Monte Carlo simulations with indefinite and complex-valued measures (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.

Monte Carlo Simulations with Complex-Valued Measure

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.

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.

TACKLING THE SIGN PROBLEM

International Journal of Modern Physics C, 1994

To tackle 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.

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.

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 Novel Complex Probability Paradigm Applied to Monté Carlo Methods

B P International, 2024

Monte Carlo methods were central to the simulations required for the Manhattan Project, though severely limited by the computational tools at the time. 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. Quantum Monte Carlo, and more specifically Diffusion Monte Carlo methods can also be interpreted as a mean field particle Monte Carlo approximation of Feynman-Kac path integrals. 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 in R, M, and C. 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. It is important to state here that one essential and very well-known probability distribution was taken into consideration in the current chapter which is the uniform and discrete probability distribution as well as a specific generator of uniform random numbers, knowing that the original CPP model can be applied to any generator of uniform random numbers that exists in literature. This will yield certainly to analogous results and conclusions and will confirm without any doubt the success of my innovative theory. 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.

Monte Carlo simulations with indefinite and complex-valued measures (original) (raw)

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