A finitely additive version of Poincare's recurrence theorem

2006 ◽  
pp. 159-162
Author(s):  
H. Francke ◽  
D. Plachky ◽  
W. Thomsen
Keyword(s):  
Recurrence Theorem ◽  
Additive Version

Probability Theory for Fuzzy Quantum Spaces with Statistical Applications

2017 ◽  
Author(s):  
Renáta Bartková ◽  
Beloslav Riečan ◽  
Anna Tirpáková
Keyword(s):  
Probability Theory ◽  
Fuzzy Sets ◽  
Ergodic Theorem ◽  
Intuitionistic Fuzzy Sets ◽  
Mathematics Students ◽  
Intuitionistic Fuzzy ◽  
Individual Ergodic Theorem ◽  
Recurrence Theorem ◽  
The Individual ◽  
Atanassov Intuitionistic Fuzzy Sets

The reference considers probability theory in two main domains: fuzzy set theory, and quantum models. Readers will learn about the Kolmogorov probability theory and its implications in these two areas. Other topics covered include intuitionistic fuzzy sets (IF-set) limit theorems, individual ergodic theorem and relevant statistical applications (examples from correlation theory and factor analysis in Atanassov intuitionistic fuzzy sets systems, the individual ergodic theorem and the Poincaré recurrence theorem). This book is a useful resource for mathematics students and researchers seeking information about fuzzy sets in quantum spaces.

Hamilton-Jacobi Theory

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Bbgky Hierarchy ◽  
Distribution Functions ◽  
Symplectic Integrators ◽  
Partition Functions ◽  
Von Neumann ◽  
Equipartition Theorem ◽  
Recurrence Theorem ◽  
Phase Amplitude ◽  
Canonical Ensembles

This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

Quantum Recurrence Theorem

1957 ◽  
Vol 107 (2) ◽  
pp. 337-338 ◽  
Author(s):  
P. Bocchieri ◽  
A. Loinger
Keyword(s):  
Recurrence Theorem

scholarly journals Shock-wave model of the earthquake and Poincaré quantum theorem give an insight into the aftershock physics.

2018 ◽  
Vol 62 ◽  
pp. 03006
Author(s):  
Vladimir Kuznetsov
Keyword(s):  
Shock Wave ◽  
Wave Model ◽  
Particle Systems ◽  
Many Body ◽  
Initial State ◽  
Recurrence Theorem ◽  
Shock Wave Model ◽  
Quantum Phenomena ◽  
First Time ◽  
Insight Into

A fundamentally new model of aftershocks evident from the shock-wave model of the earthquake and Poincaré Recurrence Theorem [H. Poincare, Acta Mathematica 13, 1 (1890)] is proposed here. The authors (Recurrences in an isolated quantum many-body system, Science 2018) argue that the theorem should be formulated as “Complex systems return almost exactly into their initial state”. For the first time, this recurrence theorem has been demonstrated with complex quantum multi-particle systems. Our shock-wave model of an earthquake proceeds from the quantum entanglement of protons in hydrogen bonds of lithosphere material. Clearly aftershocks are quantum phenomena which mechanism follows the recurrence theorem.

scholarly journals Pointwise characteristic factors for Wiener–Wintner double recurrence theorem

2015 ◽  
Vol 36 (4) ◽  
pp. 1037-1066 ◽  
Author(s):  
IDRIS ASSANI ◽  
DAVID DUNCAN ◽  
RYO MOORE
Keyword(s):  
Full Measure ◽  
Orthogonal Complement ◽  
Ergodic System ◽  
Absolute Value ◽  
Recurrence Theorem ◽  
The Absolute ◽  
Image Position ◽  
Null Set

In this paper we extend Bourgain’s double recurrence result to the Wiener–Wintner averages. Let $(X,{\mathcal{F}},{\it\mu},T)$ be a standard ergodic system. We will show that for any $f_{1},f_{2}\in L^{\infty }(X)$, the double recurrence Wiener–Wintner average $$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)e^{2{\it\pi}int}\end{eqnarray}$$ converges off a single null set of $X$ independent of $t$ as $N\rightarrow \infty$. Furthermore, we will show a uniform Wiener–Wintner double recurrence result: if either $f_{1}$ or $f_{2}$ belongs to the orthogonal complement of the Conze–Lesigne factor, then there exists a set of full measure such that the supremum on $t$ of the absolute value of the averages above converges to $0$.

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