Complexity as thermodynamic depth
A measure of complexity for the macroscopic states of physical systems is defined. Called depth, the measure is universal: it applies to all physical systems. The form of the measure is uniquely fixed by the requirement that it be a continuous, additive function of the processes that can result in a state. Applied to a Hamiltonian system, the measure is equal to the difference between the system's coarse- and fine-grained entropy, a quantity that we call thermodynamic depth. The measure satisfies the intuitive requirements that wholly ordered and wholly random systems are not thermodynamically deep and that a complex object together with a copy is not much deeper than the object alone. Applied to systems capable of computation, the measure yields a conventional computational measure of complexity as a special case. The relation of depth and thermodynamic depth to previously proposed definitions of complexity is discussed, and applications to physical, chemical, and mathematical problems are proposed.