Emergence of hydrodynamics in many-body systems: new rigorous avenues from functional analysis

现代科学最深刻的思想之一是涌现论。在一个由大量成分（如原子或分子）组成的系统中，即使有简单的相互作用定律，也很难描述在大尺度上发生的事情，因为大多数物理相关的观察都发生在大尺度上。从短尺度的微观运动到大尺度的涌现的集体行为的转变是现代理论物理和数学物理中一些最重要问题的核心。稳定水面上的局部扰动--比如一根手指触摸它--会在微观距离上产生水分子的复杂重排。但是，对于任何足够远的局部探测器（比如附近的一片漂浮的树叶），最强的影响发生在从局部扰动传播出来的表面波撞击它的时候，表面波是一种涌现的行为，有它自己的新动力学。在这种情况下，它是由Navier-Stokes方程的线性响应得到的。类似地，在一大类多体系统中，强烈的相关性预计会发生在与弹道或缓慢衰减模式（如表面水波或声波）的传播相关的沿着轨迹上，而流体力学是它们的涌现理论。尽管上述例子很简单，但对流体力学如何从牛顿基本运动定律中涌现的完整数学理解，或者他们在量子力学和相对论中的改进，仍然没有。从第一性原理出发，探索在长时间和大距离上观察到的行为，以及涉及大量粒子的行为，是一项具有深远意义的艰巨任务。除了非常具体的模型，目前还没有严格的证明，在强相互作用系统的动力学是哈密尔顿或更一般的可逆和确定性的流体动力学方程。鉴于流体力学的基本原理和思想的普遍性和明显的普遍适用性，这是数学物理学最重要的挑战之一。本项目旨在探索这一问题的新途径，这为严格和普遍的处理提供了希望。主要的假设是，泛函分析的数学，从根本上说是一种关于无限大物体的理论，为多体系统的统计力学描述提供了正确的框架。而不是试图描述特定的模型，通过这种通用语言的任务分为两个：第一，一个提取基本属性作为一组公理，并试图从中推导出流体力学;第二，一个表明，这些属性在家庭的模型。最近，在文件[arXiv：2011.00611]，首席研究员成功地以这种方式展示了一维多体量子自旋系统大时间运动的一些基本方面，包括投影到流体动力学模式和一般形式的线性化欧拉方程的出现.本项目旨在进一步发展这一理论，其目标不仅是在任意维数的一般系统中建立一些基本结果，而且探索这一新观点为流体动力学的严格证明提供的可能性.

One of the deepest ideas of modern science is that of emergence. In a system composed of a very large number of constituents, such as atoms or molecules, even with simple laws of interaction, it can be very difficult to describe what happens at large scales, where most physically relevant observation occur. the passage from short-scale, microscopic motion to large-scale, emergent collective behaviours is at the heart of some of the most important questions in modern theoretical and mathematical physics.Take the example of travelling surface-water waves. A local disturbance on a steady water surface - say a finger touching it - produces a complicated rearrangement of water molecules at microscopic distances. But the strongest effect on any local probe that is far enough away - say a nearby floating leaf - occurs when the surface wave, propagating out of the local disturbance, hits it. The surface wave is an emergent behaviour, with its own, new dynamics. In this case, it is obtained by linear response from the Navier-Stokes equations. Similarly, in a large class of many-body systems, strong correlations are expected to occur along trajectories associated with the propagation of ballistic, or slowly decaying modes, such as surface water waves or sound waves, and hydrodynamics is their emergent theory.Despite the simplicity of the above example, a full mathematical understanding of how hydrodynamics emerge from Newton's basic laws of motion, or their refinements in quantum mechanics and relativity, is still missing. Probing, from first principles, the behaviours seen at long times and large distances, and involving a large number of particles, is a monumental task of deep significance. Except for very specific models, there is currently no rigorous proof of hydrodynamic equations in strongly interacting systems whose dynamics is Hamiltonian or more generally reversible and deterministic. Given the ubiquity and apparent universal applicability of the fundamental principles and ideas of hydrodynamics, this is one of the most important challenges of mathematical physics.This project aims at exploring new avenues in this problem, which offer the hope of a rigorous and general treatment. The main hypothesis is that the mathematics of functional analysis, which is fundamentally a theory about infinitly large objects, offers the right framework for emergence in the statistical mechanics description of many-body systems. Instead of attempting to describe specific models, via this universal language one divides the task in two: first, one extracts essential properties as a set of axioms and attempts to derive hydrodynamics from them; second, one shows that such properties hold in families of models.Recently, in the paper [arXiv:2011.00611], the principal investigator has succeeded in showing, in this way, a number of fundamental aspects of the large-time motion of many-body quantum spin systems in one dimension, including the projection onto hydrodynamic modes and the emergence of the linearised Euler equation in a general form.This project aims at developing further this theory, with the goal not only of establishing at some fundamental results in general systems systems of arbitrary dimensionality, but also of exploring the possibilities offered by this new viewpoint for rigorous proofs of hydrodynamics.