# Broad Histogram Algorithm Comparison for the Square Well Fluid

# Broad Histogram Algorithm Comparison for the Square Well Fluid

#### Tanner Simpson

Thermodynamic properties of systems are often investigated computationally. Traditionally, thermal physics simulations are limited by their very small energy ranges and slow convergence. Broad histogram algorithms are a class of Monte Carlo algorithms that can explore an entire energy (and temperature) range in one ther- mal physics simulation: potentially saving months of compute time. In this paper, we investigate broad histogram methods designed inside (SAD, TMI, and TOE) and outside (Wang-Landau, Transition Matrix Monte Carlo, Wang Landau Transition Matrix Monte Carlo, and Stochastic Approximation Monte Carlo) of the Roundy re- search group. The square well potential of thermodynamics serves as the algorithm testing platform. This thesis covers the motivation and theory behind each histogram method. We then investigate algorithm performance on two sets of system configura- tions by analyzing their uncertainty and error when computing each system’s entropy over time.

Overall, three algorithms developed in group, SAD, TMI, and TOE consistently converged to low errors. In addition, SAD, TMI, and TOE were straightforward to prepare for simulation, as there aren’t any user defined parameters. Wang-Landau Transition Matrix Monte Carlo (WLTMMC) converged to low error and low uncer- tainty rapidly, but required a predefined energy range. This research demonstrates that the popular Wang-Landau algorithm (while collecting lots of independent sam- ples) does not work well for all systems and can converge to incorrect values of a system’s entropy.