Achieve the next level of sophistication in this two-piece set gown. The beaded crop top features a bateau neckline with deep scoop back. More beads around the waist with the mermaid skirt run down the full-length finish.


THIS EASTER TREAT YOURSELF WITH GORGEOUS LIPS THAT WONT SMUDGE OR WEAR OFF!
PAY ONLY $600 - NORMALLY $800!
Purchase treatment during February and you have until August 2019 to get it done!
Have a wrinkle free Christmas this year  with 10% OFF Anti-Wrinkle Injections until the 19th December!
Offer available until 19th December 2018 with our clinical nurse only.
dont. miss. this - party season sales!
OUR MOST POPULAR SERVICES ARE GOING TO GET YOU PARTY READY!
PRICES AVAILABLE TO THE 15TH DECEMBER!
ALL BOOKINGS REQUIRE A NON REFUNDABLE DEPOSIT!
YOU DONT WANT TO MISS THIS Triple Treat Facial offer!! SAVE $68!
An Intense Power Facial is customised to your skins needs to improve your over all skin texture and appearance!
Book your triple treat facial today!
The Triple Treat... "Power of 3" is an Intense Power Facial designed to tighten, refine, smooth and clear!
$169.00 for a limited time!


Edythe Hatter-Williams, CEO, CAMW! Capital Area Michigan Works! stops by the Michigan Business Network to announce that coming in April 2019, Lansing will welcome MiCareerQuest to the capital region for the first time ever.
● MiCareerQuest Capital Area and is an employer-driven, student focused, hands-on career exploration event happening on April 26, 2019, at the Lansing Center.
● This model has been successful for four years in Grand Rapids, Kalamazoo and starting this year in Southeast Michigan.
● During this unique experience, thousands of local middle and high school students will find out what it's like to be a nuclear machinist, mechanical engineer, agri-scientist, network technician and more by interacting with the professionals who do these jobs every day here in the Capital Area.
● One of the features that sets MiCareerQuest apart from other job or career fairs is its focus on new economy industries as opposed to individual employers and/or organizations.
● For this event to be successful, T3: Teach. Talent. Thrive. needs of help from local businesses.
● Support from the region's employers ensures local students will have hands-on interaction with real-world job skills and the ability to talk to the engineers, researchers, designers and more who fill those jobs. It will help students see the wide world of career possibilities in our region.
● T3 is seeking sponsors for all four quadrants within the event, representing Engineering, Manufacturing & Industrial Technology; Health Sciences & Human Services; Business, Information Technology, Arts & Communication; and Natural Resources & Agriscience.
● There are four sponsorship levels per quadrant. Those interested in learning more about MiCareerQuest Capital Area or sponsorship of the event, visit micareerquestca.org and find the Eventbrite link under sponsorship.
Click below to listen to full interview!


With mobile printing it's simple to connect smartphones and tablets to the i-SENSYS MF8280Cw, from virtually anywhere in the office. Print emails, web pages and other documents directly from your iPhone or iPad using the support of Apple AirPrint, which automatically detects the printer on your network. For the freedom of printing from wherever you are, via the Internet, using smartphone, tablet or desktop, the i-SENSYS MF8280Cw supports Google cloud print.
Easy-to-use, with intuitive navigation and a clear color display, the network and Wi-Fi ready i-SENSYS MF8280Cw makes it simple to share color printing, copying, scanning and faxing across workgroups. For more productive working, the i-SENSYS MF8280Cw prints and copies at 14 pages per minute, for both color and mono documents and has a 50-sheet automatic document feeder. There's also little time lost waiting for output thanks to Canon's Quick First-Print technologies and a fast first copy out time. Advanced image quality, with up to 1200 x 1200 dpi resolution, ensures bright, vibrant output every time.
One-touch solution keys and intuitive menus make it straightforward for anyone to operate and maintain. Productivity is boosted by easy scanning and eco-copying, which combines several pages into a single print-out saving paper and money. A backlit, 7-line color screen clearly displays all the essential information users need. The all-in-one cartridge simplifies maintenance and increases reliability.
Conveniently share documents by scanning across a network connection or straight to USB memory key. Save costs without compromising on performance, while cutting your carbon footprint, using Canon's On-Demand fixing technology. ENERGY STAR compliant, the i-SENSYS MF8280Cw is among the most energy-efficient multifunctional in its class with a market-leading low energy consumption of just 1 W in sleep mode.
Increase team productivity with this compact, stylish multifunctional, which supports mobile printing and fits neatly on any office desktop. Easy-to-use, with intuitive navigation and a clear color display, the network and Wi-Fi ready i-SENSYS MF8280Cw makes it simple to share color printing, copying, scanning and faxing across workgroups.
For more productive working, the i-SENSYS MF8280Cw prints and copies at 14 pages per minute, for both color and mono documents and has a 50-sheet automatic document feeder. There's also little time lost waiting for output thanks to Canon's Quick First-Print technologies and a fast first copy out time. Advanced image quality, with up to 1200 x 1200 dpi resolution, ensures bright, vibrant output every time.
One-touch solution keys and intuitive menus, make it straightforward for anyone to operate and maintain. Productivity is boosted by easy scanning and eco-copying, which combines several pages into a single print-out saving paper and money. A backlit, 7-line color screen clearly displays all the essential information users need. The all-in-one cartridge simplifies maintenance and increases reliability.
Save costs without compromising on performance, while cutting your carbon footprint, using Canon's On-Demand fixing technology. ENERGY STAR compliant, the i-SENSYS MF8280Cw is among the most energy-efficient multifunctional in its class with a market-leading low energy consumption of just 1 W in sleep mode.


