The approach to equilibrium in a macroscopic quantum system for a typical nonequilibrium subspace
The approach to equilibrium in a macroscopic quantum system for a typical nonequilibrium subspace
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典型非平衡子空间宏观量子系统的平衡方法
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
H. Tasaki
中科院分区：
文献类型：
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作者：
S. Goldstein;T. Hara;H. Tasaki
研究了抽象环境下宏观量子系统的平衡逼近问题。我们证明了，对于一个典型的“非平衡子空间”的选择，任何初始状态（来自能量壳层）热化，事实上，热化速度非常快，在玻尔兹曼时间$ au__mathrm{B}:=h/(k_mathrm{B}T)$的阶上。这个看似不现实，但在数学上严谨的结论具有重要的物理意义，即在现实中观察到的适度缓慢的衰变在目前的环境中并不典型。
We study the problem of the approach to equilibrium in a macroscopic quantum system in an abstract setting. We prove that, for a typical choice of "nonequilibrium subspace", any initial state (from the energy shell) thermalizes, and in fact does so very quickly, on the order of the Boltzmann time $ au__mathrm{B}:=h/(k_mathrm{B}T)$. This apparently unrealistic, but mathematically rigorous, conclusion has the important physical implication that the moderately slow decay observed in reality is not typical in the present setting.
The fact that macroscopic systems approach thermal equilibrium may seem puzzling, for example, because it may seem to conflict with the time-reversibility of the microscopic dynamics. According the present result, what needs to be explained is, not that macroscopic systems approach equilibrium, but that they do so slowly.
Mathematically our result is based on an interesting property of the maximum eigenvalue of the Hadamard product of a positive semi-definite matrix and a random projection matrix. The recent exact formula by Collins for the integral with respect to the Haar measure of the unitary group plays an essential role in our proof.