We study a system of hard rods of finite size in one space dimension, which move by Brownian noise while avoiding overlap. We consider a scaling in which the number of particles tends to infinity while the volume fraction of the rods remains constant; in this limit the empirical measure of the rod positions converges almost surely to a deterministic limit evolution. We prove a large-deviation principle on path space for the empirical measure, by exploiting a one-to-one mapping between the hard-rod system and a system of non-interacting particles on a contracted domain. The large-deviation principle naturally identifies a gradient-flow structure for the limit evolution, with clear interpretations for both the driving functional (an ‘entropy’) and the dissipation, which in this case is the Wasserstein dissipation. This study is inspired by recent developments in the continuum modelling of multiple-species interacting particle systems with finite-size effects; for such systems many different modelling choices appear in the literature, raising the question how one can understand such choices in terms of more microscopic models. The results of this paper give a clear answer to this question, albeit for the simpler one-dimensional hard-rod system. For this specific system this result provides a clear understanding of the value and interpretation of different modelling choices, while giving hints for more general systems.

我们研究了一个空间维度中有限尺寸的硬棒系统，它通过布朗噪声移动，同时避免重叠。我们考虑一个尺度，其中粒子的数量趋于无穷大，而棒的体积分数保持不变；在这个极限中，杆位置的经验测量几乎肯定会收敛到确定性极限演化。我们通过利用硬棒系统和收缩域上的非相互作用粒子系统之间的一对一映射，证明了经验测量的路径空间的大偏差原理。大偏差原理自然地确定了极限演化的梯度流结构，对驱动函数（“熵”）和耗散（在这种情况下是 Wasserstein 耗散）都有明确的解释。这项研究的灵感来自于具有有限尺寸效应的多物种相互作用粒子系统的连续介质建模的最新进展；对于这样的系统，文献中出现了许多不同的建模选择，这就提出了一个问题，即如何根据更微观的模型来理解这些选择。尽管针对更简单的一维硬棒系统，本文的结果对这个问题给出了明确的答案。对于这个特定系统，该结果提供了对不同建模选择的价值和解释的清晰理解，同时为更一般的系统提供了提示。提出了一个问题，人们如何从更微观的模型来理解这种选择。尽管针对更简单的一维硬棒系统，本文的结果对这个问题给出了明确的答案。对于这个特定系统，该结果提供了对不同建模选择的价值和解释的清晰理解，同时为更一般的系统提供了提示。提出了一个问题，人们如何从更微观的模型来理解这种选择。尽管针对更简单的一维硬棒系统，本文的结果对这个问题给出了明确的答案。对于这个特定系统，该结果提供了对不同建模选择的价值和解释的清晰理解，同时为更一般的系统提供了提示。