  Energy landscapes are high-dimensional surfaces representing the dependence
of system energy on variable configurations, which determine crucially the
system's emergent behavior but are difficult to be analyzed due to their
high-dimensional nature. In this article, we introduce an approach to reveal
the complete energy landscapes of small spin glasses and Boolean satisfiability
problems, which also unravels their non-equilibrium dynamics at an arbitrary
temperature for an arbitrarily long time. In contrary to our common belief, our
results show that it can be less likely to identify the ground states when
temperature decreases, due to trapping in individual local minima, which ceases
at different time, leading to multiple abrupt jumps with time in the
ground-state probability. Simulations agree well with theoretical predictions
on these remarkable phenomena. Finally, for large systems, we introduce a
variant approach to extract partially the energy landscapes and observe both
analytically and in simulations similar phenomena. This work introduces new
methodology to unravel the non-equilibrium dynamics of glassy systems, and
provides us with a clear, complete and new physical picture on their long-time
behaviors inaccessible by modern numerics.
