A thermodynamic uncertainty relation for a system with memory

We introduce an example of thermodynamic uncertainty relation (TUR) for systems modeled by a generalised Langevin dynamics with memory, determining the motion of a micro-bead driven in a complex fluid. Contrary to TURs typically discussed in the previous years, our observables and the entropy production rate are one-time variables. The bound to the signal-to-noise ratio (SNR) of such state-dependent observables can only in some cases be mapped to the entropy production rate. For example, this is true for steady states and for a subclass of Markovian systems. In fact, the presence of memory in the system complicates the thermodynamic interpretation of the uncertainty relation.