# A Brief Introductory Note on the Possible Chaotic Dynamics of the Muon Time Series of Cosmic Rays Measured at Sea Level by a Simple GMT Detector

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## Abstract

**:**

## 1. Introduction

## 2. A Preliminary Survey on Cosmic Rays

^{−6}s) into three new particles: an electron, an electron antineutrino and a muon neutrino, according to the following scheme of reaction (Figure 1):

#### 2.1. Muon within the Standard Model of Fundamental Particles

#### 2.2. Standard Model and Measurement of the Muon Lifetime Value

_{w}using some experimental data and the standard model. In the SM, the well-known electromagnetic interaction is related to the weak interaction that is responsible for particle decays, such as muon decay. In the SM, the strength of the instability interaction is denoted by g

_{w}and determines how long a muon exists before it breaks up. The theoretical lifetime formula permits the discovery of the weak coupling constant g

_{w}.

_{w}and on masses of W-boson M

_{W}and muon m

_{μ}.

_{w}, we have:

_{W}c

^{2}= 80.4 GeV and of muon m

_{μ}c

^{2}= 105.7 MeV given by experimental data, it has been estimated that one of the fundamental constants of nature, the dimensionless strength of the weak force g

_{w}= 0.680, with a 4% error when compared with the accepted value of 0.653.

## 3. Experimental Setup

#### Details of Our Digital System and Data Acquisition

^{2}and with this setup, the relative average rate is usually around 2 counts per minute (cpm). Signals from the GMT sensors are processed by a simple electronic board. The electronics extend the pulse length from each sensor by a certain amount of time (in the order of milliseconds), which is the time window for the coincidence signal. Then, the two signals are sent into a digital “and” gate that provides the coincident pulse. The time window is adjustable to tune different GMTs with different technical parameters and to facilitate different experiments. In the end, the signal is a squared pulse sent to a USB card connected to a personal computer. Every pulse is interpreted and counted as a cosmic ray (muon); our own software called AstroRad gathers the pulses and records several sets of data files at different time resolutions, and a set of data with a time stamp for each particle. The main time-resolution data are cpm, count/5 s, the average count for one whole day, and a dosimetry dataset. Despite the small detection area, over the years the AMD5 detector has proved to be very reliable, and now many instruments of this sort are in use in an Italian CR network called the Astroparticle Detector Array (educational project) [7]. The detectors are set in a vertical position, measuring with a zenithal angle equal to zero. The measurements in time of the muons give the origin to a time series of muons that, stored in a computer, may be studied by the methodology of chaos analysis.

## 4. Method of Analysis

#### 4.1. CHAOS Analysis of Muon Time Series

_{X}

^{(x)}and p

_{Y}

^{(y)}are the marginal probability density functions and p

_{(X,Y)}

^{(x,y)}the joint probability density function. In Figure 11, we report the results of the AMI obtained by the muon time series for raw data and surrogate data. The obtained time delay is 4 as the first minimum of the function [8,9,10,11,12]. For the ACI, the time delay is the first zero of the ACI function, and for the AMI, the time delay is the first minimum of this function. Our criterion to verify nonlinear presence is as follows: for the surrogate data, it is stated as nonlinearity if the mutual difference between the original data and surrogate data is greater than 10%. We have obtained differences varying between the 11 and 15%. Therefore, the nonlinearity of the data seems to be confirmed.

#### 4.2. Phase Space Reconstruction

_{2}+ 1, and where D

_{2}is the fractal dimension of the system [10,11,12]. To reconstruct the attractor, the estimation of the embedding dimension m and the embedding delay τ are essential. The method of autocorrelation function is used for the determination of the delay, τ, as the delay causing the value of (1 − 1/e) is its initial of the autocorrelation function [10,11,12,13]. Soon afterwards, the AMI criterion must be used for the determination of time delay in the presence of nonlinear dynamics. The embedding dimension m was estimated using the Grassberger–Procaccia algorithm method proposed by Grassberger [11,14], which is suitable for shorter time series.

_{2}and it is calculated by the following formula:

- Data of muon time series in 2019: CD = 2.425 ± 0.921
- Data of surrogate of muon time series in 2019: CD = 2.736 ± 1.132
- Data of muon time series in 2020: CD = 1.613 ± 0.276
- Data of surrogate of muon time series in 2020: CD = 2.829 ± 1.183
- Data of muon time series in 2021: CD = 2.621 ± 0.973
- Data of surrogate of muon time series in 2021: CD = 3.513 ± 1.620

#### 4.3. Results of Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Probability distribution of the raw data for muon time series in the years (

**a**) 2019, (

**b**) 2020 and (

**c**) 2021.

**Figure 11.**Muon series and surrogate series: (

**a**) AMI of the muon time series in 2019; (

**b**) AMI of the shuffled–surrogate data of muon time series in 2019; (

**c**) AMI of the muon time series in 2020; (

**d**) AMI of the shuffled–surrogate data in 2020; (

**e**) AMI of the muon time series in 2021; (

**f**) AMI of the shuffled–surrogate data in 2021.

**Figure 22.**Lyapunov exponent (the minimum initial separation between two data points); (

**a**) Muon Data Series of 2019—Dominant Lyapunov exponent; (

**b**) Surrogate–shuffled data of muon time series of 2019; (

**c**) Muon Data Series of 2020—Dominant Lyapunov exponent; (

**d**) Surrogate–shuffled data of muon time series of 2020; (

**e**) Muon Data Series of 2021—Dominant Lyapunov exponent; (

**f**) Surrogate–shuffled data of muon time series of 2021.

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**MDPI and ACS Style**

Conte, E.; Sala, N.; Arcani, M.
A Brief Introductory Note on the Possible Chaotic Dynamics of the Muon Time Series of Cosmic Rays Measured at Sea Level by a Simple GMT Detector. *Symmetry* **2023**, *15*, 659.
https://doi.org/10.3390/sym15030659

**AMA Style**

Conte E, Sala N, Arcani M.
A Brief Introductory Note on the Possible Chaotic Dynamics of the Muon Time Series of Cosmic Rays Measured at Sea Level by a Simple GMT Detector. *Symmetry*. 2023; 15(3):659.
https://doi.org/10.3390/sym15030659

**Chicago/Turabian Style**

Conte, Elio, Nicoletta Sala, and Marco Arcani.
2023. "A Brief Introductory Note on the Possible Chaotic Dynamics of the Muon Time Series of Cosmic Rays Measured at Sea Level by a Simple GMT Detector" *Symmetry* 15, no. 3: 659.
https://doi.org/10.3390/sym15030659