Gas radiation has no thermal equilibrium, and the second law of thermodynamics is invalid. The following pictures are all from the content of heat transfer and university physics, combined together, it is found that the second law of thermodynamics is invalid.
Please refer to the picture for details.
Even if some considerations are correct, the conclusions are completely wrong. You start from a non equilibrium conditions and assume that things happens only locally. To be correct one have to write the kinetic equations for the molecules and the photons solving the problem self-consistently. The problem must be divided in two aspects. From the mathematical point of view, given an isolated system with a fixed value of particle inside and a fixed total energy, the question is: does a solution of the stationary problem exists? the answer is yes and it is the equilibrium state. Now the physical problem is: if we start from a non equilibrium condition, does the system reach equilibrium? Even if your considerations are correct, particle motion will bring the system towards a thermodynamic equilibrium. Due to the large speed of photons, even in non-homogeneous systems, the photon density will reach soon an homogeneous distribution.
When a photon is emitted, its frequency in the laboratory reference is not exactly the equal to the difference of level energy due to doppler shift, which is more pronounced where the temperature is larger, and natural broadening due to the Heisenberg principle. Moreover there are not only internal transitions induced by photons, but also elastic collisions which slightly change the photon frequency. A closed system with molecules interacting with radiation is more complex than that described in the figure, and neglecting all these aspects can bring to wrong conclusions.
Another consideration about the microscopic dynamics. When a photon is absorbed or emitted, eq. 2.1 is not valid, because you are changing the radiation with internal states that deviates from the Boltzmann distribution, while the temperature is measured as the translational motion. Therefore, you need collisions to that transfer energy from the internal degree of freedom to the translational one.
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