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第68行：
第68行： − 如果在混合过程的早期观察到大容器，则可能发现它只是部分混合。我们可以合理地得出这样的结论: 在没有外界干预的情况下，液体之所以达到这种状态，是因为在过去，当分离程度较大时，液体的有序程度较高，而在未来，液体的无序程度或混合程度较高。+ − − 第76行：
第74行： − 现在想象一下这个实验重复进行，这次只有几个分子，也许十个，在一个非常小的容器里。人们可以很容易地想象，通过观察分子的随机碰撞，它可能发生---- 仅仅是偶然---- 分子整齐地分离开来，所有的染料分子在一边，所有的水分子在另一边。这种情况可以不时地发生，这可以从涨落定理中得出结论; 因此分子彼此分离并不是不可能的。然而，对于大量的分子来说，它是如此的不可能，以至于人们不得不等待，平均而言，要比宇宙的年龄长很多倍的时间才能发生。因此，如果一部电影像上面描述的那样展示了大量的分子自我分离，这看起来是不现实的，人们会倾向于说这部电影是以相反的方式播放的。把玻尔兹曼第二定律看作是无序定律。+ −
无编辑摘要
If the large container is observed early on in the mixing process, it might be found only partially mixed. It would be reasonable to conclude that, without outside intervention, the liquid reached this state because it was more ordered in the past, when there was greater separation, and will be more disordered, or mixed, in the future.
If the large container is observed early on in the mixing process, it might be found only partially mixed. It would be reasonable to conclude that, without outside intervention, the liquid reached this state because it was more ordered in the past, when there was greater separation, and will be more disordered, or mixed, in the future.
如果观察大容器的早期混合过程，则可能发现其内部只是部分混合。由此可以合理地得出这样的结论: 在没有外界干预的情况下，液体达到这种状态是因为在过去分离程度较大时，液体的有序程度较高，而在未来，液体的无序程度或混合程度更高。
Now imagine that the experiment is repeated, this time with only a few molecules, perhaps ten, in a very small container. One can easily imagine that by watching the random jostling of the molecules it might occur — by chance alone — that the molecules became neatly segregated, with all dye molecules on one side and all water molecules on the other. That this can be expected to occur from time to time can be concluded from the [[fluctuation theorem]]; thus it is not impossible for the molecules to segregate themselves. However, for a large numbers of molecules it is so unlikely that one would have to wait, on average, many times longer than the age of the universe for it to occur. Thus a movie that showed a large number of molecules segregating themselves as described above would appear unrealistic and one would be inclined to say that the movie was being played in reverse. See Boltzmann's [[Ludwig Boltzmann#The Second Law as a law of disorder|Second Law as a law of disorder]].
Now imagine that the experiment is repeated, this time with only a few molecules, perhaps ten, in a very small container. One can easily imagine that by watching the random jostling of the molecules it might occur — by chance alone — that the molecules became neatly segregated, with all dye molecules on one side and all water molecules on the other. That this can be expected to occur from time to time can be concluded from the [[fluctuation theorem]]; thus it is not impossible for the molecules to segregate themselves. However, for a large numbers of molecules it is so unlikely that one would have to wait, on average, many times longer than the age of the universe for it to occur. Thus a movie that showed a large number of molecules segregating themselves as described above would appear unrealistic and one would be inclined to say that the movie was being played in reverse. See Boltzmann's [[Ludwig Boltzmann#The Second Law as a law of disorder|Second Law as a law of disorder]].
Now imagine that the experiment is repeated, this time with only a few molecules, perhaps ten, in a very small container. One can easily imagine that by watching the random jostling of the molecules it might occur — by chance alone — that the molecules became neatly segregated, with all dye molecules on one side and all water molecules on the other. That this can be expected to occur from time to time can be concluded from the fluctuation theorem; thus it is not impossible for the molecules to segregate themselves. However, for a large numbers of molecules it is so unlikely that one would have to wait, on average, many times longer than the age of the universe for it to occur. Thus a movie that showed a large number of molecules segregating themselves as described above would appear unrealistic and one would be inclined to say that the movie was being played in reverse. See Boltzmann's Second Law as a law of disorder.
Now imagine that the experiment is repeated, this time with only a few molecules, perhaps ten, in a very small container. One can easily imagine that by watching the random jostling of the molecules it might occur — by chance alone — that the molecules became neatly segregated, with all dye molecules on one side and all water molecules on the other. That this can be expected to occur from time to time can be concluded from the fluctuation theorem; thus it is not impossible for the molecules to segregate themselves. However, for a large numbers of molecules it is so unlikely that one would have to wait, on average, many times longer than the age of the universe for it to occur. Thus a movie that showed a large number of molecules segregating themselves as described above would appear unrealistic and one would be inclined to say that the movie was being played in reverse. See Boltzmann's Second Law as a law of disorder.
现在想象重复这个实验，这一次在一个非常小的容器里只有几个分子，也许是十个分子。容易想象，通过观察分子的随机碰撞，可能会发现这种情况——分子整齐地分开，所有的染料分子在一边，所有的水分子在另一边，而这仅仅是偶然的。根据涨落定理可以得出这种情况时有发生的结论；因此，分子是可能自行分离的。然而，对于大量的分子来说，这种现象不太可能发生，以至于人们平均需要等待比宇宙年龄长很多倍的时间才能发生这种情况。因此，上述大量分子相互分离不切实际，人们可能会倾向于说这部电影是在倒放。把波尔兹曼第二定律看作无序定律。
==Mathematics of the arrow==
==Mathematics of the arrow==