Critical Dynamics under the Vlassov‐Poisson Equations: Critical Exponents and Scaling of the Distribution Function near the Point of a Marginal Stability

Abstract

Using direct integration of the Vlassov equation in configurational space, nonlinear dynamics of a model one-dimensional periodic self-gravitating system are investigated near the point of a marginal stability. The critical velocity dispersion σ, corresponding to a marginal stability of the Jeans mode with the least k, is assumed as the critical point. The peak amplitude of the Jeans mode computed in a run is assumed as the order parameter. In the neighborhood of the critical point, the dynamics can be described by a set of the power laws typical for a system undergoing a second-order phase transition. For the order parameter, this is A ∝ -θβ, where β = 1.907 ± 0.006 and θ = (σ2 - σ)/σ 0. Under this accuracy, these critical exponents satisfy the equality γ± = β(δ - 1). For a gravitating system, its existence is a direct consequence of scaling invariance of the distribution function at |θ| 1; i.e., f(λt, x, λv, λθ, λA0, λF1) = λf(t, x, v, θ, A0, F1). These critical exponents indicate a wide critical area where critical phenomena may determine macroscopic dynamics. The random external drive field of a small amplitude causes anomalous growth of the marginally stable Jeans perturbation into a spatially coherent structure.