Energy landscape

An energy landscape assigns an energy to each possible state and describes its variation between neighbouring states. Its local minima and intervening energy barriers help distinguish favourable states from states that a specified dynamics can reach quickly. In a Hopfield network, the assigned energy is an objective function and need not be a conserved physical quantity.

At fixed environmental constraints, integrate out unobserved coordinates to obtain an equilibrium probability density for a chosen reaction coordinate $x$. Relative to a specified coordinate measure, the displayed free energy reproduces that density. A change of coordinate can add a Jacobian term, so the profile is not independent of its coordinate convention. Minima identify probable basins, while barriers relative to an initial basin contribute to activated kinetics; a kinetic prefactor still requires dynamics. A conjugate force tilts the profile by $-Fx$.

A disconnectivity graph summarizes how local minima of an energy landscape become connected as the allowed maximum energy increases. Each terminal branch represents a basin around a local minimum. Two branches merge at the lowest threshold for which a permitted path joins their basins without exceeding that threshold. Vertical height therefore carries barrier information; horizontal spacing is usually chosen for readability.

For example, two low-energy states may sit at similar energies but merge only high in the graph, indicating a difficult transition between them. A cluster of branches merging low down instead suggests easy rearrangement within that group. The graph is useful for separating energetic preference from kinetic accessibility in thermal activation. It does not determine transition rates by itself: the allowed dynamics, path widths and kinetic prefactors remain necessary. Changing the permitted moves changes which paths exist and can therefore change the graph.

An energy barrier is an increase in energy that must be overcome along an allowed route between states. Its value depends on which paths or single-step moves are allowed; specifying only the energies of the endpoints does not determine the barrier.