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In this note, we discuss the low and high temperature contribution ofThermodynamic Bethe Ansatz (TBA) dressed excitations in the thermodynamics andenergy-magnetization relaxation within the Generalized Hydrodynamics approachin the linear response regime. In particular, we show how the temperaturedependent dispersions of the excitations reproduce well known behavior of thespecific heat, magnetic susceptibility, spin and energy Drude weights. In thiscontext, we derive a further formulation of the Drude weights from the finitewavevector relaxation. Furthermore, we contrast the TBA description ofthermodynamics and dynamics in terms of a multitude of string excitations tothat in terms of a single quasi-particle in low energy effective theories.

Dressed excitations, thermodynamics and relaxation in the 1D XXZ Heisenberg model