Theory of incompatible substructure problem

Schedule

• SWARM2015 paper: camera ready version submitted on 1st September, 2015.
• Submit extended version by December 2015.

Parameters

• Components
  • Radius of each component: 25 mm
  • Thickness of each component: 8 mm
  • Radius of magnet: 1.5 mm
  • Strength of magnet: N48
  • Polyamide

• Container
  • Radius of container: 125 mm
  • Depth of container: 9 mm
  • Material: Acrylic

• Shaker
  • Mode: Orbital shaking
  • Speed: 300 rpm
  • Duration for shaking: until stable structures are formed

Variable(s)

• Number of components used in each experiment

Experiments

• Simulated experiments
  • Probability of the formation of 1, 2, 3, 4 and 5 targets for increasing number of components
• Physical experiments
  • Probability of the formation of 1, 2, 3, 4 and 5 targets for increasing number of components

Particulars of simulation experiments (10000 trials each)

• Probability of the formation of 1 target structure in experiments with increasing number of components from 8 - 40
• Probability of the formation of 2 target structure in experiments with increasing number of components from 16 - 40
• Probability of the formation of 3 target structure in experiments with increasing number of components from 24 - 40
• Probability of the formation of 4 target structure in experiments with increasing number of components from 32 - 40
• Probability of the formation of 5 target structure in experiments with 40 components

Particulars of physical experiments (10 trials each)

• Photos of initial condition and final condition
• Assembly of one target structure
  • Data to be collected: number of targets and number of incompatible substructures, time for steady state
• Assembly of two target structures
  • Data to be collected: number of targets and number of incompatible substructures, time for steady state
• Assembly of three target structures
  • Data to be collected: number of targets and number of incompatible substructures, time for steady state
• Assembly of four target structures
  • Data to be collected: number of targets and number of incompatible substructures, time for steady state
• Assembly of five target structures
  • Data to be collected: number of targets and number of incompatible substructures, time for steady state

• Results to show
  • The simulated probability of the formation of 1, 2, 3, 4 and 5 target structures in self-assembly experiments
  • Comparison of simulated probability vs. probability achieved in physical experiments
  • Reasons for similarities and/or differences between simulated probability vs. probability of physical experiments

Open problems

Using the model of Hosokawa 1995. The system starts as a set of n components. Then the following step is repeated until no longer possible: pick a random pair of structures with total size at most the size of a target structure. Combine them. The process stops when no such pair exists, i.e. when the two smallest structures have total size more than the size of a target structure.

Consider a system of n components that form target structures of size k. Let f(n, k, i) be the probability that i target substructures form.

• Probabilities of yields.
  • f(n, k, 1) + f(n, k, 2) + ...: What is the probability that at least one target structure forms?
  • f(n, k, floor(n/k)): What is the probability that the maximum number of target substructures form?
  • f(n, k, 1) * 1 + f(n, k, 2) * 2 + ...: What is the average number of target substructures that form?
  • General closed form for f(n, k, i): what is the probability that i substructures form for some fixed i?

• Unexpected behavior.
  • Prove that f(n, k, m) is non-increasing as a function of n.

Meeting points

• In the case of the experiment where 24 components are used, if 2 target structures are formed (16 components consumed), then the third target structure will always form. This means that if the experiment is continued till the system reaches a steady state, then in the experiment with 24 components, there will never be a case where 2 target structures are formed.
  • Doesn't this skew the probability graph? Also, how would this be reflected in the analysis?
  • Obviously, this applies to the cases of 2, 4, 5 target structures (16, 32, 40 components) as well.