Final answer:
A non-equilibrium cascade in ecological systems serves as an example of a non-thermal equilibrium relationship where A influences B and B influences C, but A does not directly influence C, deviating from the principles of the Zeroth law of thermodynamics which assumes thermal equilibrium to be transitive.
Step-by-step explanation:
The question seeks an example where two systems, A and B, are in some form of non-thermal equilibrium with each other, and B has the same relationship with another system C, but A does not have that relationship with C.
This is an interesting scenario because it deviates from the concept outlined by the Zeroth law of thermodynamics, which states that if A is in thermal equilibrium with B, and B is with C, then A must also be in thermal equilibrium with C, assuming a transitive relationship due to the matching temperatures among the systems.
An example of a non-thermal equilibrium relationship could be a non-equilibrium cascade. In this case, let's consider three interconnected ecological systems where A affects B through some form of a cascading effect, and B affects C in the same manner.
However, A does not directly influence C in this way. This could occur in a predator-prey relationship where the presence of predator A controls the population of prey B, and prey B controls the resource species C, but predator A does not directly affect the resource species C.