Critical Scaling of Finite-Size Fluctuations around Marginal Stability in Long-Range Hamiltonian Systems

TL;DR

This paper develops a phenomenological theory predicting anomalous finite-size fluctuation scaling near marginal stability in long-range Hamiltonian systems, confirmed by numerical simulations, revealing a broad critical window with slow shrinking.

Contribution

It introduces a new phenomenological framework for understanding finite-size fluctuations near marginal stability in long-range systems, with predictions validated numerically.

Findings

Finite-size fluctuations scale anomalously near marginal stability.

The critical window for fluctuations shrinks as N^{-1/5}.

Numerical simulations confirm the theoretical predictions.

Abstract

Finite size fluctuations are a crucial ingredient in kinetic theory of long-range interacting collisionless systems. In this Letter, we introduce a phenomenological theory which predicts an anomalous scaling close to marginal stability for these fluctuations. It also pinpoints the critical window inside which the fluctuations are anomalous, and outside which they are Gaussian. Shrinking very slowly as $N^{-1/5}$, this critical window encompasses a wide region around marginal stability. We confirm our predictions through extended numerical simulations on two different simplified models.