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

多体系统中流体动力学的出现：功能分析的新严格途径

基本信息

批准号：
EP/W000458/1
负责人：
Benjamin Doyon
金额：
$ 10.04万
依托单位：
King's College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW000458%2F1
关键词：
Emergence hydrodynamics many body systems

项目摘要

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.

现代科学最深奥的思想之一就是涌现。在一个由大量组成成分（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，而大尺度是大多数物理相关观察发生的地方。从短尺度、微观运动到大规模、涌现的集体行为的过渡是现代理论和数学物理中一些最重要问题的核心。以传播的表面水波为例。在稳定的水面上，一个局部的扰动——比如一根手指碰到水面——会在微观距离上产生复杂的水分子重排。但是，对任何距离足够远的局部探测器——比如附近的一片漂浮的叶子——最强烈的影响发生在表面波从局部扰动传播到它身上的时候。表面波是一种突发行为，有它自己的新动态。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，强相关性预计会沿着与弹道传播或缓慢衰减模式（如表面水波或声波）相关的轨迹发生，而流体力学是它们的新兴理论。尽管上面的例子很简单，但对于流体力学是如何从牛顿的基本运动定律中产生的，或者它们在量子力学和相对论中的改进，仍然缺乏一个完整的数学理解。从第一性原理出发，探索在长时间、远距离、涉及大量粒子的行为，是一项具有深远意义的艰巨任务。除了非常特殊的模型外，目前还没有严格的证据证明在动力学是哈密顿的或更普遍的可逆和确定性的强相互作用系统中的流体动力学方程。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，这是数学物理最重要的挑战之一。该项目旨在探索解决这一问题的新途径，为严谨和普遍的治疗提供了希望。主要的假设是，泛函分析的数学，基本上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。不是试图描述具体的模型，而是通过这种通用语言将任务分为两部分：首先，提取基本属性作为一组公理，并试图从中推导流体力学；第二，一个证明了这些属性在模型族中是成立的。最近，在论文[arXiv:2011.00611]中，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，包括对流体动力模式的投影和线性化欧拉方程的一般形式的出现。本项目旨在进一步发展这一理论，其目标不仅是在任意维的一般系统中建立一些基本结果，而且还探索这一新观点为流体力学的严格证明提供的可能性。