Stochastic boundary approaches to many-particle systems coupled to a particle reservoir

Stochastic boundary conditions for interactions with a particle reservoir are discussed in many-particle systems. We introduce the boundary conditions with the injection rate and the momentum distribution of particles coming from a particle reservoir in terms of the pressure and the temperature of the reservoir. It is shown that equilibrium ideal gases and hard-disk systems with these boundary conditions reproduce statistical-mechanical properties based on the corresponding grand canonical distributions. We also apply the stochastic boundary conditions to a hard-disk model with a steady particle current escaping from a particle reservoir in an open tube, and discuss its nonequilibrium properties such as a chemical potential dependence of the current and deviations from the local equilibrium hypothesis.

Figure 1

The probability density functions $F_x\left(p_x\right)$ and $F_y\left(p_y\right)$ of the components $p_x$ and $p_y$ of the particle momentum $(p_x,p_y)$, respectively, in ideal gases with the PISB conditions, for the chemical potentials $\mu =-7.74,-7.26$, and $-6.74$. All these probability density functions of the momentum components are almost indistinguishable with each other. The black thin line is the corresponding Maxwell momentum distribution. Here, and in all figures hereafter, we use dimensionless units with $m=1,h=1$, and $k_{B}T=1$.

Figure 2

The probabilities $G\left(\mathcal{N}\right)$ of the number $\mathcal{N}$ of particles in ideal gases with the PISB conditions, for the chemical potentials $\mu =-7.74$ (red solid line), $-7.26$ (blue dashed line), and $-6.74$ (green dotted line). The Poisson distribution (12) is also shown as the black thin line in each chemical potential case, which is almost indistinguishable from the corresponding numerical results.

Figure 3

The pressure $P$ of ideal gases with the PISB conditions as a function of the chemical potential $\mu$ (red circles). The black line is the pressure (7) based on the corresponding grand canonical distribution.

Figure 4

The average number $N$ of disks (red circles) in hard-disk systems with the PISB conditions, as well as the graph of Eq. (18) (black solid line), as a function of the chemical potential $\mu$. Here, the graph of Eq. (8) for the corresponding ideal gases is also shown by the blue dashed line. Inset: the deviation ${\mathrm{\Delta }N\equiv N-N|}_{r=0}$ (red circles) of the average number $N$ from its ideal-gas case, as well as the graph of Eq. (22) (black solid line), as functions of $\mu$.

Figure 5

The variance $\langle {(\mathcal{N}-N)}^2\rangle$ of numbers of particles (red circles) as a function of the chemical potential $\mu$ in hard-disk systems with the PISB conditions. Here, the graph of Eq. (19) (black solid line), as well as the graph of the variance of numbers of particles in the corresponding ideal-gas cases (blue dashed line), given from the corresponding grand canonical distributions, are also shown. Inset: the deviation $\mathrm{\Delta }\mathrm{{\rm Y}}\equiv \langle {(\mathcal{N}-N)}^2\rangle -\langle {(\mathcal{N}-N)}^2\rangle {|}_{r=0}$ (red circles) and the graph of Eq. (23) (black solid line) as functions of $\mu$.

Figure 6

The pressure $P$ of hard-disk systems with the PISB conditions (red circles), and the graphs of Eq. (17) (black solid line) and Eq. (7) (blue dashed line), as functions of the chemical potential $\mu$. Inset: the difference ${\mathrm{\Delta }P\equiv P-P|}_{r=0}$ between the pressures $P$ and ${P|}_{r=0}\equiv 2\pi m{e}^{\beta \mu }/\left({\beta }^2{h}^2\right)$ (red circles), and the graph of Eq. (24) (black solid line), as functions of $\mu$.

Figure 7

Hard disks with the radius $r$ in an open-end tube with the length $L_x$ and the width $L_y$. The left end of the tube is connected to a particle reservoir (gray region) with the temperature $T$ and the chemical potential $\mu$ via the boundary $\mathcal{S}$, and the right end of the tube is open. Hard disks injected from the particle reservoir can leave not only from the left end, but also from the right end after moving inside the tube.

