Simple observables such as the number of particles might be difficult to describe when a quantum composite system encounters a violent perturbation, for instance in collisions of atomic nuclei. It requires a predictive many-body theory to describe the time evolution of the system accounting for indiscernibility of identical particles. Our goal is rather simple: predicting the probability for a number of transferred particles in such a collision.
To keep the amount of work for the physicist and his computer to a reasonable level, approximations are considered rather than solving the full Schrödinger equation. The Balian-Veneroni variational principle provides a useful theoretical framework to build dynamical microscopic models. To solve this variational principle, one has to choose one particular type of observable of interest. For expectation values of one-body operators, like the average number of transferred particles, this leads to the time-dependent Hartree-Fock (TDHF) theory. For their fluctuations, the Balian-Vénéroni variational principle leads to an equation equivalent to the time-dependent Random Phase Approximation. Examples of applications will be taken within the nuclear physics context.