论热性的起源
On the origin of thermality
摘要：
众所周知，一个弱耦合到总微正则系综中的大能量池的小系统会发现自己处于（近似）热态，最近已经表明，如果总态是一个能量范围很窄的随机纯态，那么小系统仍将以很高的概率近似热态（关于“Haar测量”）。我们问什么样的条件下，需要类似于这些“传统”和“现代”的热力学结果仍然保持系统和能量浴的大小相当。在第一部分中，我们证明了，对于给定的系统和能量池态密度s_S（e）和s_B（e），热性一般不成立，正如我们所说明的，当两者都以能量的幂的形式增加时，但在某些近似意义上，在传统和现代框架中，当两者都以exp（be）或exp（qe^2）的形式增加时，热性确实成立，我们计算了这些情况下的系统熵。在他们的“现代”版本中，我们的结果依赖于新的量，我们引入并称之为S和B的“modapprox”密度算子，我们声称，这将以很高的概率，当系统的总状态加上能量浴是一个随机的纯态，能量在一个狭窄的范围内时，给出系统和能量浴的约化密度算子的近似。在第2部分中，我们澄清了这些modapprox密度算子的含义，并为我们的主张提供论据。量子黑洞是非小热系统的主要例子。在这里和两个配套文件中，我们认为，目前的弦理论推导的黑洞熵和热性质是不完整的，在信息损失的问题上，不确定的。然而，我们认为，这些不足之处是弥补与修改后的方案，依赖于我们的方法和结果的现代链在这里，是基于我们以前的“物质引力纠缠假说”。
英文摘要：
It is well-known that a small system weakly coupled to a large energy bath in a total microcanonical ensemble will find itself in an (approximately) thermal state and, recently, it has been shown that, if the total state is, instead, a random pure state with energy in a narrow range, then the small system will still be approximately thermal with a high probability (wrt `Haar measure'). We ask what conditions are required for something resembling these 'traditional' and 'modern' thermality results to still hold when system and energy bath are of comparable size. In Part 1, we show that, for given system and energy-bath densities of states, s_S(e) and s_B(e), thermality does not hold in general, as we illustrate when both increase as powers of energy, but that it does hold in certain approximate senses, in both traditional and modern frameworks, when both grow as exp(be) or as exp(qe^2) and we calculate the system entropy in these cases. In their 'modern' version, our results rely on new quantities, which we introduce and call the S and B 'modapprox' density operators, which, we claim, will, with high probability, give a close approximation to the reduced density operator for the system and energy bath when the total state of system plus energy bath is a random pure state with energy in a narrow range. In Part 2 we clarify the meaning of these modapprox density operators and give arguments for our claim. The prime examples of non-small thermal systems are quantum black holes. Here and in two companion papers, we argue that current string-theoretic derivations of black hole entropy and thermal properties are incomplete and, on the question of information loss, inconclusive. However, we argue that these deficiencies are remedied with a modified scenario which relies on the modern strand of our methods and results here and is based on our previous 'matter-gravity entanglement hypothesis'.
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