Boltzmann Bridges

Jordan Scharnhorst, David Wolpert, Carlo Rovelli
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Abstract

It is often stated that the second law of thermodynamics follows from the condition that at some given time in the past the entropy was lower than it is now. Formally, this condition is the statement that $E[S(t)|S(t_0)]$, the expected entropy of the universe at the current time $t$ conditioned on its value $S(t_0)$ at a time $t_0$ in the past, is an increasing function of $t $. We point out that in general this is incorrect. The epistemic axioms underlying probability theory say that we should condition expectations on all that we know, and on nothing that we do not know. Arguably, we know the value of the universe's entropy at the present time $t$ at least as well as its value at a time in the past, $t_0$. However, as we show here, conditioning expected entropy on its value at two times rather than one radically changes its dynamics, resulting in a unexpected, very rich structure. For example, the expectation value conditioned on two times can have a maximum at an intermediate time between $t_0$ and $t$, i.e., in our past. Moreover, it can have a negative rather than positive time derivative at the present. In such "Boltzmann bridge" situations, the second law would not hold at the present time. We illustrate and investigate these phenomena for a random walk model and an idealized gas model, and briefly discuss the role of Boltzmann bridges in our universe.
波尔兹曼桥
人们常说,热力学第二定律源于这样一个条件:在过去的某个特定时间,熵比现在低。从形式上看,这个条件是这样的:$E[S(t)|S(t_0)]$,即当前时间$t$下宇宙的预期熵,以过去某一时间$t_0$下的值$S(t_0)$为条件,是$t$的递增函数。概率论所依据的认识论公理指出,我们应该把我们知道的一切作为期望的条件,而不应该把我们不知道的任何事物作为期望的条件。可以说,我们对当前时间 $t$ 的宇宙熵值的了解至少与对过去时间 $t_0$ 的宇宙熵值的了解一样多。然而,正如我们在这里所展示的,以两个时间而不是一个时间的熵值作为预期熵的条件,会从根本上改变其动力学,从而产生意想不到的、非常丰富的结构。例如,以两个时间为条件的期望值可以在 $t_0$ 和 $t$ 之间的中间时间(即我们的过去)达到最大值。此外,它在现在的时间导数可能是负数而不是正数。在这种 "玻尔兹曼桥 "情况下,第二定律在当前时间并不成立。我们对随机行走模型和理想化气体模型的这些现象进行了说明和研究,并简要讨论了玻尔兹曼桥在我们宇宙中的作用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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