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In this study, the theoretical maximum efficiency $\eta_{max}$ and theBoltzmann factor for photosynthetic systems under irradiation with arbitraryphoton flux density $n_\gamma(\lambda)$, solid angle $\Omega$, and degree ofpolarization P, are formulated in the most fundamental and general way throughthe energy and entropy flow analysis, using reversibility and the first-orderevaluable condition by the photon number change which is a quasi-equilibriumcondition between the radiation and the system, as essential conditions. Theradiation temperature for the diluted monochromatic light as non-equilibrium,obtained by the fundamental formulation of this study is found to agree withthe conventional radiation temperature, often called the effective temperature,for a given photon flux density (light intensity), provided that $\Omega$ and Pof the radiation are 4$\pi$ and 0, respectively. The reason for this agreementis discussed in the final section. The formulation in this study allowsquantitative analyses that are not possible with conventional radiationtemperature. As examples, the formulation of $\eta_{max}$ taking into accountentropy changes due to photochemical reactions such as glucose production byphotosynthesis, and various quantities under irradiation at arbitrary $\Omega$and P are presented. In Appendix C, a specific and rigorous proof, usingelementary geometry, of the fact that the entropy of radiation diluted on itsway from the Sun to the Earth does not increase and remains unchanged from itsoriginal value, until it is scattered by the Earth's atmosphere, which isguaranteed only in general terms by Liouville's theorem, is given.

Generalized formulation for ideal light-powered systems through energy and entropy flow analysis Part1. Based on the first-order evaluation