### Prof. Benjamin Doyon (King’s College London)

## The emergence of hydrodynamics in many-body systems

One of the most important problems of modern science is that of emergence. How do laws of motion emerge at large scales of space and time, from much different laws at small scales? Hydrodynamics offers a basic but very relevant example. Molecules in air simply go along their journey following Newton’s equations. But when there are very many of them, the solution to these equations becomes extremely complicated; seeking the knowledge of each molecule’s individual trajectory is completely impractical. Happily it is also unnecessary. At our human scale, new, different equations emerge for aggregate quantities: those of hydrodynamics. And these are apparently all we need to know in order to understand the weather! Despite its conceptual significance, the passage from microscopic dynamics to hydrodynamics remains a notorious open problem of mathematical physics. This goes much beyond molecules in air: similar principles hold very generally, including in other physical systems such as quantum gases and spin lattices, where the resulting equations themselves can be very different. In particular, integrable models, where an extensive mathematical structure allows us to make progress, admit an entirely new universality class of hydrodynamic equations. In this talk, I will discuss in a pedagogical fashion the general problem and principles, and some recent advances in our understanding, including those obtained in integrable models.