Developing a thermodynamic theory of computation is a challenging task at the interface of nonequilibrium thermodynamics and computer science. In particular, this task requires dealing with difficulties such as stochastic halting times, unidirectional (possibly deterministic) transitions, and restricted initial conditions, features common in real-world computers. Here, we present a framework which tackles all such difficulties by extending the martingale theory of nonequilibrium thermodynamics to generic nonstationary Markovian processes, including those with broken detailed balance and/or absolute irreversibility. We derive several universal fluctuation relations and second-law-like inequalities that provide both lower and upper bounds on the intrinsic dissipation (mismatch cost) associated with any periodic process—in particular, the periodic processes underlying all current digital computation. Crucially, these bounds apply even if the process has stochastic stopping times, as it does in many computational machines. We illustrate our results with exhaustive numerical simulations of deterministic finite automata processing bit strings, one of the fundamental models of computation from theoretical computer science. We also provide universal equalities and inequalities for the acceptance probability of words of a given length by a deterministic finite automaton in terms of thermodynamic quantities, and outline connections between computer science and stochastic resetting. Our results, while motivated from the computational context, are applicable far more broadly.

中文翻译：

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具有绝对不可逆性、单向跃迁和随机计算时间的计算热力学

在非平衡热力学和计算机科学的交叉领域，发展热力学计算理论是一项具有挑战性的任务。特别是，这项任务需要处理诸如随机停止时间、单向（可能是确定性）转换和受限初始条件等困难，这些都是现实世界计算机中常见的特征。在这里，我们提出了一个框架，通过将非平衡热力学的鞅理论扩展到一般的非平稳马尔可夫过程，包括那些具有破坏的详细平衡和/或绝对不可逆性的过程，来解决所有这些困难。我们推导出了几种通用涨落关系和类第二定律不等式，它们提供了与任何周期性过程（特别是所有当前数字计算背后的周期性过程）相关的内在耗散（失配成本）的下限和上限。至关重要的是，即使过程具有随机停止时间（就像在许多计算机中那样），这些界限也适用。我们通过确定性有限自动机处理位串的详尽数值模拟来说明我们的结果，这是理论计算机科学的计算基本模型之一。我们还提供了确定性有限自动机在热力学量方面对给定长度单词的接受概率的普遍等式和不等式，并概述了计算机科学和随机重置之间的联系。我们的结果虽然是受计算环境启发，但适用范围更广。