Construction of a Pathway Map on a Complicated Energy Landscape

How do we search for the entire family tree of possible intermediate states, without unwanted random guesses, starting from a stationary state on the energy landscape all the way down to energy minima? Here we introduce a general numerical method that constructs the pathway map, which guides our understanding of how a physical system moves on the energy landscape. The method identifies the transition state between energy minima and the energy barrier associated with such a state. As an example, we solve the Landau–de Gennes energy incorporating the Dirichlet boundary conditions to model a liquid crystal confined in a square box; we illustrate the basic concepts by examining the multiple stationary solutions and the connected pathway maps of the model.

Figure 1

Illustration of (a) an energy landscape of a 2D energy surface and (b) the pathway map starting from the maximum, $A_1$ down to minima $C_1$ and $C_2$. Two saddle points $B_1$ and $B_2$ are connected to $A_1$. The indices labeled in (b) are those according to the Morse definition.

Figure 2

Pathway maps found from a confined 2D LdG model with parallel orientational alignment at the boundaries. From (a) to (f), the evolution of the pathway maps are illustrated as $\alpha$, a system parameter proportional to the area of the confining square, increases from (a) $\alpha =5$, (b) 7, (c) 15, (d) 25, (e) 30, to (f) 50. The color of the nodes specifies the Morse index of saddle points. The solid arrows indicate how lower-index saddles can be deduced from higher ones. An example of upward search is represented by the dashed arrows. The height of a node approximately corresponds to its energy. The seventeen defect states are illustrated on the right panel, where the color represents the relative magnitude of directional ordering and the white bars the nematic field directions [31]. The insets further explain the map details discussed in the text.