Finite-$N$ corrections to Vlasov dynamics and the range of pair interactions

We explore the conditions on a pair interaction for the validity of the Vlasov equation to describe the dynamics of an interacting $N$-particle system in the large $N$ limit. Using a coarse graining in phase space of the exact Klimontovich equation for the $N$-particle system, we evaluate, neglecting correlations of density fluctuations, the scalings with $N$ of the terms describing the corrections to the Vlasov equation for the coarse-grained one-particle phase space density. Considering a generic interaction with radial pair force $F\left(r\right)$, with $F\left(r\right)\sim 1/r^{\gamma }$ at large scales, and regulated to a bounded behavior below a “softening” scale $\varepsilon$, we find that there is an essential qualitative difference between the cases $\gamma <d$ and $\gamma >d$, i.e., depending on the the integrability at large distances of the pair force. In the former case, the corrections to the Vlasov dynamics for a given coarse-grained scale are essentially insensitive to the softening parameter $\varepsilon$, while for $\gamma >d$ the amplitude of these terms is directly regulated by $\varepsilon$, and thus by the small scale properties of the interaction. This corresponds to a simple physical criterion for a basic distinction between long-range $\left(\gamma \le d\right)$ and short-range $\left(\gamma >d\right)$ interactions, different from the canonical one ($\gamma \le d+1$ or $\gamma >d+1$) based on thermodynamic analysis. This alternative classification, based on purely dynamical considerations, is relevant notably to understanding the conditions for the existence of so-called quasistationary states in long-range interacting systems.

Figure 1

Schema, in one dimension, of the coarse graining of phase space, showing also the different regions into which the domain of integration is divided for our analysis of the integrals in the force variance: $\mathcal{C}$ is the coarse-grained cell with center at $(\mathbf{x},\mathbf{v}),\mathcal{S}$ (in white) is the “stripe” enclosing all points in $[\mathbf{x}-\frac{{\lambda }_x}{2},\mathbf{x}+\frac{{\lambda }_x}{2}]$; ${\partial }_{\varepsilon }\mathcal{S}$ is the region containing particles within a distance $\varepsilon$ of the boundary of the stripe.

Figure 2

The variance of the force on particles in a phase space cell due to particles in the corresponding stripe, plotted as a function of $\varepsilon$, for the different values of $\gamma$. The crosses indicate the points obtained from the numerical simulation described in the text and dotted lines are the corresponding analytical results, Eq. (56). We use units in which ${\lambda }_x=1$ and $g=1$.