We compare the convergence of several flat-histogram methods applied to the 2D Ising model, including the recently introduced stochastic approximation with a dynamic update factor (SAD) method. We compare this method with the Wang-Landau (WL) method, the $1/t$ variant of the WL method, and standard stochastic approximation Monte Carlo (SAMC). In addition, we consider a procedure WL followed by a "production run" with fixed weights that refines the estimation of the entropy. To our knowledge, this work is the first to test this approach against other methods. We find that WL followed by a production run \emph{does} converge to the true density of states, in contrast to pure WL. Three of the methods converge robustly: SAD, $1/t$-WL, and WL followed by a production run. Of these, SAD does not require \emph{a priori} knowledge of the energy range. This work also shows that WL followed by a production run performs superior to other forms of WL while ensuring both ergodicity and detailed balance.
We show in this paper the results on the phase transition of the so-called fully frustrated simple cubic lattice with the Ising spin model. We use here the Monte Carlo method with the flat energy-histogram Wang-Landau technique which is very powerful to detect weak first-order phase transition. We show that the phase transition is clearly of first order, providing a definite answer to a question raised 25 years ago.
We extended the Broad Histogram Method in order to obtain spectral degeneracies for systems with multiparametric Hamiltonians. As examples we obtained the critical lines for the square lattice Ising model with nearest and next-nearest neighbor interactions and the antiferromagnetic Ising model in an external field. For each system, the entire critical line is obtained using data from a single computer run. We also discuss the accuracy and efficiency of our method.
In this letter we present a flat histogram algorithm based on the pruned and enriched Rosenbluth method (PERM). This algorithm incorporates in a straightforward manner microcanonical reweighting techniques, leading to "flat histogram" sampling in the chosen parameter space. As an additional benefit, our algorithm is completely parameter free and, hence, easy to implement. We apply this algorithm to interacting self-avoiding walks (ISAW), the generic lattice model of polymer collapse.
The diagrammatic Monte Carlo (Diag-MC) method is a numerical technique which samples the entire diagrammatic series of the Green's function in quantum many-body systems. In this work, we incorporate the flat histogram principle in the diagrammatic Monte method and we term the improved version "Flat Histogram Diagrammatic Monte Carlo" method. We demonstrate the superiority of the method over the standard Diag-MC in extracting the long-imaginary-time behavior of the Green's function, without incorporating any a priori knowledge about this function, by applying the technique to the polaron proble…
We discuss a sampling algorithm which generates flat histogram in energy. In combination with transition matrix Monte Carlo, the density of states and derived quantities such as entropy and free energy as a function of temperature can be computed in a single simulation.
We study the performance of Monte Carlo simulations that sample a broad histogram in energy by determining the mean first-passage time to span the entire energy space of d-dimensional ferromagnetic Ising/Potts models. We first show that flat-histogram Monte Carlo methods with single-spin flip updates such as the Wang-Landau algorithm or the multicanonical method perform sub-optimally in comparison to an unbiased Markovian random walk in energy space. For the d=1,2,3 Ising model, the mean first-passage time τscales with the number of spins N=L^d as τ\propto N^2L^z. The critical exponent z is fo…
We demonstrate the use of a new algorithm called the Flat Histogram sampling algorithm for the simulation of lattice polymer systems. Thermodynamics properties, such as average energy or entropy and other physical quantities such as end-to-end distance or radius of gyration can be easily calculated using this method. Ground-state energy can also be determined. We also explore the accuracy and limitations of this method. Key words: Monte Carlo algorithms, flat histogram sampling, HP model, lattice polymer systems
We compare the performance of extremal optimization (EO), flat-histogram and equal-hit algorithms for finding spin-glass ground states. The first-passage-times to a ground state are computed. At optimal parameter of tau=1.15, EO outperforms other methods for small system sizes, but equal-hit algorithm is competitive to EO, particularly for large systems. Flat-histogram and equal-hit algorithms offer additional advantage that they can be used for equilibrium thermodynamic calculations. We also propose a method to turn EO into a useful algorithm for equilibrium calculations. Keywords: extremal o…
We propose a new Monte Carlo technique in which the degeneracy of energy states is obtained with a Markovian process analogous to that of Metropolis used currently in canonical simulations. The obtained histograms are much broader than those of the canonical histogram technique studied by Ferrenberg and Swendsen. Thus we can reliably reconstruct thermodynamic functions over a much larger temperature scale also away from the critical point. We show for the two-dimensional Ising model how our new method reproduces exact results more accurately and using less computer time than the conventional h…
In any valid Monte Carlo sampling that realizes microcanonical property we can collect statistics for a transition matrix in energy. This matrix is used to determine the density of states, from which most of the thermodynamical averages can be calculated, including free energy and entropy. We discuss single-spin-flip algorithms, such as flat-histogram and equal-hit algorithms, that can be used for simulations. The flat-histogram algorithm realizes multicanonical ensemble. We demonstrate the use of the method with applications to Ising model and show its efficiency of search for spin-glass grou…
We propose a method for efficient simulations in extended ensembles, useful, e.g., for the study of problems with complex energy landscapes and for free energy calculations. The main difficulty in such simulations is the estimation of the a priori unknown weight parameters needed to produce flat histograms. The method combines several complementary techniques, namely, a Gibbs sampler for the parameter moves, a reweighting procedure to optimize data use, and a Bayesian update allowing for systematic refinement of the free energy estimate. In a certain limit the scheme reduces to the 1/t algorit…
Interacting Growth Walks is a recently proposed stochastic model for studying the coil-globule transition of linear polymers. We propose a flat energy histogram version for Interacting Growth Walk. We demonstrate the algorithm on two dimensional square and triangular lattices by calculating the density of energy states of Interacting Self Avoiding Walks.
