# Nonequilibrium protection effect and spatial localization of noise-induced
fluctuations under gas flow scattering on partially penetrable obstacle

S. P. Lukyanets  0000-0003-1169-2826 lukyan[at]iop.kiev.ua O. V.
Kliushnichenko  0000-0003-4806-8971 kliushnychenko[at]iop.kiev.ua Institute of
Physics, NAS of Ukraine, Prospect Nauky 46, 03028 Kyiv, Ukraine

###### Abstract

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.

##  I Introduction

The evolution of many dissipative or out-of-equilibrium systems results in the
formation of steady states, in particular NESSs, implying the system
decomposition into parts. Some of these parts or some of the resulting states
may become insensitive to the external noise, fluctuations, or perturbations
in the system. Often, such a decomposition is related to topological
phenomena, e.g., non-Hermitian skin effect, edge correlation effect,
topological or symmetry protection effects [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].

In the equilibrium case, the protection phenomenon can manifest itself in the
formation of a quasi-particle-like structure in similarity to polaron [23,
*Landau1948, 25, 26] or solvated ion [27, 28] dressed in a polarization coat
of the surrounding media. Another example of such a spatial structure can be a
complex of a colloidal particle and a nonlinear “deformation coat” of liquid
crystal formed around it [29, 30, 31, 32]. One of the features of such a
quasi-particle formation is the strong coupling or correlation occurring
between the particle and its nearest surrounding, such that polarization or
deformation coat might protect the inner state of this complex from external
fluctuations.

We are interested in a quasi-particle-like structure in the nonequilibrium
case, particularly, in the formation of a similar structure induced by the gas
flow scattering on an obstacle.

Steady nonequilibrium structures caused by the scattering of a forced flow on
obstacles or impurities represent a broad class of NESSs of many-body systems
[33, 34, 35, 36, 37, 38, 39]. The behavior of such systems admits
nonequilibrium phase transitions when the spatial NESS structure undergoes a
dramatic change at certain critical parameters such as flow velocity, obstacle
characteristics, mean gas density, etc. [33, 34, 35, 36, 37, 38, 39]. The
properties of these structures are significantly dependent on whether linear
or nonlinear mechanism of scattering is dominant. The same applies to flow-
induced forces acting on obstacles and flow-induced nonequilibrium interaction
between them, which is usually non-reciprocal, like wake-mediated or
nonequilibrium depletion forces [40, 41, 42, 43]. In the case of a gas, the
nonlinear mechanism of scattering can be provoked by the short-range repulsion
between particles leading to the blockade effect at certain conditions [44].

We focus on nonequilibrium phase transitions leading to the formation of
nonequilibrium nonlinear structures (like “strata” [45, 46, 47, 48, 49, 50,
51, 52]) provoked by the blockade effect in a gas. In the linear case when one
can neglect interparticle interaction (low gas concentration, small gas flow,
high penetration of obstacle), the gas density perturbation (wake shape)
formed by flow scattering on the obstacle differs slightly from the
equilibrium density and can be described in terms of the linear response
leading to the asymmetric Green function. In the particular case of advection-
diffusion systems, the Green function is given by the Yukawa-like form with
asymmetric behavior of screening length that can be interpreted as dynamic
screening length111The term dynamic screening is borrowed from the realm of
collective energy losses by swift ions in condensed matter [53, 54, 55].
determined by flow velocity [56, 57, 58]. The asymptotic behavior of the wake
profile exhibits exponential growth of gas density ahead of the obstacle and
power-law decay of rarefied gas tail behind it [42], at least in 2D and 3D.

When the interparticle interaction is significant, due to locally increasing
gas density, especially ahead of the obstacle, the nonlinear dynamic screening
regime becomes dominant, manifesting itself as the blockade effect for a gas
with short-range repulsion. This leads to the qualitative change of wake shape
or gas density profile, which takes a kink-like form ahead of the obstacle.
This corresponds to the formation of a dense gas phase with a sharp boundary.
For example, it can be the formation of strata or bow-shocks near a void in
dusty plasmas [45, 46, 47, 48, 49] or the formation of a kink-like density
profile describing a dense-gas phase ahead of obstacles embedded into the
driven hard-core interacting gas or into the driven lattice gas [59, 60, 42,
61].

Qualitatively, the growth of a dense gas phase with a sharp boundary (a
stratum) can be viewed as the growth of a “nucleus” adjacent to the obstacle
that serves as a nucleation center. One can suppose that the nucleus surface
(domain wall between depleted and dense gas phases), being an additional
scatterer for a gas flow, has to protect the state of the nucleus–obstacle
complex from the external noise and fluctuations in the rest of the gas.

In this paper, we demonstrate that the nonequilibrium transition to a two-
domain gas structure, with a dense phase ahead of the obstacle, is accompanied
by the emergence of local invariants which behave like local first integrals.
These invariants describe the conservation of the gas particle number inside
the obstacle and the collective state of the obstacle’s nearest gas
surrounding. As a result, this transition leads to the spatial decomposition
of the system and to the formation of a quasi-particle-like structure
(obstacle complex) caused by the gas flow. We show that the dense gas phase
protects the state of the obstacle complex from the noise of external driving
field. In addition, noise-induced gas fluctuations are localized near the
domain wall. We associate this effect with possible nonequilibrium protection
effect [12, 13, 14, 15, 15, 1, 62, 63].

To this end, we resort to a simple, though specific, case of one-dimensional
driven lattice gas with ring topology. We consider gas scattering on a single
partially penetrable impurity, supposing that a driving field that generates
the gas flow is non-conservative. We refer to the mean-field Smoluchowsky
equations [64, 65, 66] for the mean gas occupation numbers of lattice sites.
This mean- field model is related to the so-called ASEP (asymmetric simple
exclusion process) [67, 38, 68, 69]. In our approximation, we neglect short-
range correlations [70] in a gas and possible blockade effect due to local
fluctuations in a gas [64].

Our paper is organized as follows. In Sec. II.1, we introduce the mean-field
ASEP model on a ring with a partially penetrable obstacle. In contrast to the
well-known Lebowitz-Janowsky blockage model [69, 51, 52, 50], an obstacle is
given by an impurity site with reduced possible value of vacancies for gas
particles. In Sec. II.2, we describe the nonequilibrium phase transition
between two classes of spatial NESS structures. The first class describes a
nonordered gas phase characterized by slight accumulation of gas particles
near obstacle’s front side. The second one describes an ordered phase
corresponding to two-domain gas structures with the kink form of density
profile. The transition into two-domain structures occurs when the values of
mean gas concentration, driving field, and impurity penetration (or capacity)
exceed certain critical values. We show that this transition is accompanied by
the spatial decomposition of the system. The occupation of the impurity site
and the total occupation of its edges exhibit invariant behavior becoming
insensitive to variation of main system parameters such as driving field, mean
gas concentration, the number of particles, ring size. In Sec. III, we
consider a particular case of a time-fluctuating driving field to demonstrate
that the emerged invariants behave like the local first integrals and/or
adiabatic invariants. We also show that gas density fluctuations induced by
the external field noise are suppressed near the impurity and are strongly
localized near the interface between the depleted and dense gas phases, Sec.
IV. This spatial decomposition of the system points at the possible
nonequilibrium protection effect. The transitions between different NESSs at
the sudden change of the driving field are considered in Sec. V. We show that
the relaxation rates from one NESS to another are different, except for the
states with the opposite direction of the driving field, and they take real
and complex values in the over- and subcritical regions, respectively. Such an
inter-state transitions are realized by one-step or multi-step scenarios
implying the sequential generation of shock waves in time that leads to a
step-by-step reconfiguration of the system. Concluding remarks are given in
Sec. VI.

##  II Nonequilibrium transition to a two-domain gas structure on a ring

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
$\bar{n}=0.3$, $U=0.6$, and $g=0.02,0.1,0.18<g_{c}$. (c) Typical NESSs in
overcritical regime at $\bar{n}=0.3$, $U=0.6$, 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 numerical distributions in (b) and (c) were
obtained 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
$\max_{k}[n_{k}(\tau)-n_{k}(\tau-\Delta\tau)]\leq 10^{-30}$ with
$\Delta\tau=0.01$, where $\tau=\nu t$.

###  II.1 Mean-field ASEP model on a ring with partially penetrable obstacle

Our consideration is based on the simplest mean-field Smoluchowsky-type
equations of a two-component driven lattice gas. One component represents
mobile gas particles jumping between nearest lattice sites while the other
(heavy) component is the static one describing distribution of obstacles or
impurities. The kinetics of mobile gas component is supposed to be governed by
the asymmetric simple exclusion processes (ASEP). In the mean-field
approximation, the system dynamics is described in terms of the mean
occupation numbers of lattice site $k$ by gas particles $n_{k}(t)$ and by
impurities $U_{k}$ so that $0\leq n_{k}+U_{k}\leq 1$ for each site, see, e.g.,
[66, 59, 42]. Under this approximation, we neglect short-range correlations,
local memory effect [70], and possible transition to blockade regime induced
by local fluctuations in a gas [64].

To demonstrate features of nonequilibrium transition to blockade regime in a
gas caused by flow scattering on impurities we resort to the specific
1D-lattice case with ring topology and penetrable inclusions. We preserve the
symmetry of the mean-field equations for a two-component lattice gas, used for
2D or 3D case, see, e.g., [42, 61], within present 1D consideration. A
peculiar side effect of such approximations is that particle species can
effectively pass through each other even in the strictly 1D-case. This
circumstance enables description only of partially “transparent” obstacles.

Unlike in the known Lebowitz–Janowsky blockage model, that is ASEP on a ring
with a slow bond [50, 51, 52, 69], we implemented the obstacle as a partially
transparent impurity site with finite capacity $1-U$, the maximum possible
occupation value, for gas particles. The impurity manifests itself in
decreasing of occupation probability of this site by a gas particle and, as a
result, in the reduction of transition rates to this site from nearest
neighbor sites. Qualitatively, this can be associated with an additional
potential centered at the impurity site, as illustrated in Fig. 1a.

