Sets of the potentials generating identical structures of non-crystalline systems

We discuss here the problem of restoring an inter-particle potential using the known atomic structure. Are these cases unique or it is possible to find a way to create, for example, the set of various inter-particle potentials which can generate the same structures at computer modeling? Using the correct form of the Born–Green–Bogoliubov equation we manage to show that there is a set of inter-particle potentials capable to generate (by molecular dynamics method) the same structure of non-crystalline system at temperature T > 0. This set of potentials is a linear combination between some potential generating the given structure at temperature T > 0 and the potential generating the same structure at absolute zero. This theorem is checked up on example of one-component system model with repulsive potential (3998 particles in the basic cube with periodic boundary conditions). This means also that in the systems with dense structure the coincidence of pair correlation functions provides also coincidence or ternary correlation functions. The similar set of potentials can be constructed using the potentials which generate identical structures at two different temperatures. This result is checked up on examples of liquid Ga.