Low speed efficiency limit

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A relevant question about gemstone gum minifactories is how efficient they can work.
A quantity of interest is thermalized energy per atom deposition (or atom abstraction) operation.
The core problem is that if to less energy is thermalized one risks that mechanosynthesis runs backward.

Why an attraction force alone makes no bond

Advanced mechanosynthesis (machine phase chemistry) is quite far from "normal" solution phase chemistry but both underlie deeper laws of physics. For chemical reactions to run in forward direction the for the observed situation appropriate thermodynamic potential must decrease. In typical solution phase chemistry the appropriate thermodynamic potential is the Gibbs free energy potential.

Bouncing off

To understand this from a different deeper lying perspective let's look at an isolated attempted to cause a bonding reaction that fails. Assuming two reactant atoms A and B (for simplicity of same mass) attract each other by some interaction force. They are both placed in a vacuum. We start with the two atoms far apart from one another. So far that the interaction force is near zero. They move with a decent (thermal) speed towards each other. First they accelerate towards each other converting their potential energy into kinetic energy. Lets assume they collide elastically that is there are no photons emerging that would carrying away some energy. After the collision the speeds just change direction but keep their magnitudes. The atoms move apart from another again. They decelerate converting kinetic energy in potential energy. Finally they are far apart from each other and still move away from another with the decent speed from the beginning. Although the two atoms attract each other they cannot form a permanent bond.

Sticking

Now let's add to one of the reactant atoms (atom B1) another atom (atom B2) in a bond state. That is B1 and B2 are in rest relative to one another. When the experiment is repeated the collision of atom A with the pair of B's can pump energy in the relative speed between B1 and B2. So much in fact that all the speeds get reduced to a degree that none of three atoms can move out and leave forever. This is only temporary though. Three bodies form a complex chaotic system and after a shorter or longer while (depending on the exact initial conditions) at least one of the atoms gets so much energy again that it leaves forever.

To extend the time until this random "reejection" happens we can add a third atom B3 and a fourth and a fifth and so on until one ends up with a crystal surface B* as reaction partner for the atom A (or a liquid but crystals fit better in the context of advanced in vacuum APM). What happens here is essentially that the kinetic impact energy of A gets chaotically distributed to all the atoms of B* (it gets thermalized/devaluated). For small crystals with only a handful of atoms the time until "reejection" of the introduced energy is measurable but with rising crystal size the time to "reejection" quickly rises to times far longer than the age of our universe. At this point we can justifyably say that a permanent bond has been formed.

The distribution (thermalisation/devaluation) of the kinetic energy of atom A into crystal B* is of course equivalent to the warming of the crystal B*.

Crystals at room temperature already have lots of thermal energy distributed in them. So beside needing a multi atom crystal for a successful binding reaction to happen the binding strength need to be sufficient such that a random jost from the inherent vibrations does not break the newly formed bond loose in a short period of time (E_binding >> k_B*T).

All this holds true both for normal chemistry and advanced mechanosynthesis.

Advantages of advanced mechanosynthesis

There are two major advantages advanced mechanosynthesis has over normal solution phase chemistry. Those are energy recuperation and the sharing of the required thermalisation rate.

Recuperation of energy

Energy recuperation is possible since when moving an atom that is attracted to a surface towards that surface with a tooltip the pulling force on that tooltip can be used to do useful work. Especially if the mechanics of many tooltips that work in parallel are connected in the background.

In "normal" solution phase chemistry often the hole bonding energy gets thermalized. Many enzymes in biology do better though.

Recuperation 100% of the binding energy is not possible since then there would be no energy devaluation/thermalisation wich is (as explained above) unconditionally necessary to archive permanent bonds. Recuperation rate can go arbitrary near 100% though. (TODO: search for fundamental limits)

Sharing of the thermalization rate

Main article: Dissipation sharing

To ensure mechanosynthesis runs forward energy needs to be thermalized at a sufficient rate. By coupling the mechanics of many tooltips together the individual bonding reactions do not need to thermalize energy >>k_B*T each.

In contrast to "normal" solution phase chemistry where the minimal temerature is given by reaction slowdown and ultimately the solutions freezing point mechanosynthesis can be performed at very low temperatures reducing the k_B*T factor. Practical is most likely a factor of 10.

(TODO: figure out per what exatly 1k_B*T needs to be dissipated to ensure highly reliable forward motion of mechanosynthesis)

If the mechanics in the background of advanced mechanosynthesis mills would be infinitely stiff it would behave as just one single degree of freedom. In reality the mechanics in the background have some elasticity though. Coupling mechanosynthesis mills together which are too far apart will likely include modes of elastic thermal oscillations making it behave as more than one degree of freedom containing more than one thermal energy packet of k_B*T. Gearing may help in coupling things together that are farther apart since gearing can create a higher stiffness in a virtual way.

Via phase shifts reaction coupling can be done in time too.

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