View source for 1D diffusion transport of crystolecules

'''The (questionably feasible) idea:''' <br>
To transport crystolecules along a track instead of using a chain for active dragging <br>
could it be possible to use:
* thermal motion plus 
* valve like acting boundary points
Points where necessarily some energy needs to be dissipated <br>
to enforce directed motion ([[arrow of time]], [[brownian ratchet]])?

A clear prerequisite is that the energy barriers (from atomic corrugations) against translation are not much bigger than kT but rather smaller than kT. <br>
This is quite easily achievable even for quite low temperatures. <br>
See: [[Superlubricity]] & [[Atomically precise slide bearing]]s & [[Friction in gem-gum technology]]

The [[equipartitioning theorem]] says that every degree of freedom gets ~ kT worth of energy. <br>
With crystolecules being comprised of quite a lot of atoms that kT worth of energy dilutes down to all of these atoms.

For very short distances diffusion speeds are close to speed of thermal motion & speed of sound. <br>
This is the maximum speed that diffusion can ever reach. <br>
But that is dropping off rapidly when one increases the distance between valve like acting boundaries.

Effective thermal transport speeds (across minimal distances) are for individual atoms and diatomic molecules are on the order of several 100m/s. <br>
Closely relates is speed of sound in air 343m/s.

To get the new shortest-distance-maximal-diffusion speed: <br>
<math> m_1 v_1^2/2 = kT = m_2 v_2^2/2 </math> <br>
<math>  v_2 = v_1 \sqrt{m_1/m_2}  </math> <br>
Note that a crystolecule (plus carrying sled) 100x bigger than an N<sub>2</sub> molecule has 1000000x the volume and mass. <br>
So it diffuses 1000x slower. 343mm/s absolute max.

Adding reasonable inter-valve distances will drop speeds by another huge factor. {{todo|figure that out}} <br>
This should become expressed in the diffusion constant C.

'''To remain competitive with machine phase:''' 
* Transport speeds >=1mm/s should remain.
* Dissipation losses should remain under the level that machine phase transport incurs.
Beside the former being questionable the latter seems very unlikely too since:
* Machine phase transport without manipulation stations can go through free space so losses are only incurred at the occasional support  sprocket.
* Machine phase transport has no need for closely spaced anti back-reaction dissipation valves to enforce a preferred direction of transport. <br>Yes, it still needs to dissipate to get an [[arrow of time]] but it can do [[dissipation sharing]]. 

== The preventing stick-together challenge ==

For this kind of transport crystolecules on tracks need to behave like a "one dimensional crystolecule gas".
* Parts are not in machine phase along the deree of freedom that sliding along the rack constitutes.
* Parts are in machine phase with respect to motions normal to the tracks constraint. These are not allowed.

When a crystolecule carrying sleds hits the one in front or the one behind then it must not stick to it. <br>
To that end inter-sled VdW attraction must be kept <kT. <br>
Flat on flat surfaces are a no-go. These easily attract with energies >>kT. <br>
E.g. Sleds could be designed to collide on a sharp spike. <br>
But will this get attraction forces down far enough?

== Math ==

* s_trav … distance betwene valve type transition points
* k_valve = 1/s_trav … spacial frequency of valve type transition points
* f_valve … average frequency of valve crossings
* E_cross … energy dissipated per valve crossing
* m … crystolecule mass
* t_trav … average time to travel from valve to next valve 
* v_eff … effective transport speed
* P_loss … energy dissipation from valve crossings
* D … diffusion constant
----

{{wikitodo|review below math, check units}}

Expectation value for the position in diffusion: (For 1D motion q=2) <br>
<math> < s_{trav}^2 > = q D t_{trav} </math> <br>

How does the expectation value change when one direction blocked? <br>
The following is likely not right (off by some factor {{wikitodo|fixit}}) but <br>
it should give the correct order of magnitude: <br>
<math> t_{trav} = s_{trav}^2 / (2D) </math> <br>
Or equivalently: <br>
<math> f_{valve} = 2D / s_{trav}^2 </math> <br>
----
Let's find the effective diffusion speed across many valves: <br>
<math> v_{eff} = s_{trav} / t_{trav} </math> <br>
<math> v_{eff}(s_{trav}) = 2D/s_{trav} </math> <br>
Diffusion speed drops linearly with the distance between the valve points. <br>
A more useful question for system design is: <br>
'''What valve spacing do we need to attain a desired speed:''' <br>
<math> s_{trav}(v_{eff}) = 2D/v_{eff} </math> <br>
----
The power-dissipation here is dissipation per crossing times frequency of crossings: <br>
<math> P_{loss} = E_{cross} f_{valve} </math> <br>
Substituting for f_{valve} <br>
<math> P_{loss}(s_{trav}) =  E_{cross} 2D / s_{trav}^2 </math> <br>
Substituting for s_trav <br>
<math> P_{loss}(s_{trav}) =  E_{cross} 2D / (2D/v_{eff})^2 </math> <br>
We get: ??? <br>
Now we get the answer to: <br>
'''What power losses will be present at a chosen speed:''' <br>
<math> P_{loss}(s_{trav}) =  E_{cross} v_{eff}^2 /(2D) </math> <br>
Note that this also decides the s_trav implicitely.

For prevention of excessive backwards diffusion <br>
<math>E_{cross} > kT </math> <br>
For highly reliable prevention of backwards diffusion <br>
<math>E_{cross} >> kT </math> <br>
Assuming active probing at the output of a transport track <br>
there's no need to go excessively high.
----
The diffusion coefficient D depends on crystolecule mass and thus on size: <br>
<math>D(m) = TODO </math> <br>

== Discussion ==

=== Things to consider ===

* E_cross being low enough to allow some amount of back-flow against valve points
* Deviation of diffusion behavior from free unconstrained 1D diffusion due to one side being blocked off (less or more strongly)
* crystolecule carrying sleds can't pass each other they elastically collide (does this have an influence)

=== Checking questionable practicability ===

* High necessary energy dissipation per valve crossing 
* and short inter-valve-distance for reasonable speeds of crystolecules
may lead to too high power losses for practicability. <br>
To be confirmed in math above …

It seems that something like [[dissipation sharing]] is fundamentally impossible here.
Tying crossings together to share [[arrow of time]] giving dissipation leads to 
* bigger masses lower speeds and 
* machine phase
which is mo longer diffusion transport.

=== Inapplicability for cases with regular stations ===

When something like atomically precise manipulation is needed <br>
then the crystolecule needs to be fished back into [[machine phase]] for that.

Fishing crystolecule out of "one dimensional crystolecule gas phase" means (all equivalent):
* the unconstrained degree of freedom along the track is getting "squeezed out" 
* an increase in certainty about the position of the crystolecule
* a reduction of entropy in position space 
* a push of entropy from position space to phase space <br>=> this leads to a release of thermal energy that gets dissipated (if not recuperated by heatpump)

This makes diffusion transport of crystolecules for [[assembly line assembly]] <br>
like in [[molecular mills]] completely unfeasible.

== Related ==

* [[Diffusion transport]]
* [[Machine phase]]