  Here we numerically solve the equations derived in part I of this two-part
series and verify their validity. In particular we use the functional
Renormalisation Group (fRG) flow equations to obtain effective potentials for
initially highly anharmonic and non-polynomial potentials, including potentials
with multiple trapping wells and barriers, and at different temperatures. The
numerical computations determining the effective action are much faster than
the direct simulation of the stochastic dynamics to which we compare our fRG
results. We benchmark our numerical solutions to the flow equations by
comparing the first two equilibrium cumulants from the fRG against the
Boltzmann distribution. We obtain excellent agreement between the two methods
demonstrating that numerical solutions for the effective potential can be
accurately obtained in all the highly unharmonic cases we examined. We then
assess the utility of the effective potential to describe the equilibrium
2-point correlation function $\langle x(0)x(t) \rangle$ and the relevant
correlation time. We find that when Wavefunction Renormalisation is also
utilized, these are obtained to percent accuracy for temperatures down to the
typical height of the potentials' barriers but accuracy is quickly lost for
lower temperatures. We also show how the fRG can offer strong agreement with
direct numerical simulation of the nonequilibrium evolution of average position
and variance. Also, the fRG solution represents the whole ensemble average,
further adding to its convenience over other techniques, such as direct
numerical simulations or solving the Fokker-Planck diffusion equation, which
require multiple solutions with different initial conditions to construct
averages over an ensemble.
