Universal constraints on energy flow and SYK thermalization

We study the dynamics of a quantum system in thermal equilibrium that is suddenly coupled to a bath at a different temperature, a situation inspired by a particular black hole evaporation protocol. We prove a universal positivity bound on the integrated rate of change of the system energy which holds perturbatively in the system-bath coupling. Applied to holographic systems, this bound implies a particular instance of the averaged null energy condition. We also study in detail the particular case of two coupled SYK models in the limit of many fermions using the Schwinger-Keldysh non-equilibrium formalism. We solve the resulting Kadanoff-Baym equations both numerically and analytically in various limits. In particular, by going to low temperature, this setup enables a detailed study of the evaporation of black holes in JT gravity.