Minimal dissipation with viscoelastic baths in weakly driven processes

We investigate the thermodynamics of overdamped systems weakly driven by time-dependent protocols while interacting with viscoelastic heat baths. Using a generalized Langevin equation with memory, we derive the conditions under which the friction kernel ensures thermodynamic consistency, notably requiring the addition of a Dirac delta. Within linear response theory, we compute the relaxation function and relaxation time for two classes of protocols: moving and stiffening harmonic traps. Surprisingly, we find that viscoelastic memory does not always hinder relaxation; in certain cases, it accelerates it by reducing the effective relaxation time, leading to lower dissipation. We also derive optimal protocols that minimize the irreversible work and show how they are modified by the presence of the persistence time of the viscoelastic heat bath. Our results reveal that memory effects in the overdamped regime leave measurable thermodynamic signatures, depending on the protocol, with direct implications for controlling complex systems.