Monte Carlo Simulations with Complex-Valued Measure (original) (raw)

Monte Carlo simulation of systems with complex-valued measures

Nuclear Physics B - Proceedings Supplements, 1998

A simulation method based on the RG blocking is shown to yield statistical errors smaller than that of the crude MC using absolute values of the original measures. The new method is particularly suitable to apply to the sign problem of indefinite or complex-valued measures. We demonstrate the many advantages of this method in the simulation of 2D Ising model with complex-valued temperature.

Simulations with complex measure

Nuclear Physics B, 1998

A method is proposed to handle the sign problem in the simulation of systems having indefinite or complex-valued measures.

Monte Carlo simulations with indefinite and complex-valued measures

Physical Review E, 1994

A method is presented to tackle the sign problem in the simulations of systems having inde nite or complex-valued measures. In general, this new approach is shown to yield statistical errors smaller than the crude Monte Carlo using absolute values of the original measures. Exactly solvable, one-dimensional Ising models with complex temperature and complex activity illustrate the considerable improvements and the workability of the new method even when the crude one fails.

Simulations with complex measures

Nuclear Physics B - Proceedings Supplements, 1994

Towards a solution to the sign problem in the simulations of systems having inde nite or complex-valued measures, we propose a new approach which yields statistical errors smaller than the crude Monte Carlo using absolute values of the original measures. The 1D complex-coupling Ising model is employed as an illustration.

A Monte Carlo Sampling Scheme for the Ising Model

Journal of Statistical Physics, 2000

In this paper we describe a Monte Carlo sampling scheme for the Ising model and similar discrete state models. The scheme does not involve any particular method of state generation but rather focuses on a new way of measuring and using the Monte Carlo data.

Monte Carlo technique for very large ising models

Journal of Statistical Physics, 1982

Rebbi's multispin coding technique is improved and applied to the kinetic Ising model with size 600 * 600 * 600. We give the central part of our computer program (for a CDC Cyber 76), which will be helpful also in a simulation of smaller systems, and describe the other tricks necessary to go to large lattices. The magnetization M at T= 1,4 * T c is found to decay asymptotically as exp(-t/2.90) if t is measured in Monte Carlo steps per spin, and M(t = O) = 1 initially.

Monte Carlo methods in sequential and parallel computing of 2D and 3D ising model

Because of its complexity, the 3D Ising model has not been given an exact analytic solution so far, as well as the 2D Ising in non zero external field conditions. In real materials the phase transition creates a discontinuity. We analysed the Ising model that presents similar discontinuities. We use Monte Carlo methods with a single spin change or a spin cluster change to calculate macroscopic quantities, such as specific heat and magnetic susceptibility. We studied the differences between these methods. Local MC algorithms (such as Metropolis) perform poorly for large lattices because they update only one spin at a time, so it takes many iterations to get a statistically independent configuration. More recent spin cluster algorithms use clever ways of finding clusters of sites that can be updated at once. The single cluster method is probably the best sequential cluster algorithm. We also used the entropic sampling method to simulate the density of states. This method takes into account all possible configurations, not only the most probable. The entropic method also gives good results in the 3D case. We studied the usefulness of distributed computing for Ising model. We established a parallelization strategy to explore Metropolis Monte Carlo simulation and Swendsen-Wang Monte Carlo simulation of this spin model using the data parallel languages on different platform. After building a computer cluster we made a Monte Carlo estimation of 2D and 3D Ising thermodynamic properties and compare the results with the sequential computing. In the same time we made quantitative analysis such as speed up and efficiency for different sets of combined parameters (e.g. lattice size, parallel algorithms, chosen model).

The Monte Carlo Techniques and the Complex Probability Paradigm

IntechOpen, 2020

The concept of mathematical probability was established in 1933 by Andrey Nikolaevich Kolmogorov by defining a system of five axioms. This system can be enhanced to encompass the imaginary numbers set after the addition of three novel axioms. As a result, any random experiment can be executed in the complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. We aim here to incorporate supplementary imaginary dimensions to the random experiment occurring in the “real” laboratory in R and therefore to compute all the probabilities in the sets R,M, and C. Accordingly, the probability in the whole set C = R + M is constantly equivalent to one independently of the distribution of the input random variable in R, and subsequently the output of the stochastic experiment in R can be determined absolutely in C. This is the consequence of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. We will apply this innovative paradigm to the well-known Monte Carlo techniques and to their random algorithms and procedures in a novel way.

The paradigm of complex probability and Monte Carlo methods

Systems Science & Control Engineering, OA, 2019

In 1933, Andrey Nikolaevich Kolmogorov established the system of five axioms that define the concept of mathematical probability. This system can be developed to include the set of imaginary numbers and this by adding a supplementary three original axioms. Therefore, any experiment can be performed in the set C of complex probabilities which is the summation of the set R of real probabilities and the set M of imaginary probabilities. The purpose here is to include additional imaginary dimensions to the experiment taking place in the ‘real’ laboratory in R and hence to evaluate all the probabilities. Consequently, the probability in the entire set C = R +M is permanently equal to one no matter what the stochastic distribution of the input random variable in R is, therefore the outcome of the probabilistic experiment in C can be determined perfectly. This is due to the fact that the probability in C is calculated after subtracting from the degree of our knowledge the chaotic factor of the random experiment. This novel complex probability paradigm will be applied to the classical probabilistic Monte Carlo numerical methods and to prove as well the convergence of these stochastic procedures in an original way.