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。由于这种相互作用，混沌使得预测粒子的轨迹变得困难，实际上是不可能的。这是正确的，即使只有三个粒子，特别是有大量的粒子。但是，对于大量的粒子，还有另一种简化：如果我们忘记单独的轨迹，转而看看当我们从远处看到时会发生什么，这个系统又变得容易描述了。从本质上讲，轨迹是平均的，在大的观测尺度上出现的东西更简单、更平滑，并且用更少的有效自由度来描述。不需要知道水分子的所有轨迹来确定波是如何传播的：波动方程要简单得多。这就是流体动力学，波是自由的浮现程度。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概念：它描述了金属中的电子，现代实验中的准一维量子超冷Rb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的必要条件。对于“可积”的系统--这是一种数学性质，意味着只需要很少的粒子，轨迹就可以完全计算出来，而且不会出现混沌--流体力学的思想仍然适用。只是出现了更多涌现的“浪潮”。这就是广义流体动力学理论。事实证明，这是适用于准一维超冷量子原子气体的理论，也适用于描述(经典！)某些湍流态的孤子气体。浅水。这个项目将使用并进一步扩展流体力学理论，以便评估相互作用的多体系统中的精确量，否则这些多体系统是无法访问的。它将对可积系统使用特别广义的流体力学，因为那里有许多强大的数学技术可用，但对于不可积系统，也会使用传统的流体力学，其中现象学可能会非常不同。该项目的基础理论是由PI及其合作者于2018年提出的“弹道涨落理论”(BFT)。这给了我们一个完全基于流体动力学的理解，即多体系统是如何在非常大的空间和时间尺度上波动的。例如，波动编码了系统的许多深层次特性，这些特性不能仅通过观察波的传播来看到。这个理论实际上是对久负盛名的热力学理论的“动力学”概括。该项目的目标是首先确认BFT，并通过与计算机模拟进行比较，向不同领域的更广泛的研究人员解释它；进一步开发该框架；并提取其最重要的结果。结果将包括对相关性衰减和统计累积量增长的预测。这些量的准确计算在多体物理中是一个长期存在的问题，特别是在可积性的背景下。该项目还将通过分析扩散的影响，并结合成功的、更老的“宏观涨落理论”，以及可积性破缺和(量子)玻尔兹曼方程的影响，进一步发展BFT。

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.