  By combining the Baeriswyl wavefunction with equilibrium and time-dependent
variational principles, we develop a non-equilibrium formalism to study quantum
quenches for two dimensional spinless fermions with nearest-neighbour hopping
and repulsion. The variational ground state energy and the short time dynamics
agree convincingly with the results of numerically exact simulations. We find
that depending on the initial and final interaction strength, the quenched
system either exhibits undamped oscillations or relaxes to a time independent
steady state. The time averaged expectation value of the CDW order parameter
rises sharply when crossing from the steady state regime to the oscillating
regime, indicating that the system, being non-integrable, shows signs of
thermalization with an effective temperature above or below the equilibrium
critical temperature, respectively.
