# A Photon Force and Flow for Dissipative Structuring: Application to Pigments, Plants and Ecosystems

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## Abstract

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## 1. Introduction

## 2. Derivation of the Planck Equation for the Entropy of an Arbitrary Photon Spectrum

## 3. Derivation of Planck’s Radiation Law for the Spectrum of a Black-Body

#### 3.1. Planck’s Radiation Law

#### 3.2. Energy Emitted by a Black Body at Temperature T

## 4. A Generalized Photon Force and Flow for the Photon Dissipation Process

## 5. Entropy Production of an Arbitrary Photon Beam Interacting with Material

## 6. Photon Force

**Figure 4.**Phytoplankton pigment absorption spectra (m${}^{2}$ mg${}^{-1}$) from Bricaud et al. [14]. Absorption spectra of photosynthetic and nonphotosynthetic carotenoids are shown in red and blue, respectively. All pigments (including chlorophyll) have a wavelength of maximum absorption of $430<{\lambda}_{max}<550$ nm, exactly where entropy production due to photon dissipation is maximal for our solar spectrum (see Figure 5). Reprinted with permission from the American Geophysical Union.

## 7. Comparison of the Entropy Production of Inorganic and Dissipatively Structured Organic Material

## 8. Discussion

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CIT | Classical Irreversible Thermodynamic theory |

UV | ultraviolet light |

UVC | light in the region 100–285 nm (the region 210–285 nm being relevant to dissipative |

structuring on Archean Earth) |

## References

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**Figure 1.**Diagram of incident radiation (energy $i(\lambda ,\theta ,\varphi )$ or entropy $j(\lambda ,\theta ,\varphi )$) in solid angle $d\Omega $ over a surface element $dA$.

**Figure 2.**The light arriving in a parallel beam from a source ${i}^{i}\left(\lambda \right)$ and captured by the slab of material, with a fraction $a\left(\lambda \right)$ absorbed and emitted isotropically as a black- or grey-body spectrum ${i}^{e}\left(\lambda \right)$ and a portion $1-a\left(\lambda \right)$ reflected or transmitted ${i}^{r,t}\left(\lambda \right)$ with the same wavelength dependence as the incident spectrum. $\theta $ is the angle of the incident, emitted, reflected or transmitted beams (the last two assumed to be Lambertian), with respect to the surface normal.

**Figure 3.**The thermodynamic force $F\left(\lambda \right)$ (Equation (50)—blue line) as a function of wavelength $\lambda $ for the irreversible process of photon dissipation of an incident beam arriving parallel to a slab of material (e.g., a leaf) with a fraction $a\left(\lambda \right)=0.90$ absorbed and emitted isotropically as a grey-body spectrum (emissivity ${\u03f5}_{avg}=0.980$) and a portion $1-a\left(\lambda \right)=0.10$ reflected or transmitted with the same wavelength dependence as the incident spectrum. The contributions of the emitted (red line), reflected plus transmitted beams (green line) are assumed Lambertian. The grey line corresponds to the negative contribution of the incident light. Plotted on the right y-axis (black line) is the contribution to the entropy production, $F\left(\lambda \right)\xb7N\left(\lambda \right)$ (Equation (49)), per m${}^{2}$ per second integrated over a wavelength bin of 1 nm at the given wavelength for an incident energy spectrum ${i}^{E}\left(\lambda \right)$ of the Sun at Earth’s surface ( grey-body, absorption $a\left(\lambda \right)=0.90$, emissivity $\u03f5=0.98$, solar constant 1000 Wm${}^{-2}$). Entropy production peaks at 502 nm for our solar spectrum. All major plant and phytoplankton pigments (chlorophyll a, b, carotenes, Xanthins, etc.) have their peak absorption between 430 and 550 nm (see Figure 4) and this is exactly where entropy production (black line) is maximized, whereas photosynthetic efficiency is maximal between 600 and 700 nm. This is a strong indication that plants have evolved to optimize photon dissipation rather than photosynthesis.

**Figure 5.**Plotted on the left y-axis (blue line) is the number of photons $N\left(\lambda \right)$ incident at Earth’s surface per m${}^{2}$ per second for an incident energy spectrum ${i}^{E}\left(\lambda \right)$ of the Sun at Earth’s surface (grey-body, absorption $a\left(\lambda \right)=0.90$, emissivity $\u03f5=0.98$, solar constant 1000 Wm${}^{-2}$). Plotted on the right y-axis (black line) is the contribution to the entropy production, $F\left(\lambda \right)\xb7N\left(\lambda \right)$, per m${}^{2}$ per second integrated over a wavelength bin of 1 nm at a given wavelength. Entropy production peaks at 502 nm, while photon intensity (number of photons) peaks at 635 nm for our solar spectrum.

**Figure 6.**The absorption (black line) and emissivity (grey line) of water of 100 m thickness. Plotted also are the incident surface spectrum (blue) and the reflected and transmitted (violet) and emitted (red) spectra on a log–log scale (right y-axis). The data for the absorption are approximated from Chaplin [15] and the data for the emissivity are taken from Huang et al. [16].

