Scaling of the distribution function and the critical exponents near the point of a marginal stability under the Vlasov-Poisson equations

A model system, described by the consistent Vlasov-Poisson equations under periodical boundary conditions, has been studied numerically near the point of a marginal stability. The power laws, typical for a system, undergoing a second-order phase transition, hold in a vicinity of the critical point: (i) A ∝ -θβ, β=1.907±0.006 for θ ≤ 0, where A is the saturated amplitude of the marginally-stable mode; (ii) χ ∝ θ-γ as θ → 0, γ=γ-=1.020±0.008 for θ < 0, and γ=γ+=0.995±0.020 for θ > 0, where χ=∂A/∂F1 at F1 → 0 is the susceptibility to external drive of the strain F1; (iii) at θ=0 the system responds to external drive as A ∝ F11/δ, and δ=1.544±0.002. θ=(-)/ is the dimensionless reduced velocity dispersion. Within the error of computation these critical exponents satisfy to equality γ=β(δ-1), known in thermodynamics as the Widom equality, which is direct consequence of scaling invariance of the Fourier components fm of the distribution function f at |θ| << 1, i.e. fm(λatt, λavv, λaθθ, λaA0A0, λaFF1)=λfm(t, v, θ, A0, F1) at θ approx. = 0. On the contrary to thermodynamics these critical indices indicate to a very wide critical area. In turn, it means that critical phenomena may determine macroscopic dynamics of a large fraction of systems. (author)