Analytical solution for the long- and short-range every-pair-interactions system

Many physical, biological, and social systems exhibit emergent properties arising from their components’ interactions (cells). In this study, we systematically treat every-pair interactions (a) that exhibit power-law dependence on the Euclidean distance and (b) act in structures that can be characterized using fractal geometry. It can represent the two-body interaction potential, the heat flux between two parts of a structure, friendship strength between two people, etc.. We analytically derive the average intensity of influence that one cell has on the others or, conversely, receives from them. This quantity is referred to as the mean interaction field of the cells, and we find that (i) in a long-range interaction regime, the mean interaction field increases following a power-law with the size of the system, (ii) in a short-range interaction regime, the field saturates, and (iii) in the intermediate range it follows a logarithmic behavior. To validate our analytical solution, we perform numerical simulations. For long-range interactions, the theoretical calculations align closely with the numerical results. However, for short-range interactions, we observe that discreteness significantly impacts the continuum approximation used in the derivation, leading to incorrect asymptotic behavior in this regime. To address this issue, we propose an expansion that substantially improves the accuracy of the analytical expression. We discuss applications of the every-pair interactions system proposed, and one of them is to explore a framework for estimating the fractal dimension of unknown structures. This approach offers an alternative to established methods such as box-counting or sandbox methods. Overall, we believe that our analytical work will have broad applicability in systems where every-pair interactions play a role.