In this setting, the gas kinetics is given by equations for mean occupation
numbers $n_{k}(t)$:

$\dot{n}_{k}=J_{k-1,k}-J_{k,k+1},$ (1)

where $J_{k,k+1}$ is bond current between $k$ and $k+1$ sites:

$J_{k,k+1}=\nu^{+}n_{k}(1-n_{k+1}-U_{k+1})-\nu^{-}n_{k+1}(1-n_{k}-U_{k}),$ (2)

where $\nu^{\pm}=\nu\pm\delta\nu$ are forward-backward particle hopping rates
between nearest sites. The asymmetry $\pm\delta\nu=\pm\nu g$ is caused by the
action of nonconservative driving force or field $g$. The nonconservative
force is usually inherent for transport problems on a ring, as it is for
colloidal particle in a periodic potential, see [71]. In the case of an
infinite chain, driving field can be given by conservative force that leads to
asymmetric particle jumps
$\nu_{ji}\approx\nu[1+\mathbf{g}\cdot(\mathbf{r}_{i}-\mathbf{r}_{j})/\ell]$
where $|\mathbf{g}|=\ell|\mathbf{G}|/(2kT)<1$, $\ell$ is the lattice constant,
and $\mathbf{G}$ is external drive, see [42]. This form corresponds to the one
of the wind force often encountered in electromigration of adsorbed atoms on a
solid-state substrate, see, e.g., [72, 73, 74].

For simplicity, we consider a particular case of single inclusion and neglect
effects entailed by collective scattering. The latter would lead to a number
of specific features caused by the additional induced correlations between
impurities [75]. The impurity is placed at zeroth lattice site
$U_{k}=U\delta_{k,0}$ , and the numbering of lattice sites is taken as $-L\leq
k\leq L$, see Fig. 1(a). The total number of ring sites is $L_{0}=2L+1$.
Equations (1) are supplemented by periodic boundary conditions
$n_{L+1}=n_{-L}$.

###  II.2 Nonequilibrium transition to two-domain gas structure

We consider the nonequilibrium transition emerging between the main two types
(or classes) of nonequilibrium, steady-state, spatial gas structures which can
be formed by flow scattering on the obstacle at $t\rightarrow\infty$. These
NESSs are determined by the system’s parameters: the driving field $g$, the
mean gas concentration $\bar{n}=N/L_{0}$ ($N$ is the particle number), and
impurity capacity $1-U$. We show that the transition to the two-domain gas
structure with dense and depleted phases is accompanied by the emergence of
quantities which become insensitive to variations of the main parameters
driving the system, i.e., behave like invariants. We also estimate critical
values $(\bar{n}_{c},g_{c},U_{c})$ and changes in the asymptotic behavior of
order parameters, which characterize this transition.

In the steady-state case, $\dot{n}_{k}=J_{k-1,k}-J_{k,k+1}=0$, the system has
an additional conserved quantity, besides the particle number $N$ (or mean gas
concentration $\bar{n}$). It is a current $J_{k,k+1}=J=\mathrm{const}$, Eq.
(2), flowing through each bond $(k,k+1)$ of the ring

$\nu^{+}n_{k}(1-n_{k+1}-U\delta_{k+1,0})-\nu^{-}n_{k+1}(1-n_{k}-U\delta_{k,0})=J$
(3)

written for the case of single impurity site at $k=0$, i.e.,
$U_{k}=U\delta_{k,0}$.

One can qualitatively estimate the value of $J$. For a sufficiently large ring
$L\gg 1$, one can suppose that far from the impurity, $|k|\gg 1$, the site-
occupancies are approximately equal, i.e., $n_{-L}\approx n_{L}\approx
n_{\infty}$, that gives

$J_{\infty}=J_{L,-L}\approx(\nu^{+}-\nu^{-})n_{\infty}(1-n_{\infty}).$ (4)

There are two main classes of nonequilibrium steady structures formed in the
system at $t\rightarrow\infty$. The first one is resulted by flow scattering
on obstacle at relatively small values of $\bar{n}$, $g$, and $U$, see Fig.
1(b). The second class is described by kink-type spatial structures of NESS,
and corresponds to emergence of two-domain gas structure when parameters
$(\bar{n},g,U)$ exceed certain critical values $(\bar{n}_{c},g_{c},U_{c})$,
see Fig. 1(c). Note that such a structure is the effect of the ring topology.
In contrast, in the case of an infinite chain, the gas density perturbation
caused by flow scattering can be located around the obstacle and dumps to
equilibrium state at infinity.

Below, we restrict our consideration mainly to the case $\bar{n}<1/2$.
Analogous description in $\bar{n}>1/2$ regime is straightforward by virtue of
wake inversion effect [42, 44], stemming from particle-hole symmetry, when the
kink profile describes a cavity behind the obstacle. This case corresponds to
what is called high-density (HD) phase in ASEP models [69]. In particular, the
symmetry relative to $\bar{n}\leftrightarrow 1-\bar{n}$ interchange is evident
from numerical calculations in the $\bar{n}\in[0\ 1]$ range, as is shown in
this subsection later.

Subcritical region. In the equilibrium case (in the absence of driving field,
$g=0$), the current $J=0$. On the average, the gas has no structure, akin to
quasi-one-dimensional liquid in equilibrium [76]. The equilibrium distribution
of mean occupation numbers $n_{k}$ is given by expression

$\displaystyle n_{k}$ $\displaystyle=$
$\displaystyle(1-U)n^{e}\delta_{k,0}+n^{e}(1-\delta_{k,0}),$ (5)

where $n^{e}=(1-\bar{U})^{-1}\bar{n}$, and $\bar{U}=U/L_{0}$. At small driving
field $g\ll 1$, and/or small mean gas concentration $\bar{n}\ll 1$, and/or
hight impurity penetrability $U\ll 1$, the distribution of gas density
perturbation $\delta n_{k}$, caused by flow scattering on the obstacle, weakly
deviates from its equilibrium value $n^{e}$ (or mean gas concentration
$\bar{n}$) for a large ring, $L_{0}\gg 1$. The scattering leads to the
relatively small accumulation of gas particles ahead of the obstacle (the
impurity site), see Fig. 1(b).

This class of steady-state structures or solutions can be easily estimated by
resorting to the continuum limit of Eqs. (1) for gas density $n(x)$ outside
impurity site, see Appendix A. In the lowest order of the long-wave
approximation, $n(x)$ is governed by the Burgers equation.

Representing gas concentration $n(x)=\bar{n}+\delta n(x)$ outside the
inclusion and occupancy of impurity site $n_{0}=\bar{n}+\delta n_{0}$ as small
deviations from mean gas concentration $\bar{n}$ one can obtain linearized
equations corresponding to Eq. (3)

$-\partial_{x}\delta n+2g\bar{n}(1-\bar{n})\delta n=\delta J,$ (6a)
$\displaystyle\delta
n_{-1}\left(\nu^{+}(1-U-\bar{n})+\nu^{-}\bar{n}\right)-\delta
n_{0}\bar{n}+\nu^{-}\left(\nu^{+}(1-\bar{n})\right)$ $\displaystyle=\delta
J+\nu^{+}U\bar{n},$ (6b) $\displaystyle\delta
n_{0}\left(\nu^{+}(1-\bar{n})+\nu^{-}\bar{n}\right)-\delta
n_{1}\left(\nu^{+}\bar{n}+\nu^{-}(1-U-\bar{n})\right)$ $\displaystyle=\delta
J-\nu^{-}U\bar{n}.$ (6c)

Here we change coordinate system for dimensionless variable $-2L<x<0$ so that
occupancies on impurity edges are written as $\delta n_{-1}=\delta n(0-)$ and
$\delta n_{1}=\delta n(-2L+0)$. These equations are supplemented by one for
conservation of the particle number (or $\bar{n}$) to obtain unknown constants
$\delta n_{-1}$, $\delta n_{1}$, $\delta n_{0}$ and $\delta
J=J-2g\bar{n}(1-\bar{n})$.

In zeroth order in $\lambda/L_{0}\ll 1$, see Appendix A, behavior of $\delta
n(x)$ takes the simplest form

$\displaystyle\delta n\approx\delta n_{-1}\exp(\lambda x),$ (7)

where $\lambda^{-1}=2g\bar{n}(1-\bar{n})$ is dynamic screening length. This
approximation neglects the difference between $n_{\infty}$ and $\bar{n}$ far
away from the obstacle (actually $n_{\infty}<\bar{n}$ ). As a result, $\delta
J\approx\delta n_{1}\approx 0$. Nevertheless, obtained results for $\delta
n(x)$ as well as for $\delta n_{0}(\bar{n},g,U)$ and $\delta
n_{-1}(\bar{n},g,U)$ [expressions for which are given by Eqs. (65) and (66)]
are in more-or-less good agreement with asymptotical numerical results
obtained from Eqs. (1) at long times, see Figs. 2 and 3.

Figure 2:  Behavior of density distribution (NESS profile) ahead of the
obstacle in subcritical regime at $\bar{n}=0.1$, $U=0.6$, and $g=0.2<g_{c}$:
comparison of numerical result and analytical asymptotics, Eq. (7). Here $x$
corresponds to continuous coordinate, $k$ is the site index, and $L_{0}=401$.

In subcritical domain, the class of nonequilibrium steady structures of the
gas or the class of stationary solution of the system Eqs. (1) correspond to
the linear dynamic screening regime. The spatial structure of the gas exhibits
exponential growth region, due to particle accumulation ahead of the obstacle,
with characteristic length $\lambda\approx[2g\bar{n}(1-\bar{n})]^{-1}$, while
behind the obstacle, we have small density deviation from the equilibrium
value $n^{e}\approx\bar{n}$. Such particle accumulation near the obstacle edge
is the simplest manifestation of non-Hermitian skin effect, see [1, 17].

Overcritical region. If the system parameters $\bar{n}$, $U$, and $g$ exceed
certain critical values $(\bar{n}_{c},U_{c},g_{c})$, the system undergoes
transition to two-domain structure with depleted and dense gas phases Fig.
1(c). The spatial structures of NESS behave similarly to a kink solution.

To find critical estimates of $(\bar{n}_{c},U_{c},g_{c})$ and to assess the
behavior of the main characteristics of this transition, we resort to the
rough approximation of a solution representing it as a kink with sharp step

$n_{k}=\left\\{\begin{array}[]{l}n_{\infty},\quad\quad\ \ -L\leq k<-S\\\
1-n_{\infty},\quad-S\leq k<0\\\ n_{0},\quad\quad\quad k=0\\\
n_{\infty},\quad\quad\ \ 0<k\leq L.\end{array}\right.$ (8)

In the case of a single impurity, the class of the approximate kink-type
solutions, Eq. (8), is determined by concentration $n_{\infty}$, the mean
occupation number of the impurity site $n_{0}$ and the length of dense phase
(stratum) $S$ corresponding to the lattice site $k=k_{s}=-S$.

Note that at $k_{s}=-S=-L$ and $n_{0}=(1-U)/2$, Eq. (8) gives exact particular
single-kink solution for a finite chain. Generally, result is sensitive to the
choice of boundary condition for a finite unclosed chain.

