内容来源：https://www.quantamagazine.org/theory-of-fluids-enters-the-21st-century-20260817/

内容总结：

流体理论迎来21世纪变革：从微观世界重塑物理学基石

20世纪下半叶，一场概念性的浪潮席卷了物理学界。科学家发现，我们的世界源于分子构成的微观世界，而分子又源于更微观的亚原子粒子世界——这一认知革命促使人们对物质理论进行了全面改写。

然而，这场革命并未触及流体力学。描述流体运动的纳维-斯托克斯方程仍保持着19世纪的经典形式。这套方程在预测流体流动和漩涡方面极为成功，但却无法解释构成物质的微观粒子群体的存在。

如今，物理学家们终于构建出了一套全新的流体理论。这是历时20年、从零开始重建流体力学理论的成果。在此过程中，物理学家基于被称为“对称性”的基本属性，提出了一套全新的流体定义方式，并证明了纳维-斯托克斯方程实际上是对称性的必然结果——这解释了该方程为何呈现其特定的形式。

通过理解纳维-斯托克斯方程的起源，研究人员找到了超越该方程的途径，重新定义了流体的本质，并预测出源于微观粒子运动的新行为。

颇具讽刺意味的是，这条通往理解流体作为微观世界产物的道路，最初是由研究宇宙最大尺度的物理学家开辟的。最终是研究黑洞和宇宙整体的学者们的洞见，将流体力学带入了现代纪元。

流体力学的黎明

几个世纪以来，科学家们对流体的基本原理已有了解。18世纪50年代，数学家莱昂哈德·欧拉将牛顿第二运动定律（即F=ma）应用于预测液体运动。欧拉方程对“完美”流体完全适用——这类流体由于没有粘度（一种内在的黏滞性）来减缓流动，电流可以永远流动。

19世纪初，克劳德-路易·纳维和乔治·加布里埃尔·斯托克斯对欧拉方程进行了升级。新的纳维-斯托克斯方程能够处理任何流体，无论是否完美。它可以处理一种流体在另一种流体中消散的方式（如墨水滴扩散填满一杯水），以及流体（包括空气，技术上属于流体）如何经历摩擦。

如今，物理学家和工程师使用纳维-斯托克斯方程设计机翼和游艇螺旋桨；预测飓风登陆地点；模拟气候变化如何增加干旱风险；以及预测熔岩流、火山灰云甚至恒星内部的运动行为。

但他们也知道，流体的奥秘远不止欧拉、纳维和斯托克斯所认识到的那些。

对称性超越物质本身

自20世纪60年代以来，物理学家一直对金属的共性感到困惑。当金属冷却时，其原子最终会排列整齐，导致磁化。神秘的是，无论铁、镍、钴还是其他材料，这种集体排列总是以完全相同的速度发生。

肯尼斯·威尔逊的方法最终揭示了原因。关键在于观察材料的对称性。

你可以将对称性视为一种无关紧要的变化。正方形具有某种对称性——旋转90度后无人察觉。圆形具有更多对称性——可以随意旋转任意角度而不会产生任何后果。

威尔逊利用对称性精确计算了描述任何材料的数学如何随缩放而变化。他提出了一个两步过程：首先在最微观层面识别系统的对称性；其次向宏观层面缩放，威尔逊的数学机制告诉你理论中每一项在缩放过程中会增长还是缩小。他发现，缩放时大多数项几乎缩小到零——这与你放大观察时数学视野变得模糊有关。单个原子或小团原子相关的细节变得太小而不值得关注，只有追踪大趋势的项才能存活下来，最终得到一个简洁的磁性理论。

威尔逊的工作还阐明了一个问题：为什么像磁化温度这样的一些性质在不同磁体间差异巨大，即使它们具有相同的自旋对称性。这些性质与少数存活项的大小有关。对称性告诉你保留大项的总体形状，但并不精确告诉你它们多大。这些项的大小就像细线一样，将宏观世界与微观世界轻轻系在一起。

在接下来的几十年里，威尔逊的机制渗透到物理学的许多领域。但当涉及到流体时，科学家们仍然停滞不前——流体的定义对称性尚不明确。

如何定义流体

第一个突破出现在2000年代，当时一组宇宙学家正在用威尔逊式思维为整个宇宙发展有效场论。在这个过程中，他们偶然发现了一个关键洞见：宇宙的膨胀打破了时空中的关键对称性。

一般来说，空间和时间没有参照点来测量速度。如果你在一艘没有窗户的宇宙飞船里，你无法判断自己是快速移动、缓慢移动还是完全静止。时空在速度方面具有对称性。

但在膨胀的宇宙中，存在一个特殊参照点，你可以据此辨别运动。流体同样打破了这种对称性。如果你置身于静止的液体中，你可以判断自己处于静止状态。如果你开始游泳，你会感受到流体的阻力。静止的流体就像膨胀的宇宙一样，缺乏时空固有的速度对称性。

