Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram

• Published in: Physical review. E, Statistical, nonlinear, and soft matter physics Vol. 64; no. 5 Pt 2; p. 056101
• Main Authors: Wang, F, Landau, D P
• Format: Journal Article
• Language: English
• Published: United States 01.11.2001
• ISSN: 1539-3755
• DOI: 10.1103/physreve.64.056101

We describe an efficient Monte Carlo algorithm using a random walk in energy space to obtain a very accurate estimate of the density of states for classical statistical models. The density of states is modified at each step when the energy level is visited to produce a flat histogram. By carefully controlling the modification factor, we allow the density of states to converge to the true value very quickly, even for large systems. From the density of states at the end of the random walk, we can estimate thermodynamic quantities such as internal energy and specific heat capacity by calculating canonical averages at any temperature. Using this method, we not only can avoid repeating simulations at multiple temperatures, but we can also estimate the free energy and entropy, quantities that are not directly accessible by conventional Monte Carlo simulations. This algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability. As with the multicanonical Monte Carlo technique, our method overcomes the tunneling barrier between coexisting phases at first-order phase transitions. In this paper, we apply our algorithm to both first- and second-order phase transitions to demonstrate its efficiency and accuracy. We obtained direct simulational estimates for the density of states for two-dimensional ten-state Potts models on lattices up to 200 x 200 and Ising models on lattices up to 256 x 256. Our simulational results are compared to both exact solutions and existing numerical data obtained using other methods. Applying this approach to a three-dimensional +/-J spin-glass model, we estimate the internal energy and entropy at zero temperature; and, using a two-dimensional random walk in energy and order-parameter space, we obtain the (rough) canonical distribution and energy landscape in order-parameter space. Preliminary data suggest that the glass transition temperature is about 1.2 and that better estimates can be obtained with more extensive application of the method. This simulational method is not restricted to energy space and can be used to calculate the density of states for any parameter by a random walk in the corresponding space.