Ergodicity, large-scale correlations and hydrodynamics in many-body systems

Abstract: Long-time behaviours in statistical ensembles of many-body systems are notoriously difficult to access. Hydrodynamics, which is the theory for the emergent large-scale dynamics, gives a lot of information, such as exact asymptotics of correlation functions. It turns out that at the Euler scale, the emergent theory for extended systems is largely universal. I will discuss a number of such universal results in one dimension of space. Some can be shown rigorously: a notion of ergodicity at all frequencies hold for correlation functions in stationary states of all short-range quantum spin chains. The Boltzmann-Gibbs principle, where local observables project onto hydrodynamic modes in two-point functions, and the linearised Euler equations, are also established. The complete space of hydrodynamic modes is the space of ``extensive conserved quantities”, which is defined unambiguously. I will illustrate these concepts in integrable systems, using generalised hydrodynamics. For correlations in non-stationary states, much less is established. I will describe a macroscopic fluctuation theory for the Euler scale which provides a framework for these. In particular, I will explain how a new type of long-range correlations, hitherto not observed, appear when the system is subject to fluid motion, which breaks the paradigm that separate fluid cells are not correlated.

mathematical physics

Audience: researchers in the discipline

One world IAMP mathematical physics seminar

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