“在对称性层面，它们是相同的。”现在哥伦比亚大学工作的物理学家阿尔贝托·尼科利斯说。

这群研究人员还需最后一个要素——通过考虑在不改变能量前提下对流体的改变方式（另一组对称性）找到了答案。他们意识到，你可以免费交换流体的两块区域，也可以随机洗牌三块、四块或任意数量。相比之下，固体的任何区域交换都需要以破裂或严重内应力形式付出能量代价，因此固体缺乏这种交换对称性。

该小组确定了流体拥有的无限数量的对称性。他们考虑了这些对称性要求理论中包含哪些项，然后效仿威尔逊精神，缩放观察并看着微观细节被冲刷干净。他们准确得到了欧拉方程——对完美流体而言完美适用，源自基本对称性原理。

黑洞的引领

新的有效场论加速了研究黑洞的科学家群体的努力，黑洞与流体有着奇特的联系。

1990年代末的理论突破确立了在特殊条件下，可以将球形黑洞视为平坦的量子汤。理论家将汤的粘度（使其能够耗散能量）与黑洞吞噬并隐藏能量的能力联系起来。“东西可以掉进黑洞，”尼科利斯说，“那是一种耗散形式。”

物理学家们已经有了黑洞的有效场论：爱因斯坦的引力理论。如果黑洞有一个秘密身份——一种不完美的、耗散能量的粘性流体——那么也应该存在一个描述不完美、耗散能量粘性流体的有效场论。多个团队竞相寻找。

麻省理工学院的物理学家刘洪领导了其中一组。该小组注意到黑洞表面具有类似交换对称性——你可以交换黑洞的斑块而不会扰乱其结构。

但仅凭这种对称性不足以描述不完美流体。为了达到这一目标，刘的团队（包括研究员迈克尔·克罗斯利和保罗·格洛里奥索）求助于量子力学的老技巧。他们在理论中复制了物质——实际上添加了第二个流体，其时钟向后走，而第一个流体的时钟向前走。随时比较两种流体，研究人员可以跟踪随机变化。

团队还有最后一个问题。他们知道缩放后的流体必须服从热力学定律：它必须具有随地点变化的温度，并且热斑块倾向于随时间与冷斑块融合。同时，他们知道在放大水平上，原子的嗡嗡声不区分过去和未来。有没有办法将流体的热力学与时间的两种运作方式联系起来？

经过反复试验，该小组找到了一个有效的对称性。这种变换切换两种流体、逆转它们的时钟，并以特定方式调整温度。如果再次发生，流体会回到原始状态。时间的不同概念及其最终兼容性源于这种对称性——这是物理学家以前从未见过的。“这是一种有趣的对称性，”加拿大维多利亚大学的物理学家克里斯坦·延森说，“而且它是必不可少的。”

2015年，刘和他的团队发布了他们的新有效场论。这是一部杰作，长达110页。“它让我叹为观止。”兰德里说。

后续发展

新理论使得理论家能够更高效地工作并推进计算。特别是，他们能够找出纳维-斯托克斯方程中被忽视的微小项——这些项捕捉了随机分子运动的某些效应。“我们做的事情不仅仅是流体动力学的替代版本，”兰德里说，“它实际上严格来说更正确。”

芝加哥大学的卢卡·德拉克雷茨使用新理论更精确地计算了热量在液体中的传播方式——发现由于随机抖动，在最初时刻热量传播比预期更慢。这种效应在水中太小无法测量，但在由有限数量粒子组成的量子系统中变得更加显著。“在混沌量子多体系统中做精密物理听起来很疯狂，但现在可能了。”德拉克雷茨说。

其他研究人员使用该理论推导了奇异物质态（如分形物质）的纳维-斯托克斯类比。大致来说，分形物质由粒子群组成，这些粒子群集体表现为一个粒子。单个“分形子”被锁定在原地；只有当分形子聚集在一起时才能移动。

2020年，科罗拉多大学博尔德分校的物理学家安德鲁·卢卡斯及其合作者在分形相中发现了新的对称性，并使用有效场论框架推导了分形版的纳维-斯托克斯方程。这项工作为理解全新类别的类液体物质相打开了大门。

2024年，卢卡斯和他的团队进一步扩展了刘的理论，明确了一些构建基础。他们利用新发现的“强”对称性——与流体中粒子总数始终不变的事实相关。由于特定区域中粒子精确数量可能时刻变化，这种对称性被部分打破为“弱”对称性。他们发现，这种破缺对称性的结果是漫长而缓慢的扩散——这有助于解释为什么墨水滴需要几秒钟才能在由皮秒级激烈碰撞的粒子组成的流体中扩散。

卢卡斯和合作者于2024年将该对称性整合到刘的有效场论中。“我们的主要贡献是证明了刘所做工作的合理性，”卢卡斯说，“对我来说，这是一个故事的完结。”

