Statistics of Long-Range Force Fields in Random Environments: Beyond Holtsmark

TL;DR

This paper introduces a new statistical law for long-range force fields in random environments that accounts for finite size effects, revealing bi-scaling and a transition in force moments.

Contribution

It presents a novel scaling law for force fields that extends beyond the classical Le9vy-Holtsmark distribution, incorporating finite size effects and bi-scaling behavior.

Findings

Discovered bi-scaling with a sharp transition at the order d/e9psilon

Derived a new statistical law for large fields considering finite size effects

High-order moments are described by the new framework, applicable to many systems

Abstract

Since the times of Holtsmark (1911), statistics of fields in random environments have been widely studied, for example in astrophysics, active matter, and line-shape broadening. The power-law decay of the two-body interaction, of the form $1/|r|^\delta$, and assuming spatial uniformity of the medium particles exerting the forces, imply that the fields are fat-tailed distributed, and in general are described by stable L\'evy distributions. With this widely used framework, the variance of the field diverges, which is non-physical, due to finite size cutoffs. We find a complementary statistical law to the L\'evy-Holtsmark distribution describing the large fields in the problem, which is related to the finite size of the tracer particle. We discover bi-scaling, with a sharp statistical transition of the force moments taking place when the order of the moment is $d/\delta$, where $d$ is the dimension. The high-order moments, including the variance, are described by the framework presented in this paper, which is expected to hold for many systems. The new scaling solution found here is non-normalized similar to infinite invariant densities found in dynamical systems.