  In the real world, one almost never knows the parameters of a thermodynamic
process to infinite precision. Reflecting this, here we investigate how to
extend stochastic thermodynamics to systems with uncertain parameters,
including uncertain number of heat baths / particle reservoirs, uncertainty in
the precise values of temperatures / chemical potentials of those reservoirs,
uncertainty in the energy spectrum, uncertainty in the control protocol, etc.
We formalize such uncertainty with an (arbitrary) probability measure over all
transition rate matrices satisfying local detailed balance. This lets us define
the effective thermodynamic quantities by averaging over all LDB-obeying rate
matrices. We show that the resultant effective entropy violates the second law
of thermodynamics. In contrast to the effective entropy though, the expected
stochastic entropy, defined as the ensemble average of the effective
trajectory-level entropy, satisfies the second law. We then and explicitly
calculate the second-order correction to the second law for the case of one
heat bath with uncertain temperature. We also derive the detailed fluctuation
theorem for expected effective trajectory entropy production for this case, and
derive a lower bound for the associated expected work. Next, to ground these
formal considerations with experimentally testable bounds on allowed
energetics, we derive a bound on the maximal work that can be extracted from
systems with arbitrarily uncertain temperature. We end by extending previous
work on "thermodynamic value of information", to allow for uncertainty in the
time-evolution of the rate matrix.