虽然有效场论框架催化了新的计算，但真正的回报可能更多是概念性的：对流体的本质给出新的定义——流体是任何具有特定对称性集合的物质，无论它由什么构成。

“流体动力学有一些神奇之处，”尼科利斯说，“油、水、水银——这些在微观上是非常不同的东西，但当你看它们如何流动时，在实践中却表现得非常相似。”

中文翻译：

流体理论迈入21世纪

20世纪下半叶，一场概念性的海啸席卷了物理学界。人们发现，我们的世界源于一个由分子构成的微观世界，而分子世界又源于一个更微观的亚原子粒子世界（亚原子粒子世界又源于更奇异的东西），这一发现促使我们重写了关于物质的理论。

但这场革命并未触及流体。描述流体的方程仍然保持着它们简单而古朴的形式：纳维-斯托克斯方程，最早诞生于19世纪。纳维-斯托克斯方程在预测流体的流动和涡旋方面极为成功。但它无法解释构成物质的微观粒子群的存在。

如今，物理学家们有了一套能够做到这一点的理论。这是二十年来从头重建流体理论的成果。在这一过程中，物理学家们提出了一套全新的方式来定义“何为流体”，其基础是被称为对称性的基本性质，并证明了纳维-斯托克斯方程是对称性的一个结果，这解释了为什么这些方程会呈现它们现有的形式。

通过理解纳维-斯托克斯方程的起源，研究人员找到了一条超越这些方程的途径，重新定义了“何为流体”，并预测了源于微观粒子运动的新行为。

颇具讽刺意味的是，将流体理解为微观世界产物的这条道路，最初是由一些思考现实中最宏大尺度的物理学家开辟的。最终需要借助研究黑洞和整个宇宙的研究人员的洞见，才能将流体带入现代时代。

流体的黎明

几个世纪以来，科学家们已经理解了流体的基本知识。

18世纪50年代，数学家莱昂哈德·欧拉将牛顿第二运动定律——就是给出F=ma的那条定律——加以改造，用于预测液体的运动。欧拉方程对“完美”流体完美适用，在这种流体中，水流可以永远流动，因为流体没有黏性——一种内在的黏滞性——来使其减速。

19世纪初，克劳德-路易·纳维和乔治·加布里埃尔·斯托克斯对欧拉方程进行了升级。新的纳维-斯托克斯方程可以处理任何流体，无论完美与否。它们可以处理一种流体在另一种流体中扩散的方式，比如一滴墨水扩散开来填满一杯水，以及流体（包括空气，从技术上讲空气也是流体）受到摩擦的方式。

如今，物理学家和工程师使用纳维-斯托克斯方程来设计机翼和游艇螺旋桨；预测飓风将在何处登陆；模拟气候变化将如何增加干旱风险；以及预测熔岩流、火山灰云甚至恒星内部的运动行为。

他们也知道，流体所包含的内容远超欧拉、纳维和斯托克斯当年的认知。

纳维-斯托克斯方程假定流体是连续物质，无论你多么放大一个点，流体都能完美平滑地流动。但实际上，流体是分子和原子的混合体。随着你不断放大，你最终会在流动中看到由于这种颗粒状本质而产生的小小扰动，而这些扰动正是经典流体方程所忽略的。

“纳维-斯托克斯方程在很大程度上是一个近似，”麻省理工学院的物理学家迈克尔·兰德里说。“它不是一个精确的方程。”

从这个意义上说，我们的流体理论是一个异类。在20世纪，随着我们在越来越小的尺度上揭示世界的结构，物理学家们改造了许多物质理论，以将原子等的存在纳入考虑。

新的理论也是近似，因为追踪每一个原子的运动是不可能的。但在几十年的时间里，物理学家们找到了一种重写这些理论并以更有利于原子的方式重新推导它们的方法。

20世纪70年代，康奈尔大学的物理学家肯尼斯·威尔逊将所有碎片收集起来，放入一个数学包中。威尔逊通过严谨的计算证明，为什么较小尺度的效应以极为有限的方式渗透到我们的层面。他开发了一种构建高层理论的崭新方法，称为有效场论，这将成为物理学大部分领域的新基础。这项工作为他赢得了诺贝尔奖。

一切始于磁铁。

对称性胜于实体

自20世纪60年代以来，物理学家们一直在思考金属的一个共同性质。当金属被冷却时，其原子最终会对齐，使金属磁化。神秘的是，无论是铁、镍、钴还是其他材料，这种集体对齐总是以完全相同的速度发生。