Under approximation Eq. (8), $n_{\infty}$ and $n_{0}$ can be obtained from
equations $J_{-1,0}=J_{\infty}$ and $J_{0,1}=J_{\infty}$, see Eq. (3),

$\nu^{+}n_{-1}(1-n_{0}-U)-\nu^{-}n_{0}(1-n_{-1})=J_{\infty},$ (9a)
$\nu^{+}n_{0}(1-n_{1})-\nu^{-}n_{1}(1-n_{0}-U)=J_{\infty},$ (9b)

where $n_{-1}=1-n_{\infty}$ and $n_{1}=n_{\infty}$ are the occupation numbers
of the nearest sites from the left and to the right of the impurity,
correspondingly, and the current $J_{\infty}$ is given by Eq. (4). Equations
(9) can be rewritten in the form

$\left\\{\nu^{+}(1-n_{\infty})+\nu^{-}n_{\infty}\right\\}(1-U-2n_{0})=0,$
(10a)
$\displaystyle\left\\{\nu^{+}(1-n_{\infty})-\nu^{-}n_{\infty}\right\\}(1-U)$
$\displaystyle=2(\nu^{+}-\nu^{-})n_{\infty}(1-n_{\infty}).$ (10b)

It follows directly from Eq. (10a) that the mean occupancy of impurity site by
gas particles

$n_{0}=\frac{1-U}{2}$ (11)

is determined only by the impurity capacity $(1-U)$. It does not depend on the
number or mean concentration $\bar{n}$ of particles, on the value of driving
field $g$ and as a result on the current $J_{\infty}$ flowing through the
obstacle, in contrast to the subcritical regime, Figs. 3 and 5.

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,
$\bar{n}=0.3$, $U=0.6$, and $L_{0}=401$. Figure 4:  (a) Plot shows the
existence of critical concentration value $\bar{n}_{c}$ above which
$n_{\infty}$ no longer depends on $\bar{n}$, signifying the transition into
two-domain structure at $\bar{n}>n_{c}$ (for $g=0.5$). (b) Dependence of
concentration $n_{\infty}$ on field $g$. Analytical solution for $n_{\infty}$
is given by Eq. (12). (c) Dependence of domain wall position $k_{s}=-S(g)$ or
the spatial width of dense phase $S(g)$ on the driving field $g$ (numerical
against Eq. (14)). The numerical values of $S$ were taken as the closest site
to the domain wall $k_{s}$ defined by $n(k_{s})=1/2$, cf. [61] for domain wall
stopping criterion. (d) Inter-particle correlations at the nearest sites
$\overline{\delta n_{k}\delta n_{k+1}}$, where $\delta n_{k}=n_{k}-\bar{n}$,
and $\overline{(\dots)}=L_{0}^{-1}\sum_{k}(\dots)$; analytical curve is given
by Eq. (17). Gas concentration $\bar{n}=0.3$ for (a), (c), and (d); $U=0.6$
and $L_{0}=401$ for all panels.  Figure 5:  (Upper row) Two-dimensional
projections of the critical surface for the phase diagram given in
$(U,\bar{n},g)$ parameter space: (a) $(U,\bar{n})$-projection at $g=0.5$, (b)
$(g,U)$-projection at $\bar{n}=0.3$, and (c) $(g,\bar{n})$-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)
$U(\bar{n})=1-[4\bar{n}(1-\bar{n})]/(3-4\bar{n})$, (b)
$U(g)=1-[4gn(1-n)]/(1-2n+g)$, and (c)
$g(\bar{n})=(1-U)(1-2\bar{n})/[4\bar{n}(1-\bar{n})+U-1]$. Ring length is
$2L=400\ell$ for all cases. Panels (a) and (c) features inversion symmetry in
$\bar{n}$ between high-density (HD) and low-density (LD) phases known in ASEP
models [69].

The value of $n_{0}$ saturates at the nonequilibrium transition and
corresponds to the half-filling of the impurity site. Note that half-filling
is a property of the center of domain wall separating dense and depleted
phases. In addition, the value of $n_{0}$ also does not depend on amplitude
$1-2n_{\infty}$ and length $S$ of the kink.

In other words, saturation or half-filling of the impurity $n_{0}/(1-U)=1/2$
indicates the emergence of ordered two-domain structure in the system and can
serve as invariant that characterizes the class of kink-type solutions,
despite their form that is determined by three parameters $\bar{n},U,g$, see
Fig. 1(c).

Equation (10) has two distinct solutions from which only one,

$n_{\infty}=\frac{1}{2}\left\\{1+\frac{n_{0}(U)}{g}-\left[1-2n_{0}(U)+\left(\frac{n_{0}(U)}{g}\right)^{2}\right]^{\frac{1}{2}}\right\\},$
(12)

satisfies the condition $0\leq n_{\infty}\leq 1$. Where $n_{0}(U)$ is given by
Eq. (11), and $\nu^{\pm}=\nu(1\pm g)$ is taken into account. As is seen,
concentration $n_{\infty}$, determining the amplitudes of two gas phases,
loses its dependence on the total number of particles $N$ in the system or
mean gas density $\bar{n}$ (for $\bar{n}>n_{c}$), Fig. 4(a). Figure 4(b) is
the comparison of $n_{\infty}(g)$ estimate against numerical result showing a
good agreement.

Using Eq. (12) one can easily estimate the length $S$ of dense gas phase or
position of the second domain wall (the first one coincides with impurity
site). Taking into account the particle number conservation and our
approximation for $n_{k}$ Eq. (8) one can write

$\sum_{k=-L}^{L}n_{k}=(2L-S)n_{\infty}+S(1-n_{\infty})+n_{0}=N.$ (13)

It enables to write expression for fraction of the ring $S/L_{0}$ occupied by
dense phase

$\frac{S}{L_{0}}=\frac{\bar{n}-n_{\infty}}{1-2n_{\infty}}+\frac{1}{L_{0}}\frac{n_{\infty}-n_{0}}{1-2n_{\infty}}\approx\frac{\bar{n}-n_{\infty}}{1-2n_{\infty}}.$
(14)

For a large ring, $L_{0}\gg 1$, the part of the ring occupied by dense phase
does not depend on its size and is determined only by mean concentration
$\bar{n}$ and concentration at infinity $n_{\infty}$. This gives a good
agreement with numerical data, Fig. 4(c). The latter is the result of strong
correlations between gas particles in the presence of driving field within
ring topology — when external field $g$ gathers gas particles from all over
the system towards impurity.

To demonstrate enhancement of inter-particles correlation after the
nonequilibrium transition we consider correlation at the nearest sites

$\overline{\delta n_{k}\delta n_{k+1}}=\overline{n_{k}n_{k+1}}-\bar{n}^{2},$
(15)

where $\overline{(\dots)}=L_{0}^{-1}\sum_{k}(\dots)$, and $\delta
n_{k}=n_{k}-\bar{n}$. Taking into account

$\overline{J_{k,k+1}}=J_{\infty}=(\nu^{+}-\nu^{-})n_{\infty}(1-n_{\infty}),$

it is easy to show that

$\displaystyle\overline{n_{k}n_{k+1}}$ $\displaystyle=$
$\displaystyle\bar{n}-n_{\infty}(1-n_{\infty})-$ (16) $\displaystyle-$
$\displaystyle\frac{1}{\nu^{+}-\nu^{-}}\overline{\left(\nu^{+}n_{k}U_{k+1}-\nu^{-}n_{k+1}U_{k}\right)}.$

In case of single impurity ($U_{k}=\delta_{k,0}$), the last term in Eq. (16)
reads $(U/2gL_{0})\left[(1-2n_{\infty})+g\right]$. For a large ring ($L_{0}\gg
1$), this term can be neglected and expression (15) takes the form analogous
to gas compressibility, see [77],

$\displaystyle\overline{\delta n_{k}\delta n_{k+1}}$ $\displaystyle\approx$
$\displaystyle\bar{n}(1-\bar{n})-n_{\infty}(1-n_{\infty}).$ (17)

As it follows from Fig. 4(d), expression (17) gives a good agreement with the
numerical results for overcritical domain.

Expressions (12) or (10) allow us also to obtain the simple analytical
expression of the form $B_{c}=B(\bar{n},U,g)=0$ for the critical interface
separating two different classes of NESS in the space of system parameters
$(\bar{n},U,g)$ (the transition phase diagram). The two-domain structure
becomes unstable when $n_{\infty}$ approaches to the equilibrium concentration
$n_{e}\approx\bar{n}$, see Figs. 4(b). Setting $n_{\infty}\approx\bar{n}$ in
Eq. (12) or Eq. (10) we obtain approximate equation for critical surface
$B_{c}$:

$g\approx\frac{(1-U)(1-2\bar{n})}{4\bar{n}(1-\bar{n})-(1-U)},$ (18)

which gives more-or-less good agrement with the numerical results, see Fig. 5.

Note that expressions $n_{0}=(1-U)/2$ and $n_{-1}+n_{1}=1$ are exact and
satisfy Eqs. (10). This follows from the equation for impurity site occupancy
$\partial_{t}n_{0}=J_{-1,0}-J_{0,1}=0$ that expresses $n_{0}$ in terms of gas
states on impurity edges:

$n_{0}=\frac{1-U}{2}\left(1+\frac{n_{-1}+n_{1}-1}{g(n_{-1}-n_{1})+1}\right).$
(19)

This means that after the transition to the steady two-domain gas structure,
the system possesses two invariants — impurity occupancy $n_{0}$ and the total
occupation of its edges $n_{-1}+n_{1}=1$ which are insensitive to variations
of the main system parameters, Fig. 3. Such a behavior of impurity edges can
be considered as strong correlation of gas states at impurity edges, and the
nonequilibrium transition can be characterized by switching between the
diffusive skin effect and the edge-edge correlation one. Note that we exploit
the rough model of asymmetric hopping transport. Our consideration is limited
by the coordinate representation and loses the spectral properties of the
system which are basic in the description of many exotic topological phenomena
in non-Hermitian systems, see e.g., [4, 6, 7, 5, 62, 78].

The universal behavior of the local invariants enables us to suppose that
impurity state and correlated state of its edges can behave as local first
integrals, i.e., $n_{0}(t)=\text{const}$ and $n_{+1}(t)+n_{-1}(t)=1$, after
nonequilibrium phase transition ($g>g_{c},U>U_{c},\bar{n}>\bar{n}_{c}$) and,
as a result, can become insensitive to the noise of external field $\delta
g(t)$ and to the fluctuations of the number of particles $\delta N(t)$ in the
system. To qualitatively demonstrate the emergence of local first integrals we
resort to the particular case of the time-dependent external non-conservative
force $g(t)=g+\delta g(t)$.

