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

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.