Energy landscape of a spin-glass model: exploration and characterization

Complete realization of energy landscapes and non-equilibrium trapping dynamics in small spin glass and optimization problems

Energy landscapes are high-dimensional surfaces underlie all physical systems, which determine crucially the energetic and behavioral dependence of the systems on variable configurations, but are difficult to be analyzed due to their high-dimensional nature. Here we introduce an approach to reveal for the complete energy landscapes of spin glasses and Boolean satisfiability problems with a small system size, and unravels their non-equilibrium dynamics at an arbitrary temperature for an arbitrarily long time. Remarkably, our results show that it can be less likely for the system to attain ground states when temperature decreases, due to trapping in individual local minima, which ceases at a different time, leading to multiple abrupt jumps in the ground-state probability. For large systems, we introduce a variant approach to extract partially the energy landscapes and observe both semi-analytically and in simulations similar phenomena. This work introduces new methodology to unravel the energy landscapes and non-equilibrium dynamics of glassy systems, and provides us with a clear, complete and new physical picture on their long-time behaviors inaccessible by existing approaches.

Memory effect and phase transition in a hierarchical trap model for spin glasses

We introduce an efficient dynamical tree method that enables us to explicitly demonstrate the thermoremanent magnetization memory effect in a hierarchical energy landscape. Our simulation nicely reproduces the nontrivial waiting-time and waiting-temperature dependences in this nonequilibrium phenomenon. We further investigate the condensation effect, in which a small set of microstates dominates the thermodynamic behavior in the multilayer trap model. Importantly, a structural phase transition of the multilayer tree model is shown to coincide with the onset of the condensation phenomenon. Our results underscore the importance of hierarchical structure and demonstrate the intimate relation between the glassy behavior and structure of barrier trees.

Exploring experimental fitness landscapes for chemical synthesis and property optimization

The topology of experimental fitness landscapes for chemical optimization objectives is assessed through svr-based HDMR modeling.

Energy landscapes of resting-state brain networks

During rest, the human brain performs essential functions such as memory maintenance, which are associated with resting-state brain networks (RSNs) including the default-mode network (DMN) and frontoparietal network (FPN). Previous studies based on spiking-neuron network models and their reduced models, as well as those based on imaging data, suggest that resting-state network activity can be captured as attractor dynamics, i.e., dynamics of the brain state toward an attractive state and transitions between different attractors. Here, we analyze the energy landscapes of the RSNs by applying the maximum entropy model, or equivalently the Ising spin model, to human RSN data. We use the previously estimated parameter values to define the energy landscape, and the disconnectivity graph method to estimate the number of local energy minima (equivalent to attractors in attractor dynamics), the basin size, and hierarchical relationships among the different local minima. In both of the DMN and FPN, low-energy local minima tended to have large basins. A majority of the network states belonged to a basin of one of a few local minima. Therefore, a small number of local minima constituted the backbone of each RSN. In the DMN, the energy landscape consisted of two groups of low-energy local minima that are separated by a relatively high energy barrier. Within each group, the activity patterns of the local minima were similar, and different minima were connected by relatively low energy barriers. In the FPN, all dominant local minima were separated by relatively low energy barriers such that they formed a single coarse-grained global minimum. Our results indicate that multistable attractor dynamics may underlie the DMN, but not the FPN, and assist memory maintenance with different memory states.

Convergence of the Equi-Energy Sampler and Its Application to the Ising Model

We provide a complete proof of the convergence of a recently developed sampling algorithm called the equi-energy (EE) sampler (Kou, Zhou, and Wong, 2006) in the case that the state space is countable. We show that in a countable state space, each sampling chain in the EE sampler is strongly ergodic a.s. with the desired steady-state distribution. Furthermore, all chains satisfy the individual ergodic property. We apply the EE sampler to the Ising model to test its efficiency, comparing it with the Metropolis algorithm and the parallel tempering algorithm. We observe that the dynamic exponent of the EE sampler is significantly smaller than those of parallel tempering and the Metropolis algorithm, demonstrating the high efficiency of the EE sampler.

Random walk over basins of attraction to construct ising energy landscapes

An efficient algorithm is developed to construct disconnectivity graphs by a random walk over basins of attraction. This algorithm can detect a large number of local minima, find energy barriers between them, and estimate local thermal averages over each basin of attraction. It is applied to the Sherrington-Kirkpatrick (SK) spin glass Hamiltonian where existing methods have difficulties even for a moderate number of spins. Finite-size results are used to make predictions in the thermodynamic limit that match theoretical approximations and recent findings on the free energy landscapes of SK spin glasses.

Efficient exploration of discrete energy landscapes

Many physical and chemical processes, such as folding of biopolymers, are best described as dynamics on large combinatorial energy landscapes. A concise approximate description of the dynamics is obtained by partitioning the microstates of the landscape into macrostates. Since most landscapes of interest are not tractable analytically, the probabilities of transitions between macrostates need to be extracted numerically from the microscopic ones, typically by full enumeration of the state space or approximations using the Arrhenius law. Here, we propose to approximate transition probabilities by a Markov chain Monte Carlo method. For landscapes of the number partitioning problem and an RNA switch molecule, we show that the method allows for accurate probability estimates with significantly reduced computational cost.