# Evidence of Large-Scale Quantization in Space Plasmas

^{1}

^{2}

^{*}

## Abstract

**:**

_{*}in non-equilibrium space plasmas that connects the energy of particles in a Debye sphere to an equivalent wave frequency. In particular, while there is no a priori reason to expect a single value of h

_{*}across plasmas, we find a very similar value of h

_{*}≈ (7.5 ± 2.4)×10

^{−22}J·s using four independent methods: (1) Ulysses solar wind measurements, (2) space plasmas that typically reside in stationary states out of thermal equilibrium and spanning a broad range of physical properties, (3) an entropic limit emerging from statistical mechanics, (4) waiting-time distributions of explosive events in space plasmas. Finding a quasi-constant value for the phase space minimum in a variety of different plasmas, similar to the classical Planck constant but 12 orders of magnitude larger may be revealing a new type of quantization in many plasmas and correlated systems more generally.

## 1. Introduction

## 2. Large-Scale Phase Space Quantization

## 3. Is There Large-Scale Quantization in Non-Equilibrium Plasmas?

**Figure 1.**Phase space portion ${\hslash}_{*}$ calculated for the solar wind ion-electron plasma measurements and using Equation (1). (

**a**) Diagram $({\epsilon}_{C};{t}_{C})$ (on a log-log scale), constructed from Ulysses daily measurements. (

**b**) The product ${\hslash}_{*}=2{\epsilon}_{C}{t}_{C}$ is depicted as a function of heliocentric distance r. (a) and (b) are two-dimensional normalized histograms. (

**c**) Normalized histogram of the values of the values of $\mathrm{log}{\hslash}_{*}$. The fitted line in (a) has slope -1 and intercept $\cong -22.22$ (= $\mathrm{log}({\scriptscriptstyle \frac{1}{2}}{\hslash}_{*})$). The weighted mean of $\mathrm{log}{\hslash}_{*}$ values in (b) is found to be $\mathrm{log}{\hslash}_{*}(\text{J}\cdot \text{s})\cong -21.92\text{\hspace{0.17em}}\pm 0.15$. (For details on the statistical method, see Appendix A).

**Figure 2.**Phase space portion ${\hslash}_{*}$ in non-equilibrium space plasmas. (

**a**) Types of space plasmas that are typically out of thermal equilibrium, across a broad range of electron density and temperature. (

**b**) These plasmas produce a linear relation in $({\epsilon}_{C};{t}_{C})$ diagram (log-log scale) with slope -1, implying a relation ${\epsilon}_{C}{t}_{C}~$ constant. (

**c**) The constancy of ${\hslash}_{*}$ is also indicated by the respective log ${\hslash}_{*}$ values. (For details on the statistical method, see Appendix B. Details of plasma parameters used are provided in the Appendix C. Acronyms: CH: Corona Holes; CO: Corona; IH: Inner Heliosheath; IS: Interstellar; MA: Magnetosphere; MS: Magnetosheath; PS: Plasma Sheet; RC: Ring Current; TL: Tail Lobe; WH: solar Wind - Helios; WU: solar Wind-Ulysses).

## 4. Statistical Mechanics of Non-Equilibrium Plasmas

^{th}uncorrelated cluster, with average ${N}_{D}=N/M$. Replacing h with ${({c}_{M,\text{\hspace{0.17em}}N}/N\text{\hspace{0.05em}}!)}^{1/(3N)}\cdot {h}_{*}\cong {({N}_{D}/e)}^{-{\scriptscriptstyle \frac{1}{3}}}\cdot {h}_{*}$, we obtain the thermal parameter $\sigma \cong {(n/{N}_{D})}^{{\scriptscriptstyle \frac{1}{3}}}{h}_{*}/m\theta $. For quasineutral plasmas of ions and electrons, the thermal parameter is $\sigma \cong {(n/{N}_{D})}^{{\scriptscriptstyle \frac{1}{3}}}{h}_{*}/\sqrt{{m}_{i}{m}_{e}{\theta}_{i}{\theta}_{e}}$ that can be written as $\sigma \cong {6}^{{\scriptscriptstyle \frac{1}{3}}}{\pi}^{{\scriptscriptstyle \frac{2}{3}}}{2}^{-{\scriptscriptstyle \frac{1}{2}}}{k}_{\text{B}}^{-1}{\epsilon}^{-{\scriptscriptstyle \frac{1}{2}}}{q}_{e}{({m}_{i}{m}_{e})}^{-{\scriptscriptstyle \frac{1}{4}}}{\hslash}_{*}\text{\hspace{0.17em}}{n}^{{\scriptscriptstyle \frac{1}{2}}}{({T}_{i}\text{\hspace{0.17em}}{T}_{e})}^{-{\scriptscriptstyle \frac{3}{4}}}{({T}_{i}\text{\hspace{0.17em}}+{T}_{e})}^{{\scriptscriptstyle \frac{1}{2}}}$.

