Nonequilibrium protection effect and spatial localization of noise-induced fluctuations: Quasi-one-dimensional driven lattice gas with partially penetrable obstacle

We consider a nonequilibrium transition that leads to the formation of nonlinear steady-state structures due to the gas flow scattering on a partially penetrable obstacle. The resulting nonequilibrium steady state (NESS) corresponds to a two-domain gas structure attained at certain critical parameters. We use a simple mean-field model of the driven lattice gas with ring topology to demonstrate that this transition is accompanied by the emergence of local invariants related to a complex composed of the obstacle and its nearest gas surrounding, which we refer to as obstacle edges. These invariants are independent of the main system parameters and behave as local first integrals, at least qualitatively. As a result, the complex becomes insensitive to the noise of external driving field within the overcritical domain. The emerged invariants describe the conservation of the number of particles inside the obstacle and strong temporal synchronization or correlation of gas states at obstacle edges. Such synchronization guarantees the equality to zero of the total edge current at any time. The robustness against external drive fluctuations is shown to be accompanied by strong spatial localization of induced gas fluctuations near the domain wall separating the depleted and dense gas phases. Such a behavior can be associated with nonequilibrium protection effect and synchronization of edges. The transition rates between different NESSs are shown to be different. The relaxation rates from one NESS to another take complex and real values in the sub- and overcritical regimes, respectively. The mechanism of these transitions is governed by the generation of shock waves at the back side of the obstacle. In the subcritical regime, these solitary waves are generated sequentially many times, while only a single excitation is sufficient to rearrange the system state in the overcritical regime.

Figure 1

(a) Periodic chain with ring topology and the impurity site at the origin (below): ${\nu }^{\pm }=\nu (1\pm g)$ are particle hopping rates along and against the direction of nonconservative field $g$. Qualitatively, this “hopping transport” model can be associated with a potential energy landscape as illustrated (above). (b) Typical NESSs in subcritical regime, at $\overline{n}=0.3, U=0.6$, and $g=0.02,0.1,0.18<g_c$. (c) Typical NESSs in overcritical regime at $\overline{n}=0.3, U=0.6$, and $g=0.6,0.7,0.8>g_c$. The number of lattice sites $L_0=401$, the ring length $2L=400\ell$, with lattice constant $\ell$. The distributions in panels (b) and (c) were obtained from direct numerical solutions of the mean-field Eqs. (4) and (5) as steady-state profiles established after $\approx 1.6\times {10}^7$ time steps of evolution since the driving field was switched from $g_0=0$ to $g$ at $t_0=0$. The resulting NESS was regarded as finally established if ${max}_k[n_k\left(\tau \right)-n_k(\tau -\mathrm{\Delta }\tau )]\le {10}^{-30}$ with $\mathrm{\Delta }\tau =0.01$, where $\tau =\nu t$; see Appendix pp1 on details of numerical implementation.

Figure 2

Behavior of density distribution (NESS profile) ahead of the obstacle in subcritical regime at $\overline{n}=0.1, U=0.6$, and $g=0.2<g_c$: comparison of numerical result and analytical asymptotics, Eq. (10). Here $x$ corresponds to continuous coordinate, $k$ is the site index, and $L_0=401$. The distribution of $n_k$ was obtained from the numerical solution of mean-field Eqs. (4) and (5) in the same way as in Fig. 1; see also Appendix pp1 for details.

Figure 3

Occupations of impurity site $n_0$, its edges $n_{\pm 1}$, and their half-sum as a function of external field $g$. Above critical field $g_c$, determining the phase transition to kink-profile NESS, quantities $n_0$ and $n_{-1}+n_1$ are no longer dependent on $g$. Here, $\overline{n}=0.3, U=0.6$, and $L_0=401$. The results are obtained based on the numerical solution of mean-field Eqs. (4) and (5) in the same way as in Fig. 1; see also Appendix pp1 for details.

Figure 4

(a) Plot shows the existence of critical concentration value ${\overline{n}}_c$ above which $n_{\infty }$ no longer depends on $\overline{n}$, signifying the transition into two-domain structure at $\overline{n}>n_c$ (for $g=0.5$). (b) Dependence of concentration $n_{\infty }$ on field $g$. Analytical solution for $n_{\infty }$ is given by Eq. (15). (c) Dependence of domain wall position $k_s=-S\left(g\right)$ or the spatial width of dense phase $S\left(g\right)$ on the driving field $g$ [numerical against Eq. (17)]. The numerical values of $S$ were taken as the closest site to the domain wall $k_s$ defined by $n\left(k_s\right)=1/2$, cf. [61] for domain wall stopping criterion. (d) Interparticle correlations at the nearest sites $\overline{\delta n_k\delta n_{k+1}}$, where $\delta n_k=n_k-\overline{n}$, and $\overline{(\cdots )}=L_0^{-1}{\sum }_k(\cdots )$; analytical curve is given by Eq. (20). Gas concentration $\overline{n}=0.3$ for (a), (c), and (d); $U=0.6$ and $L_0=401$ for all panels. The results are obtained based on the numerical solution of mean-field Eqs. (4) and (5) in the same way as in Fig. 1; see also Appendix pp1 for details.