威尔逊的方法最终揭示了原因。关键在于观察材料的对称性。

你可以把对称性理解为一种无关紧要的变化。正方形有一些对称性；你可以将它旋转90度，没有人会注意到。圆形有更多的对称性；你可以将它旋转任意角度而不会产生任何后果。

威尔逊利用对称性精确计算了描述任何材料的数学在放大和缩小时会如何变化。他提出了一套两步流程。

第一步是识别你的系统在你所理解的最微观层面上的对称性。以磁铁为例。它们的原子以某种网格排列，打破了空间的底层连续对称性。（你必须将磁铁在晶格上移动一个格距才能使其看起来相同。）原子还可以自由地指向任何方向——这是另一个对称性。像这样的对称性精确地决定了哪些数学项应该出现在你的理论中。

第二步是向宏观层面缩小。威尔逊的数学机制告诉你，在缩小的过程中，你理论中的每一项会增长还是衰减。他发现的是，随着你缩小，大多数项几乎衰减到零。这与以下事实有关：随着你缩小，你的数学视野变得模糊。与单个原子或小群原子相关的细节变得太小而不值得关注。只有追踪大趋势的项才能存活下来，你最终得到一个简洁优美的磁性理论——一个有效场论——与几乎所有的原子细节脱钩。

“这就是威尔逊理解的力量，”兰德里说。“你可以直接跳到答案。”

借助这套机制，威尔逊解决了为什么许多磁铁以相同速率磁化的谜团。缩小步骤会经过一个点，在这个点上所有磁铁看起来都一样，无论它们的原子是排列成立方体还是四面体。唯一重要的是原子具有允许它们指向任何方向的对称性。（例如，原子被限制在一个平面内旋转的磁铁，磁化速率则不同。）

威尔逊的工作还阐明了为什么少数性质，如磁化发生的温度，在不同磁铁之间差异巨大，即使它们具有相同的自旋对称性。这些性质与少数存活的项的大小有关。

对称性告诉你那些保持较大的项的整体形状，但不告诉你它们具体有多大。这些项的大小就像细线一样，将宏观世界与微观世界轻轻相连。

在接下来的几十年里，威尔逊的机制渗透到物理学的许多领域。物理学家们利用威尔逊的计算来为此前模糊不清的量子场论机制提供依据，量子场论将每个粒子视为波，同时冲淡了最小振动的不太重要的效应。威尔逊的贡献也塑造了某些物质相（如固体）的现代理论。

但当涉及流体时，科学家们仍然卡住了。流体的定义性对称性尚不清楚。

如何定义流体

此案的第一个突破出现在2000年代，当时一群宇宙学家正在运用威尔逊式思维为整个宇宙发展一个有效场论。在这个过程中，他们偶然发现了一个关键洞见：宇宙的膨胀打破了时空中的一个关键对称性。

一般来说，空间和时间没有可以用来测量速度的参考点。如果你在一艘没有窗户的宇宙飞船里，你无法判断自己是在快速移动、缓慢移动还是根本没动。时空具有关于速度的对称性。

但在膨胀的宇宙中，有一个特殊的参考点，可以据此辨别运动：那就是空间自身的膨胀使星系均匀地远离你的那个参考点。如果你离开你的星系乘坐宇宙飞船，沿着与这种宇宙退行相反的方向朝某个特定方向行进，你会看到前方的星系比后方的星系退行得更慢。

这个小组指出，流体打破了同样的对称性。如果你浸在静止的液体中，你能感觉到自己是静止的。如果你开始游动，你会感受到流体从你身边流过时产生的阻力。静止的流体，就像膨胀的宇宙一样，缺乏时空底层的速度对称性。同时，它拥有其他标准的时空对称性；旋转和平移不会改变流体。

“在对称性层面上，它们是相同的，”现任职于哥伦比亚大学的物理学家阿尔贝托·尼科利斯说，他曾参与宇宙有效场论的研究。

这种相似性让这个小组陷入了思考，但他们还需要一个要素。他们通过考虑可以对流体做出哪些不改变其能量的改变——这是另一组对称性——找到了答案。他们意识到，你总是可以免费交换流体的两个块。你也可以打乱三个块、四个块或任意数量的块。相比之下，你不能交换固体的任何区域而不付出能量代价，要么通过破裂，要么通过严重的内部应力，所以固体缺乏这些交换对称性。

这个小组确认了流体具有无限多的对称性。他们考虑了这些对称性要求在理论中包含哪些项，然后秉承威尔逊的精神，向外缩小，观察微观细节被冲刷殆尽。他们正好落在了欧拉方程上，这些方程从基本的对称性原理出发推导出来，完美适用于完美流体。

由此产生的理论——最初埋藏在2005年的一篇宇宙学论文中，并在2012年的一篇流体动力学论文中得到重点阐述——是第一个将完整的有效场论处理应用于流体的理论。“这是这一切的奠基文本，”兰德里说。

下一步是将非完美流体也纳入其中。

黑洞的引领

新的有效场论加速了研究黑洞的科学家群体中的一项努力，黑洞与流体有着自身奇特的联系。

20世纪90年代末的一个理论突破确立了，在特殊条件下，你可以将一个球形黑洞视为一种平坦的量子汤。理论家们将这种汤的黏性——使其能够耗散能量——与黑洞吞噬能量并将其隐藏的能力联系了起来。“东西可以掉进黑洞，”尼科利斯说。“那是一种耗散形式。”