**Figure 7.**The absorption (black line) and emissivity (grey line) of water of 2 m thickness. Plotted also are the incident surface spectrum (blue) and the reflected and transmitted (violet) and emitted (red) spectra on a log–log scale (right y-axis). The data for the absorption are approximated from Chaplin [15] and the data for the emissivity are taken from Huang et al. [16].

**Figure 8.**The absorption (black line) and emissivity (grey line) of water of 235 μm thickness. Plotted also are the incident surface spectrum (blue) and the reflected and transmitted (violet) and emitted (red) spectra on a log–log scale (right y-axis). The data for the absorption are taken from Chaplin [15] and the data for the emissivity are taken from Huang et al. [16].

**Figure 9.**The absorption (black line) and emissivity (grey line) of a dry sand and pebble desert. Plotted also are the incident surface spectrum (blue) and the reflected and transmitted (violet) and emitted (red) spectra on a log–log scale (right y-axis). The data for the absorption are approximated from the experimental data of Pinker and Karnieli et al. [17] for a semi-arid region of the Sahara and the data for the emissivity are taken from Mattar et al. [18] for the Atacama desert.

**Figure 10.**The absorption (black line) and emissivity (grey line) of a leaf. Plotted also are the incident surface spectrum (blue) and the reflected and transmitted (violet) and emitted (red) spectra on a log–log scale (right y-axis). The data for the absorption are approximated from Gates [19] and the data for the emissivity are taken from Ribeiro and Crowley [20] for a Cornus florida (Dogwood) leaf.

**Figure 11.**The absorption (black line) and emissivity (grey line) of a forest. Plotted also are the incident surface spectrum (blue) and the reflected and transmitted (violet) and emitted (red) spectra on a log–log scale (right y-axis). The data for the absorption are approximated from Rautiainen et al. [21] for an 80-year old coniferous spruce forest and the data for the emissivity are taken from Huang et al. [16].

**Table 1.**Entropy production due to the interaction of the solar spectrum at Earth’s surface with different materials. The incident solar spectrum is that of a grey-body at T $=5779$ K with emissivity 0.986 and adjusted for atmospheric absorption by reducing the solar radius (see text) to give a solar constant at Earth’s surface of 1000 W m${}^{-2}$. Effective temperatures ${T}_{eff}$ are obtained by using energy balance equations and the Stefan–Boltzmann law. Material temperature ${T}_{e}$ is obtained by including the average emissivity ${\u03f5}_{avg}$ (wavelength weighted by emitted photon energy). ${J}^{i}$ is the incident entropy flow, ${J}^{r,t}$ is the reflected and transmited entropy flow, ${J}^{e}$ is the red-shifted emitted entropy flow and $\mathbf{J}$ is the total entropy production per square meter of the material.

Material | Slab | H${}_{2}$O | H${}_{2}$O | H${}_{2}$O | Desert | Leaf | Forest |
---|---|---|---|---|---|---|---|

BB Ideal | 100 m | 2 m | 235 $\mathsf{\mu}$m | Sand+Stone | 235 $\mathsf{\mu}$m | Conifer | |

${a}_{avg}$ | 1.00 | 0.988 | 0.582 | 0.071 | 0.517 | 0.540 | 0.944 |

${\u03f5}_{avg}$ | 1.00 | 0.951 | 0.945 | 0.919 | 0.926 | 0.984 | 0.994 |

${T}_{eff}$ (K) | 364.57 | 363.53 | 318.48 | 188.46 | 309.16 | 312.57 | 359.34 |

${T}_{e}$ (K) (with ${\u03f5}_{avg}$) | 364.57 | 368.10 | 322.99 | 192.50 | 315.15 | 313.79 | 359.83 |

Entropy Flux | |||||||

${J}^{i}$ (W m${}^{-2}$ K${}^{-1}$) | 0.233 | 0.233 | 0.233 | 0.233 | 0.233 | 0.233 | 0.233 |

${J}^{r,t}$ (W m${}^{-2}$ K${}^{-1}$) | 0.00 | 0.008 | 0.256 | 0.769 | 0.490 | 0.484 | 0.065 |

${J}^{e}$ (W m${}^{-2}$ K${}^{-1}$) | 7.312 | 7.250 | 4.875 | 1.009 | 4.456 | 4.608 | 7.002 |

$\mathbf{J}$ (W m${}^{-2}$ K${}^{-1}$) | 7.080 | 7.026 | 4.898 | 1.545 | 4.714 | 4.859 | 6.834 |

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**MDPI and ACS Style**

Michaelian, K.; Cano Mateo, R.E. A Photon Force and Flow for Dissipative Structuring: Application to Pigments, Plants and Ecosystems. *Entropy* **2022**, *24*, 76.
https://doi.org/10.3390/e24010076

**AMA Style**

Michaelian K, Cano Mateo RE. A Photon Force and Flow for Dissipative Structuring: Application to Pigments, Plants and Ecosystems. *Entropy*. 2022; 24(1):76.
https://doi.org/10.3390/e24010076

**Chicago/Turabian Style**

Michaelian, Karo, and Ramón Eduardo Cano Mateo. 2022. "A Photon Force and Flow for Dissipative Structuring: Application to Pigments, Plants and Ecosystems" *Entropy* 24, no. 1: 76.
https://doi.org/10.3390/e24010076