Kinetic theory for a simple modeling of a phase transition: Dynamics out of local equilibrium

This is a continuation of previous work [J. Stat. Phys., 172, 880 (2018)] that introduces the presumably simplest model of kinetic theory for phase transition. Here the main concern is to clarify the stability of uniform equilibrium states in the kinetic regime rather than that in the continuum limit. It is found by linear stability analysis that the linear neutral curve is invariant with respect to the Knudsen number, although the transition process is dependent on the Knudsen number. In addition, numerical computations of the (nonlinear) kinetic model are performed to investigate the transition processes in detail. Numerical results show that (unexpected) incomplete transitions may happen as well as clear phase transitions.

Figure 1

Diagrams of phase transition. (a) Neutral curve of the linear stability. Markers with L, C, R, D, and A in region U indicate cases $(c,\tilde{b})=(3.8,4/15)$, (3.8,$1/3)$, $(3.8,2/5)$, $(3.4,1/3)$, and $(3.45,1/3)$, respectively. In region U above the neutral curve $c=1/\left\{2\tilde{b}{(1-\tilde{b})}^2\right\}$, uniform equilibrium states are linearly unstable. Markers with B1, B2, and B3 indicate cases $(c,\tilde{b})=(3.4,4/15)$, $(3.3,1/3)$, and $(3.4,2/5)$, respectively. $◯$ ($\times$) indicates that the phase transition is observed (not observed) in numerical computations. Note that $\tilde{b}$ is a measure of the volume fraction of molecules to the total volume. (b) Equiarea rule for determining two densities in stationary states after phase transition. The curve is the one for case C in (a), i.e., $(c,\tilde{b})=(3.8,1/3)$.

Figure 2

Time evolution of the density profile for cases C and B2 with $\tilde{\kappa }/\tilde{b}=5\times {10}^{-4}$. (a) $(\sqrt{\pi }/2)\mathrm{Kn}=0.1$, (b) $(\sqrt{\pi }/2)\mathrm{Kn}=1$, and (c) $(\sqrt{\pi }/2)\mathrm{Kn}=10$ for case C, while (d) $(\sqrt{\pi }/2)\mathrm{Kn}=1$ for case B2. The top and bottom dashed lines in panels (a)–(c) indicate ${\tilde{\rho }}_H$ and ${\tilde{\rho }}_L$ predicted by the equiarea rule in Fig. 1.

Figure 3

Time evolution of the density profile for case C for $(\sqrt{\pi }/2)\mathrm{Kn}=1$, (a) $\tilde{\kappa }/\tilde{b}=1\times {10}^{-3}$ and (b) $\tilde{\kappa }/\tilde{b}=1\times {10}^{-4}$. See also Fig. 2 for comparison. Top and bottom dashed lines in each panel indicate ${\tilde{\rho }}_H$ and ${\tilde{\rho }}_L$ predicted by the equiarea rule in Fig. 1.

Figure 4

Time evolution of the density profile for cases L and R for $(\sqrt{\pi }/2)\mathrm{Kn}=1$ and $\tilde{\kappa }/\tilde{b}=5\times {10}^{-4}$, (a) case L and (b) case R. Top and bottom dashed lines in each panel indicate ${\tilde{\rho }}_H$ and ${\tilde{\rho }}_L$ predicted by the equiarea rule in Fig. 1. See also Fig. 2 (case C) for comparison.

Figure 5

Time evolution of $\tilde{\mathcal{M}}$, ${\tilde{\mathcal{M}}}_A$, and ${\tilde{\mathcal{M}}}_{\ln }$. Panels (a)–(c) are case C with $\tilde{\kappa }/\tilde{b}=5\times {10}^{-4}$ for different Knudsen numbers: (a) $(\sqrt{\pi }/2)\mathrm{Kn}=0.1$, (b) $(\sqrt{\pi }/2)\mathrm{Kn}=1$, and (c) $(\sqrt{\pi }/2)\mathrm{Kn}=10$. Panel (d) is case C with $\tilde{\kappa }/\tilde{b}=3\times {10}^{-4}$ and $(\sqrt{\pi }/2)\mathrm{Kn}=0.1$, while panel (e) is case B2 with $\tilde{\kappa }/\tilde{b}=5\times {10}^{-4}$ and $(\sqrt{\pi }/2)\mathrm{Kn}=1$. Panel (f) is case C with $\tilde{\kappa }/\tilde{b}=5\times {10}^{-4}$ for $(\sqrt{\pi }/2)\mathrm{Kn}=0.01,0.05,0.1,1$, and 10.

Figure 6

Time evolution of the density and functional $\tilde{\mathcal{M}}$ in case A. (a) $\tilde{\rho }$ for $\tilde{\kappa }/\tilde{b}=5\times {10}^{-4}$, (b) $\tilde{\rho }$ for $\tilde{\kappa }/\tilde{b}=1\times {10}^{-4}$, and (c) $\tilde{\mathcal{M}}$ for both values of $\tilde{\kappa }/\tilde{b}$, where $\mathrm{Kn}$ is commonly set as $(\sqrt{\pi }/2)\mathrm{Kn}=1$. Top and bottom dashed lines in (a) and (b) indicate ${\tilde{\rho }}_H$ and ${\tilde{\rho }}_L$, respectively.

Figure 7

Influence of the domain periodicity: time evolution of the density profile for case D for $(\sqrt{\pi }/2)\mathrm{Kn}=0.1$. (a) $\tilde{\kappa }/\tilde{b}=1\times {10}^{-4}$ with period 1, (b) $\tilde{\kappa }/\tilde{b}=1\times {10}^{-3}$ with period 1, (c) $\tilde{\kappa }/\tilde{b}=1\times {10}^{-3}$ with period 2. In (c), the sinusoidal initial perturbation has a period 2. Top and bottom dashed lines in each panel indicate ${\tilde{\rho }}_H$ and ${\tilde{\rho }}_L$, respectively.