物理学家们已经拥有了一个黑洞的有效场论：爱因斯坦的引力理论。如果黑洞有一个秘密身份，是一种非完美、耗散能量的黏性流体，那么一种用于非完美、耗散能量的黏性流体的有效场论也应该存在。多个团队竞相寻找它。

麻省理工学院的物理学家刘洪领导了其中一个小组。该小组注意到，黑洞表面具有一种交换对称性，类似于尼科利斯及其同事用来定义流体的那种对称性；你可以交换黑洞上的区域而不扰动黑洞的结构。

但仅凭这种对称性不足以描述非完美流体。为了实现这一点，刘的小组——包括研究人员迈克尔·克罗斯利和保罗·格洛里奥索——求助于量子力学中的一个老技巧。他们复制了理论中的物质，实际上添加了第二个流体，其时钟向后走，而第一个流体的时钟向前走。在任何给定时刻比较这两种流体，使研究人员能够跟踪随机变化。（真实的物理流体本质上是这两种数学流体的平均。）

这个团队还有最后一个问题。他们知道，缩小后的流体必须服从热力学定律：它必须有随位置变化的温度，并且随着时间的推移，任何热区都有与冷区混合的趋势。同时，他们知道，在放大的层面上，原子的嗡嗡声不区分过去和未来。（你无法判断一段原子运动的视频是正放还是倒放。）有没有办法将流体的热力学与时间的两种运作方式联系起来？

经过一些试错，这个小组找到了一个能解决问题的对称性。这个变换交换了两种流体，反转了它们的时钟，并以一种特定方式调整了温度。如果它再次发生，流体会回到其原始状态。不同的时间概念及其最终兼容性，都源自这个对称性，它与物理学家们以前见过的任何对称性都不同。“这是一个有趣的对称性，”现任职于加拿大维多利亚大学的物理学家克里斯坦·延森说，他也为这项从黑洞到流体的工作做出了贡献。“而且它是必不可少的。在那之前，有大量的项。然后（一切）坍缩下来，你恰好得到已知的现象，不多也不少。”

他们完成了。从这些对称性出发，他们可以写出一个理论，然后向外缩小，推导出纳维-斯托克斯方程。2015年，刘和他的团队发布了他们新的有效场论。这是一部杰作，长达110页。“它让我大开眼界，”兰德里说。

接下来，物理学家们将把这一理论付诸应用。

流体的后续

如何将这一高度技术性的理论用于具体应用，并非立竿见影。但刘和格洛里奥索在2018年以更容易理解的方式重写了这些方程，计算开始慢慢涌现。

新的工作使理论学家们能够更高效地工作，并将计算推得更远。特别是，他们能够找出纳维-斯托克斯方程中此前被忽略的微小项，这些项捕捉了随机分子运动的某些效应。“我们正在做的事情不仅仅是流体动力学的某种替代版本，”兰德里说。“它在严格意义上更为正确。”

芝加哥大学的卢卡·德拉克雷塔兹使用新理论更精确地计算了热量如何在液体中传播——并发现热量在最初几瞬间的传播速度比预期的要慢，这是由于随机的抖动。这种效应在水中小到无法测量，但在由有限数量粒子组成的量子系统中变得更加显著。“在混沌的量子多体系统中做精密物理听起来很疯狂，但现在这是可能的，”德拉克雷塔兹说。

其他研究人员使用这一理论推导了奇异物质态的纳维-斯托克斯类比，如分形子物质。粗略地说，分形子物质由粒子群组成，这些粒子群集体表现得像一个粒子。单个“分形子”被困在原位；只有当分形子聚集在一起时，它们才能移动。

2020年，科罗拉多大学博尔德分校的物理学家安德鲁·卢卡斯及其合作者在分形子相中发现了新的对称性，并使用有效场论框架推导出了纳维-斯托克斯方程的分形子版本。这项工作为理解全新类别的液态物质相打开了大门。“只需要几行代数，把它插入（刘）开发的 formalism 中，然后看看什么是可能的，”卢卡斯说。

2024年，卢卡斯和他的团队在刘的理论基础上进一步发展，明确了其一些构建的基础。他们使用了一种新发现的“强”对称性，这与流体中粒子总数始终保持不变这一事实有关。由于流体特定区域中的确切粒子数每时每刻都可能变化，这种对称性被部分打破，成为一种“弱”对称性。他们发现，这种对称性被打破的后果是缓慢的长程扩散。这有助于解释为什么一滴墨水需要许多秒才能在一流体中扩散开来，而该流体中的粒子从皮秒到皮秒都在激烈碰撞。

