  We investigate the quench dynamics of the transverse field Ising model on a
finite fully connected lattice as a prime example of non-equilibrium mean field
dynamics. Using a rate function approach we compute the leading order
corrections to the mean field behavior analytically. Our focus is threefold: i)
We analyze the validity of the mean field approximation and observe that
deviations can occur quickly even for large systems. ii) We study the variance
of the order parameter and identify four qualitative different regions that
cannot be distinguished by solely looking at its mean. iii) We derive the
complete entanglement Hamiltonian for a bipartition of the lattice, which
remarkably turns out to be a time-dependent harmonic oscillator within the
validity of the mean field analysis.