Figure 5

(Upper row) Two-dimensional projections of the critical surface for the phase diagram given in $(U,\overline{n},g)$ parameter space: (a) $(U,\overline{n})$-projection at $g=0.5$, (b) $(g,U)$-projection at $\overline{n}=0.3$, and (c) $(g,\overline{n})$-projection at $U=0.6$. (Lower row) Corresponding behaviors of ${\eta }_0=n_0/(1-U)$ as a function of the same pairs of parameters. The plateaux at ${\eta }_0=0.5$ correspond to the kink-phase (two spatial domains). Analytically estimated phase boundaries (solid lines) are given by (a) $U\left(\overline{n}\right)=1-\left[4\overline{n}(1-\overline{n})\right]/(3-4\overline{n})$, (b) $U\left(g\right)=1-\left[4gn\right(1-n\left)\right]/(1-2n+g)$, and (c) $g\left(\overline{n}\right)=(1-U)(1-2\overline{n})/[4\overline{n}(1-\overline{n})+U-1]$. Ring length is $2L=400\ell$ for all cases. Panels (a) and (c) features inversion symmetry in $\overline{n}$ between high-density (HD) and low-density (LD) phases known in ASEP models [69]. The results are obtained based on the numerical solution of mean-field Eqs. (4) and (5) in the same way as in Fig. 1; see also Appendix pp1 for details.

Figure 6

Numerical illustration of the emergence of local invariants (local first integrals) $n_0\left(\tau \right)=(1-U)/2=\mathrm{const}$ and $n_{\pm 1}\left(\tau \right)=n_1\left(\tau \right)+n_{-1}\left(\tau \right)\approx 1$ after nonequilibrium transition at $g>g_c$ for the case of driving-field noise $g\left(\tau \right)=g+\delta g\left(\tau \right)$. Exploited noise samples are shown at the top panel, both have switching frequency $\lambda =0.02$. For both kinds of noises the drive $g\left(t\right)$ fluctuates around $g=0.2$ (subcritical regime) or $g=0.8$ (overcritical regime) with maximum amplitude $\left|\delta g\right(t\left)\right|=0.1$. Here, ring length $2L=400\ell$, average density (filling fraction) $\overline{n}=0.3$, and $U=0.6$. Enlarged segment of $n_1\left(\tau \right)$ shows relaxation between $|g=0.7\rangle \rightleftarrows |g=0.9\rangle$ with dominant asymptotic behavior $\sim {\mathrm{e}}^{-\gamma \tau }$, implying decay rate $\gamma \gg \lambda$. The results shown are obtained based on the numerical solution of mean-field Eqs. (4) and (5) with $\delta g\left({\tau }_k\right)$ being a sampled random realization of a stochastic process (either telegraphic or random periodic noise); see Appendix pp1 for details.

Figure 7

Time invariance of quantity $n_{\pm k}=n_{-k}+n_{+k}\approx 1$ against dynamic fluctuations of driving $g$ within $g>g_c$ domain. External drive $g\left(\tau \right)$ is governed by periodic random noise around the mean value $g=0.8>g_c$ as in Fig. 6 (see caption). In the subplot (a) the example of time behavior of $n_{\pm k}$ is shown for $k=2$. In the subplot (b) the averaging is performed over $6\times {10}^3\left[\nu t\right]$ during which $g\left(\tau \right)$ approximately reaches the mean value $g=0.8\pm 0.01$. The quantity $n_{\pm k}\left(t\right)$ holds quasi-invariant within the interval from $k=1$ up to $\approx k_s$. As can be seen from panel (c) the invariance weakens slowly until certain $k_{*}$ which can be naturally associated with the size of obstacle complex. In the interval $k_{*}<k<{\langle k_s\rangle }_t$, the quasi-invariants $n_{\pm k}\approx 1$ undergo destruction and reach complete destruction at the domain wall position ${\langle k_s\rangle }_t$, as confirmed by numerical calculation of dispersion ${\langle \mathrm{\Delta }{n}_{\pm k}^2\rangle }_t={\langle {(n_{\pm k}-{\langle n_{\pm k}\rangle }_t)}^2\rangle }_t\approx {10}^{-30}-{10}^{-9}$ for dense gas phase $0<k<40$, where ${\langle \cdots \rangle }_t=\frac{1}{T}{\sum }_{t=0}^T\cdots$ and symbol $\mathrm{\Delta }$ is used (instead of $\delta$ in the main text) to denote numerically obtained values. One can say that obstacle complex, i.e., the domain $\left|k\right|<k_{*}$, is protected from $g$ noise-induced fluctuations. The results shown are obtained based on the numerical solution of mean-field Eqs. (4) and (5) with $\delta g\left({\tau }_k\right)$ being a sampled random realization of the periodic random process; see Appendix pp1 for details.