卢卡斯和他的合作者在2024年将这一对称性整合到了刘的有效场论中。“我们的主要贡献是证明了刘所做工作的合理性，”卢卡斯说。“对我来说，这是一个故事的完结。”

虽然有效场论框架催化了新的计算，但真正的奖励可能更多是概念性的：对“何以为流体”的新定义。流体是任何具有特定一组对称性的材料，无论它由什么构成。

“流体动力学有一种神奇之处，”尼科利斯说。“有油、水、水银。这些都是非常不同的东西，在微观上非常不同，但实际上，当你观察它们如何流动时，它们的行为非常相似。”

英文来源：

Theory of Fluids Enters the 21st Century
In the second half of the 20th century, a conceptual tsunami swept through physics. The discovery that our world emerges from a microscopic world of molecules, which emerges from an even more microscopic world of subatomic particles (which in turn emerges from even stranger stuff) triggered the rewriting of our theories of matter.
But the revolution didn’t reach fluids. Their governing equations remained in their simple, vintage form: the Navier-Stokes equations, first developed in the 19th century. The Navier-Stokes equations are enormously successful at predicting how fluids flow and swirl. But they fail to account for the existence of swarms of microscopic bits that make up matter.
Now physicists have a theory that does. It is the fruit of a 20-year effort to rebuild the theory of fluids from the ground up. Along the way, physicists have come up with a whole new way of defining what it means to be a fluid, based on fundamental properties known as symmetries, and have shown that the Navier-Stokes equations are a consequence of symmetries, which explains why the equations take the forms that they do.
By understanding the origins of the Navier-Stokes equations, researchers have found a way to go beyond them, redefining what it means to be a fluid and predicting new behaviors that stem from the motions of microscopic particles.
The trail to understanding fluids as the product of the microscopic world was blazed, ironically, by physicists thinking about some of reality’s biggest scales. It would take insights from researchers studying black holes and the universe at large to finally bring fluids into the modern era.
The Dawn of Fluids
For centuries, scientists have understood the basics of fluids.
In the 1750s, the mathematician Leonhard Euler adapted Newton’s second law of motion — the same one that gives us F = ma — to predict the motion of liquids. Euler’s equations work perfectly for “perfect” fluids, in which a current can flow forever because the fluid has no viscosity — a sort of intrinsic stickiness — to slow it down.
In the early 1800s, Claude-Louis Navier and George Gabriel Stokes gave Euler’s equations an upgrade. The new Navier-Stokes equations could handle any fluid, perfect or not. They could handle the way one fluid dissipates in another, like an ink drop spreading out to fill a glass of water, and the way fluids (including air, which is technically a fluid) experience friction.
Archivio storico dell’Accademia delle Scienze; Public Domain
Today physicists and engineers use Navier-Stokes to shape aircraft wings and yacht propellers; to forecast where a hurricane will make landfall; to model how climate change will increase the risk of drought; and to predict the behavior of flows of lava, clouds of ash, and even the interiors of stars.
They also know that there’s more to fluids than Euler, Navier, and Stokes appreciated.
The Navier-Stokes equations presume that fluids are continuous substances that flow perfectly smoothly, no matter how much you zoom in on a point. But in fact, fluids are amalgamations of molecules and atoms. As you zoom in, you’ll eventually discern tiny blips in the flow due to this grainy nature, blips that the classic fluid equations lop off.
“Navier-Stokes is very much an approximation,” said Michael Landry, a physicist at the Massachusetts Institute of Technology. “It’s not an exact equation.”
In this way, our theory of fluids is an outlier. In the 1900s, as we uncovered the structure of the world at smaller and smaller scales, physicists revamped many of their theories of matter to take the existence of atoms and the like into account.
The new theories were also approximations, because tracking the motion of every last atom is impossible. But over the decades, physicists found a way to rewrite the theories and re-derive them in a more atom-friendly way.
In the 1970s, Kenneth Wilson, a physicist at Cornell University, gathered up all the pieces and put them into one mathematical package. Wilson showed, with rigorous calculations, why the smaller scales bleed through to our level in mercifully few ways. He developed a whole new method for building high-level theories, called effective field theories, that would form the new foundation for much of physics. The work would win him a Nobel prize.
It all started with magnets.
Symmetries Over Substance
Since the 1960s, physicists had been puzzling over a common property of metals. When a metal is cooled, its atoms eventually align, causing it to magnetize. Mysteriously, this collective alignment always happens at precisely the same speed, whether the metal is iron, nickel, cobalt, or another material.
Wilson’s approach would eventually show why. The key was to look at a material’s symmetries.
Keystone Pictures USA/ZUMAPRESS
You can think of a symmetry as a change that doesn’t matter. A square has some symmetry; you can rotate it by 90 degrees, and no one will notice. A circle has more symmetry; you can rotate it by any angle you like without consequences.
Wilson used symmetries to calculate exactly how the math that describes any material will change as you zoom in and out. He laid out a two-step process.
The first step is to identify the symmetries of your system at the most microscopic level you understand. Take magnets, for example. Their atoms are laid out in some kind of grid, breaking the underlying continuous symmetry of space. (You have to move the magnet by one space on the lattice for it to look the same.) The atoms also have the freedom to point in any direction — another symmetry. Symmetries like these determine exactly which mathematical terms belong in your theory.
