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Stochastic thermodynamics is formulated under the assumption of perfect knowledge of all thermodynamic parameters. However, in any real-world situation, there is nonzero uncertainty about the precise value of temperatures, chemical potentials, energy spectrum, etc.
Here we investigate how this uncertainty modifies the theorems of stochastic thermodynamics. We consider two scenarios: in the effective scenario, we fix the (unknown, randomly generated) parameters of an experimental apparatus and then repeatedly observe (stochastic) trajectories of the system for that fixed apparatus. In contrast, in the (consistently with the effective scenario) phenomenological scenario, the (unknown) apparatus is re-generated for each trajectory.
We derive expressions for thermodynamic quantities in both scenarios. For the effective scenario, we also discuss the physical interpretation of entropy production (EP) and derive the mismatch cost. To illustrate this scenario, we also provide a numerical analysis of the thermodynamics of a quantum dot implementing bit erasure with uncertain temperature.
We also analyze the protocol for changing the distribution over states in a way that maximizes work extraction, again in the effective scenario. Next, we investigate the effective thermodynamic value of information, focusing on the case where there is a delay between the initialization of the system and the start of the protocol.
Finally, we derive the detailed and integral fluctuation theorems (FTs) for the phenomenological EP. In particular, we show how the phenomenological FTs account for the fact that the longer a trajectory runs, the more information it provides concerning the precise experimental apparatus, and therefore the less EP it generates.
Our results provide a very preliminary investigation of the myriad issues that arise when one tries to expand stochastic thermodynamics to account for uncertainty in the parameters governing a physical process.