The effect of discrete-time evolution on thermalisation in a lattice

Particles subject to weak contact interactions in a finite-size lattice tend to thermalise. The Hamiltonian evolution ensures energy conservation and the final temperature is fully determined by the initial conditions. In this work we show that equilibration processes are radically different in lattices subject to discrete-step unitary evolution, which have been implemented in photonic circuits, coin polarisation walkers and time-multiplexed pulses in fibres. Using numerical simulations, we show that weak nonlinearities lead to an equilibrium state of equal modal occupation across all the bands, i.e, a thermal state with infinite temperature and chemical potential. This state is reached no matter the initial distribution of modes and independent of the bands dispersions and gaps. We show that by engineering the temporal periodicity inherent to these systems, equilibration can be accelerated through resonant Floquet channels. Our results demonstrate that discrete time plays a crucial role in wave dynamics, fundamentally altering thermalisation processes within weak wave turbulence theory.