Drift effect and “negative” mass transport in an inhomogeneous medium: Limiting case of a two-component lattice gas

The mass transport in an inhomogeneous medium is modeled as the limiting case of a two-component lattice gas with excluded volume constraint and one of the components fixed. In the long-wavelength approximation, the density relaxation of mobile particles is governed by diffusion and interaction with a medium inhomogeneity represented by the static component distribution. It is shown that the density relaxation can be locally accompanied by density distribution compression, i.e., the local mass transport directed from low-to high-density regions. The origin of such a “negative” mass transport is shown to be associated with the presence of a stationary drift flow defined by the medium inhomogeneity. In the quasi-one-dimensional case, the compression dynamics manifests itself in the hoppinglike motion of packet front position of diffusing substance due to staged passing through inhomogeneity barriers, and it leads to fragmentation of the packet and retardation of its spreading. The root-mean-square displacement reflects only the averaged packet front dynamics and becomes inappropriate as the transport characteristic in this regime. In the stationary case, the mass transport throughout the whole system may be directed from the boundary with lower concentration towards the boundary with higher concentration. Implications of the excluded volume constraint and particle distinguishability for these effects are discussed.

Figure 1

(a) Density distribution $m\left(x,t\right)$ for different values of $N$ at $t=319$ (time is given in units of ${\nu }_m^{-1}$); (b) $m_1\left(x\right)$ is the density distribution given by equation ${\dot{m}}_1={\partial }_x\left[\left(1-n\right){\partial }_x{m}_1\right]$, $m\left(x\right)$ obeys Eq. (9). The inserts display the corresponding behavior of the function $\tilde{\zeta }\left(\text{ln}t\right)=\text{ln}R\left(t\right)/\text{ln}t$, where $R\left(t\right)$ is given by Eq. (8). The initial amplitude $M=0.1$, the initial half-width $\sqrt{2}l=14.14$, and $k_0=0.1$ are used in both cases (a) and (b) (values of $x$ and $l$ are given in units of lattice constant $a$).

Figure 2

Packet dynamics on time intervals: (a) $\left[t_0,t_1\right]$, (b) $\left[t_1,t_2\right]$; ($t_0=0,t_1=14\times {10}^5,t_2=45\times {10}^5$ are given in units of ${\nu }_m^{-1}$). The initial amplitude is $M=0.5$, the amplitude of the frozen component is $N=0.7$, the initial half-width is $\sqrt{2}l=10.5$, and $k_0=0.168$. Values of $x$ and $l$ are given in units of lattice constant $a$. The shaded pattern shows the density distribution profile of the frozen component $n\left(x\right)$ given by Eq. (11), the arrows denote whether the subpacket is squeezing (growing) $(\uparrow )$ or spreading $(\downarrow )$ on a certain time interval. The dashed lines $x_f^i$ denote the position of packet front $x_f$ on the corresponding time intervals.

Figure 3

Time dependence of the packet front $x_f\left(t\right)$ defined as a point where $\dot{m}=0$, and root-mean-square displacement $R\left(t\right)$ defined by Eq. (8) for different values of $N$. Curve labels correspond to the values of $N$. The initial amplitude is $M=0.1$, the initial half-width is $\sqrt{2}l=25.5$, and $k_0=0.06$ for each curve. Values of $x$, $l$, $x_f\left(t\right)$, and $R\left(t\right)$ are given in units of lattice constant $a$, time $t$ is given in units of ${\nu }_m^{-1}$. The dotted lines $x_f^i$ correspond to the dotted lines in Figs. 2a, 2b.

Figure 4

Central peak squeezing (growth) of the mobile component density profile on the initial time interval $\left[t_0,t_1\right]$, ($t_0=0,t_1=3\times {10}^6$ are given in units of ${\nu }_m^{-1}$). The initial amplitude is $M=0.1$, $N=0.9$. The initial half-width is $\sqrt{2}l=17$, and $k_0=0.1$ Values of $x$ and $l$ are given in units of lattice constant $a$. The shaded pattern shows the density distribution profile of the frozen component $n\left(x\right)$ given by Eq. (11), the arrows indicate the directions of packet evolution. The filled region in the insert is determined by inequality [Eq. (19)] and condition $N<1$, and it corresponds to the negative velocity of packet motion.