The second step is to zoom out toward the macroscopic level. Wilson’s mathematical machinery tells you whether each term in your theory will grow or shrink as you do so. What he found was that as you zoom out, most terms shrink nearly to zero. This has to do with the fact that as you zoom out, your mathematical vision blurs. Details related to individual atoms, or small groups of atoms, become too small to care about. Only the terms tracking broad trends survive, and you land on a short and sweet theory of magnetism — an effective field theory — disconnected from almost all the atomic details.
“This is the power of Wilson’s understanding,” Landry said. “You can just skip to the answer.”
Holly Reynolds
With this machinery, Wilson solved the mystery of why many magnets magnetize at the same rate. The zooming-out step passes through a point at which all magnets look the same, whether their atoms are arranged in cubes or tetrahedra. All that matters is that the atoms have a symmetry that lets them point in any direction. (Magnets whose atoms are pinned down to spin in a plane, for instance, magnetize at a different rate.)
Wilson’s work also clarified why a few properties, like the temperature at which the magnetization takes place, vary wildly from magnet to magnet, even when they have the same spin symmetry. These properties are related to the size of the few surviving terms.
The symmetries tell you the overall shape of the terms that stay large, but not exactly how large they are. The sizes of these terms act like threads lightly tethering the macroscopic world to the microscopic one.
Over the following decades, Wilson’s machinery seeped into many areas of physics. Physicists used Wilson’s calculations to justify the previously murky mechanics of quantum field theory, which treats each particle as a wave while washing out the less significant effects of the smallest vibrations. Wilson’s contributions also shaped modern theories of certain phases of matter, like solids.
But when it came to fluids, scientists remained stuck. Their defining symmetries weren’t yet clear.
How To Define a Fluid
The first break in the case came in the 2000s, when a group of cosmologists was using Wilsonian thinking to develop an effective field theory for the universe as a whole. As they did so, they stumbled upon a key insight: The universe’s expansion breaks a crucial symmetry in space-time.
In general, space and time have no reference point against which you can measure speed. If you’re in a windowless spaceship, you can’t tell whether you’re moving quickly, slowly, or not at all. Space-time has a symmetry with respect to speed.
But in the expanding universe, there is a special reference point against which you can discern a motion: It’s the one in which the expansion of space itself moves galaxies uniformly away from you. If you were to leave your galaxy in a spaceship and travel against this cosmic recession in a particular direction, you would see the galaxies in front of you recede more slowly than the galaxies behind you.
Fluids, the group noted, break the same symmetry. If you’re immersed in a resting liquid, you can tell you’re at rest. And if you start to swim, you’ll feel the drag of the fluid as it moves past you. The resting fluid, like the expanding universe, lacks the underlying speed symmetry of space-time. Meanwhile, it has other standard space-time symmetries; rotations and translations don’t change the fluid.
“At the level of the symmetries, they are the same,” said Alberto Nicolis, a physicist now at Columbia University, who worked on the effective field theory of the cosmos.
Anna DeBeer
The resemblance got the group thinking, but they needed one more ingredient. They found it by considering what changes they could make to a fluid without changing its energy — another set of symmetries. They realized you could always swap two parcels of a fluid for free. You could also shuffle three parcels, or four, or any number. In contrast, you can’t exchange any regions of a solid without paying an energy toll, through rupture or serious internal stress, so solids lack these swapping symmetries.
The group had identified an unlimited number of fluid symmetries. They considered what terms these symmetries would require in the theory and then, channeling the spirit of Wilson, zoomed out and watched the microscopic details wash away. They landed right on the Euler equations, perfect for perfect fluids, derived from fundamental symmetry principles.
The resulting theory — initially buried in a cosmology paper in 2005 and highlighted in a hydrodynamics paper in 2012 — was the first to apply the full effective field theory treatment to fluids. “It’s the foundational text of all of this,” Landry said.
The next step would be to include the imperfect fluids, too.
A Black Hole Lead
The new effective field theory accelerated an effort within a community of scientists studying black holes, which have their own strange connection to fluids.
A theoretical breakthrough from the late 1990s had established that, under special conditions, you could view a spherical black hole as a flat quantum soup. Theorists had connected the viscosity of the soup, which enabled it to dissipate energy, to the black hole’s ability to gobble up energy and hide it. “Things can fall into the black hole,” Nicolis said. “That’s a form of dissipation.”
Physicists already had an effective field theory of black holes: Einstein’s theory of gravity. If a black hole had a secret identity as an imperfect, energy-dissipating, viscous fluid, then an effective field theory for imperfect, energy-dissipating, viscous fluids should exist, too. Multiple teams raced to find it.
