The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. Hence, we will apply this novel paradigm to the law of large numbers in order to demonstrate it in an innovative way and to prove as well in an original way an important property at the foundation of statistical physics.The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. Hence, we will apply this novel paradigm to the law of large numbers in order to demonstrate it in an innovative way and to prove as well in an original way an important property at the foundation of statistical physics.

Book Publisher International (a part of SCIENCEDOMAIN International)Feb 21, 20240 citations

Some trends and problems in quantum probability

Luigi Accardi

Lecture notes in mathematicsJan 1, 198433 citations

Concepts of convergence for a quantum law of large numbers

Wilhelm Ochs

Reports on Mathematical PhysicsFeb 1, 19803 citations

Generalized Boltzmann distributions for systems strongly coupled to large finite bath -- a microcanonical approach

Yu-Chen Cheng, Wenning Wang, Zhiyue Lu +1 more

The theory of probability shows that, as the fraction $X_n/Y\to 0$ , the conditional probability for $X_n$ , given $X_n+Y \in h_δ:=[h, h+δ]$ , has a limit law $f_{X_n}(x)e^{-ψ_n(h_δ)x}$ , where $ψ_n(h_δ) $ equals to $[\partial \ln P(Y \in y_δ)/\partial y]_{y=h}$ plus an additional term, contributed from the correlation between $X_n$ and bath $Y$ . By applying this limit law to an isolated composite system consisting of two strongly coupled parts, a system of interest and a large but finite bath, we derive the generalized Boltzmann distribution law for the system of interest in the exponential form of a redefined Hamiltonian and corrected Boltzmann temperature that reflects the modification due to strong system-bath coupling and the large but finite bath.The theory of probability shows that, as the fraction $X_n/Y\to 0$, the conditional probability for $X_n$, given $X_n+Y \in h_δ:=[h, h+δ]$, has a limit law $f_{X_n}(x)e^{-ψ_n(h_δ)x}$, where $ψ_n(h_δ) $ equals to $[\partial \ln P(Y \in y_δ)/\partial y]_{y=h}$ plus an additional term, contributed from the correlation between $X_n$ and bath $Y$. By applying this limit law to an isolated composite system consisting of two strongly coupled parts, a system of interest and a large but finite bath, we derive the generalized Boltzmann distribution law for the system of interest in the exponential form of a redefined Hamiltonian and corrected Boltzmann temperature that reflects the modification due to strong system-bath coupling and the large but finite bath.

arXiv:1811.11321Nov 28, 20180 citations

Analysis of quantum probability theory. II

James van Aken

Journal of Philosophical LogicAug 1, 19861 citation

Indeterminate Probabilities and the Weak Quantum Law of Large Numbers

Fedor Herbut

The quantum probabilistic convergence in measurement, distinct from mathematical convergence, is derived for indeterminate probabilities from the weak quantum law of large numbers. This is presented in three theorems. The first establishes the necessary bridge between ensemble theory and experiment. The second analyzes the most important theoretical ensemble entity: the eigen-projector of the relative frequency operator. Its physical meaning is the experimental relative frequency. The third theorem formulates the quantum probabilistic convergence, which is the final result of this investigation.The quantum probabilistic convergence in measurement, distinct from mathematical convergence, is derived for indeterminate probabilities from the weak quantum law of large numbers. This is presented in three theorems. The first establishes the necessary bridge between ensemble theory and experiment. The second analyzes the most important theoretical ensemble entity: the eigen-projector of the relative frequency operator. Its physical meaning is the experimental relative frequency. The third theorem formulates the quantum probabilistic convergence, which is the final result of this investigation.

arXiv:1512.00474Dec 1, 20150 citations

The theory of probability shows that, as the fraction

X_n/Y\to 0

, the conditional probability for

X_n

, given

X_n+Y \in h_δ:=[h, h+δ]

, has a limit law

f_{X_n}(x)e^{-ψ_n(h_δ)x}

, where $ψ_n(h_δ) $ equals to

[\partial \ln P(Y \in y_δ)/\partial y]_{y=h}

plus an additional term, contributed from the correlation between

X_n

and bath

Y

. By applying this limit law to an isolated composite system consisting of two strongly coupled parts, a system of interest and a large but finite bath, we derive the generalized Boltzmann distribution law for the system of interest in the exponential form of a redefined Hamiltonian and corrected Boltzmann temperature that reflects the modification due to strong system-bath coupling and the large but finite bath.