Determining the interparticle force laws in amorphous solids from a visual image

We consider the problem of how to determine the force laws in an amorphous system of interacting particles. Given the positions of the centers of mass of the constituent particles we propose an algorithm to determine the interparticle force laws. Having $n$ different types of constituents we determine the coefficients in the Laurent polynomials for the $n(n+1)/2$ possibly different force laws. A visual providing the particle positions in addition to a measurement of the pressure is all that is required. The algorithm proposed includes a part that can correct for experimental errors in the positions of the particles. Such a correction of unavoidable measurement errors is expected to benefit many experiments in the field.

Figure 1

An example of a typical configuration of an amorphous solid made of 1000 particles of two diameters. The external forces are those between the walls and the particles adjacent to the walls, marked in red short lines. The force chains, made from the 20% strongest forces, are indicated by black lines between particles whose width is proportional to the strength of the interparticle forces. The particle “diameters” used in this figure correspond to the minima of the $AA$ and $BB$ interactions. The $AB$ interaction force vanishes at shorter distances so that a lot of $AB$ pairs appear more compressed than they really are.

Figure 2

Comparison between the predicted (scalar) forces of interaction (green circles), and the ones used in the simulation (black squares). Panels (a), (b), and (c) present the $AA,BB$, and $AB$ force laws, respectively. In the present comparison the exact particle positions are provided.

Figure 3

An example of the effect of errors in particle positions and the correction. Panel (a): the $AB$ force as predicted from the erroneous data. Panel (b): the same force as predicted after the initial errors in the particle positions are corrected with the help of the pseudo gradient-descent algorithm.