Fluctuations and correlations at large scales from emergent hydrodynamics: integrable systems and beyond

气体和流体由大量相互作用的粒子组成。由于相互作用，混沌使得预测粒子的轨迹变得困难，事实上几乎不可能。即使只有三个粒子，情况也是如此，更不用说有大量粒子了。但是，对于大量粒子，会出现另一种简化：如果我们忘记单个轨迹，而是看看“从远处”看时会发生什么，系统就会再次变得易于描述。从本质上讲，轨迹是平均的，在大观察尺度上出现的结果更简单、更平滑，并且通过减少的有效自由度来描述。无需知道水分子的所有轨迹即可确定波如何传播：波动方程要简单得多。这就是流体动力学，波是涌现的自由度。令人惊讶的是，流体动力学是一套远远超出水和其他简单流体的思想：它描述了金属中的电子、现代实验中的准一维量子超冷铷原子、磁性材料中的自旋等等。事实上，更令人惊讶的是，最近发现混沌并不是流体动力学发生所必需的。对于“可积”系统（一种数学性质，意味着只要很少的粒子，就可以完全计算轨迹并且不存在混沌），流体动力学的思想仍然适用。只是涌现的“浪潮”更多了。这就是广义流体力学理论。事实证明，这是准一维超冷量子原子气体的正确理论，也是描述（经典！）浅水某些湍流状态的孤子气体的理论。该项目将使用并进一步扩展流体动力学理论，以评估相互作用的多体系统中无法获得的精确量。它将特别使用广义流体动力学，用于可积系统，因为那里有许多强大的数学技术可用，但也使用传统流体动力学，用于不可积系统，其现象学可能非常不同。该项目基础的理论是 PI 及其合作者于 2018 年引入的“弹道涨落理论”(BFT)。这仅基于流体动力学，可以理解多体系统如何在非常大的情况下波动。空间和时间的尺度。例如，波动编码了系统的许多深层特性，这些特性仅通过观察波的传播是无法看到的。该理论实际上是成熟的热力学理论的“动力学”概括。该项目的目标是首先通过与计算机模拟进行比较来确认 BFT 并向各个领域的更广泛的研究人员解释它；进一步发展该框架；并提取其最重要的后果。后果将包括对相关性衰减和统计累积量增长的预测。这些量的精确评估是多体物理学中长期存在的问题，特别是在可积性的背景下。该项目还将通过分析扩散效应并与成功的、较古老的“宏观波动理论”联系起来，进一步发展拜占庭容错理论；以及可积性破坏和（量子）玻尔兹曼方程的影响。

Gases and fluids are composed of a very large number of particles that interact with each other. Because of the interaction, chaos makes it difficult, in fact practically impossible, to predict the particles' trajectories. This is true even if there were just three particles, a fortiori with a large number of them. But, with a large number of particles, there's another simplification that occurs: if we forget about the individual trajectories and instead look at what happens when seen "from far", the system becomes again simple to describe. Essentially, trajectories average out, and what emerges, at large observation scales, is simpler, smoother, and described by a reduced number of effective degrees of freedom. No need to know all trajectories of water molecules in order to determine how waves propagate: the wave equations are much simpler. This is hydrodynamics, and waves are the emergent degrees of freedom.Surprisingly, hydrodynamics is a set of ideas that goes much beyond water and other simple fluids: it describes eletrons in metal, quasi-one-dimensional quantum ultracold Rubidium atoms in modern experiments, spins in magnetic materials, and much more. In fact, even more surprisingly, it was found recently that chaos is not necessary for hydrodynamics to occur. For systems that are "integrable" - a mathematical property that implies that with few particles, the trajectoris can be fully calculated and there is no chaos - still the ideas of hydrodynamics apply. It's just that there are more emergent "waves". This is the theory of generalised hydrodynamics. It is, it turns out, the right theory for quasi-one-dimensional ultracold quantum atomic gases, and also the theory for soliton gases describing certain turbulent states of (classical!) shallow water.This project will use and further expand the theory of hydrodynamics in order to evaluate exact quantities in interacting many-body systems that are otherwise inaccessible. It will use especially generalised hydrodynamics, for integrable systems, as there are many strong mathematical techniques available there, but also conventional hydrodynamics, for non-integrable systems, where the phenomenology can be very different.The theory at the basis of this project is the "ballistic fluctuation theory" (BFT), introduced by the PI and his collaborators in 2018. This gives an understanding, based solely on hydrodynamics, for how the many-body system fluctuates at very large scales of space and time. Fluctuations encode many deep properties of the system which cannot be seen just by looking at wave propagations, for instance. This theory is in effect a "dynamical" generalisation of the well-established theory of thermodynamics. The goal of the project is to first confirm the BFT and explain it to a wider audience of researchers in various fields, by comparing with computer simulations; to further develop the framework; and to extract its most non-trivial consequences.The consequences will include predictions for the decay of correlations and the growth of statistical cumulants. The exact evaluation of these quantities is a long-standing problem in many-body physics, and especially in the context of integrability. The project will also develop further the BFT by analysing the effects of diffusion and connecting with the successful, older, "macroscopic fluctuation theory"; and the effects of integrability breaking and the (quantum) Boltzmann equation.