Stochastic thermodynamics in uncertain environments

Conventional stochastic thermodynamics assumes perfect knowledge of the parameters governing how a system interacts with its environment (number of baths, their temperatures, chemical potentials, precise rate matrices, etc.). In reality, none of these parameters is known precisely. We explore the thermodynamic implications of such uncertainty in the parameter vector. We assume a probability distribution over parameter vectors, which is sampled before the process begins, and the system then evolves according to that (unknown) parameter vector. We first investigate whether the modified definitions of the standard thermodynamic quantities such that the parameter-averaged state distribution evolving under the parameter-averaged parameter vector will obey the standard laws of thermodynamics. As we show, the first law is still obeyed with parameter-averaged definitions of heat, work, and internal energy. However, the Shannon entropy of the parameter-averaged state distribution can violate the second law. On the other hand, we can use the parameter-averaged stochastic (i.e., trajectory-level) entropy to construct two quantities fulfilling the second law at the ensemble level. We investigate the relation between them and establish a connection to observable quantities in an experiment.