Hong Liu, a physicist at MIT, led one of the groups. The group noticed that the surface of a black hole had a swapping symmetry akin to the one Nicolis and company had used to define a fluid; you could exchange patches of a black hole with each other without disturbing the black hole’s structure.
But that symmetry wasn’t enough to describe an imperfect fluid. To achieve that, Liu’s group — which included the researchers Michael Crossley and Paolo Glorioso — resorted to an old trick from quantum mechanics. They duplicated the substance in their theory, effectively adding a second fluid with a clock that ticked backward while the first fluid’s clock ticked forward. Comparing the two fluids at any given moment let the researchers keep track of random variations. (The real physical fluid was essentially an average of the two mathematical fluids.)
The team had one last problem. They knew that a zoomed-out fluid must obey the laws of thermodynamics: It has to have a temperature that varies from place to place, and a tendency for any hot patches to blend with cold patches as time ticks along. At the same time, they knew that at the zoomed-in level, the buzzing of atoms makes no distinction between past and future. (You wouldn’t be able to tell if a video of atomic motion was playing forward or backward.) Was there a way to tie the thermodynamics of the fluid to the two ways that time worked?
After some trial and error, the group found a symmetry that did the trick. The transformation switched the two fluids, reversed their clocks, and fiddled with the temperature in a particular way. If it happened a second time, the fluid would return to its original state. The different notions of time, and their ultimate compatibility, flowed from this symmetry, which was unlike any that physicists had seen before. “It’s a funny symmetry,” said Kristan Jensen, a physicist now at the University of Victoria in Canada, who also contributed to the black-holes-to-fluids effort. “And it’s essential. Before that, there are a ton of terms. And then [everything] collapses down and you just get known phenomena, nothing more and nothing less.”
They were done. From these symmetries they could write down a theory and zoom out to derive the Navier-Stokes equations. In 2015, Liu and his group posted their new effective field theory. It was a magnum opus, spanning 110 pages. “It blew my mind,” Landry said.
Next, physicists would put the theory to work.
Fluid Follow-up
It wasn’t immediately obvious how to harness the highly technical theory for specific applications. But Liu and Glorioso rewrote the equations in a more accessible way in 2018, and the calculations started to trickle out.
The new work allowed theorists to work more efficiently and push their calculations further. In particular, they were able to ferret out tiny terms in Navier-Stokes equations that had previously been ignored, terms that capture some of the effects of random molecular motion. “The stuff that we’re doing is more than just sort of an alternate version [of fluid dynamics],” Landry said. “It is actually strictly more correct.”
Luca Delacrétaz at the University of Chicago used the new theory to calculate more precisely how heat spreads through a liquid — and found that it moves more slowly than expected during its first few moments, due to random jitters. The effect is far too small to measure in water, but it becomes more substantial in quantum systems made from a limited number of particles. “Doing precision physics in chaotic quantum many-body systems sounds crazy, but now it’s possible,” Delacrétaz said.
Other researchers used the theory to derive Navier-Stokes analogues for exotic states of matter, like fracton matter. Roughly speaking, fracton matter is composed of groups of particles that collectively act like one particle. Individual “fractons” stay trapped in place; only when fractons band together can they move about.
In 2020, Andrew Lucas, a physicist at the University of Colorado, Boulder, and his collaborators spotted new symmetries in the fracton phase and used the effective field theory framework to derive a fracton version of the Navier-Stokes equations. The work opened the door to understanding whole new classes of liquidlike phases of matter. “It’s just a few lines of algebra to kind of plug it into the formalism that [Liu] developed and ask what’s possible,” Lucas said.
In 2024, Lucas and his group built on Liu’s theory by making explicit the basis for some of its construction. They used a newly discovered “strong” symmetry, related to the fact that the total number of particles in a fluid is always the same. That symmetry is partially broken, becoming a “weak” symmetry, due to the fact that the exact number of particles in a specific region of a fluid can vary from moment to moment. The consequence of this broken symmetry, they found, is a long, slow diffusion. This helps explain why it takes many seconds for an ink drop to spread through a fluid made of particles furiously colliding from picosecond to picosecond.
Lucas and his collaborators integrated this symmetry into Liu’s effective field theory in 2024. “Our main contribution was to justify what Liu was doing,” Lucas said. “To me it’s a closure of a story.”
While the effective field theory framework has catalyzed new calculations, the real prize may be more conceptual: a new definition of what makes a fluid a fluid. A fluid is any material with a particular set of symmetries, no matter what it’s made of.
“There’s something magical about fluid dynamics,” Nicolis said. “There’s oil, water, mercury. These are all very different, microscopically, very different things that, in practice, behave very similarly when you look at how they flow.”