We present Monte Carlo simulations of lattice models of polymers. These simulations are intended to demonstrate the strengths of a powerful new flat histogram algorithm which is obtained by adding microcanonical reweighting techniques to the pruned and enriched Rosenbluth method (PERM).
We determine the optimal scaling of local-update flat-histogram methods with system size by using a perfect flat-histogram scheme based on the exact density of states of 2D Ising models.The typical tunneling time needed to sample the entire bandwidth does not scale with the number of spins N as the minimal N^2 of an unbiased random walk in energy space. While the scaling is power law for the ferromagnetic and fully frustrated Ising model, for the +/- J nearest-neighbor spin glass the distribution of tunneling times is governed by a fat-tailed Frechet extremal value distribution that obeys expo…
We study transition matrices for projected dynamics in the energy-magnetization space, magnetization space and energy space. Several single spin flip dynamics are considered such as the Glauber and Metropolis canonical ensemble dynamics and the Metropolis dynamics for three multicanonical ensembles: the flat energy-magnetization histogram, the flat energy histogram and the flat magnetization histogram. From the numerical diagonalization of the matrices for the projected dynamics we obtain the sub-dominant eigenvalue and the largest relaxation times for systems of varying size. Although, the pr…
We presented an efficient algorithm, fast adaptive flat-histogram ensemble (FAFE), to estimate the density of states (DOS) and to enhance sampling in large systems. FAFE calculates the means of an arbitrary extensive variable $U$ in generalized ensembles to form points on the curve $β_{s}(U) \equiv \frac{\partial S(U)}{\partial U}$, the derivative of the logarithmic DOS. Unlike the popular Wang-Landau-like (WLL) methods, FAFE satisfies the detailed-balance condition through out the simulation and automatically generates non-uniform $(β_{i}, U_{i})$ data points to follow the real change rate of…
Using the Wang-Landau flat histogram Monte Carlo (FHMC) simulation technique, we were able to study two types of triangulated spherical surface models in which the two-dimensional extrinsic curvature energy is assumed in the Hamiltonian. The Gaussian bond potential is also included in the Hamiltonian of the first model, but it is replaced by a hard-wall potential in the second model. The results presented in this paper are in good agreement with the results previously reported by our group. The transition of surface fluctuations and collapsing transition were studied using the canonical Metrop…
We present a model for the dynamics in energy space of multicanonical simulation methods that lends itself to a rather complete analytic characterization. The dynamics is completely determined by the density of states. In the \pm J 2D spin glass the transitions between the ground state level and the first excited one control the long time dynamics. We are able to calculate the distribution of tunneling times and relate it to the equilibration time of a starting probability distribution. In this model, and possibly in any model in which entering and exiting regions with low density of states ar…
We present a method for estimating the density of states of a classical statistical model. The algorithm successfully combines the Wang-Landau flat histogram method with the N-fold way in order to improve efficiency of the original single spin flip version. We test our implementation of the Wang-Landau method with the two-dimensional nearest neighbor Ising model for which we determine the tunneling time and the density of states. Furthermore, we show that our new algorithm performs correctly at right edges of an energy interval over which the density of states is computed. This removes a disad…
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. This algorithm is especially useful for complex systems with a rough landscape since all possible energy levels are visited with the same probability. In this paper,…
We present a Monte Carlo method that efficiently computes the density of states for spin models having any number of interaction per spin. By combining a random-walk in the energy space with collective updates controlled by the microcanonical temperature, our method yields dynamic exponents close to their ideal random-walk values, reduced equilibrium times, and very low statistical error on the density of states. The method can host any density of states estimation scheme, including the Wang-Landau algorithm and the transition matrix method. Our approach proves remarkably powerful in the numer…
A new approach known as flat histogram method is used to study the +/-J Ising spin glass in two dimensions. Temperature dependence of the energy, the entropy, and other physical quantities can be easily calculated and we give the results for the zero-temperature limit. For the ground-state energy and entropy of an infinite system size, we estimate e0 = -1.4007 +/- 0.0085 and s0 = 0.0709 +/- 0.006, respectively. Both of them agree well with previous calculations. The time to find the ground-states as well as the tunneling times of the algorithm are also reported and compared with other methods.
