In the present note, a summary of selected aspects of timedependent
mean-field theory is first recalled. This approach is optimized to
describe one-body degrees of freedom. A special focus is made on how this microscopic
theory can be reduced to a macroscopic dynamic for a selected set of
collective variables. Important physical phenomena like adiabaticity/diabaticity,
one-body dissipation or memory effect are discussed. Special aspects related
to the use of a time-dependent density functional instead of a time-dependent
Hartree-Fock theory based on a bare hamiltonian are underlined. The absence
of proper description of complex internal correlations however strongly impacts
the predictive power of mean-field. A brief overview of theories going beyond
the independent particles/quasi-particles theory is given. Then, a special attention
is paid for finite fermionic systems at low internal excitation. In that case,
quantum fluctuations in collective space that are poorly treated at the meanfield
level, are important. Several approaches going beyond mean-field, that are
anticipated to improve the description of quantum fluctuations, are discussed:
the Balian-V´en´eroni variational principle, the Time-Dependent Random Phase
Approximation and the recently proposed Stochastic Mean-Field theory. Relations
between these theories are underlined as well as their advantages and
shortcomings.
Keywords: Density functional theory, Correlations, Quantum fluctuations.