The iPhone itself is a marvellous piece of modern convenience, but it wouldn't work its wonders if it's broken. Unfortunately, people aren't very good at keeping their iPhones intact, making services like iPhone screen repair or replacement a must. In an article posted at the PC Advisor website, it was highlighted in a report that a person could damage his or her iPhone within 10 weeks, from minor nicks to full-blown cracked screens. In fact, it was also reported that users are more likely to damage newer iPhone models compared to old ones. Most people can place their phones at risk of damage in a variety of settings. A survey by a U.S.-based warranty provider indicated that people accidentally ruin their iPhones most in the kitchen, followed a close second by in the living room and the bathroom. All in all, about 51 percent of iPhone accidents happen indoors rather than outdoors, which almost entirely puts the blame on the owners' carelessness.


If you need parts to correct car trouble near Evesham Township, the auto parts experts of Auto Parts Connection would like to help get you back on the road quickly and safely.
We are a preferred choice for auto parts services in Evesham Township, NJ. Some of the services that we provide the families of Evesham Township are Batteries, Counter Services, Custom Paint Services, Hoses, Machine Shop, Paint Brands, Paint Supplies & Services and Refurbishing Supplies. Take a moment to browse our website, and get to know us at Auto Parts Connection.
If you don't know what parts your car, truck, or van needs, or need further information, call our trained staff at 609-261-5070.


text:
Embraer earning reasults 2nd quarter 2020
Vicky Karantzavelou / 06 Aug 2020 09:24 1592
2Q20 adjusted net loss (excluding special items and deferred income tax and social contribution) was US$ (198.8) million, with Adjusted loss per ADS of US$ (1.08).
Embraer delivered four commercial jets and 13 executive jets (nine light / four large) in 2Q20, and the company's firm order backlog at the end of 2Q20 was US$ 15.4 billion.
Excluding special items, adjusted EBIT and EBITDA were US$ (140.5) million and US$ (120.4) million, respectively, negatively impacted by weak Commercial Aviation results, yielding adjusted EBIT margin of -26.2%% and adjusted EBITDA margin of -22.4%.
The 2Q20 results include total net negative non-cash special items of US$ 202 million: 1) additional negative provisions for expected credit losses during the Covid-19 pandemic of US$ 16.1 million, 2) an impairment loss on the Commercial Aviation business unit of US$ 91.1 million, 3) recognition of previous period depreciation and amortization expense in the Commercial Aviation business of US$ 101.2 million, and 4) a positive valuation mark to market of US$ 6.5 million on the company's stake in Republic Airways shares.
Embraer reported Free cash flow of US$ (476.2) million in 2Q20, still affected by working capital needs in Commercial Aviation, but an improvement versus the US$ (676.5) million usage of free cash flow reported in 1Q20, due to several initiatives to reduce investments, control working capital, and minimize fixed costs.
Embraer's liquidity remains solid as the company finished the quarter with total cash of US$ 2.0 billion and major debt maturities starting in 2022 onwards. The company also finalized the terms of contracts for working capital and export financing with export credit agencies in Brazil and the United States and private and public banks, adding a total of up to US$ 700 million to its total liquidity. Disbursements of these new financing lines are expected to be completed in the third quarter of 2020 reinforcing Embraer's cash position in the second half of 2020 and into 2021.
Due to continued uncertainty related to the COVID-19 pandemic, financial and deliveries guidance for the Company's 2020 results remains suspended at this point.
Vicky Karantzavelou
Co-Founder & Chief Editor
Vicky is the co-founder of TravelDailyNews Media Network where she is the Editor-in Chief. She is also responsible for the daily operation and the financial policy. She holds a Bachelor's degree in Tourism Business Administration from the Technical University of Athens and a Master in Business Administration (MBA) from the University of Wales. She has many years of both academic and industrial experience within the travel industry. She has written/edited numerous articles in various tourism magazines.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
\label{s:int}
\par\noindent
The efficiency of transport of active matter in microscopic systems is an
issue of paramount importance in a number of fields of science including
biology, chemistry, and logistics.
Looking particularly at drug--delivery design scenarios  \cite{Kosmidis},
ion moving in molecular cytosol \cite{ABC14,ABCM14,ABCM11},
percolation of aggressive acids through reactive porous media \cite{Desirable},
the traffic of pedestrians in regions with drastically reduced visibility
(e.g., in the dark or in the smoke) \cite{CM12,CM13,MCKB} (see also the
problem of traffic of cars on single--lane highways \cite{Speer}),
we see that the efficiency of a medical treatment,
the properties of ionic currents thorough cellular membranes,
the durability of a highly permeable material,
or the success of the evacuation of a crowd of humans, strongly depends on
the time spent by the individual particle (colloid, ion, acid molecule, or
human being) in the constraining  geometry (body, molecule, fabric,
or corridor).

