Assessing thermalization and estimating the Hamiltonian with output data only

I consider the generic situation where a finite number of identical test systems in varying (possibly unknown) initial states are subjected independently to the same unknown process. I show how one can infer from the output data alone whether or not the process in question induces thermalization, and if so, which constants of the motion characterize the final equilibrium states. In case thermalization does occur and there is no evidence for constants of the motion other than energy, I further show how the same output data can be used to estimate the test systems' effective Hamiltonian. For both inference tasks I devise a statistical framework inspired by the generic techniques of factor and principal component analysis. I illustrate its use in the simple example of qubits.