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

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.

Figures (7)

• Figure 1: Schematic figure of scalings around marginal stability. $A$ is an order parameter, $t$ is the time, and $\lambda$ is the real growth rate. Critical scaling of finite size fluctuations is crucial in the critical region. $\rho\simeq 4/5$ and $\nu\simeq 1/5$. (Non-)Gaussian fluctuations are for $A(t)$.
• Figure 2: Variance $V_{M}$ computed for $t\in [20,100]$ on the stable side of the bifurcation, for the HMF model: $\lambda=0, -0.03, -0.05, -0.1,$ and $-0.2$ from top to bottom. Error bars, the standard deviation over $100$ realizations, are smaller than the symbols. The guide lines are computed by the least square method for $\lambda=0$ (red, slope $-0.7755$) and $\lambda=-0.2$ (blue, slope $-0.9674$) for $N\in [10^{2},10^{6}]$. Inset shows $\lambda$ dependence of $\rho$ corresponding to slopes, which are computed in the interval $N\in [10^{2},10^{4}]$ (green circles), $[10^{3},10^{5}]$ (blue triangles), and $[10^{4},10^{6}]$ (red squares). The red and blue horizontal lines mark $\rho=4/5$ and $1$ respectively.
• Figure 3: Probability distribution function $P(M)$ for $N=10^{3}$ at $\lambda=0$ in the HMF model. PDFs are averaged for $t\in [20,t_{\rm max}]$ and $10^{2}$ realizations. $t_{\rm max}=10^{2},10^{3},10^{4},10^{5}$ from top left to bottom right; in parentheses is the estimated value of $\mu$. Inset shows $t_{\rm max}$ dependence of the exponent $\mu$, which is the average (points) and standard deviation (error bars) over $10^{3}$ resamples by the bootstrap method (see Appendix D). The red and blue horizontal lines mark $\mu=5/2$ and $4$ respectively.
• Figure 4: Scaled power spectral density $N^{\rho-\nu}S_{M}$ as a function of $N^{\nu}f$ in the HMF model. $\rho=0.775, \nu=0.2$. The parameter $\epsilon=N^{\nu}\lambda$ is (a) $\epsilon=-1$, (b) $-0.3$, (c) $0$, and (d) $0.3$. $N=10^{5}$ (green), $10^{6}$ (blue), and $10^{7}$ (orange). The gray curve is $N^{\nu}\lambda=0$ with $N=10^{7}$ for comparison. Average over $100$ realizations. Time series are taken in the time region $t\in [180.9, 1000]$ to avoid an initial transient regime and for convenience of the fast Fourier transform.
• Figure 5: Test of the phenomenological equation \ref{['eq:scaled_pheno']} at the critical point in the Euler-like model Eq. \ref{['eq:Euler']}, computed from $t\in [20,100]$ of $100$ realizations. (a) Variance of $W$ at the critical point. The red line represents the best fit for the four points in $N\in [10^{4},10^{7}]$. The slope is $-0.8892$ while the prediction is $-\rho=-4/5$. (b) Probability distribution function for $N=10^{5}, 10^{6}$, and $10^{7}$ from bottom right to top left; in parentheses is the estimated value of $\mu$ whose theoretical value is $\mu=5/2$.