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In fact, if all the microscopic physical processes are reversible (see discussion below), then the Second Law of Thermodynamics can be proven for any isolated system of particles with initial conditions in which the particles states are uncorrelated. To do this, one must acknowledge the difference between the measured entropy of a system—which depends only on its macrostate (its volume, temperature etc.)—and its information entropy, which is the amount of information (number of computer bits) needed to describe the exact microstate of the system. The measured entropy is independent of correlations between particles in the system, because they do not affect its macrostate, but the information entropy does depend on them, because correlations lower the randomness of the system and thus lowers the amount of information needed to describe it. Therefore, in the absence of such correlations the two entropies are identical, but otherwise the information entropy is smaller than the measured entropy, and the difference can be used as a measure of the amount of correlations.

In fact, if all the microscopic physical processes are reversible (see discussion below), then the Second Law of Thermodynamics can be proven for any isolated system of particles with initial conditions in which the particles states are uncorrelated. To do this, one must acknowledge the difference between the measured entropy of a system—which depends only on its macrostate (its volume, temperature etc.)—and its information entropy, which is the amount of information (number of computer bits) needed to describe the exact microstate of the system. The measured entropy is independent of correlations between particles in the system, because they do not affect its macrostate, but the information entropy does depend on them, because correlations lower the randomness of the system and thus lowers the amount of information needed to describe it. Therefore, in the absence of such correlations the two entropies are identical, but otherwise the information entropy is smaller than the measured entropy, and the difference can be used as a measure of the amount of correlations.

事实上，如果所有的微观物理过程都是可逆的(见下面的讨论) ，那么对于任何一个孤立的粒子系统，只要其初始条件中的粒子状态是不相关的，那么热力学第二定律就可以被证明。要做到这一点，我们必须认识到一个系统测量熵和信息熵（描述系统精确微状态所需的信息量，即计算机位数）的差别ーー这个差别仅仅取决于它的宏观状态(体积、温度等)。测量到的熵与系统中粒子之间的相关性无关，因为它们不影响系统的宏观状态，但熵确实依赖于它们，因为相关性降低了系统的随机性，从而降低了描述系统所需的信息量。因此，在没有这种相关性的情况下，两个熵是相同的，否则信息熵比测量的熵要小，这种差异可以用来衡量相关性的程度。

事实上，如果所有的微观物理过程都是可逆的(见下面的讨论) ，那么对于任何一个孤立的粒子系统，只要其初始条件中的粒子状态是不相关的，那么热力学第二定律就可以被证明。要做到这一点，我们必须认识到一个系统测量熵和信息熵（描述系统精确微状态所需的信息量，即计算机位数）的差别ーー这个差别仅仅取决于它的宏观状态(体积、温度等)。测量熵与系统中粒子之间的相关性无关，因为它们不影响系统的宏观状态，但熵确实依赖于粒子间的相关性，因为相关性降低了系统的随机性，从而降低了描述系统所需的信息量。因此，在没有这种相关性的情况下，两种熵是相同的，否则信息熵比测量熵小，这一差值可以用来衡量相关性程度。

Now, by Liouville's theorem, time-reversal of all microscopic processes implies that the amount of information needed to describe the exact microstate of an isolated system (its information-theoretic joint entropy) is constant in time. This joint entropy is equal to the marginal entropy (entropy assuming no correlations) plus the entropy of correlation (mutual entropy, or its negative mutual information). If we assume no correlations between the particles initially, then this joint entropy is just the marginal entropy, which is just the initial thermodynamic entropy of the system, divided by Boltzmann's constant. However, if these are indeed the initial conditions (and this is a crucial assumption), then such correlations form with time. In other words, there is a decreasing mutual entropy (or increasing mutual information), and for a time that is not too long—the correlations (mutual information) between particles only increase with time. Therefore, the thermodynamic entropy, which is proportional to the marginal entropy, must also increase with time  (note that "not too long" in this context is relative to the time needed, in a classical version of the system, for it to pass through all its possible microstates—a time that can be roughly estimated as <math>\tau e^S</math>, where <math>\tau</math> is the time between particle collisions and S is the system's entropy. In any practical case this time is huge compared to everything else). Note that the correlation between particles is not a fully objective quantity. One cannot measure the mutual entropy, one can only measure its change, assuming one can measure a microstate. Thermodynamics is restricted to the case where microstates cannot be distinguished, which means that only the marginal entropy, proportional to the thermodynamic entropy, can be measured, and, in a practical sense, always increases.