##  III Emergent Local first integrals

Figure 6: Numerical illustration of the emergence of local invariants (local
first integrals) $n_{0}(\tau)=(1-U)/2=\mathrm{const}$ and $n_{\pm
1}(\tau)=n_{1}(\tau)+n_{-1}(\tau)\approx 1$ after nonequilibrium transition at
$g>g_{c}$ for the case of driving field noise $g(\tau)=g+\delta g(\tau)$.
Exploited noise samples are shown at the top panel, both have switching
frequency $\lambda=0.02$. For both kinds of noises the drive $g(t)$ fluctuates
around $g=0.2$ (subcritical regime) or $g=0.8$ (overcritical regime) with
maximum amplitude $|\delta g(t)|=0.1$. Here, ring length $2L=400\ell$, average
density (filling fraction) $\bar{n}=0.3$, and $U=0.6$. Enlarged segment of
$n_{1}(\tau)$ shows relaxation between
$|g=0.7\rangle\rightleftarrows|g=0.9\rangle$ with dominant asymptotic behavior
$\sim\mathrm{e}^{-\gamma\tau}$, implying decay rate $\gamma\gg\lambda$.
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 $g(\tau)$ 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 $6\times 10^{3}\ [\nu t]$ during which $g(\tau)$
approximately reaches the mean value $g=0.8\pm 0.01$. The quantity $n_{\pm
k}(t)$ holds quasi-invariant within the interval from $k=1$ up to $\approx
k_{s}$. As can be seen from (c) the invariance weakens slowly until certain
$k_{\ast}$ which can be naturally associated with the size of obstacle
complex. In the interval $k_{\ast}<k<{\langle k_{s}\rangle}_{t}$, 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 ${\langle\Delta n^{2}_{\pm
k}\rangle}_{t}={\left\langle(n_{\pm k}-{\langle n_{\pm
k}\rangle}_{t})^{2}\right\rangle}_{t}\approx 10^{-30}-10^{-9}$ for dense gas
phase $0<k<40$, where
${\langle\dots\rangle}_{t}=\tfrac{1}{T}\sum_{t=0}^{T}\dots$. One can say that
obstacle complex, i.e., the domain $|k|<k_{\ast}$, is protected from $g$
noise-induced fluctuations.

Generally, the problem of obtaining invariants and local first integrals for
the dynamical system (1)–(2) is of independent interest, see, e.g., [79, 80,
81]. Here we qualitatively consider this question supplementing our reasoning
by numerical results. For the numerical simulations, we will consider discrete
changes of the field $g(t)$ in time, that is similar to telegraph-like
processes.

To qualitatively illustrate emergence of local first integrals in the
overcritical domain $(U>U_{c},\bar{n}>\bar{n}_{c},g>g_{c})$, we resort to the
simplest case when the field $g(t)$ jumps from $g=\text{const}>g_{c}$ to
$g^{\prime}=g+\delta g=\text{const}>g_{c}$ at time moment $t_{0}=0$ supposing
that previously the system is in steady state
$\vec{n}^{g}=(n^{g}_{-L},\ldots,n^{g}_{L})^{T}$ (the superscript $T$ denotes
transposition). In fact, we are interested in time-relaxation of the system
from one steady state $\vec{n}^{g}$ (corresponding to field value $g$) to
another one $\vec{n}^{g^{\prime}}$. For this simple case, we show that
occupation of the impurity site $n_{0}(t)=(1-U)/2$ and of its edges
$n_{1}(t)+n_{-1}(t)=1$ can behave like local first integrals in the vicinity
of noncritical point $\vec{n}^{g}$. Numerically, we demonstrate the emergence
of these invariants as adiabatic ones at least for the system under the action
of the time-fluctuating field $g(t)$.

For convenience, we rewrite the dynamical system (1)–(2) in the form

$\partial_{t}\vec{n}=\vec{f}^{g^{\prime}}(\vec{n}),$ (20a)
$f^{g^{\prime}}_{i}(\vec{n})=J^{g^{\prime}}_{i-1,i}-J^{g^{\prime}}_{i,i+1},$
(20b)

with initial condition

$\vec{n}\big{|}_{t=0}=\vec{n}^{g}.$ (21)

Here $\vec{n}=(n_{-L},\dots,n_{L})^{T}$, $n_{k}$ is the mean occupation number
of $k$th site, $\vec{n}^{g}$ is nonequilibrium steady state corresponding to
field $g$, the bond flow $J^{g^{\prime}}_{i,i+1}$ given by expression (3) is
written for field $g^{\prime}=g+\delta g$, i.e., $\nu^{\pm}=\nu(1\pm
g^{\prime})$.

The formal solution of (20a) can be represented in the Lie-series form, see,
e.g., [82, 79],

$\vec{n}(t)=e^{tD^{g^{\prime}}}\vec{n}\big{|}_{\vec{n}=\vec{n}^{g}},$ (22)
$D^{g^{\prime}}=\vec{f}^{g^{\prime}}(\vec{n})\tfrac{\partial}{\partial\vec{n}}=\sum_{i}\left(J^{g^{\prime}}_{i-1,i}-J^{g^{\prime}}_{i,i+1}\right)\frac{\partial}{\partial
n_{i}}.$ (23)

Taking into account that $\vec{f}^{g}(\vec{n}^{g})=\vec{0}$ for steady state
$\vec{n}^{g}$, and as a result $D^{g}\vec{n}\big{|}_{\vec{n}=\vec{n}^{g}}=0$,
it is convenient to rewrite operator $D^{g^{\prime}}$ in the form:

$D^{g^{\prime}}=D^{g}+\delta D=\left(\vec{f}^{g}+\delta
g\vec{\phi}\right)\frac{\partial}{\partial\vec{n}},$ (24)

where

$\phi_{i}=(1-2n_{i}-U\delta_{i,0})(n_{i-1}-n_{i+1})+Un_{i}(\delta_{i,-1}-\delta_{i,1}).$
(25)

From (25) and (24) it follows that

$D^{g^{\prime}}n_{0}\big{|}_{\vec{n}=\vec{n}^{g}}=\phi_{0}\big{|}_{n^{g}_{0}=(1-U)/2}=0.$
(26)

This means that $n^{g}_{0}=(1-U)/2$ behaves as local first integral at least
in the vicinity of non critical point $\vec{n}=\vec{n}^{g}$
[$\vec{f}^{g^{\prime}}(\vec{n}^{g})\neq\vec{0}$], see Fig. 6. Recall that
$n_{0}^{g}=(1-U)/2$ is the exact steady state solution for impurity site
only.222It would be desirable to show that
$\exp(tD^{g^{\prime}})n_{0}\big{|}_{\vec{n}^{g}}=n^{g}_{0}=(1-U)/2$. Note, one
can show that $\exp(t\delta
D)n_{0}\big{|}_{\vec{n}^{g}}=\left[n_{0}+(1-U-2n_{0})\xi(\vec{n})\right]\big{|}_{\vec{n}^{g}}=n^{g}_{0}=(1-U)/2$
as well as $\exp(tD^{g})\exp(t\delta D)n_{0}\big{|}_{\vec{n}^{g}}=n^{g}_{0}$,
where $\xi$ is some function. However, to prove the general statement, using,
for instance, Trotter’s product formula, the exact steady state solution
$\vec{n}^{g}$, i.e., for all the sites, is needed.

Note that if $n_{0}=1-U-n_{0}=(1-U)/2$ were integral of motion then quantity
$n_{\pm 1}(t)=n_{-1}(t)+n_{1}(t)$ would be the integral of motion as well that
immediately follows from equation for $n_{0}(t)$:

$0=\partial_{t}n_{0}=J_{-1,0}-J_{0,1}=n_{0}(\nu^{+}+\nu^{-})[n_{-1}(t)+n_{1}(t)-1].$

To qualitatively show that quantity $n_{\pm 1}(t)\approx 1$ behaves like local
invariant we use our approximate solution for steady state distribution
$n_{k}^{g}$ with sharp step form (8). Action of Lie operator gives

$\displaystyle D^{g^{\prime}}(n_{-1}+n_{1})\big{|}_{\vec{n}=\vec{n}^{g}}=$
(27) $\displaystyle\left(\delta
g/g\right)\left[(n^{g}_{-1}+n^{g}_{1})-(n^{g}_{-2}+n^{g}_{2})\right]\Big{|}_{\begin{subarray}{c}n^{g}_{-1}=n^{g}_{-2}\approx
1-n_{\infty}\\\ n^{g}_{1}=n^{g}_{2}\approx n_{\infty}\end{subarray}}=0$

in the vicinity of our approximate solution, see Fig. 6. This approximation
gives a series of such quasi-invariants $n_{\pm k}(t)=n_{-k}(t)+n_{k}(t)=1$
which undergo destruction near site $k=k_{s}$ corresponding to the position of
interface between dense-depleted gas phases (domain wall). This is in
qualitative agreement with numerical solution, Fig. 7. However, numerical
results suggest that quantities $n_{\pm k}$ behave like adiabatic invariants
$\langle n_{\pm k}\rangle_{t}\approx 1$ undergoing weakening as $k$ tends to
$k=k_{*}<\langle k_{s}\rangle_{t}$ and complete destruction near
characteristic wall position $\langle k_{s}\rangle_{t}$ as it follows from the
case of fluctuating field $g(t)$, see Fig. 7. Here $\langle\ldots\rangle_{t}$
denotes time averaging. The value $k_{\ast}$ can be related to the size of
obstacle complex (the region $|k|<k_{\ast}$) which is protected from
$g$-noise-induced fluctuations. In contrast, the system loses the invariants
and correlated time-behavior for sites near the impurity in the subcritical
domain ($g(t)<g_{c}$, $U>U_{c}$, $\bar{n}>\bar{n}_{c}$). The state of the
impurity and its surroundings are sensitive to fluctuations of the driving
field $g(t)$ as is shown in Figs. 6 and 7.

Note that the impurity filling conservation in time,
$n_{0}=(1-U)/2=\text{const}$, requires equality of the incoming and outgoing
currents at impurity, that means strong gas correlation at opposite its edges.
Synchronized behavior of the gas states at the opposite impurity edges
$n_{-1}(t)+n_{1}(t)=1$ guarantees that incoming and outgoing currents at the
impurity are equal
$J_{-1,0}(t)=J_{0,1}(t)=n_{0}\left[\nu^{+}(t)n_{-1}(t)-\nu^{-}(t)n_{1}(t)\right]=\nu
n_{0}\left[n_{-1}(t)-n_{1}(t)+g(t)\right]$, that is a necessary condition for
impurity filling conservation. Such synchronized dynamics can be formally
viewed as a manifestation of the edge correlation effect.

