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Finally, in the 1970s [[E.G.D. Cohen]] and J. R. Dorfman proved that a systematic (power series) extension of the Boltzmann equation to high densities is mathematically impossible. Consequently, [[non-equilibrium statistical mechanics|nonequilibrium statistical mechanics]] for dense gases and liquids focuses on the [[Green–Kubo relations]], the [[fluctuation theorem]], and other approaches instead.

Finally, in the 1970s [[E.G.D. Cohen]] and J. R. Dorfman proved that a systematic (power series) extension of the Boltzmann equation to high densities is mathematically impossible. Consequently, [[non-equilibrium statistical mechanics|nonequilibrium statistical mechanics]] for dense gases and liquids focuses on the [[Green–Kubo relations]], the [[fluctuation theorem]], and other approaches instead.

最后，在20世纪70年代科恩 E.G.D. Cohen和多夫曼 J. R. Dorfman证明了玻尔兹曼方程在高密度上的系统(幂级数)推广在数学上是不可能的。因此，稠密气体和液体的非平衡统计力学侧重于格林-久保亮五关系、涨落定理和其他方法。

最后，在20世纪70年代科恩 E.G.D. Cohen和多夫曼 J. R. Dorfman证明了玻尔兹曼方程在高密度上的系统(幂级数)推广在数学上是不可能的。因此，稠密气体和液体的'''非平衡统计力学 [[non-equilibrium statistical mechanics|nonequilibrium statistical mechanics]]'''侧重于'''格林-久保亮五关系  [[Green–Kubo relations]]'''、'''涨落定理 [[fluctuation theorem|Fluctuation theorem]]''' 和其他方法。

In particular, it was Boltzmann's attempt to reduce it to a [[stochastic]] collision function, or law of probability following from the random collisions of mechanical particles. Following Maxwell,<ref>Maxwell, J. (1871). Theory of heat. London: Longmans, Green & Co.</ref> Boltzmann modeled gas molecules as colliding billiard balls in a box, noting that with each collision nonequilibrium velocity distributions (groups of molecules moving at the same speed and in the same direction) would become increasingly disordered leading to a final state of macroscopic uniformity and maximum microscopic disorder or the state of maximum entropy (where the macroscopic uniformity corresponds to the obliteration of all field potentials or gradients).<ref>Boltzmann, L. (1974). The second law of thermodynamics. Populare Schriften, Essay 3, address to a formal meeting of the Imperial Academy of Science, 29 May 1886, reprinted in Ludwig Boltzmann, Theoretical physics and philosophical problem, S. G. Brush (Trans.). Boston: Reidel. (Original work published 1886)</ref> The second law, he argued, was thus simply the result of the fact that in a world of mechanically colliding particles disordered states are the most probable. Because there are so many more possible disordered states than ordered ones, a system will almost always be found either in the state of maximum disorder – the macrostate with the greatest number of accessible microstates such as a gas in a box at equilibrium – or moving towards it. A dynamically ordered state, one with molecules moving "at the same speed and in the same direction", Boltzmann concluded, is thus "the most improbable case conceivable...an infinitely improbable configuration of energy."<ref>Boltzmann, L. (1974). The second law of thermodynamics. p. 20</ref>

In particular, it was Boltzmann's attempt to reduce it to a [[stochastic]] collision function, or law of probability following from the random collisions of mechanical particles. Following Maxwell,<ref>Maxwell, J. (1871). Theory of heat. London: Longmans, Green & Co.</ref> Boltzmann modeled gas molecules as colliding billiard balls in a box, noting that with each collision nonequilibrium velocity distributions (groups of molecules moving at the same speed and in the same direction) would become increasingly disordered leading to a final state of macroscopic uniformity and maximum microscopic disorder or the state of maximum entropy (where the macroscopic uniformity corresponds to the obliteration of all field potentials or gradients).<ref>Boltzmann, L. (1974). The second law of thermodynamics. Populare Schriften, Essay 3, address to a formal meeting of the Imperial Academy of Science, 29 May 1886, reprinted in Ludwig Boltzmann, Theoretical physics and philosophical problem, S. G. Brush (Trans.). Boston: Reidel. (Original work published 1886)</ref> The second law, he argued, was thus simply the result of the fact that in a world of mechanically colliding particles disordered states are the most probable. Because there are so many more possible disordered states than ordered ones, a system will almost always be found either in the state of maximum disorder – the macrostate with the greatest number of accessible microstates such as a gas in a box at equilibrium – or moving towards it. A dynamically ordered state, one with molecules moving "at the same speed and in the same direction", Boltzmann concluded, is thus "the most improbable case conceivable...an infinitely improbable configuration of energy."<ref>Boltzmann, L. (1974). The second law of thermodynamics. p. 20</ref>

