A family of integrable Hamiltonian systems of classical particles with arbitrary two-body scattering shifts
Benjamin Doyon (King’s College)
Some special systems of classical particles are integrable in the sense of Liouville: there are as many conserved quantities, including the Hamiltonian itself, as there are particles. When this happens, the system is typically not chaotic, and many of its properties can be evaluated exactly. It is interesting to find many-body systems that are integrable: interaction potentials that can written for any number of particles N, and such that the system is integrable for all N. When the interaction is local enough, such a many-body integrable system has the property of elastic and factorised scattering: in any scattering event, the sets of in- and out-momenta are the same. The trajectories are still shifted from linear trajectories due to the interaction, but these shifts are « factorised » into two-body shifts, as if the many-body scattering happened as a succession of separate two-body scattering events. Amongst the known locally interacting integrable models, only very specific functions of momenta do play the role of two-body shifts. I will overview a new family of many-body integrable systems for which the scattering two-body function can be taken as any (differentiable, positive) function. It can be seen as a certain flow on interactions, starting from free particles, and produced by perturbing by bilinear terms in conserved densities and currents. If time permits, I will explain how we evaluate the thermodynamics of this system, which gives the classical version of the so-called thermodynamic Bethe ansatz, and its hydrodynamics, which gives generalised hydrodynamics.