DISSIPATION IN A SEQUENCE OF RELAXATIONS: THE LADDER THEOREM

We consider one relaxation: the complete equilibration of a system to a bath. We show that replacing one relaxation with two relaxations starting and ending at the same states but reaching an intermediate equilibrium along the way always produces less entropy than the single relaxation. We present a completely general proof of this Ladder Theorem, which asserts that the entropy production in a relaxation process is decreased when the relaxation proceeds via intermediate steps.
This is indeed what is observed in the central carbon metabolism of the microbes in coral ecosystems. The microbes in the coral- dominated, energy-limited environments have an enrichment in genes encoding for the Embden Meyerhoff Parnas (EMP) pathway, a central carbon catabolic pathway that has 12 steps involved in the breakdown of sugars to pyruvate. On the other hand, microbes in algal dominated, energy surplus environments have an enrichment for alternative central carbon metabolism pathways such as the Pentose Phosphate (PP) pathway and the Entner Doudoroff (ED) pathway, which contain fewer steps than the EMP pathway.