Entropy production for velocity-dependent macroscopic forces: The problem of dissipation without fluctuations

In macroscopic systems, velocity-dependent phenomenological forces F(v) are used to model friction, feedback devices or self-propulsion. Such forces usually include a dissipative component which conceals the fast energy exchanges with a thermostat at the environment temperature T, ruled by a microscopic Hamiltonian H. The mapping (H,T) -> F(v)- even if effective for many purposes - may lead to applications of stochastic thermodynamics where an incomplete fluctuating entropy production (FEP) is derived. An enlighting example is offered by recent macroscopic experiments where dissipation is dominated by solid-on-solid friction, typically modelled through a deterministic Coulomb force F(v). Through an adaptation of the microscopic Prandtl-Tomlinson model for friction, we show how the FEP is dominated by the heat released to the T-thermostat, ignored by the macroscopic Coulomb model. This problem, which haunts several studies in the literature, cannot be cured by weighing the time-reversed trajectories with a different auxiliary dynamics: it is only solved by a more accurate stochastic modelling of the thermostat underlying dissipation.