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Systems Science & Control Engineering

The paradigm of complex probability and Ludwig Boltzmann's entropy

Abdo Abou Jaoudé

Published Jan 1, 2018• 14 citations
Publisher

Abstract

The Andrey Nikolaevich Kolmogorov's classical system of probability axioms can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Hence, any experiment can thus be executed in what is now the complex probability set 𝒞 which is the sum of the real set ℛ with its corresponding real probability and the imaginary set ℳ with its corresponding imaginary probability. The objective here is to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Whatever the probability distribution of the input random variable in ℛ is, the corresponding probability in the whole set 𝒞 is always one, so the outcome of the random experiment in 𝒞 can be predicted totally and perfectly. The result indicates that chance and luck in ℛ is replaced now by total determinism in 𝒞 . This is the consequence of the fact that the probability in 𝒞 is got by subtracting the chaotic factor from the degree of our knowledge of the stochastic system. This novel complex probability paradigm will be applied to Ludwig Boltzmann's classical concept of entropy in thermodynamics and in statistical mechanics.

DOI
10.1080/21642583.2018.1471427

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The Andrey Nikolaevich Kolmogorov's classical system of probability axioms can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Hence, any experiment can thus be executed in what is now the complex probability set 𝒞 which is the sum of the real set ℛ with its corresponding real probability and the imaginary set ℳ with its corresponding imaginary probability. The objective here is to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Whatever the probability distribution of the input random variable in ℛ is, the corresponding probability in the whole set 𝒞 is always one, so the outcome of the random experiment in 𝒞 can be predicted totally and perfectly. The result indicates that chance and luck in ℛ is replaced now by total determinism in 𝒞 . This is the consequence of the fact that the probability in 𝒞 is got by subtracting the chaotic factor from the degree of our knowledge of the stochastic system. This novel complex probability paradigm will be applied to Ludwig Boltzmann's classical concept of entropy in thermodynamics and in statistical mechanics.
The set of imaginary numbers is taken into account by extending the probability system of five axioms of Andrey Nikolaevich Kolmogorov which was put forward in 1933. This is achieved by adding three new and supplementary axioms. Hence, any random experiment can thus be performed in the extended complex probability set C = R + M which is the sum of the real set R of real probabilities and the imaginary set M of imaginary probabilities. The objective here is to determine the complex probabilities by encompassing and considering additional new imaginary dimensions to the event that occurs in the “real” laboratory. The outcome of the stochastic phenomenon in C can be foretold perfectly whatever the probability distribution of the input random variable in R is since the corresponding probability in the whole set C is permanently and constantly equal to one. Thus, the consequence that follows indicates that randomness and chance in R is substituted now by absolute determinism in C. This is the result 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. This novel complex probability paradigm will be applied to a newly defined logic that I called “Dynamic Logic”.
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 Isaac Newton’s classical mechanics and to prove as well in an original way an important property at the foundation of statistical physics.
The system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real set R with its corresponding real probability, and the imaginary set M with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set C instead of the real set R. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally in C. Subsequently, it follows that, chance and luck in R is replaced by total determinism in C. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in C. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C = R + M as well as to the quantum uncertainty principle in order to verify it and to extend it to the universes M and C.
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 Central Limit Theorem and to prove as well its convergence in a novel way.
Problem statement: The five basic axioms of Kolmogorov define the probability in the real set of numbers and do not take into consideration the imaginary part which takes place in the complex set of numbers, a problem that we are facing in many engineering systems. Approach: Evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the "real" laboratory. The Kolmogorov's system of axioms can be extended to encompass the imaginary set of numbers and this by adding to the original five axioms of Kolmogorov an additional three axioms. Hence, any experiment can thus be executed in what is now the complex set C which is the sum of the real set R with its corresponding real probability and the imaginary set M with its corresponding imaginary probability. Results: Whatever the probability distribution of the random variable in R is, the corresponding probability in the whole set C is always one, so the outcome of the random experiment in C can be predicted totally. Conclusion: The result indicated that, the chance and luck in R is replaced now by total determinism in C. This is the consequence of the fact that the probability in C is got by subtracting the chaotic factor from the degree of our knowledge of the system.
In the year 1933, the Russian mathematician Andrey Nikolaevich Kolmogorov put forward the system of axioms of modern probability theory. By adding to Kolmogorov’s original five axioms and an additional three axioms, this established system can be extended to encompass the imaginary set of numbers. Accordingly, the complex probability set C will be created and which is the sum of its corresponding real probability belonging to the real set R and of its corresponding imaginary probability belonging to the imaginary set M. Thus, all random phenomena do not occur now in the real set R but in the general complex set C that encompasses both R and M. Hence, we take into consideration supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory to evaluate the complex probabilities. This is consequently the objective of this novel paradigm. Subsequently, the outcome of the stochastic experiments that follow any probability distribution in R is now predicted perfectly and totally in C and the corresponding probability in the whole set C is always equal to one. Afterward, it follows that luck and chance in R are substituted by absolute determinism in C. Therefore, we evaluate the probability of any probabilistic phenomenon in C by subtracting the chaotic factor from the degree of our knowledge of the random system. My groundbreaking Complex Probability Paradigm (or CPP) will be applied to the well-known theory of Markov Chains Transition Matrices in order to express it perfectly and absolutely in a deterministic way in the universe C = R + M as well as to extend it to the probabilities’ universes M and C.