  We reformulate stochastic thermodynamics in terms of noise realizations for
Langevin systems in contact with multiple reservoirs and investigated the
structure of the second laws of thermodynamics. We derive a hierarchy of
fluctuation theorems when one degree of freedom of the system is affected by
multiple reservoirs simultaneously, that is, when noise mixing occurs. These
theorems and the associated second laws of thermodynamics put stricter bounds
on the thermodynamics of Langevin systems. We apply our results to a stochastic
machine in noise-mixing environments and demonstrate that our new bounds play a
crucial role in determining the potential function and performance of the
machine.