We discuss the conceptual differences between the Broad Histogram (BHM) and reweighting methods in general, and particularly the so-called Multicanonical (MUCA) approaches. The main difference is that BHM is based on microcanonical, fixed-energy averages which depends only on the good statistics taken {\bf inside} each energy level. The detailed distribution of visits among different energy levels, determined by the particular dynamic rule one adopts, is irrelevant. Contrary to MUCA, where the results are extracted from the dynamic rule itself, within BHM any microcanonical dynamics could be a…
We apply the Broad Histogram Method to an Ising system in the context of the recently reformulated Generalized Thermostatistics, and we claim it to be a very efficient simulation tool for this non-extensive statistics. Results are obtained for the nearest-neighbour version of the Ising model for a range of values of the $q$ parameter of Generalized Thermostatistics. We found an evidence that the 2D-Ising model does not undergo phase transitions at finite temperatures except for the extensive case $q=1$.
A Monte Carlo method to estimate the Joint Density of States g(E,M) of the Ising and Ising-like models is presented. The method is applied to the well-known 2D Ising model, and is shown to be accurate, efficient, and embarrassingly parallel. The method presented offers major improvements over existing approaches. Furthermore, we obtain g(E,M) estimates for the spin S Ising model, with the spin number S = {1/2, 1, 3/2, 2}, thus showing that the algorithm can handle larger and more complex (E, M) phase spaces.
We introduce a rejection-free, flat histogram, cluster algorithm to determine the density of states of hard-core lattice gases. We show that the algorithm is able to efficiently sample low entropy states that are usually difficult to access, even when the excluded volume per particle is large. The algorithm is based on simultaneously evaporating all the particles in a strip and reoccupying these sites with a new appropriately chosen configuration. We implement the algorithm for the particular case of the hard-core lattice gas in which the first k next-nearest neighbors of a particle are exclud…
A generalization of the microcanonical ensemble suggests a simple strategy for the simulation of first order phase transitions. At variance with flat-histogram methods, there is no iterative parameters optimization, nor long waits for tunneling between the ordered and the disordered phases. We test the method in the standard benchmark: the Q-states Potts model (Q=10 in 2 dimensions and Q=4 in 3 dimensions), where we develop a cluster algorithm. We obtain accurate results for systems with more than one million of spins, outperforming flat-histogram methods that handle up to tens of thousands of…
We analyse the steady state regime of a one dimensional Ising model under a tapping dynamics recently introduced by analogy with the dynamics of mechanically perturbed granular media. The idea that the steady state regime may be described by a flat measure over metastable states of fixed energy is tested by comparing various steady state time averaged quantities in extensive numerical simulations with the corresponding ensemble averages computed analytically with this flat measure. The agreement between the two averages is excellent in all the cases examined, showing that a static approach is …
We study the convergence of the density of states and thermodynamic properties in three flat-histogram simulation methods, the Wang-Landau (WL) algorithm, the 1/t algorithm, and tomographic sampling (TS). In the first case the refinement parameter f is rescaled (f -> f/2) each time the flat-histogram condition is satisfied, in the second f ~ 1/t after a suitable initial phase, while in the third f is constant (t corresponds to Monte Carlo time). To examine the intrinsic convergence properties of these methods, free of any complications associated with a specific model, we study a featureless e…
We study the role of the quench temperature $T_f$ in the phase-ordering kinetics of the Ising model with single spin flip in $d=2,3$. Equilibrium interfaces are flat at $T_f=0$, whereas at $T_f>0$ they are curved and rough (above the roughening temperature in $d=3$). We show, by means of scaling arguments and numerical simulations, that this geometrical difference is important for the phase-ordering kinetics as well. In particular, while the growth exponent $z=2$ of the size of domains $L(t)\sim t^{1/z}$ is unaffected by $T_f$, other exponents related to the interface geometry take different v…
We show in this paper by using the Wang-Landau flat-histogram Monte Carlo method that the phase transition in the XY stacked triangular antiferromagnet is clearly of first-order, confirming results from latest Monte Carlo simulation and from a nonperturbative renormalization group, putting an end to a long-standing controversial issue.
By using the Wang-Landau flat-histogram Monte Carlo (MC) method for very large lattice sizes never simulated before, we show that the phase transition in the frustrated Heisenberg stacked triangular antiferromagnet is of first-order, contrary to results of earlier MC simulations using old-fashioned methods. Our result lends support to the conclusion of a nonperturbative renormalization group performed on an effective Hamiltonian. It puts an end to a 20-year long controversial issue.
