Why does entropy drive evolution equations?

`Entropy' appears as driving force in many different evolution equations, both deterministic and stochastic, and in these equations this `entropy' also takes different forms. We show how all these examples can be understood as different instances of a common principle: Entropy drives evolutions because it characterizes the invariant measure of an underlying stochastic process. This interpretation explains the appearance of entropy, the different forms that entropy takes in these equations, and how entropy `drives' these evolution equations. We illustrate this common structure with examples from stochastic processes, gradient flows, and GENERIC systems.