In 1961 Roger Landauer argued that any logically irreversible computation must produce a minimum amount of heat. The argument at its most basic: consider a particle in one of two equivalent but distinct states called "0" and "1". If that particle is then forced to be in just the "0" state, effectively "erasing" or "zeroing" the particle, that operation reduces the particle's number of states by half, decreasing its entropy by $\ln(2)$. The entropy of the universe must have increased by at least $\ln(2)$ in kind, and this is envisioned by Landauer as a transfer of at least $k T \ln(2)$ joules of heat into the universe.
Let's put aside whether this is true or not in the general case. I want to make a model of a machine that does operate at this limit, what I'm calling a "Landauer machine".
There are many models out there of reversable computers: ones that generate no heat at all. There are fewer models of Landauer machines, but they do exist. [todo: examples? Maybe talk about Landauer's own model?]
The model described here goes back to our thermodynamic roots. It does computation on gas particles in containers by compressing and expanding them in various ways.
Some papers I've read hand-wave away some details of how a computer like this would work. You have to be careful though. You can't just attach any old "sensor" to the boxes and decide what operations to perform, because that implies a form of computation, and you'd have to argue that process is itself reversible.
So I'm deliberate in my choice of primitives:
Probably the simplest example of a reversible operation is NOT:
[image]
[todo: explain]
The simplest example of an irreversible operation is RST:
[image]
This is a classic thermodynamics example. The dividing wall is removed between the two halves, doing a free expansion on the gass. Pressure is halved, volume doubled, temperature remains the same. Then a piston isothermally compresses into the correct half, leaving behind a vaccum in the other. This compression performs $k T ln(2)$ joules of work on the gass. But since we're keeping a constant temperature, the energy of our monoatomic gass is also contant, and so that $k T ln(2)$ joules radiates off as heat, exactly as the Landauer limit predicts.
$$S = k \ln \left( V \left( \frac{4 \pi m U}{3 h^2} \right)^{3/2} \right) + \frac{5}{2}$$
$$\Delta S = k \ln(2)$$
The simplest reversible opeation if the CNOT or Toffoli gate:
[image]
The cooresponding 2-input irreversible operation is the controlled set / controlled reset:
[image]
CSET is equivalent to OR.
CRST is equivalent to the negative converse gate, which I don't know any cool abbreviation for. Negating the the A input results in a AND gate.
The construction of these gates has the property that work is done, and therefore heat produced, only when A = 1.
The construction
https://worrydream.com/refs/Landauer_1961_-_Irreversibility_and_Heat_Generation_in_the_Computing_Process.pdf
https://www.feynmanlectures.caltech.edu/I_46.html