Dr. Jianyuan Yin, National University of Singapore

Jie Xu and Weiying Zheng

Construction of Solution Landscape on a Complicated Energy Landscape

15:30-17:30 June 19 (Wednesday) 2024, N733

The energy landscape has been widely applied to many physical and chemical systems. Transition pathways on the energy landscape represent the most probable routes of phase transition between different stable states, and index-1 saddle points on the transition pathways correspond to the transition states. On the energy landscape of a complex system, there could exist multiple metastable states, and accordingly, multiple transition pathways connect these states as a connected graph. How do we search for a comprehensive graph of possible transition states on a complicated energy landscape, without unwanted random guesses of transition states or prior knowledge of transition pathways? Here I will introduce a general numerical method that constructs the solution landscape, which is a pathway map of stationary points of the energy, starting from a saddle point all the way down to multiple energy minima. The key idea of the solution landscape is downward search, which provides convenience for finding multiple transition states systematically. We develop a general and efficient saddle dynamics method to implement downward search to find a saddle point with a given index, and generalize it to constrained cases and dynamical systems. An upward search method is also developed to find a starting point of downward search. The solution landscape guides our understanding of how a physical system moves on the energy landscape and identifies multiple transition states between energy minima.

Several numerical examples will be presented to demonstrate the effectiveness of the method. First, we solve the Landau-de Gennes energy to model nematic liquid crystals confined in a shallow square well, and multiple equilibrium configuration is identified with defect structures. We also identify non-axisymmetric critical points of the Onsager model. We consider phase transition between crystals and quasicrystals with a Lifshitz-Petrich free energy; Two possible transition pathways are identified to connect two-dimensional hexagonal crystalline and dodecagonal quasicrystalline phases, involving a lamellar quasicrystalline phase as a metastable state or a moving interface. As a constrained example, we identify vortex states of two-dimensional rotational Bose-Einstein condensates and reveal four excitation mechanisms. The successive quantum phase transitions with increasing rotational frequencies can be explained in our results.