Figure 8

The square-root dispersion ${\langle \mathrm{\Delta }{n}_k^2\rangle }_t^{1/2}={\langle {(n_k-{\langle n_k\rangle }_t)}^2\rangle }_t^{1/2}$ of the noise-induced fluctuations of the $k\mathrm{th}$ site occupation number (a) for periodic random noise with $\lambda =0.02$, such as in Figs. 6 and 7, below $g_c$ (${\langle g\rangle }_t=0.2$) and above $g_c$ (${\langle g\rangle }_t=0.8$). (Symbol $\mathrm{\Delta }$ is used, instead of $\delta$ in the main text, to denote numerically obtained values.) In the subcritical domain ($g<g_c$), the fluctuations induced by the multiplicative noise are mostly distributed near the impurity with the accumulation ahead of it, the site $k=0$. On the contrary, in overcritical domain ($g>g_c$), the noise-induced fluctuations are totally suppressed in the impurity, $k=0$, and strongly localized near the defect (the domain wall) position ${\langle k_s\rangle }_t$. This local enhancement of density-fluctuations intensity nearby the domain wall (phase coexistence layer) is caused by its back/forward trembling or, in other words, by noise-induced floating of the domain wall position with time, Fig. 9. Panel (b) shows the zoomed in region from panel (a) near the impurity, $k\in [-2020]$. Panels (c) and (d) show the same as panel (a) above $g_c$ (${\langle g\rangle }_t=0.8, \delta g=0.1$) but for telegraphic noises with $\lambda =0.02$ and $\lambda =0.002$, respectively. Other parameters: $2L=400\ell , \overline{n}=0.3$, and $U=0.6$. The results shown are obtained based on the numerical solution of mean-field Eqs. (4) and (5) with $\delta g\left({\tau }_k\right)$ being a sampled random realization of a stochastic process (either telegraphic or random periodic noise); see Appendix pp1 for details.

Figure 9

Schematic representation of $g$-noise-induced wall shifting approximation.

Figure 10

Overcritical regime (within the domain $g>g_c$) Numerically obtained $\gamma$ for transitions $|g=0.7\rangle \rightleftarrows |g=0.9\rangle$. Generally, $n_1\left(\nu t\right)$ decays according to Eq. (45), where exponential behavior is decomposed into a fast decay with ${\gamma }^{\prime }$ at initial times and a slower decay with ${\gamma }^{″}$ at moderate times, see figure insets; for estimate on upper plot (solid line) $b=0.0025, \alpha =0.3$. As a rough estimate, we can use dominant (fastest) asymptotic behavior of transition rate ${\gamma }^{\prime }$, and associate characteristic relaxation time with $\tau =\frac{1}{{\gamma }^{\prime }}$. The results shown are obtained based on the numerical solution of mean-field Eqs. (4) and (5); see Appendix pp1 for details.

Figure 11

Subcritical regime (within the domain $g<g_c$): (a) Numerically obtained decay rate ${\gamma }^{\prime }$ for transition from $g=0.1$ to $g=0.3$. $n_1\left(\tau \right)$ decays as $\sim A{\mathrm{e}}^{-{\gamma }^{\prime }\tau }cos\left({\omega }^{\prime }\tau \right)$ with exponent describing the envelope behavior, and $A=n_1^{\left(0.3\right)}-n_1^{\left(0.1\right)}=-0.0025$. (b) The same for reverse transition. $n_1\left(\tau \right)$ decays as $\sim A{\mathrm{e}}^{-{\gamma }^{\prime }\tau }cos\left({\omega }^{\prime }\tau \right)$, with $A=n_1^{\left(0.1\right)}-n_1^{\left(0.3\right)}=0.0025$ and exponent describing corresponding envelope behavior. The results shown are obtained based on the numerical solution of mean-field Eqs. (4) and (5); see Appendix pp1 for details.

Figure 12

Transition rate ${\gamma }^{\prime }(g,g^{\prime })$ (numerical) corresponding to the fast-time part of relaxation from $|g\rangle \to |g^{\prime }\rangle$, see Fig. 10, solid lines plotted for different values of initial field $g=0.6,0.7,0.8,0.9$. The transition rate $\gamma (g,g^{\prime })$, Eq. (50), is estimated as the decay rate of the state $|g\rangle$ at sudden field switching $g\to g^{\prime }$, dashed lines for the same parameters. The order of plotted lines, both solid and dashed, corresponds to the descending value of $g$, in accordance with the legend: lines higher on the plot correspond to larger values of $g$. Empty circles correspond to $g\equiv g^{\prime }$ case. $\gamma \left(g^{\prime }\right)$ is relaxation rate to steady state $|g^{\prime }\rangle$ regardless of an initial state $|g\rangle$, estimated by Eq. (33), thick solid line. The results shown are obtained based on the numerical solution of mean-field Eqs. (4) and (5); see Appendix pp1 for details.

Figure 13

Schematic illustration of quasi-1D lattice gas corresponding to narrow channel system. Black circles correspond to mobile particles and red (gray) ones to impurity (“heavy”) particles. The cell $i+1$ is partially occupied by impurity particles, that corresponds to partially penetrable site $i+1$ for 1D lattice.

Figure 14

Continuum representation for the ring, where $n(-0)=n_{-1}$ and $n(-2L+0)=n_1$ correspond to site occupations for lattice ring, respectively.