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

Assessing thermalization and estimating the Hamiltonian with output data only