# Re: This Week's Finds in Mathematical Physics (Week 235)

1. Jul 18, 2006

### John Baez

In article <e9b9vh$r0g$1@gemini.csx.cam.ac.uk>,
Ben Rudiak-Gould <br276deleteme@cam.ac.uk> wrote:

>John Baez wrote:

>> In short: for a fundamentally analogue physical system to keep
>> acting digital, it must dispose of irrelevant information, which
>> amounts to pumping out waste heat.

>> In fact, Rolf Landauer showed back in 1961 that getting rid of
>> one bit of information requires putting out this much energy
>> in the form of heat:
>>
>> kT ln(2)

>It's easy to understand where this formula comes from. Getting rid of a bit
>means emitting one bit of entropy, which is k ln 2 in conventional units.
>The associated quantity of heat is ST = kT ln 2.

Thanks; I should have said that. Maybe I'll add it to the version
on the web.

Landauer's analysis showing that "forgetting information" costs
energy is still interesting, and it was surprising at the time.
There had been a number of other analyses of why Maxwell's demon
can't get you something for nothing, by Szilard and others, but
none (I think) had focussed on the key importance of resetting
the demon's memory to its initial state.

>But it seems to me that you're conflating two different issues here. One is
>the cost of forgetting a bit, which only affects irreversible computation,
>and the other is the cost of keeping the computation on track, which affects
>reversible computation also. Landauer's formula tells you the former, but I
>don't think there's any lower bound on the latter.

Not in principle: with a perfectly tuned dynamics, an analogue
system can act perfectly digital, since each macrostate gets mapped
perfectly into another one with each click of the clock. But with
imperfect dynamics, dissipation is needed to squeeze each macrostate
down enough so it can get mapped into the next - and the dissipation
makes the dynamics irreversible, so we have to pay a thermodynamic cost.

If I were smarter I could prove an inequality relating the "imperfection
of the dynamics" (how to quantify that?) to the thermodynamic cost
of computation, piggybacking off Landauer's formula.