Construction of a Pathway Map on a Complicated Energy Landscape

Abstract

How do we search for the entire family tree without unwanted random guesses, starting from a high-index and high-energy stationary state on the energy landscape? Here we introduce a general numerical method that constructs the pathway map clearly connecting all stationary points branched from a single parent state. The map guides our understanding of how a physical system moves on the energy landscape. In particular, the method allows us to identify the transition state between energy minima and the energy barrier associated with such a state. As an example, we solve the Landaude Gennes energy incorporating the Dirichlet boundary conditions to model a liquid crystal confined in square box; we illustrate the basic concepts by examining the multiple stationary solutions and the connected pathway maps of the model.

Dynamic tuning of the director field in liquid crystal shells using block copolymers

“…The experimentally observable states are modeled by the locally stable points, i.e., local or global minimizers, of the LdG free energy. We point out that there are typically multiple local energy minimizers for highly nonlinear and nonconvex problems such as the LdG minimization problem and while the global minimizer may be most frequently observed, local minimizers also have a basin of attraction and stability [65]. In particular, in an experiment, a local minimizer will not relax to the global minimizer with minimum free energy because all stable states are separated by an energy barrier.…”

“…Secondly, compared to the experimental shells, the simulated shells are small and relatively thick, since computational resolution renders it difficult to extract structural information for thinner shells. Finally, these complex systems have multiple stable configurations [65] and the gradient flow algorithm may converge to a locally stable configuration. Indeed, our direct simulations of hybrid director fields in Fig.…”

Computable EarlyC. elegansEmbryo with a Data-driven Phase Field Model

“…However, as the cell number increases with the C. elegans development, it will become computationally too expensive if we want to enumerate all possible stable developmental paths. This problem might be solved by the recently proposed computational approach of constructing the solution landscape to identify all possibilities (Yin et al, 2020). Second, the morphogenesis of a compressed embryo may involve more subtle and complicated regulations such as contraction and polarization when the number of cells is large (Lee et al, 2006;Tao et al, 2020).…”

Modelling and computation of liquid crystals

“…In recent years we have incorporated the spectral method with energy-minimization techniques and successfully applied our approach to different confined LC systems, including three-dimensional spherical droplets (Hu, Qu and Zhang 2016), three-dimensional cylinders (Hu et al 2016, Han, Hu, Zhang and, nematic shells (Qu, Wei and Zhang 2017), nematic wells (Yin et al 2020a) and LC colloids , Tong, Wang and Zhang 2017, Wang, Zhang and Chen 2018. From a computational perspective, as an efficient numerical method with high accuracy, the spectral method makes an accurate free-energy calculation for 3D problems possible and enables us to determine the phase diagram of some complicated LC systems .…”

“…In a recent work (Yin, Zhang and Zhang 2019), high-index saddle dynamics was proposed to efficiently compute any-index saddle points. By applying the high-index optimization-based shrinking dimer method, a solution landscape, which is a pathway map of all connected solutions, can be constructed for NLCs confined on a square domain (Yin et al 2020a).…”