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

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.