Ballistic macroscopic fluctuation theory and long-range correlations
Evaluating fluctuations and correlations at large scales of space and time in quantum and classical many-body systems, in and out of equilibrium, is one of the most important problems of emergent physics. I will explain how basic hydrodynamic principles in fact give access to exact results at the ballistic scale, solely from the data of the Euler-scale hydrodynamic equations of the many-body system. This includes the large-deviation theory for fluctuations of space-time-integrated currents and densities, and the Euler scaling limits of multi-point correlation functions. In particular, a new theory that we developed recently (with G. Perfetto, T. Sasamoto and T. Yoshimura), dubbed the ``ballistic macroscopic fluctuation theory” (BMFT), is an adaptation of the well-known macroscopic fluctuation theory that has been very successful for purely diffusive systems. A surprising new result from the BMFT is that generically, long-range spatial correlations develop over time if the initial state of the many-body system is spatially inhomogeneous. Therefore, the ``fluid cells” of Euler hydrodynamics are in fact generically correlated amongst each other, something which had not been appreciated or even observed before. I will give examples of integrable systems based on generalised hydrodynamics, and present numerical confirmations of the results in the hard rod gas.