  At finite temperature, the field along a linear stretch of correlation length
size is supposed to trace the shortest path in the field space given the two
end point values, known as the Geodesic rule. In this study, we compute the
probability that, the field variations over distances of correlation length
follow this rule in theories with $O(2)$ global symmetry. We consider a simple
ferromagnetic $O(2)$ spin-model and a complex $\phi^4$ theory. The computations
are carried out on an ensemble of equilibrium configurations, generated using
Monte Carlo simulations. The numerical results suggest significant deviation to
the Geodesic rule, relevant for formation of topological defects during quench
in 2nd order phase transition. Also for the case of $O(2)-$spins in two
dimensions, distribution and density of vortices, have been studied. It is
found that, for quench temperatures close to the transition point, the
Kibble-Zurek Mechanism underestimates equilibrium density of defects. The
exponents corresponding to width of the distributions, are found to be smaller
than Kibble Mechanism estimates and match only when there is no deviation from
the geodesic rule.
