Algorithmic design of self-assembling structures

Abstract

We study inverse statistical mechanics: how can one design a potential function so as to produce a specified ground state? In this article, we show that unexpectedly simple potential functions suffice for certain symmetrical configurations, and we apply techniques from coding and information theory to provide mathematical proof that the ground state has been achieved. These potential functions are required to be decreasing and convex, which rules out the use of potential wells. Furthermore, we give an algorithm for constructing a potential function with a desired ground state.

Fig. 1.

Two potential functions under which the regular dodecahedron minimizes energy: the blue one uses potential wells, and the green one is convex and decreasing.

Fig. 2.

The paths of points converging to the regular dodecahedron under the green potential function from Fig. 1. Only the front half of the sphere is shown.

Fig. 3.

The hypercube, drawn in four-point perspective.

Fig. 4.

The Schlegel diagram for the regular 120-cell (with a dodecahedral facet in red).

Fig. 5.

Gaussian energy on the space of two-dimensional lattices (red means high energy).