具体来讲，玻尔兹曼试图将其简化为随机碰撞函数，或机械粒子随机碰撞后的概率定律。继麦克斯韦之后，玻尔兹曼把气体分子模拟成一个盒子里碰撞的台球，并指出每一次碰撞非平衡态速度分布(多组分子运动的速度和方向相同)会越来越无序，并导致最终的宏观均匀和微观最无序或最大熵状态(宏观均匀的状态对应所有场势或梯度的消失)。因此，他认为，热力学第二定律描述了这样一个事实结果：在一个机械碰撞粒子的世界里，无序状态是最有可能的。因为可能的无序状态比有序状态要多得多，一个系统几乎总是会处于最无序状态——具有最多可接近的微观状态的宏观状态，如处于平衡状态的盒子里的气体——或向最无序状态移动。因此，玻尔兹曼总结道，一个动态有序的状态，即分子以“相同的速度和相同的方向”运动，是“最不可思议的可能情况…… 一种无限不可能的能量配置。

具体来讲，玻尔兹曼试图将其简化为随机碰撞函数，或机械粒子随机碰撞后的概率定律。继麦克斯韦之后，玻尔兹曼把气体分子模拟成一个盒子里碰撞的台球，并指出每一次碰撞非平衡态速度分布(多组分子运动的速度和方向相同)会越来越无序，并导致最终的宏观均匀和微观最无序或最大熵状态(宏观均匀的状态对应所有场势或梯度的消失)。因此，他认为，热力学第二定律描述了这样一个事实结果：在一个机械碰撞粒子的世界里，无序状态是最有可能的。因为可能的无序状态比有序状态要多得多，一个系统几乎总是会处于最无序状态——具有最多可接近的微观状态的宏观状态，如处于平衡状态的盒子里的气体——或向最无序状态移动。因此，玻尔兹曼总结道，一个动态有序的状态，即分子以“相同的速度和相同的方向”运动，是“最不可思议的可能情况…… 一种无限不可能的能量配置”。

Boltzmann accomplished the feat of showing that the second law of thermodynamics is only a statistical fact. The gradual disordering of energy is analogous to the disordering of an initially ordered [[pack of cards]] under repeated shuffling, and just as the cards will finally return to their original order if shuffled a gigantic number of times, so the entire universe must some-day regain, by pure chance, the state from which it first set out. (This optimistic coda to the idea of the dying universe becomes somewhat muted when one attempts to estimate the timeline which will probably elapse before it spontaneously occurs.)<ref>"[[Collier's Encyclopedia]]", Volume 19 Phyfe to Reni, "Physics", by David Park, p. 15</ref> The tendency for entropy increase seems to cause difficulty to beginners in thermodynamics, but is easy to understand from the standpoint of the theory of probability. Consider two ordinary [[dice]], with both sixes face up. After the dice are shaken, the chance of finding these two sixes face up is small (1 in 36); thus one can say that the random motion (the agitation) of the dice, like the chaotic collisions of molecules because of thermal energy, causes the less probable state to change to one that is more probable. With millions of dice, like the millions of atoms involved in thermodynamic calculations, the probability of their all being sixes becomes so vanishingly small that the system ''must'' move to one of the more probable states.<ref>"Collier's Encyclopedia", Volume 22 Sylt to Uruguay, Thermodynamics, by Leo Peters, p. 275</ref> However, mathematically the odds of all the dice results not being a pair sixes is also as hard as the ones of all of them being sixes{{Citation needed|date=January 2019}}, and since statistically the [[data]] tend to balance, one in every 36 pairs of dice will tend to be a pair of sixes, and the cards -when shuffled- will sometimes present a certain temporary sequence order even if in its whole the deck was disordered.