Thus, the nonlinear dynamic screening effect caused by gas flow scattering on
the impurity leads to the formation of gas dense phase near the impurity. This
effect leads to decomposition of the system with emergence of invariant
behavior of impurity state and strong correlated behavior of a gas on the
impurity edges in overcritical domain. The growth of dense gas phase near
impurity can be speculatively considered as the growth of a nucleus around
impurity serving as a nucleation centre the “size” of which is larger than the
critical one that formally corresponds to the condition $U>U_{c}$. In
addition, the boundary of nucleus (dense gas phase) being an additional
scatterer protects it’s own center (impurity state) against fluctuations in a
gas.

We can say that the interface appearing between dense and depleted gas phases
(the domain wall) results in protection of the state of the “impurity-dense
gas” complex against fluctuations in the rest of the gas and against noise of
external field. We can associate such a decomposition with nonequilibrium
protection effect by analogy with topological protection effect, and edge
correlation one (see, e.g., [13, 14, 15]).

##  IV Spatial localization of induced fluctuations

As we have seen, the reaction of the system to external noise $\delta g(t)$
below the phase transition ($g<g_{c}$) and above ($g>g_{c}$) is qualitatively
different, see Fig. 6. Within the subcritical region, noise-induced
fluctuations of density are distributed over all the system, reaching the
maximum at impurity site, Fig. 8,

Figure 8:  The square root dispersion ${\langle\Delta
n^{2}_{k}\rangle}_{t}^{1/2}={\left\langle(n_{k}-{\langle
n_{k}\rangle}_{t})^{2}\right\rangle}_{t}^{1/2}$ of the noise-induced
fluctuations of the $k$th site occupation number (a) below $g_{c}$ (${\langle
g\rangle}_{t}=0.2$) and above $g_{c}$ (${\langle g\rangle}_{t}=0.8$). 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. (b) shows the zoomed in region from (a) near the
impurity, $k\in[-20\ 20]$. (c) and (d) shows the same as (a) above $g_{c}$
(${\langle g\rangle}_{t}=0.8$) but for telegraphic noises with $\lambda=0.02$
and $\lambda=0.002$, respectively.

see also Fig. 7. Contrary, above the transition ($g>g_{c}$), the fluctuations
are suppressed near impurity. In addition, the main part of the fluctuations
in a gas are localized near nucleus boundary (domain wall), Figs. 8 and 7.

Here, we make an attempt to explain qualitatively the localization of induced
fluctuations at the domain-wall position and to estimate the characteristic
amplitude of density fluctuations within depleted gas phase and in the
vicinity of nucleus boundary.

In general case, the system response to external-field noise $\delta g(t)$ is
determined by the system of nonlinear stochastic equations Eq. (1) for the
dynamics of mean occupation numbers $n_{k}(t)$ at lattice sites $k$, where
$\nu^{\pm}(t)=\nu\pm\nu[g+\delta g(t)]$.

We consider the perturbation of occupation numbers (concentration) at sites
$n_{k}(t)=n^{g}_{k}+\delta n_{k}(t)$, caused by external noise $\delta g(t)$,
close to NESS $n^{g}_{k}$ corresponding to field $g$ (supposing $|\delta g|\ll
g$ and $|\delta n_{k}|\ll n^{g}_{k}$). As we have seen, the steady-state
concentration distribution $n^{g}_{k}$ in depleted phase behind the impurity
($k\geq 1$) up to the position of domain wall ($k<-S$) can be treated as
constant $n^{g}_{k}\approx n_{\infty}$, Eq. (8). Then, to estimate
characteristic amplitude of fluctuations $\sqrt{\langle\delta
n_{k}^{2}\rangle}$ in the depleted phase it is sufficient to consider behavior
of concentration $n_{1}(t)=n^{g}_{1}+\delta n_{1}(t)$ at the nearest to the
impurity site $k=1$. Stochastic equation for $n_{1}(t)$ reads:

$\displaystyle\dot{n}_{1}(t)$ $\displaystyle=$ $\displaystyle
J_{0,1}-J_{1,2}=\nu^{+}n_{0}\tilde{n}_{1}(t)-\nu^{-}n_{1}(t)\tilde{n}_{0}$
(28) $\displaystyle-$
$\displaystyle[\nu^{+}n_{1}(t)\tilde{n}_{2}(t)-\nu^{-}n_{2}(t)\tilde{n}_{1}(t)],$

where $\nu^{\pm}=\nu\pm\nu(g+\delta g(t))$, $\tilde{n}_{1(2)}=1-n_{1(2)}$, and
$n_{0}$ is taken into account to be the local first integral,
$n_{0}=\bar{n}_{0}=1-U-n_{0}=(1-U)/2$. Assuming that the behaviors of mean
occupation numbers $n_{1}(t)=n_{1}^{g}+\delta n_{1}(t)$ and
$n_{2}(t)=n_{2}^{g}+\delta n_{2}(t)$ at neighboring sites $1$ and $2$ do not
differ significantly, $n_{2}(t)\approx n_{1}(t)=n^{g}_{1}+\delta n_{1}$,
within linear approximation, we obtain closed Langevin equation on the
deviation of mean occupation number $\delta n_{1}(t)$ from its steady-state
value $n^{g}_{1}=n_{\infty}$ at the nearest to the impurity site:

$\delta\dot{n}_{1}\approx-\gamma\delta n_{1}-A\delta g,$ (29)

where

$\gamma=2\nu[n_{0}+g(1-2n_{1})]=2\nu[n_{0}+g(1-2n_{\infty})],$ (30)

and

$A=-\nu[n_{0}-2n_{1}\bar{n}_{1}]=\nu n_{0}(1-2n_{\infty})/g.$ (31)

In the latter expression we have accounted for Eq. (10).

The rate $\gamma$, Eq. (30), is the rate of relaxation towards a NESS
$|g,U,\bar{n}\rangle$ corresponding to the kink configuration determined by
the external field $g>g_{c}$, impurity state $U>U_{c}$ ($n_{0}=(1-U)/2$), and
mean gas concentration (or particle number) $\bar{n}>\bar{n}_{c}$.

It is worth to note that relaxation rate, Eq. (30), depends on $g$ and $U$
only, $\gamma=\gamma_{g,U}$, and does not depend on mean concentration
$\bar{n}$. This means, that transition rates $\gamma_{1,2}$ and $\gamma_{2,1}$
between two NESSs, $|1\rangle$ and $|2\rangle$, with different $g_{1}\neq
g_{2}$ and/or $U_{1}\neq U_{2}$, are different,
$\gamma_{1,2}\neq\gamma_{2,1}$, see Sec. V for details.

It is easy to write the solution of Eq. (29) and the expression for
autocorrelation function or dispersion of fluctuations of the mean occupation
number $\delta n_{1}$ close to the NESS $n^{g}_{1}=n_{\infty}(g)$:

$\delta n_{1}=\mathrm{e}^{-\gamma
t}\displaystyle\int\limits_{0}^{t}\mathrm{d}t_{1}\mathrm{e}^{\gamma
t_{1}}A\delta g(t_{1}),$ (32) $\left\langle\delta
n_{1}^{2}\right\rangle=\mathrm{e}^{-2\gamma}\displaystyle\int\limits_{0}^{t}\mathrm{d}t_{1}\int\limits_{0}^{t}\mathrm{d}t_{2}\mathrm{e}^{\gamma(t_{1}+t_{2})}A^{2}\langle\delta
g(t_{1})\delta g(t_{2})\rangle.$ (33)

Here, we assume that at the initial moment the system was in a NESS
$n_{1}(t=0)=n^{g}_{1}=n_{\infty}(g)$.

For simplicity, we consider the ordinary telegraphic process [83, 84], see
Fig. 6, where $\delta g(t)$ takes two values, $\pm\delta g$, with amplitude
$\delta g$ and switching rate $\lambda$, for which the correlation function
takes the form $\langle\delta g(t_{1})\delta g(t_{2})\rangle=(\delta
g)^{2}\displaystyle\exp(-2\lambda|t_{1}-t_{2}|)$.

For the parameter values used in numerical solution (Figs. 6 and 8),
$n_{0}=0.2$, $n_{\infty}\approx 0.22$, $g=0.8$, noise switching rate
$\lambda\sim\leavevmode\nobreak\ 2\times 10^{-2}\nu$, and noise amplitude
$\delta g\sim 10^{-1}$, we obtain the decay rate $\gamma\sim 1.3\nu$ with Eq.
(30), that is consistent with numerical results, cf. inset in Fig. 7. It is
easy to notice, $\gamma\gg\lambda$. In this case, dispersion Eq. (33) takes
the form

$\displaystyle\left\langle\delta
n_{1(\infty)}^{2}\right\rangle\approx\left(\frac{\delta
gA}{\gamma}\right)^{2}$ $\displaystyle=\left[\frac{n_{0}(1-2n_{\infty})\delta
g}{2g[n_{0}+g(1-2n_{\infty})]}\right]^{2}\propto 1.16\times 10^{-4},$ (34)

which also agrees well with numerical calculations for characteristic
dispersion of concentration outside the impurity and domain wall, see Figs. 8.

Let us estimate the dispersion of fluctuations around characteristic position
of domain wall. Note, condition $\gamma\gg\lambda$ implies that the system
asymptotically reaches its NESS over the time between two consecutive switches
of the field $\delta g(t)$. In other words, the system follows the field and
adapts to its new values with a slight delay. That facilitates the problem —
assuming fluctuations in the system $\delta n_{1}(t)$ to be governed by the
telegraphic process as well. The system stays on average (equally-likely) in
two NESSs, $|g+\delta g\rangle$ and $|g-\delta g\rangle$ described by the
kinks at the fields $g+\delta g$ and $g-\delta g$, see Fig. 9.

Figure 9: Schematic representation of $g$-noise-induced wall shifting
approximation.