Now, by Liouville's theorem, time-reversal of all microscopic processes implies that the amount of information needed to describe the exact microstate of an isolated system (its information-theoretic joint entropy) is constant in time. This joint entropy is equal to the marginal entropy (entropy assuming no correlations) plus the entropy of correlation (mutual entropy, or its negative mutual information). If we assume no correlations between the particles initially, then this joint entropy is just the marginal entropy, which is just the initial thermodynamic entropy of the system, divided by Boltzmann's constant. However, if these are indeed the initial conditions (and this is a crucial assumption), then such correlations form with time. In other words, there is a decreasing mutual entropy (or increasing mutual information), and for a time that is not too long—the correlations (mutual information) between particles only increase with time. Therefore, the thermodynamic entropy, which is proportional to the marginal entropy, must also increase with time  (note that "not too long" in this context is relative to the time needed, in a classical version of the system, for it to pass through all its possible microstates—a time that can be roughly estimated as <math>\tau e^S</math>, where <math>\tau</math> is the time between particle collisions and S is the system's entropy. In any practical case this time is huge compared to everything else). Note that the correlation between particles is not a fully objective quantity. One cannot measure the mutual entropy, one can only measure its change, assuming one can measure a microstate. Thermodynamics is restricted to the case where microstates cannot be distinguished, which means that only the marginal entropy, proportional to the thermodynamic entropy, can be measured, and, in a practical sense, always increases.

现在，根据刘维尔定理，所有微观过程的时间反转意味着描述孤立系统精确微观状态所需要的信息量(其信息论联合熵)在时间上是不变的。这个联合熵等于边际熵(假设没有相关性)加上相关熵(互熵，或其负互信息)。如果我们最初假设粒子之间没有相关性，那么这个联合熵就是边际熵，也就是系统的初始熵，除以玻耳兹曼常数。然而，如果这些确实是初始条件(这是一个关键的假设) ，那么这种相关性与时间形成。换句话说，存在一个减少的互熵(或增加的互信息) ，并且在不太长的时间内，粒子之间的关联(互信息)只随时间增加。因此，与边际熵成正比的熵也必须随着时间而增加(注意，在这种情况下，“不要太久”是相对于经典系统版本所需的时间，以便它通过所有可能的微观状态ーー这个时间可以粗略地估计为数学 τ e ^ s / math，其中 math tau / math 是粒子碰撞和 s 之间的时间，s 是系统的熵。在任何实际的情况下，这个时间是巨大的比其他任何事情)。注意，粒子之间的相关性并不是一个完全客观的量。我们不能测量互熵，我们只能测量它的变化，假设我们可以测量一个微观状态。热力学仅限于微观状态无法区分的情况，这意味着只有与熵成正比的边际熵才能被测量，而且，在实际意义上，总是在增加。

现在，根据刘维尔定理，所有微观过程的时间反转意味着描述孤立系统精确微观状态所需要的信息量(信息论中的联合熵)在时间上是恒定的。这个联合熵等于边际熵(假设没有相关性的熵)加上相关熵(互熵，或其负互信息)。如果我们假设最初粒子之间没有相关性，那么联合熵就是边际熵，也就是系统的初始热力学熵除以玻耳兹曼常数。然而，如果这些确实是初始条件(这是一个关键的假设) ，那么这种相关性随时间推移而形成。换句话说，互熵减少(互信息增加) ，并且在不太长的时间内，粒子之间的关联(互信息)只会随时间推移而增加。因此，与边际熵成正比的热力学熵也必须随着时间而增加(注意，在这种情况下，上文中的“不太长”是相对于经典系统所需的时间，以便它经过所有可能的微观状态ーー这段时间可以粗略地估计为      ，其中              是粒子碰撞之间的时间，S是系统的熵。在任何实际的情况下，这段时间比其他任何事情都重要)。注意，粒子之间的相关性并不是一个完全客观的量。不能测量互熵，只能在假设可以测量微观状态的情况下测量它的变化。热力学被限制在微观状态无法区分的情况下，这意味着只有与热力学熵成正比的边际熵才能测量，而且，在实际情况中，热力学熵总是在增加。