- (1)
- For systems with no correlations, the limit $\sigma ~\sqrt{\pi \text{\hspace{0.17em}}e}$ means the system approaches "quantum degeneracy", beyond which the non-quantum approach of Statistical Mechanics is no longer valid. This concept is usually described by comparing the thermal de Broglie wavelength ${\lambda}_{\text{Bg}}\equiv h\cdot {(2\pi \text{\hspace{0.17em}}{k}_{\text{B}}{m}_{p}{T}_{p})}^{-{\scriptscriptstyle \frac{1}{2}}}$ and the inter-particle distance $b\equiv {({\scriptscriptstyle \frac{4}{3}}\pi \text{\hspace{0.17em}}n)}^{-{\scriptscriptstyle \frac{1}{3}}}$: The passage of the system into the quantum regime implies ${\lambda}_{\text{Bg}}/b\le {e}^{{\scriptscriptstyle \frac{5}{6}}}{({\scriptscriptstyle \frac{4}{3}}\pi )}^{{\scriptscriptstyle \frac{1}{3}}}$ or ${n}^{{\scriptscriptstyle \frac{1}{3}}}{\lambda}_{\text{Bg}}\le {e}^{{\scriptscriptstyle \frac{5}{6}}}$.
- (2)
- For systems with correlations, the limit $\sigma ~\sqrt{\pi \text{\hspace{0.17em}}e}$ means the system transitions to "correlation degeneracy", beyond which the description through correlation clusters (Debye spheres) no longer applies. Then, the correlation clusters dissolve and the entropy increases, leading to plasma states that involve less-significant correlations.

## 5. Waiting-Time Distributions of Explosive Events in Non-Equilibrium Space Plasmas

_{D}). Substituting the Coulomb collision rate ${\nu}_{col}/{\omega}_{pl}=\mathrm{ln}(9{N}_{D})/(\sqrt{{\scriptscriptstyle \frac{2}{\pi}}}48{N}_{D})$ [15] in Eq. (3a), we find, ${\hslash}_{*}\cong 2.99\times {10}^{\text{\hspace{0.17em}}-23}\cdot \tau \cdot T\cdot [1+0.043\cdot \mathrm{ln}(0.9\text{\hspace{0.17em}}{N}_{D})]\cdot {N}_{D}^{-1}$ with units ${\hslash}_{*}$ ($\text{Js}$), τ (h), T (MK), and N

_{D}(${10}^{\text{\hspace{0.17em}}10}$).

Bursts | Refs | slope |
---|---|---|

SoFs | [23,24] | −2.38 ± 0.03; −2.40 ± 0.10 |

SoFs | [25] | −2.38 ± 0.06 |

SoFs | [26] | −2.26 ± 0.11 |

CMEs | [26] | −2.36 ± 0.11 |

StFs | [27] | −2.29 ± 0.07; −2.31 ± 0.12 |

Bursts | Refs | τ (h) | Plasma | log T(K) | log N_{D} | $\mathrm{log}\text{\hspace{0.17em}}{\hslash}_{*}(\text{J}\cdot \text{s})]$ |
---|---|---|---|---|---|---|

SoFs | [23,26,28] | 1.81 ± 0.03 | (CO) | 6.7 ± 0.6 | 10.2 ± 1.0 | −22.72 ± 1.17 |

CMEs | [26] | 5.0 ± 0.5 | (CH) | 6.0 ± 1.0 | 9.4 ± 1.6 | −22.21 ± 1.89 |

GSs | [29] | 5.5 ± 0.5 | (MA) | 6.3 ± 1.7 | 10.9 ± 2.7 | −23.31 ± 3.14 |

**Figure 3.**Waiting-time distributions of (

**a–c**) Solar Flares [23,26,28], and (

**d**) CMEs [26] (red data). The modeled distribution (blue lines) that is derived from (3) using one-particle kappa distribution of energy is well-fitted to the data (over six orders of magnitude), leading to an estimation of ${\hslash}_{*}$ consistent with other methods.

## 6. Conclusions

Method | $\mathrm{log}\text{\hspace{0.17em}}{\hslash}_{*}(\text{J}\cdot \text{s})]$ |
---|---|

Ulysses measurements (Figure 1) | −21.92 ± 0.15 |

Non-equilibrium space plasmas (Figure 2) | −21.87 ± 0.18 |

Correlation Degeneracy | −22.7 ± 0.8 |

Bursts (Table 2, Figure 3) | −22.6 ± 0.9 |

Weighted Mean (Appendix B) | −21.93 ± 0.14 |

## Acknowledgments

## Appendix A: Determination of ${\hslash}_{*}$ from Ulysses Data

_{Q}its standard deviation). Figure A1 presents the histograms of the relative errors, where the modeled quantities indeed have small variations, compared to the observational data, with respective relative errors about two orders of magnitude larger. This clearly shows that the variation of the density and the magnetic field within each of the intervals is not due to Parker’s relations and these selected data are suitable for examining whether the quasi-constancy of ${\hslash}_{*}$ is simply caused by these relations.

