Thermodynamics of used punched tape: A weak and a strong equivalence principle

We study the repeated use of a monotonic recording medium--such as punched tape or photographic plate--where marks can be added at any time but never erased. (For practical purposes, also the electromagnetic "ether" falls into this class.) Our emphasis is on the case where the successive users act independently and selfishly, but not maliciously; typically, the "first user" would be a blind natural process tending to degrade the recording medium, and the "second user" a human trying to make the most of whatever capacity is left. To what extent is a length of used tape "equivalent"--for information transmission purposes--to a shorter length of virgin tape? Can we characterize a piece of used tape by an appropriate "effective length" and forget all other details? We identify two equivalence principles. The weak principle is exact, but only holds for a sequence of infinitesimal usage increments. The strong principle holds for any amount of incremental usage, but is only approximate; nonetheless, it is quite accurate even in the worst case and is virtually exact over most of the range--becoming exact in the limit of heavily used tape. The fact that strong equivalence does not hold exactly, but then it does almost exactly, comes as a bit of a surprise.