  We formulate a statistical model of two sequential measurements and prove a
so-called J-equation that leads to various diversifications of the well-known
Jarzynski equation including the Crooks dissipation theorem. Moreover, the
J-equation entails formulations of the Second Law going back to Wolfgang Pauli.
We illustrate this by an analytically solvable example of sequential discrete
position-momentum measurements accompanied with the increase of Shannon
entropy. The standard form of the J-equation extends the domain of applications
of the quantum Jarzynski equation in two respects: It includes systems that are
initially only in local equilibrium and it extends this equation to the cases
where the local equilibrium is described by microcanononical, canonical or
grand canonical ensembles. Moreover, the case of a periodically driven quantum
system in thermal contact with a heat bath is shown to be covered by the theory
presented here. Finally, we shortly consider the generalized Jarzynski equation
in classical statistical mechanics.