**Figure A1.**Histograms of the relative errors of density (

**a**,

**c**) and magnetic field (

**b**,

**d**) for both their modeled (upper panels) and observational (lower panels) values (for the Ulysses data separated in 37 intervals of Δr = 0.1 AU, where ϑ ≥10°). Note that the range of the top panels represent only ~3% that of the bottom ones.

**Figure A2.**Parker relations modeled (black solid line) and observational values of ${\hslash}_{*}$ (red points) plotted in terms of r (for the Ulysses data separated into 37 intervals). The Parker model was produced by fitting B and n separately to the actual data and using ${\hslash}_{*}\cong 7.053\cdot {10}^{-24}\cdot {B}^{2}/{n}^{3/2}$(see text).

## Appendix B: Statistical Analyses

#### B1. Statistical Determination of ${\hslash}_{*}$ for Various Space Plasmas

_{0}) is that ${\hslash}_{*}$ is a constant for the N=11 types of plasmas. This is tested by determining the chi-square of fitting a constant ${\hslash}_{*}$ to the given data points. The values of the individual parameters are given by their logarithms, i.e., ${\{\mathrm{log}{\hslash}_{*i}\pm {\sigma}_{\mathrm{log}{\hslash}_{*i}}\}}_{i=1}^{N}$, assuming a normally distributed error of the “order of magnitude.” The result of the fitting is the weighted mean, $\overline{\mathrm{log}{\hslash}_{*}}={\displaystyle {\sum}_{i=1}^{N}{w}_{i}\mathrm{log}{\hslash}_{*i}}$, where ${w}_{i}~{\sigma}_{\mathrm{log}{\hslash}_{*i}}^{-2}/{\displaystyle {\sum}_{i=1}^{N}{\sigma}_{\mathrm{log}{\hslash}_{*i}}^{-2}}$. The variance of the mean is derived only from fitting the data ${\sigma}^{2}={\scriptscriptstyle \frac{1}{N-1}}{\displaystyle {\sum}_{i=1}^{N}{w}_{i}{(\mathrm{log}{\hslash}_{*i}-\overline{\mathrm{log}{\hslash}_{*}})}^{2}}$, and not from propagating the standard deviations because they represent variability in the data and not errors. The chi-square value that characterizes the fitting is ${\chi}_{\text{est}}^{2}={\displaystyle {\sum}_{i=1}^{N}{\sigma}_{\mathrm{log}{\hslash}_{*i}}^{-2}{(\mathrm{log}{\hslash}_{*i}-\overline{\mathrm{log}{\hslash}_{*}})}^{2}}$. Finally, we have that $\overline{\mathrm{log}{\hslash}_{*}}=-21.87\text{\hspace{0.17em}}$, $\sigma =0.18$, and ${\chi}_{\text{est}}^{2}=5.0$. This leads to the p-value $\cong 0.11$, thus H

_{0}is highly likely to be true. Figure B1 demonstrates the results of the statistical method.

**Figure B1.**(

**a**) Plot of the 11 values of $\mathrm{log}{\hslash}_{*}(\text{J}\cdot \text{s})$, estimated for the 11 types of space plasmas (Appendix C), and their weighted mean $\mathrm{log}{\hslash}_{*}(\text{J}\cdot \text{s})\cong -21.87\text{\hspace{0.17em}}\pm 0.18$. (

**b**) Chi-square minimization (fitting). (

**c**) Chi-Square distribution and p-value.

#### B2. Statistical Determination of ${\hslash}_{*}$ for the Four Methods

_{0}) is that ${\hslash}_{*}$ is a constant for the values found for the N=4 independent methods shown in Table 3 of main text. We find $\overline{\mathrm{log}{\hslash}_{*}}=-21.93$, $\sigma =0.14$, and ${\chi}_{\text{est}}^{2}=1.6$. This leads to large p-value $\cong 0.34$, thus H

_{0}is highly likely and ${\hslash}_{*}$ can be well represented by this single value, even though the range of standard deviations over the four methods span one and a half orders of magnitude. Figure B2 demonstrates the results of the statistical method.