The distribution function for $n_{1}$ can be written as follows

$\mathcal{P}(n_{1})=\frac{1}{2}\delta(n_{1}-n^{g+\delta
g}_{1})+\frac{1}{2}\delta(n_{1}-n^{g-\delta g}_{1}),$ (35)

where $n^{g\pm\delta g}_{1}$ denotes steady-state values of $n_{1}$ at the
field $g\pm\delta g$. Far away from the domain wall and impurity site, the
dispersion of induced fluctuations of concentration takes the form

$\langle\delta n_{1}^{2}\rangle=\left\langle n_{1}^{2}-\langle
n_{1}\rangle^{2}\right\rangle=\frac{(n^{g-\delta g}_{1}-n^{g+\delta
g}_{1})^{2}}{4}\approx 1.44\times 10^{-4},$ (36)

which gives a good agreement with previous estimates, Eq. (IV), and numerical
calculations, Figs. 8(a) and 8(c). The mean position of the kink-wall (site’s
number) $\langle k_{s}\rangle\approx k^{g}_{s}=-S$, see Eq. (14), and the kink
height $\approx(1-n^{g}_{1})$ are determined by the stationary value
$n^{g}_{1}$ at the field $g$. Taking into account that mean position
$k^{g+\delta g}_{s}<\langle k_{s}\rangle<k^{g-\delta g}_{s}$, Fig. 9, we can
roughly estimate the dispersion of fluctuations of site occupation nearby the
characteristic position of domain wall, namely, at the site $\langle
k_{s}\rangle$. The distribution function for $n_{\langle k_{s}\rangle}$ can be
represented as, see Fig. 9,

$\mathcal{P}\\!\left(n_{\langle
k_{s}\rangle}\right)\\!=\\!\frac{1}{2}\delta(n_{\langle
k_{s}\rangle}-(1-n^{g+\delta g}_{1}))+\frac{1}{2}\delta(n_{\langle
k_{s}\rangle}-n^{g-\delta g}_{1}),$ (37)

and the dispersion takes the form

$\langle\delta n^{2}_{\langle k_{s}\rangle}\rangle=\left\langle n_{\langle
k_{s}\rangle}^{2}-\langle n_{\langle
k_{s}\rangle}\rangle^{2}\right\rangle=\left(\frac{1}{2}-\langle
n_{1}\rangle\right)^{2},$ (38)

where $\langle n_{1}\rangle=(n^{g-\delta g}_{1}+n^{g+\delta g}_{1})/2$. From
(38) and (36), one can obtain the estimation for characteristic enhancement of
induced fluctuations of gas concentration in the vicinity of domain wall as
compared to fluctuation distant from it:

$\sqrt{\frac{\langle\delta n^{2}_{\langle k_{s}\rangle}\rangle}{\langle\delta
n^{2}_{1}\rangle}}\sim 23.57,$ (39)

which shows more-or-less good consistence with numerical results, see Fig.
8(c).

Thus, it has been shown that the external field $g$-noise ($g>g_{c}$) induces
density fluctuations only outside the impurity complex with their strong
localization near the domain wall, Figs. 8(a)–(c). The latter is caused by the
instability of the wall position, similar to the nucleus surface fluctuations
during the first-order phase transitions, see, e.g., [85, 86]. In contrast, in
the subcritical region ($g<g_{c}$), Fig. 8(a), the fluctuations are
distributed throughout the system with a maximum near the impurity the
magnitude of which is an order less than the one in supercritical region.

In the both cases, above and bellow critical point $g_{c}$, the most of the
induced fluctuations are accumulated near lattice defects which act as the
main scatterers of gas flow. In the first case ($g<g_{c}$), the induced
fluctuations are concentrated only ahead of the impurity site (the obstacle),
Fig. 8(a)-(b). The states of the sites behind the impurity are weakly affected
by fluctuations, i.e., the scatterer suppresses fluctuations behind itself. In
the second case ($g>g_{c}$), corresponding to two-domain gas structure, the
fluctuations are localized near the domain wall that can be described as the
topological defect. It is significant that gas fluctuations are totally
suppressed inside the obstacle located behind the defect. In other words, the
tightly bound structure of the defect and the obstacle state, emerging after
the phase transition, can be considered as an nonequilibrium quasiparticle
similarly to one in liquid crystal [29, 30]. The role of the defect (domain
wall) is to protect the obstacle state from the external fluctuations, in our
particular case, from ones induced by a noise of the external driving field
$g$.

##  V  Transition rates between different NESSs

As it was mentioned above, the damping rate $\gamma$ of the fluctuations,
induced by the external noise $\delta g(t)$ near NESS $|g\rangle$, depends on
the magnitude of the carrier field $g$ and impurity state $n_{0}$ (or $U$),
and it does not depend on mean particle concentration $\bar{n}$. This means
that transition rates $\gamma_{1,2}$ and $\gamma_{2,1}$ for transitions
between two NESSs $|1\rangle\rightarrow|2\rangle$ and
$|2\rangle\rightarrow|1\rangle$ with different $g_{1}\neq g_{2}$ and/or
$U_{1}\neq U_{2}$ can be different, $\gamma_{1,2}\neq\gamma_{2,1}$. It may
indicate heat production during such switching, see, e.g., [87, 71].

Here we make simple analytical estimation of these rates comparing them with
asymptotic form of relaxation time values obtained from numerical solution,
and show that $\gamma_{1,2}=\gamma_{2,1}$ if states $|1\rangle$ and
$|2\rangle$ are different in field direction $\pm g$ that corresponds to
transitions between steady states $|g\rangle\leftrightarrows|-g\rangle$.

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, $n_{1}(\nu t)$ decays
according to Eq. (47), where exponential behavior is decomposed into a fast
decay with $\gamma^{\prime}$ at initial times and a slower decay with
$\gamma^{\prime\prime}$ 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
$\tau=\tfrac{1}{\gamma^{\prime}}$.

We consider relaxation of the system to its NESS $|g^{\prime}\rangle$
supposing that initially, i.e., at $t=0$, the system was in the NESS
$|g\rangle$, i.e., the driving field $g$ abruptly (non-adiabatically) changes
its value form $g$ to $g^{\prime}$ at $t=0$. To qualitatively estimate the
relaxation rate we again consider the time behavior of mean occupation number
$n_{1}(t)$ only for the nearest to the impurity site $k=1$:

$\displaystyle\dot{n}_{1}$ $\displaystyle=$
$\displaystyle\nu^{+}_{g^{\prime}}n_{0}\tilde{n}_{1}-\nu^{-}_{g^{\prime}}n_{1}\tilde{n}_{0}-[\nu^{+}_{g^{\prime}}n_{1}\tilde{n}_{2}-\nu^{-}_{g^{\prime}}n_{2}\tilde{n}_{1}],$
(40)

with initial condition $n_{1}|_{t=0}=n_{1}^{g}$. Here
$\nu^{\pm}_{g^{\prime}}=\nu(1\pm g^{\prime})$ and $n_{1}^{g}$ is steady state
of $n_{1}$ corresponding to the field magnitude $g$.

Figure 11:  Subcritical regime (within the domain $g<g_{c}$): (a) Numerically
obtained decay rate $\gamma^{\prime}$ for transition from $g=0.1$ to $g=0.3$.
$n_{1}(\tau)$ decays as $\sim
A\mathrm{e}^{-\gamma^{\prime}\tau}\cos(\omega^{\prime}\tau)$ with exponent
describing the envelope behavior, and $A=n_{1}^{(0.3)}-n_{1}^{(0.1)}=-0.0025$.
(b) The same for reverse transition. $n_{1}(\tau)$ decays as $\sim
A\mathrm{e}^{-\gamma^{\prime}\tau}\cos(\omega^{\prime}\tau)$, with
$A=n_{1}^{(0.1)}-n_{1}^{(0.3)}=0.0025$ and exponent describing corresponding
envelope behavior.

We estimate the characteristic relaxation time $\tau=\tau_{g,g^{\prime}}$
corresponding to the transition $|g\rangle\rightarrow|g^{\prime}\rangle$ by
maximal decay rate of the state $|g\rangle$:

$\dot{n}_{1}|_{t=0}=\frac{1}{\tau}\displaystyle\int\limits_{0}^{\tau}\dot{n}_{1}\
\mathrm{d}t\approx\frac{n_{1}^{g}-n_{1}^{g^{\prime}}}{\tau},$ (41)

formally supposing that $n_{1}(t)$ has decreasing behaviour and
$n_{1}(\tau)\approx n_{1}^{g^{\prime}}$. Expression for $\dot{n}_{1}|_{t=0}$
follows from (40):

$\dot{n}_{1}|_{t=0}=\nu^{+}_{g^{\prime}}n_{0}^{g}\tilde{n}_{1}^{g}-\nu^{-}_{g^{\prime}}n_{1}^{g}\tilde{n}_{0}^{g}-\left(\nu^{+}_{g^{\prime}}n_{1}^{g}\tilde{n}_{2}^{g}-\nu^{-}_{g^{\prime}}n_{2}^{g}\tilde{n}_{1}^{g}\right).$
(42)

This expression can be simplified by representing acting field as
$g^{\prime}=g+\Delta g$ (where $\Delta g=g^{\prime}-g$) and taking into
account that $n_{1}^{g}$, $n_{2}^{g}$,
$n_{0}^{g}=n_{0}=\overline{n}_{0}=(1-U)/2$ are steady states corresponding to
field $g$:

$\dot{n}_{1}|_{t=0}=\nu\Delta
g\left[n_{0}-(n_{1}^{g}\tilde{n}_{2}^{g}+n_{2}^{g}\tilde{n}_{1}^{g})\right].$
(43)

From (40) and (43), we obtain characteristic time of transition from the state
$|g\rangle$ to the state $|g^{\prime}\rangle$:

$\tau_{g,g^{\prime}}=\nu^{-1}\frac{n_{1}^{g^{\prime}}-n_{1}^{g}}{g^{\prime}-g}\left[n_{0}-(n_{1}^{g}\tilde{n}_{2}^{g}+n_{2}^{g}\tilde{n}_{1}^{g})\right]^{-1}.$
(44)

Taking into account the fact that $n_{1}^{g}\approx n_{2}^{g}$ and Eq. (10) we
can write the rough estimation for transition rate
$\gamma_{g,g^{\prime}}\sim\tau_{g,g^{\prime}}^{-1}$ between two NESSs,

$\gamma_{g,g^{\prime}}\approx-\nu\left(\frac{n_{1}^{g^{\prime}}-n_{1}^{g}}{g^{\prime}-g}\right)^{-1}\left[\frac{n_{0}}{g}(1-2n_{1}^{g})\right].$
(45)

It directly follows from Eq. (45) that
$\gamma_{g,g^{\prime}}\neq\gamma_{g^{\prime},g}$:

$\frac{\gamma_{g,g^{\prime}}}{\gamma_{g^{\prime},g}}\approx\frac{g^{\prime}}{g}\frac{1-2n_{1}^{g}}{1-2n_{1}^{g^{\prime}}}.$
(46)

The ratio $\tfrac{\gamma_{g,g^{\prime}}}{\gamma_{g^{\prime},g}}>1$ for
transitions with $g^{\prime}>g$ as it follows from Fig. 12.

