Figure 1

A schematic diagram of two nonequilibrium steady states with contact region ${\tilde{V}}_1$ and ${\tilde{V}}_2$.Reuse & Permissions

Figure 2

Bottom panel: The plot of density $n$ vs chemical potential $\mu$ for two $120\times 120$ systems, ${\mathrm{NESS}}_1$ with $K=1$, $E=2$ (circles) and ${\mathrm{NESS}}_2$ with $K=E=2$ (triangles) with $\tilde{K}=0$. Arrows indicate how the density changes if ${\mathrm{NESS}}_1$ and ${\mathrm{NESS}}_2$ with the same initial density $n=0.30$ (denoted by middle horizontal line) are brought into contact, reaching respective final densities $n\simeq 0.33$ (denoted by top horizontal line) and $n\simeq 0.27$ (denoted by bottom horizontal line) with equal chemical potential $\mu \simeq 0.36$. Top panel: Numerical experiments to test zeroth law ($\tilde{K}=0$)—(1) ${\mathrm{NESS}}_1$ ($K=4$, $E=6$, $L=120$) with density $n_1$ (bottom red profile) equilibrated with ${\mathrm{NESS}}_2$ ($K=E=2$, $L=110$) with density $n_2$ (middle blue profile), (2) ${\mathrm{NESS}}_2$ with density $n_2$ (middle magenta profile) equilibrated with ${\mathrm{EQ}}_1$ ($K=1$, $E=0$, $L=100$) with density $n_3$ (top grey profile), and (3) ${\mathrm{NESS}}_1$ with density $n_1^{\prime }$ (bottom green profile) equilibrated with ${\mathrm{EQ}}_1$ with density $n_3^{\prime }$ (top black profile) where $n_1^{\prime }\approx n_1$ and $n_3^{\prime }\approx n_3$.Reuse & Permissions

Figure 3

Density $n$ vs chemical potential $\mu$ is plotted for a $120\times 120$ ${\mathrm{NESS}}_1$ ($K=1$, $E=2$) for the following cases. Bottom panel: ${\mathrm{NESS}}_1$ separately in contact with $120\times 120$ systems with ${\mathrm{EQ}}_1$ ($K=E=0$, red) and ${\mathrm{EQ}}_2$ ($K=1$, $E=0$, blue). Middle panel: ${\mathrm{NESS}}_1$ separately in contact with ${\mathrm{EQ}}_1$ (blue) and ${\mathrm{EQ}}_2$ (black) with $4\times 1$ contact area. Top panel: ${\mathrm{NESS}}_1$ separately in contact with ${\mathrm{EQ}}_1$ (blue) and ${\mathrm{EQ}}_2$ (black) for nonzero interaction strength $\tilde{K}=1$. In middle and top panel, $n$ vs $\mu$ plot is compared with that obtained for ${\mathrm{NESS}}_1$ in contact with ${\mathrm{EQ}}_1$ with $2\times 2$ contact area.Reuse & Permissions

Figure 4

Integrated susceptibilities $I_{\chi }$ (squares) and fluctuations $I_{\sigma }$ (circles) vs chemical potential $\mu$ are plotted for a $20\times 20$ ${\mathrm{NESS}}_1$ with $K=1$, $E=2$, separately in contact with five different $100\times 100$ reservoirs: (1) $K=E=0$ (red), (2) $K=1$, $E=2$ (green), (3) $K=1$, $E=4$ (blue), (4) $K=2$, $E=4$ (sky-blue), (5) $K=2$, $E=6$ (magenta). Inset: Same quantities are plotted for two different $20\times 20$ systems, ${\mathrm{NESS}}_1$ with $K=1$, $E=2$ (blue) and ${\mathrm{NESS}}_2$ $K=E=2$ (red), in contact with a reservoir ($K=E=0$, $L=100$).Reuse & Permissions

Figure 5

The ratio $R=[\underset{q_y\to 0}{lim}S(0,q_y)]/[\underset{q_x\to 0}{lim}S(q_x,0)]$ vs density $n$ is plotted for a NESS with $K=1$, $E=2$, $L=120$. Inset: The structure factors $S(q_x,0)$ (open points) and $S(0,q_y)$ (filled points) vs $q_x$ and $q_y$ respectively are plotted for densities $n=0.1$ (red) and $n=0.5$ (blue).Reuse & Permissions