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Re: Entropy: quenched disorder is entropy



Today in several messages, David Bowman and Leigh Palmer have been
advocating the notion that quenched disorder should not be counted as
entropy; for instance:
> The ordered
>pack has exactly the same entropy as the disordered pack,

Alas, that isn't right.

In fact, it is a standard result from the thermodynamics of computation
that quenched disorder is disorder, and it counts as entropy.

In particular, we can draw a very strict analogy:
sorted card deck <==> blank tape for Turing machine
shuffled card deck <==> "dirty" tape containing random 1s and 0s

and beautiful classic results show that the thermodynamic cost of erasing
the dirty tape is on the order of kT per bit, whereas blank tape doesn't
need to be erased, and any tape with a known (or easily computable) pattern
can be erased at arbitrarily small cost (much less than kT per bit).

The arguments advocated earlier today have no calculable
consequences; that is, they don't prove anything. On the contrary,
detailed calculations show that it doesn't matter that the tape is not part
of an ergodic system. Similarly, it doesn't matter that you cannot afford
to exhibit the entire ensemble of tapes. The dirty tape is entropic -- get
used to it.

The correct physical argument goes something like this: Imagine that bits
are represented by the position of a particle in a double-well
potential. The left well represents zero, and the right well represents
one. Imagine now a machine to erase a large number of such bits. If it
knows the bit pattern, it knows where to find each particle, so it can grab
each particle that needs to be moved and isentropically move it. OTOH if
it doesn't know which ones need to be moved, it has to guess, and each time
it guesses wrong the particle will dissipate something like kT of
energy. You can visualize trying to scoop up the particle with a scoop big
enough to encompass the zero-well and the one-well; if the scoop is that
big the particle will go "clunk" when it gets scooped. For details, please
refer to the scintillatingly clear and scholarly papers by Charlie Bennett
and Rolf Landauer.

---

A reversible computer has the interesting property that the thermodynamic
cost of doing the computation is only the cost of writing down the answer
-- and writing on blank tape is costless. Writing on dirty tape is
costly. Blank tape is more valuable than dirty tape.

These ideas are central to the work of Seth Lloyd (which as you may recall
was the original subject of this thread).