Figure 12:  Behavior of relaxation rate $\gamma(g^{\prime},g)$ as a function
of final state $\\{g\\}$ obtained numerically for several values of initial
state $g^{\prime}=0.6,0.7,0.8,0.9$. Corresponding analytical estimates from
Eq. (45) are given by gray lines and follow the same order of $g^{\prime}$ as
numerical ones. Empty circles correspond to $g\equiv g^{\prime}$ case.

The estimated values of $\gamma_{g,g^{\prime}}$ are in a agreement with one of
$\gamma^{\prime}_{g,g^{\prime}}$ corresponding to main asymptotic of

$n_{1}(t)\approx|n_{1}^{g}-n_{1}^{g^{\prime}}|\exp(-\gamma^{\prime}_{g,g^{\prime}}t)+bt^{\alpha}\exp(-\gamma^{\prime\prime}_{g,g^{\prime}}t)$
(47)

obtained from numerical solution of complete system of equations for
$n_{k}(t)$, cf. Fig. 10. As an example, for transitions
$|g=0.7\rangle\rightleftarrows|g^{\prime}=0.9\rangle$ at given $U=0.6$,
$\bar{n}=0.3$, the estimated values are $\gamma_{g,g^{\prime}}=1.286\nu$
(numerically $\gamma^{\prime}_{g,g^{\prime}}=1.389\nu$) and
$\gamma_{g^{\prime},g}=1.089\nu$ (numerically
$\gamma^{\prime}_{g^{\prime},g}=1.074\nu$).

However, $\gamma_{g,g^{\prime}}=\gamma_{g^{\prime},g}$ can be realized for
transitions between steady states with different directions of the field $g$,
i.e., $|g\rangle\leftrightarrows|-g\rangle$. This is intuitively clear as
changing of the field sign to opposite leads to the similar kink-form NESS
with gas concentration distribution $n_{k}^{-g}=n_{-k}^{g}$ obtained from
initial concentration $n_{k}^{g}$ by inversion with respect to impurity site
$k=0$. Note, that the states $|\pm g\rangle$ correspond to the states with
different sign of the winding number naturally appearing for ASEP on the ring,
see, e.g., [88, 89].

The situation is different for the transitions between steady states below
critical point ($g,g^{\prime}<g_{c}$). The dynamics of $n_{1}(t)$ during the
transition $|g\rangle\rightarrow|g^{\prime}\rangle$ is shown on Figs. 11(a)
and 11(b). The evolution of this transition in time has a damped oscillating
form. In particular, the main asymptotics for $n_{1}(t)$ behaves like
$\sim\exp(-\gamma^{\prime}_{g,g^{\prime}}+i\omega^{\prime}_{g,g^{\prime}})t+\mathrm{c.c.}$,
see Figs. 11. The behavior of $n_{\infty}(t)$, and $\overline{J}(t)$ has
similar oscillating character as well. This indicates that eigenvalues of the
system of the equations (at least for the linearized one) have complex values
in contrast to the case $g>g_{c}$, where eigenvalues are real. The nature of
these oscillations is in the generation of shock wave (solitary wave) on the
one of impurity edges, at non-adiabatic switching of the field from $g$ to
$g^{\prime}$, which many times runs around the ring. More precisely, the
system rebuilds from one steady state to another in portions by sequential
generation of solitary wave at the back side of impurity and by its traveling
around the ring. In the case of $g>g_{c}$, one solitary wave is enough to
rearrange from one kink state to another. In the latter case, the transmission
of a solitary dissipative wave through the dense phase is hindered due to
prohibited diffusion of particles in this localized region. This is another
interesting manifestation of protection phenomenon concerning the dynamics of
steady state formation. Sudden changes of the driving field, either in
subcritical or supercritical regime, always give rise to the creation of a
“quasi-particle”, namely, the generation and traveling dissipative soliton in
the wake region of impurity, see [61]. The steady state is established as a
result of solitary wave run around the system. While below critical point,
such a soliton can make several turns over the ring passing through the
impurity again and again, above the transition point we observe only one turn,
so that the transmission of the quasi-particle through impurity-nucleus
complex appears to be prohibited. This forbidden transport of dissipative
solitons can be interpreted physically by the absence of diffusion flow within
the flat density plateau of the stratum region: locally suppressed diffusion
(dissipative) processes makes the existence of dissipation assisted quasi-
particles impossible. Thus, the total diffusion flow over the ring vanishes as
well. The impurity becomes protected from the dissipative solitary
perturbations traveling over the ring. Similar effect of the forbidden
diffusion of quasi-particles was found in [12] under entanglement phase
transition induced by the non-Hermitian skin effect.

It is necessary to emphasize that magnitudes of damping rates
$\gamma_{g,g^{\prime}}$ of inter-NESS transitions before
($\\{g_{1},g_{1}^{\prime}\\}<g_{c}$) and after
($\\{g_{2},g_{2}^{\prime}\\}>g_{c}$) phase transition can differ significantly
so that $\gamma_{g_{1},g_{1}^{\prime}}\ll\gamma_{g_{2},g_{2}^{\prime}}$. In
the particular case
$\gamma^{\prime}_{(0.1),(0.3)}/\gamma^{\prime}_{(0.7),(0.9)}\propto 10^{-4}$.
This means strongly correlated behavior of the system with a domain structure
above the critical point $g_{c}$.

Note that we can consider fluctuating field $g$ as an additional “thermal
reservoir”. The fact that the transition rates between two steady states are
different, $\gamma_{g,g^{\prime}}\neq\gamma_{g^{\prime},g}$, could mean that
the heat $Q\propto\ln\gamma_{g,g^{\prime}}/\gamma_{g^{\prime},g}$ can be
transfered to system’s environment ($g$-field environment) during such
switching. This fact is related to the problem of the generation of the
housekeeping heat and the excess heat, see, e.g., [87, 90], and requires more
careful and detailed consideration that goes beyond our approximations.

##  VI Conclusions and discussion

In this work, we have focused on some features of the nonequilibrium
transition leading to the formation of nonlinear steady-state structures. The
corresponding NESS configuration results from the gas flow scattering on a
partially penetrable obstacle. The occurrence of nonlinear structures is
provoked by the blockade effect in a gas with short-range repulsion. It
manifests itself in the formation of a two-domain gas structure with a sharp
boundary of dense-phase (kink-form profile) ahead of the obstacle. This
transition takes place when the values of mean gas concentration, driving
field, and obstacle potential exceed certain critical values
$(\bar{n}_{c},g_{c},U_{c})$.

The main result of this work is that such transitions can be accompanied by
the emergence of local invariants (local first integrals) which indicate the
spatial system decomposition. The first invariant corresponds to the
conservation of the number of gas particles inside the obstacle. This
invariant is insensitive to a further increase or changes in the mean gas
concentration or in the driving field. As a result, within this overcritical
region, the obstacle state becomes robust against temporal fluctuations or
noise of the external driving field. At the same time, the rest of the system
remains dependent on the total particle number and the external field. The
necessary condition for such a decomposition is that the total current flowing
through the obstacle boundary should be equal to zero, despite the fact that
the local current fluctuates in time all over the system as well as inside the
obstacle. This means synchronization between currents or gas states at
different points of the obstacle boundary. Such a synchronized dynamics is
described by the second local invariant that can be viewed as manifestation of
the edges correlation effect. As a matter of fact, the rest of the system
serves as a reservoir for the obstacle that ensures the particle number
conservation in this region as main parameters driving the system change.

Another interesting effect is the spatial localization of the gas fluctuations
induced by the external drive noise in the overcritical regime. The domain
wall between the dense and depleted gas phases as if captures or accumulates
the fluctuations in the system, thereby protects the obstacle state against
them. The concentration of induced fluctuations near the domain wall is caused
by its instability under the action of the external noise. In some sense, this
is similar to the growth of a nucleus at the first order phase transition,
where the fluctuations of the nucleus surface are due to the active adsorbing
and desorbing processes. The formation of the dense gas phase near the
obstacle can be viewed as the nucleus growth adjusting to the obstacle as to
the nucleation center whose size is larger than the critical one ($U>U_{c}$).
One can consider this effect as the manifestation of the nonequilibrium
protection effect when the domain wall serves as an additional scatter and
protects the state of the dense gas phase and the obstacle state. Note that
the similar effect can take place not only for a single obstacle but also for
a finite-sized cluster of obstacles at least in the 1D case [75].

To demonstrate a possibility of these effects we used a simple deterministic
model of one-dimensional driven lattice gas on a ring in the mean-field
approximation. Our model is a variation of the well-known Lebowitz–Janowsky
blockage model describing ASEP on a ring with slow bond [50, 51, 52, 69].
Instead of a slow bond, we implemented the obstacle as a partially transparent
impurity site with finite capacity $(1-U)$ for gas particles. It is natural
that some of the obtained results correlate with those of earlier
considerations, e.g., Refs. [69, 50, 51, 52]. One of the important
deficiencies of our model is neglecting of fluctuations in a gas. This, in
turn, leads to neglecting of short-range correlations between gas particles,
see, e.g., [91, 42, 61], and a possibility of nonequilibrium transition to the
blockade regime caused by local gas fluctuations [64].

In this setting, we found the form of phase diagram, i.e., the critical
surface $(\bar{n}_{c},g_{c},U_{c})$, for this transition. In the overcritical
region, the two-domain gas structure is characterized by the onset of a kink-
like concentration profile with a dense phase near the impurity. The first
local invariant corresponds to the average occupation number of the impurity
site $n_{0}=(1-U)/2$ that is determined by the impurity capacity $1-U$ only,
while the shape of the kink is determined by the values of $\bar{n}$, $g$, and
$U$. In particular, $n_{0}$ does not changes at further increasing of the
external field $g$ or $\bar{n}$, despite the increasing of the current flowing
through the impurity. Formally, the value $n_{0}(1-U)^{-1}=1/2$ can serve as
an indicator (or index) for the emergence of a two-domain structure or the
kink density profile regardless of its shape. The saturation of $n_{0}$ to the
finite value $(1-U)/2$, which is less than maximum possible one, i.e.,
$n_{0}+U=1$, means that the full blockade regime in the system can never be
reached, except for the case of fully unpenetrable impurity $U=1$.

