If it were really purely functional it wouldn't even warm up. When you compile your code, referential transparency would result in your executable being nothing more than the final value produced by your code.
If we want to get really pedantic, I guess we could say that we're using one Turing machine (our computer) to emulate another equivalent model of computation (the lambda calculus), so the computer warms up because the Turing Machine is an inefficient emulator of the instantaneously-evaluating lambda calculus evaluator. But the lambda calculus evaluator doesn't change state.
...so, what we really need is to wait for someone to invent a lambda calculus evaluator. Then we wouldn't need to emulate them with these silly Turing machines, and we could get instant evaluations of our programs based solely on ϐ-reductions, with no side-effects (thermal side-effects included). That would put an end to this debate!
(Furthermore, who cares if P=NP if ϐ-reductions can be evaluated 'directly' instead of being emulated by these obsolete Turing machines? Even the hardest decidable problems would be solved instantaneously!)
Yeah, in case it wasn't clear, by my second reply I was just being facetious.
Pedantry is infinitely recursive, with no base case in sight and no tail optimization - my tolerance/stack for these things overflows rather quickly. :-)
But thanks for the link! I'll be sure to check it out later; it looks interesting.
Depends on the program. If it were computing all the twin primes it would warm up. Here the input would be round about - how long you kept the program running. And since each time a bit was erased or a non-not operation was done > klog2(T) J must be dissipated, a sufficiently sophisticated being could use the timings and heat signature to infer the state and value to extract a runnning output. Another output you could more easily get is that if it started to cool down and you had a notion to return an output after 10^(absurdly large number) successive failures then you have a pretty good confidence bound on the falsity of the conjecture..