Boltzmann accomplished the feat of showing that the second law of thermodynamics is only a statistical fact. The gradual disordering of energy is analogous to the disordering of an initially ordered [[pack of cards]] under repeated shuffling, and just as the cards will finally return to their original order if shuffled a gigantic number of times, so the entire universe must some-day regain, by pure chance, the state from which it first set out. (This optimistic coda to the idea of the dying universe becomes somewhat muted when one attempts to estimate the timeline which will probably elapse before it spontaneously occurs.)<ref>"[[Collier's Encyclopedia]]", Volume 19 Phyfe to Reni, "Physics", by David Park, p. 15</ref> The tendency for entropy increase seems to cause difficulty to beginners in thermodynamics, but is easy to understand from the standpoint of the theory of probability. Consider two ordinary [[dice]], with both sixes face up. After the dice are shaken, the chance of finding these two sixes face up is small (1 in 36); thus one can say that the random motion (the agitation) of the dice, like the chaotic collisions of molecules because of thermal energy, causes the less probable state to change to one that is more probable. With millions of dice, like the millions of atoms involved in thermodynamic calculations, the probability of their all being sixes becomes so vanishingly small that the system ''must'' move to one of the more probable states.<ref>"Collier's Encyclopedia", Volume 22 Sylt to Uruguay, Thermodynamics, by Leo Peters, p. 275</ref> However, mathematically the odds of all the dice results not being a pair sixes is also as hard as the ones of all of them being sixes{{Citation needed|date=January 2019}}, and since statistically the [[data]] tend to balance, one in every 36 pairs of dice will tend to be a pair of sixes, and the cards -when shuffled- will sometimes present a certain temporary sequence order even if in its whole the deck was disordered.

玻尔兹曼完成了证明热力学第二定律只是一个统计事实的壮举。能量的逐渐无序化类似于一叠最初有序排列的卡牌重复洗牌过程中变得无序，也正如经历了无数次洗牌而重归初始有序状态的卡牌一样，我们的宇宙最终将在某一时刻回到最初的状态。（当人们试图估计宇宙正在消亡的时间线时，这种乐观的结论就显得有些平淡了，因为时间线很可能在它自然发生之前就已经过去了。）熵增加的趋势似乎对初学热力学的人们造成困难，但从概率论的观点出发就很容易理解。考虑两个普通的骰子，都是正面朝上的。摇骰子后，发现这两个6面都朝上的概率很小(1 / 36)；因此，我们可以说，骰子的随机运动，就像分子由于热能的混乱碰撞，会导致系统从概率较小的状态变为概率较大的状态。数以百万计的骰子，就像热力学计算中数以百万计的原子一样，它们全部为6的概率变得如此之小，以至于系统必须移动到一个更可能的状态。然而，数学上出现所有骰子结果都是6或者都不是6的几率都很小，由于统计数据会趋于平衡，每36对骰子会有一对6出现，而洗过的牌有时会呈现出某种临时的顺序，即使整个牌是无序的。

玻尔兹曼完成了证明热力学第二定律只是一个统计事实的壮举。能量的逐渐无序化类似于一叠最初有序排列的卡牌重复洗牌过程中变得无序，也正如经历了无数次洗牌而重归初始有序状态的卡牌一样，我们的宇宙最终将在某一时刻回到最初的状态。（当人们试图估计宇宙正在消亡的时间线时，这种乐观的结论就显得有些平淡了，因为时间线很可能在它自然发生之前就已经过去了。）熵增加的趋势似乎对初学热力学的人们造成困难，但从概率论的观点出发就很容易理解。考虑两个普通的骰子，都是正面朝上的。摇骰子后，发现这两个6面都朝上的概率很小(1/36)；因此，我们可以说，骰子的随机运动，就像分子由于热能的混乱碰撞，会导致系统从概率较小的状态变为概率较大的状态。数以百万计的骰子，就像热力学计算中数以百万计的原子一样，它们全部为6的概率变得如此之小，以至于系统必须移动到一个更可能的状态。然而，数学上出现所有骰子结果都是6或者都不是6的几率都很小，由于统计数据会趋于平衡，每36对骰子会有一对6出现，而洗过的牌有时会呈现出某种临时的顺序，即使整个牌是无序的。

== 获奖经历与荣誉 ==

== 获奖经历与荣誉 ==