**Figure B2.**(

**a**) Plot of the four values of $\mathrm{log}{\hslash}_{*}(\text{J}\cdot \text{s})$, estimated by the four independent methods that are examined in the main text and shown in Table 3; their weighted mean $\mathrm{log}{\hslash}_{*}(\text{J}\cdot \text{s})\cong -21.93\text{\hspace{0.17em}}\pm 0.14$ is also shown. (

**b**) Chi-square minimization (fitting). (

**c**) Chi-Square distribution and p-value.

## Appendix C: Characteristics of Space Plasmas out of Thermal Equilibrium

PLASMA TYPE | Refs | log n | log T | log λ_{D} ^{b} | log N_{D} | log U_{ms} | log ε_{C} | log t_{C} | log ${\hslash}_{*}\text{\hspace{0.17em}}$ |
---|---|---|---|---|---|---|---|---|---|

Corona (CO) | [16,36,37,38] | 11.0 ± 1.0 | 6.7 ± 0.6 | −0.5 ± 0.6 | 10.2 ± 1.0 | 6.3 ± 0.5 | −14.5 ± 1.0 | −7.6 ± 0.7 | −21.7 ± 1.2 |

Corona Holes (CH) | [39,40] | 10.5 ± 1.0 | 6.0 ± 1.0 | −0.6 ± 0.7 | 9.4 ± 1.6 | 5.7 ± 0.5 | −15.7 ± 1.0 | −7.3 ± 0.9 | −22.7 ± 1.3 |

Inner Heliosheath (IH) | [7,41] | 4.2 ± 0.6 | 5.5 ± 0.3 | 2.3 ± 0.3 | 11.8 ± 0.5 | 4.9 ± 0.3 | −17.2 ± 0.6 | −4.2 ± 0.4 | −21.0 ± 0.7 |

Interstellar (IS) ^{c} | [42,43] | 5.0 ± 0.5 | 3.5 ± 0.5 | 0.9 ± 0.4 | 8.4 ± 0.8 | 4.5 ± 0.5 | −18.1 ± 1.0 | −4.6 ± 0.4 | −22.3 ± 1.1 |

Magnetosphere (MA) | [13,16,44] | 8.5 ± 1.5 | 6.3 ± 1.7 | 0.6 ± 1.1 | 10.9 ± 2.7 | 5.6 ± 0.5 | −16.0 ± 1.0 | −6.3 ± 1.4 | −22.0 ± 1.7 |

Magnetosheath (MS) | [45,46,47,48] | 7.0 ± 0.5 | 5.5 ± 0.5 | 0.9 ± 0.4 | 10.4 ± 0.8 | 5.1 ± 0.5 | −16.9 ± 1.0 | −5.6 ± 0.4 | −22.1 ± 1.1 |

Plasma Sheet (PS) | [17,49] | 5.5 ± 1.0 | 6.5 ± 0.5 | 2.2 ± 0.6 | 12.7 ± 0.9 | 5.6 ± 1.0 | −15.9 ± 2.0 | −4.8 ± 0.6 | −20.4 ± 2.1 |

Ring Current (RC) ^{d} | [17,50] | 6.3 ± 1.0 | 7.5 ± 0.5 | 2.3 ± 0.6 | 13.8 ± 0.9 | 5.9 ± 0.7 | −15.2 ± 1.4 | −5.2 ± 0.6 | −20.1 ± 1.5 |

Solar Wind - Helios (WH) ^{e} | [21,51] | 7.5 ± 0.5 | 5.6 ± 0.3 | 0.7 ± 0.3 | 10.3 ± 0.5 | 5.2 ± 0.2 | −16.6 ± 0.4 | −5.8 ± 0.3 | −22.1 ± 0.5 |

Solar Wind - Ulysses (WU) | [21,51] | 5.6 ± 0.5 | 4.9 ± 0.3 | 1.3 ± 0.3 | 10.2 ± 0.5 | 4.7 ± 0.2 | −17.6 ± 0.4 | −4.9 ± 0.3 | −22.2 ± 0.5 |

Tail Lobe (TL) | [17,52] | 5.0 ± 1.0 | 5.5 ± 0.5 | 1.9 ± 0.6 | 11.4 ± 0.9 | 5.1 ± 0.7 | −16.9 ± 1.4 | −4.6 ± 0.6 | −21.1 ± 1.5 |

^{a}Units are in SI.

^{b}Debye length λ

_{D}is calculated assuming quasineutral plasma of density n and common ion-electron temperature T, and it is given by ${\lambda}_{D}\cong \sqrt{({k}_{\text{B}}\epsilon /\text{}{q}_{e}^{2})T\text{}/(2n)}$.

^{c}IS: Local Interstellar Medium.

^{d}RC: (magnetic field ~10

^{1.5}nT).

^{e}WH & WU: Data are derived by analyses of the authors and used for the scope of this paper.

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Livadiotis, G.; McComas, D.J.
Evidence of Large-Scale Quantization in Space Plasmas. *Entropy* **2013**, *15*, 1118-1134.
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Evidence of Large-Scale Quantization in Space Plasmas. *Entropy*. 2013; 15(3):1118-1134.
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