A thermodynamic analysis on gravitational redshift

Gravitational redshift is derived from General Relativity: when a photon moves upwards in gravity, its wavelength increases, and vice versa. However, is it compatible with the second law of thermodynamics? Particularly, Maxwell’s model of double-column engine demands that in any substance, the equilibrium temperature distribution must be uniform. The main concern is that photon-photon interaction is rare and therefore, photon gas is not a thermodynamic system. In classical mechanics, it is well known that certain nonchaotic particle movements are non-thermodynamic, but they tend to be small-scale, and their energy properties are “trivial”. Yet, as a photon gas is beyond the boundary of the second law of thermodynamics, it could be macroscopic and the consequence is nontrivial: without any other effect, in a gravitational field, with a thermal bath, useful work may be produced through heat absorption from a single thermal reservoir; in an isolated setup, entropy may decrease. Such phenomena, while counterintuitive, can be analyzed on the basis of the principle of maximum entropy, with the additional constraints of nonchaoticity.