaneous symmetry breaking of the asymptotic state with the conservation of entropy under canonical transformations and the manifest symmetry of potential and initial condition of the bistable system.
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