Calculation of for the Ising model with via replica exchange simulation and macrostate Markov chain methods. (a) Twenty-four simulated energy distributions corresponding to temperatures (from left to right): 1, 1.25, 1.45, 1.6, 1.75, 1.9, 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.65, 2.8, 3.0, 3.25, 3.6, 4.0, 4.5, 5.2, 6.0, 7.5, 10, and 15. (b) Representative simulated functions along with exact results. The inset shows the associated relative errors.
Convergence of for the iterative macrostate Markov chain methods applied to the Ising model with . All methods computed from the stochastic matrix produced by the replica exchange simulation from Fig. 1. Results for both vector-matrix (bottom axis) and matrix powering (top axis) schemes are shown.
Convergence of for the Ising model obtained from different multicanonical simulation schemes. Estimations of via transition matrix techniques were performed using either the GTH or sequential detailed balance methods. “Delay” denotes that energy transitions were recorded only after in simulations involving WL updates. (a) , 32, or 64 with . (b) with computed from simulation.
Simulated and exact results for Ising models with (from bottom to top) , 32, and 64 calculated using the GTH algorithm after flip attempts with . The associated relative error functions (averaged over ten runs) are plotted in the inset.
Simulated energy distributions and results for replica exchange and multicanonical simulations of the lattice protein. (a) Replica exchange canonical ensemble energy distributions corresponding to temperatures (from left to right): 0.8, 1.35, 1.6, 2.3, 3.5, and 12.0. (b) curves generated from macrostate Markov chain methods or multiple histogram reweighting (left axis), and flat energy distributions produced from GTH-WL simulations (right axis). The caption in (b) highlights the convergence of the iterative Markov chain procedures.
Results from replica exchange simulations of the Lennard-Jones system with different values. (a) Energy distributions corresponding to temperatures (from left to right): 0.8, 0.88, 0.98, 1.1, 1.24, 1.40, 1.58, 1.78, 2.0, and 2.24. (b) Simulated functions. For clarity, the curves are displaced and nonconsecutive data points are shown. The symbols represent (from bottom to top) the GTH algorithm with (, ), (, ), (, ), (, ), (, ), and (, ), and the sequential method with and , 0.3, and 0.5. All curves are compared to multiple histogram reweighting results.
Mean relative deviations in (relative to a common energy reference) between transition matrix and histogram reweighting values plotted as a function of the macrostate ensemble for different .
Energy transition probability matrices generated from the Lennard-Jones replica exchange simulation of Fig. 6. with and (a) or (b) . For clarity, nonconsecutive points are plotted.
Representative results for (solid line and circles, left axis) and the sampled energy distribution (broken line, right axis) from GTH-WL simulations of the Lennard-Jones system with and . For clarity, nonconsecutive points are shown. Histogram reweighting data from Fig. 6 are also plotted for comparison.
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