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The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.

The Poincaré recurrence theorem considers a theoretical microscopic description of an isolated physical system. This may be considered as a model of a thermodynamic system after a thermodynamic operation has removed an internal wall. The system will, after a sufficiently long time, return to a microscopically defined state very close to the initial one. The Poincaré recurrence time is the length of time elapsed until the return. It is exceedingly long, likely longer than the life of the universe, and depends sensitively on the geometry of the wall that was removed by the thermodynamic operation. The recurrence theorem may be perceived as apparently contradicting the second law of thermodynamics. More obviously, however, it is simply a microscopic model of thermodynamic equilibrium in an isolated system formed by removal of a wall between two systems. For a typical thermodynamical system, the recurrence time is so large (many many times longer than the lifetime of the universe) that, for all practical purposes, one cannot observe the recurrence. One might wish, nevertheless, to imagine that one could wait for the Poincaré recurrence, and then re-insert the wall that was removed by the thermodynamic operation. It is then evident that the appearance of irreversibility is due to the utter unpredictability of the Poincaré recurrence given only that the initial state was one of thermodynamic equilibrium, as is the case in macroscopic thermodynamics. Even if one could wait for it, one has no practical possibility of picking the right instant at which to re-insert the wall. The Poincaré recurrence theorem provides a solution to Loschmidt's paradox. If an isolated thermodynamic system could be monitored over increasingly many multiples of the average Poincaré recurrence time, the thermodynamic behavior of the system would become invariant under time reversal.

庞加莱始态复现定理认为这是一个孤立的物理系统的理论微观描述。这可以被认为是一个热力学系统的模型后，一个热力学操作已经移除了一个内墙。在足够长的时间之后，系统将返回到一个非常接近初始状态的显微定义状态。庞加莱回归时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且敏感地依赖于被热力学操作拆除的墙体的几何形状。递归定理可能被认为是明显与热力学第二定律相矛盾的。然而，更明显的是，它只不过是一个微观模型，热力学平衡在一个孤立的系统中，通过移除两个系统之间的一道墙而形成。对于一个典型的热力学系统，重现时间是如此之大(比宇宙的生命周期长许多倍) ，以至于在所有的实际目的中，人们都不能观察到重现。然而，人们可能希望想象一下，可以等待庞加莱复发，然后重新插入被热力学操作移除的壁。然后很明显，不可逆性的出现是由于庞加莱循环的完全不可预测性，只要初始状态是热力学平衡，就像宏观热力学的情况一样。即使可以等待，也没有实际可能性选择合适的时间重新插入墙壁。庞加莱始态复现定理为 Loschmidt 悖论提供了一个解决方案。如果一个孤立的热力学系统可以被监测到在越来越多的平均 poincar 递归时间的倍数上，系统的热力学行为将在时间反转下变得不变。

庞加莱始态复现定理（又叫递归定理）考虑了孤立物理系统的理论微观描述。经过热力学作用去除隔离内壁之后，便可以认为其是热力学系统的模型。在足够长的时间后，系统将返回到非常接近初始状态的微观定义状态。庞加莱复现时间是指回归前经过的时间长度。它极其漫长，可能比宇宙的寿命还要长，并且敏感地依赖于被热力学作用拆除的墙体的几何形状。复现定理可能被认为是明显与热力学第二定律相矛盾的。但是，更显而易见的是，它只是通过移除两个系统之间的内壁而形成的隔离系统中热力学平衡的微观模型。对于典型的热力学系统而言，重复时间是如此之长（比宇宙的寿命长很多倍），以至于在所有的实际目的中，人们都无法观察到这种重复。尽管如此，还是有人会想像一个机会可以等待庞加莱的复现出现，然后重新插入被热力学作用去除的内壁。然后很明显，不可逆性的出现是由于庞加莱递归的完全不可预测性，因为仅仅给出了初始状态遵守热力学平衡的条件之一，就像宏观热力学的情况一样。即使可以等待，也没有实际可操作性来选择合适的时间重新插入内壁。庞加莱始态复现定理为 Loschmidt 悖论提供了一个解决方案。如果一个孤立的热力学系统能以多倍于平均庞加莱复现时间的长度下进行监控，则该系统的热力学行为在时间反转下将变得恒定。

 

詹姆斯·克拉克·麦克斯韦

詹姆斯·克拉克·麦克斯韦

===Maxwell's demon===

===Maxwell's demon===