Figure 8

The particle current density $\mathcal{J}$ (red circles) of hard disks in an open tube connected to a particle reservoir with the chemical potential $\mu$ and the temperature $T$, and the injected current density $\nu /L_y$ (black solid line) from the reservoir, as functions of $\mu$. Inset: the ratio $\mathcal{J}{L}_y/\nu$ between the particle current density $\mathcal{J}$ and the injected current density $\nu /L_y$ as a function of $\mu$.

Figure 9

The probability density functions $F_x^{\left(1\right)}\left(p_x\right)$ (red solid line), $F_x^{(\eta /2)}\left(p_x\right)$ (blue dashed line), and $F_x^{\left(\eta \right)}\left(p_x\right)$ (green dotted line) of $x$ components of momentum vectors of hard disks in the regions $W_1,W_{\eta /2}$, and $W_{\eta }$, respectively, in the model with hard disks in an open tube connected to a particle reservoir with the chemical potential $\mu$ and the temperature $T$. The corresponding Maxwell momentum distribution $F_x\left(p_x\right)$ with the temperature $T$, and the Gaussian distribution function ${\tilde{F}}_x^{(\eta /2)}\left(p_x\right)$ with the average and the variance given from the data for $F_x^{(\eta /2)}\left(p_x\right)$ are also shown as the black thin dashed-dotted line and the black thin dashed line, respectively.

Figure 10

The probability density functions $F_y^{\left(1\right)}\left(p_y\right)$ (red solid line), $F_y^{(\eta /2)}\left(p_y\right)$ (blue dashed line), and $F_y^{\left(\eta \right)}\left(p_y\right)$ (green dotted line) of $y$ components of momentum vectors of hard disks in the regions $W_1,W_{\eta /2}$, and $W_{\eta }$, respectively, in the model with hard disks in an open tube connected to a particle reservoir with the chemical potential $\mu$ and the temperature $T$. The corresponding Maxwell momentum distribution $F_y\left(p_y\right)$ with the temperature $T$ is shown as the black thin dashed-dotted line. The Gaussian distribution functions ${\tilde{F}}_y^{\left(1\right)}\left(p_x\right),{\tilde{F}}_y^{(\eta /2)}\left(p_x\right)$, and ${\tilde{F}}_y^{\left(\eta \right)}\left(p_x\right)$ with the averages and the variances given from the data for $F_x^{\left(1\right)}\left(p_x\right),F_x^{(\eta /2)}\left(p_x\right)$, and $F_x^{\left(\eta \right)}\left(p_x\right)$ are also shown as the black thin solid line, the black thin dashed line, and the black thin dotted line, respectively.

Figure 11

The average ${\langle p_x\rangle }^{\left(j\right)}$ (red solid line) and the variance ${\langle {(p_x-{\langle p_x\rangle }^{\left(j\right)})}^2\rangle }^{\left(j\right)}$ (blue dashed line) of the momentum component $p_x$, and the average ${\langle p_y\rangle }^{\left(j\right)}$ (green dotted line) and the variance ${\langle {(p_y-{\langle p_y\rangle }^{\left(j\right)})}^2\rangle }^{\left(j\right)}$ (purple dashed-dotted line) of the momentum component $p_y$, in the region $W_j$ for $j=1,2,\cdots ,\eta$ as functions of $x$, in the model with hard disks in an open tube connected to a particle reservoir.

Figure 12

The probability $G_j\left(\mathcal{N}\right)$ of the number $\mathcal{N}$ of particles in the region $W_j$ for $j=1,2,\cdots ,\eta$ as a function of $\mathcal{N}$ and $x$, in the model with hard disks in an open tube connected to a particle reservoir.

Figure 13

The average ${\langle \mathcal{N}\rangle }^{\left(j\right)}$ (red solid line) and the variance ${\langle {(\mathcal{N}-{\langle \mathcal{N}\rangle }^{\left(j\right)})}^2\rangle }^{\left(j\right)}$ (blue dashed line) of the number $\mathcal{N}$ of particles in the region $W_j$ for $j=1,2,\cdots ,\eta$ as functions of $x$, in the model with hard disks in an open tube connected to a particle reservoir.

Figure 14

The pressure $P^{\left(j\right)}$ on the wall of the region $W_j$ of the tube for $j=1,2,\cdots ,\eta$ as a function of $x$, in the model with hard disks in an open tube connected to a particle reservoir.