A new, efficient Monte Carlo algorithm to calculate the density of states and its applications to phase transition problems

An efficient Monte Carlo algorithm is proposed that uses 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 by calculating canonical averages at any temperature. Using this method, we not only can avoid repeating simulations at multiple temperatures, but can also estimate the free energy and entropy, quantities which 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 dissertation, we apply our algorithm to both 1st and 2nd 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 as large as 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 3D +/- 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.;The algorithm is applied to study the behavior of the critical endpoint where a second-order critical line meets, and is truncated, by a first-order phase boundary in the phase diagram. We study an Ising model with two- and three-body interactions on a 2D triangular lattice with an external field. With the 2D random walk to estimate a special 2D density of states G(E ', M'), we can estimate the thermodynamic quantities at any temperature and magnetic field without multiple simulations. Using our efficient random walk algorithm, we study the behavior in the vicinity of the critical endpoint for the Ising model with both first- and second-order phase transitions. The singularities of the curvature for the spectator phase boundary and the order parameters have been observed as predicted. We performed the first quantitative analysis of such singularities at the critical endpoint for this model. Our finite-size analysis shows that the singularity at the critical endpoint is not different from the first-order phase transitions along the spectator phase boundary. We present the first numerical evidence that the critical behaviors do not change when we approach the critical endpoint along the critical line.;This simulational method is not restricted to energy and order-parameter spaces; it can be used to calculate the density of states for any parameter by a random walk in the corresponding space.