Another emerging local invariant determines the strongly correlated behavior
of the gas states at the opposite edges of the impurity and correspond at
least to the adiabatic invariance of the total occupation of impurity edges
(adjacent sites), $n_{-1}(t)+n_{1}(t)=1$. This leads to the equality of the
incoming and outgoing currents at the impurity ensuring the particle
conservation at the impurity site $n_{0}(t)=\text{const}$. Such a strong
synchronized dynamics of currents at its opposite edges can be viewed as
manifestation of the edge correlation effect. Note that this nonequilibrium
transition can be characterized as the transformation of the diffusive skin
effect [1, 17] into the effect similar to the edge correlation one.

To demonstrate that the dense gas phase protects the impurity complex from the
external noise, we considered a particular case of fluctuating external field
$g(t)$, Fig. 6. The external field noise $g(t)$ at $\left(g>g_{c}\right)$ was
shown to induce density fluctuations only outside the impurity complex with
their strong localization near the domain wall. The latter is caused by the
fluctuations of the wall position. In contrast, in the subcritical region
$\left(g<g_{c}\right)$, the fluctuations are distributed uniformly throughout
the system with maximum localization near the impurity, cf. Figs. 7 and 8. It
is also shown that the characteristic relaxation times of the system to the
nonequilibrium stationary state above the threshold are at least an order of
magnitude higher than those below the transition. Note that the relaxation
rates of the linearized dynamical system are complex below the threshold and
are real above it.

The rate of transitions between two different NESSs corresponding to two
different values of $g$ are different, except for the case where they are
distinguished by the direction of non-conservative driving field $g$, or by
the sign of their winding number, which is similar to non-Hermitian systems,
see, e.g., [1, 62, 63]. Relaxation from one NESS to another is governed by the
one-step or multi-step mechanism caused by the sequential generation of shock
waves at one impurity edge.

The nonequilibrium transition with the formation of nonlinear structures like
stratum, and effects like protection and edge synchronization ones can take
place in the case of gas flow scattering on a finite cluster of impurities.
Note that the gas flow scattering on an impurity cluster and the total drag
force acting on the cluster significantly depend on its structure factor. As
was shown for 2D case, scattering and the drag force can be enhanced by the
decomposition of a solid obstacle into fragments or a sparse cluster of
impurities. This enhancement is due to the collective scattering effect
implying the onset of the common wake around impurities, and it is more
efficient for disordered clusters as compared to the regular ones, see, e.g.,
[61]. At the same time, the mechanism of gas scattering determines the
features of induced nonequilibrium (flow- or wake-mediated) inter-impurity
interaction [40, 41, 42, 92] which usually belongs to the non-reciprocal type
[93, 43]. Crossover between linear and nonlinear regimes of dynamic screening
can drastically change this interaction. In particular, the formation of a
common coat of gas density perturbation accompanied by the blockade effect
around impurities can result in their dissipative binding [94, 95]. In the
case of finite clusters, the collective scattering effect can lead to the
shift of critical points of the nonequilibrium transition, for example, to the
reduction of the critical value of driving field $g_{c}$. The protection
effect can manifest itself in suppressing flow-induced correlations between
obstacles after the nonequilibrium transition to the domain gas structure, at
least in 1D [75]. The quasi-one-dimensional domain gas structures can serve as
an example of nonequilibrium atomic-wire structure, whose shape is controlled
by the external drive. This might be realized by the wind effect [96] for
adsorbed atoms on the anisotropic or stepped solid-state substrate. It would
be interesting to consider a possibility for analogous system decomposition,
boundary current synchronization, and protection effect in 2D and 3D, i.e.,
when the wake around a cluster of obstacles protects its internal state or
internal dynamics against external fluctuations or perturbations.

###### Acknowledgements.

We are grateful to B.I. Lev, I.S. Gandzha, A.V. Semenov, and Ch. Riemann for
helpful discussions, comments on the manuscript, and heartfelt support.
Authors express sincere gratitude to the Krzysztof Skubiszewski Foundation for
financial support as part of the programme “Urgent Aid for the Preservation of
the Research Capabilities of Ukraine”.

##  Appendix A  Continuum limit. Linear approximation of Burgers equation.

Here we consider continuum limit for gas concentration outside the impurity
introducing continuum coordinate $k\rightarrow x$ and setting
$n_{k}\rightarrow n(x)$, see Fig. 13.

Figure 13: Continuum representation for the ring, where $n(-0)=n_{-1}$ and
$n(-2L+0)=n_{1}$ correspond to site occupations for lattice ring respectively.

Quantity $n_{k+1}$ in this approximation is written as

$n_{k+1}\rightarrow n(x+l)=e^{l\partial_{x}}n(x),$ (48)

where $\partial_{x}\equiv\partial/\partial x$. In what follows we use
dimensionless coordinate $x/l\rightarrow x$. In this representation,
stationary equation outside impurity $J_{k-1,k}-J_{k,k+1}=0$ is approximated
(taking into account the first nonvanishing terms) by the stationary Burgers
equation

$\partial_{x}\left[-\partial_{x}n+2gn(1-n)\right]=0,$ (49)

or

$-\partial_{x}n+2gn(1-n)=J=\text{const}.$ (50)

For convenience, we change the coordinate system by taking $-2L<x<0$, see Fig.
13, and suppose that $n(-0)=n_{-1}$ and $n(-2L+0)=n_{1}$ correspond to the
occupations numbers on the left and the right nearest to the impurity site
ones. Equations for currents at two impurity bonds $(-1,0)$ and $(0,1)$ are no
change

$J_{-1,0}=\nu^{+}n_{-1}\left(1-n_{0}-U\right)-\nu^{-}n_{0}\left(1-n_{-1}\right)=J,$
(51)
$J_{0,1}=\nu^{+}n_{0}\left(1-n_{1}\right)-\nu^{-}n_{1}\left(1-n_{0}-U\right)=J.$
(52)

Equations (50), (51) and (52) are supplemented by one for the conservation of
the particle number in the ring or mean gas concentration $\bar{n}$

$\int\limits_{0}^{-2L}n(x)dx+n_{0}=(2L+1)\bar{n}.$ (53)

For the large ring $L_{0}=2L+1\gg 1$, we will neglect the term $n_{0}/(2L+1)$:

$\frac{1}{L_{0}}\int\limits_{0}^{-L_{0}}n(x)dx\approx\bar{n}.$ (54)

We are interested in the subcritical case when induced by scattering gas
density perturbation weakly deviates from its equilibrium value
$n^{e}\approx\bar{n}$. Representing gas concentration $n(x)=\bar{n}+\delta
n(x)$ outside the inclusion and occupancy of impurity site
$n_{0}=\bar{n}+\delta n_{0}$ as small perturbation of mean gas concentration
$\bar{n}$ one can obtain linearized form of equations (50), (51), and (52):

$-\partial_{x}\delta n+2g\bar{n}(1-\bar{n})\delta
n=J-2g\bar{n}(1-\bar{n})=\delta J,$ (55a) $\displaystyle\delta
n_{-1}(\nu^{+}(1-U-\bar{n})+\nu^{-}\bar{n})-\delta
n_{0}\bar{n}+\nu^{-}(\nu^{+}(1-\bar{n}))=\delta J+\nu^{+}U\bar{n},$ (55b)
$\displaystyle\delta n_{0}(\nu^{+}(1-\bar{n})+\nu^{-}\bar{n})-\delta
n_{1}(\nu^{+}\bar{n}+\nu^{-}(1-U-\bar{n}))=\delta J-\nu^{-}U\bar{n}.$ (55c)

Solution of Eq. (50) can be written as

$\delta n(x)=\frac{\delta J}{\lambda}+Ce^{\lambda x},$ (56)

where $\lambda=2g\bar{n}(1-\bar{n})$, and unknown constants $\delta J$ and $C$
can be expressed through $\delta n_{-1}$ and $\delta n_{1}$.

$\delta n_{1}=\frac{\delta J}{\lambda}+Ce^{-\lambda L_{0}},$ (57) $\delta
n_{-1}=\frac{\delta J}{\lambda}+C.$ (58)

As a result, the total system of equations to obtain unknown constants $\delta
n_{-1}$, $\delta n_{0}$, $\delta n_{1}$, and $\delta J$ takes the form

$\delta n_{1}\left(1-(\lambda L_{0})^{-1}+(\lambda
L_{0})^{-1}\mathrm{e}^{-\lambda L_{0}}\right)\approx\delta
n_{-1}\left(-(\lambda L_{0})^{-1}+\left(1+(\lambda
L_{0})^{-1}\right)\mathrm{e}^{-\lambda L_{0}}\right),$ (59) $\delta
n_{-1}\left(\nu^{+}(1-\bar{n}+U)+\nu^{-}\bar{n}\right)-\delta
n_{0}\left(\nu^{+}\bar{n}+\nu^{-}(1-\bar{n})\right)=\delta J+\nu^{+}\bar{n}U,$
(60) $\delta n_{0}\left(\nu^{+}(1-\bar{n})+\nu^{-}\bar{n}\right)-\delta
n_{1}\left(\nu^{+}\bar{n}+\nu^{-}(1-\bar{n}-U)\right)=\delta
J-\nu^{-}\bar{n}U,$ (61) $\delta J=\lambda\frac{\delta
n_{1}-\textrm{e}^{-\lambda L_{0}}\delta n_{-1}}{1-\textrm{e}^{-\lambda
L_{0}}}.$ (62)

Neglecting terms of the order of $(\lambda L_{0})^{-1}$, we get simple
expressions for the concentration distribution $\delta n(x)$, the current
deviation $\delta J$, and site occupation numbers $\delta n_{\\{-1,0,1\\}}$

$\delta n(x)=\delta n_{-1}e^{\lambda x},$ (63) $\delta n_{1}\approx\delta
J\approx 0,$ (64) $\delta n_{0}\approx-\frac{(1-g)U\bar{n}}{1+(1-2\bar{n})g},$
(65) $\delta
n_{-1}\approx\frac{4}{1-U+(1-U-2\bar{n})g}\cdot\frac{(1-\bar{n})gU\bar{n}}{1+(1-2\bar{n})g}.$
(66)

In the approximation that takes into account terms of the order of $(\lambda
L_{0})^{-1}$, we get that $\delta n_{1}\approx-(\lambda L_{0})^{-1}\delta
n_{-1}$, $\delta J\approx\lambda\delta n_{1}$, and $\delta n(x)=\delta
n_{1}+(\delta n_{-1}-\delta n_{1})e^{\lambda x}$.

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