Coagulation by Random Velocity Fields as a Kramers Problem

We analyze the motion of a system of particles suspended in a fluid which has a random velocity field. There are coagulating and noncoagulating phases. We show that the phase transition is related to a Kramers problem, and we use this to determine the phase diagram in two dimensions, as a function of the dimensionless inertia of the particles, $ϵ$, and a measure of the relative intensities of potential and solenoidal components of the velocity field, $\Gamma$. We find that the phase line is described by a function which is nonanalytic at $ϵ=0$, and which is related to escape over a barrier in the Kramers problem. We discuss the physical realizations of this phase transition.

Figure 1

Evolution of uniform initial particle distribution under Eq. (1) in the unit square with periodic boundary conditions, for ${10}^4$ particles; $\Gamma =1/3$, $ϵ=0.33$ (coalescing phase), at $t=0$ (a), 10 (b), and $t=275$ (c); $\Gamma =1/3$, $ϵ=4.30$ at $t=0$ (d), 10 (e), and 275 (f). The statistics of the force is explained in the text; we chose $C(R,\Delta t)\propto \exp [-\Delta t^2/(2{\tau }^2)-R^2/(2{\xi }^2)]$ with $\xi =0.1/\pi$ and $\tau \to 0$.

Figure 2

(a) A trajectory for the Langevin equations, (7), for $ϵ=0.5$, $\Gamma =1$, and $0\le t^{\prime }\le {10}^2$. The trajectory $\mathit{x}(t)$ is most frequently located close to the origin, but occasionally it passes the unstable fixed point at $(-1/ϵ,0)$, making a large excursion before returning to the origin. The nonanalytic contribution to the Lyapunov exponent is associated with these large excursions. (b) Lyapunov exponent as a function of $ϵ$, for $\Gamma =\frac{1}{3}$ (○), 1 (◇), 3 (□): Monte Carlo simulations of (7) (lines) and integration of (1) (symbols). (c) Comparison of Monte Carlo simulation of (7) (symbols) and Eq. (19) (lines), for $\Gamma =0.85$ (○), 1 (□), 1.15 (◇) (with $\chi =0.34,0.28,0.215$). (d) Phase diagram in two dimensions: Monte Carlo simulations (○) of (7); the solid line is the function $1-\Gamma =\exp [-1/(6{ϵ}^2)]$.

Figure 3

Same as Fig. 1a, 1b, 1c, but with masses uniformly distributed on an interval of half width $\delta m=0.2⟨m⟩$.