In this framework, we focus our attention on the study of the simplest
2D scenario that mimics alike dynamics.
The {\em Gedankenexperiment} we make is the following: we
imagine a vertical strip whose top and bottom entrances are
in touch with infinite particle reservoirs at constant
densities. Assume particles are driven downwards
by the boundary densities difference and/or an
external constant and uniform field (electrical, gravitational,
generally--accepted crowd opinion, ...).
Let the \emph{residence time} be the typical time a particle
entering the strip at stationarity from the
top boundary needs to exit through the bottom one.
In this framework, under the assumption that
particles in the strip interact only via hard--core exclusion,
we study the {\em ballistic--like} versus the {\em diffusion--like}
dependence of the residence time on the
external driving force (main source of anisotropy in the system),
on the length of the strip,
on the horizontal diffusion, and, finally, on the
choice of the boundary condition at the bottom.

We recover the structure of the fluxes as well as the residence times
proven mathematically by Derrida and co--authors in \cite{DEHP} for the
asymmetric simple exclusion model on the line; see also \cite{Gorissen}
for a more recent approach.
In chemistry single file diffusion has been demonstrated for zeolite
catalysts \cite{HK} to dramatically reduce the
rate of a reaction. This happens in particular when zeolitic microporous
systems are used with linear micropores with dimensions that are similar as
the size of the molecules that are converted. Since they cannot pass single
file inhibition occurs (see, for instance, \cite{GGS}).

Additionally, we  discover new effects
that are purely due to the choice of the 2D geometry and
which are therefore
absent in a 1D lattice. The most prominent, within the precision
of our numerical simulations, is the non--monotonic
behavior in changes in the horizontal displacement
probability in the bouncing back regime reported in
Section~\ref{s:caso020}.
Under certain conditions, particles start accumulating near the bottom exit of the strip. This crowding leads to a bouncing-back effect in which particles trying to escape are reflected in the bulk.
We observe that, in such a case,
increasing the frequency of horizontal movements
help particles to overcome obstacles and to find their way to the
exit.

To investigate this model, we employ several working techniques
including Taylor series truncations for the derivation of the mean--field
equations, ODE analysis of the stationary case, estimates involving the
structure of the stationary measure for birth and death processes on a line,
as well as Monte Carlo simulations to exploit the resources offered by the
various parameter regimes. In 1D, this model has been
widely studied both by the mathematical and physical
communities, see e.g. \cite{Rez,BM,DEHP,Lebowitz,Bertini,Gorissen}.
In 2D, the situation is very much unexplored
especially in the asymmetric case. We deal with this precise
problem and we give a rather complete description of the
phenomenon. Our results, which are based on a thorough study of two
simplified models and extensive Monte Carlo simulations, open
new mathematical problems concerning the typical time a
particle need to cross a region in hard--core repulsion regime.

More precisely,
in the paper we develop two analytic approaches to predict
the mean residence time: a macroscopic Mean Field theory and a
semi--microscopic approach in which the particle motion is
imagined as that of a single particle against a prescribed
background density profile borrowed from the macroscopic Mean Field theory.
These two different predictions are very similar to each other
but for the regime described above in which the non--monotonic effect
is found. In this regime the horizontal displacement
probability tunes the system behavior from the macroscopic prediction
to the semi--microscopic one, with the two limits recovered, respectively,
in the zero and one horizontal probability displacement cases.

The importance of the exclusion rule on the time dependence of
the typical distance covered by a particle is not new in the
scientific literature.
Due to the exclusion rule,
the asymmetric process on a square lattice that we discuss here
can be considered to occur on a percolating lattice.
The symmetric exclusion case  has been widely explored for diffusion,
as for instance the ``ant in the labirinth'' by de Gennes \cite{AS}.
The distance travelled by the ant is proportional to the square root of the
time (random walk diffusion) as long as the site occupancy is low,
but, when the critical $0.5928$ site occupation is approached,
this changes to a time dependence close to cubic root of time.
Beyond this critical site occupancy the order in time rapidly drops.
The excluded volume problem in several dimensions has not yet found however
a satisfactory solution \cite{vK}.

A related but similar phenomenon occurs in the symmetric 1D
case. When site occupation increases the distance time relation becomes
the fourth root of time \cite{fedders}.
The asymmetric problem in 2D, that we are interested in,
can be more easily addressed as asymmetric exclusion with driven diffusion,
see \cite{Krug} for a paper and \cite{BE} for a complete review.
In the 1D
total asymmwtric case (particles can move only from the top
to the bottom),
using a kinematic wave theory related approach and the method of maximum
transported current \cite{BE}, it is identified a
three parameter space region as a function of the rates at which
particles would enter (from the top) or exit (from the bottom) the strip.
These regions closely relate to the particle density where percolation
sets in (on a square lattice with stochastic bond formation the percolation
threshold is 0.5).
In this paper we will be mainly concerned with the case where the relative
probability for a particle to enter the strip (from the top)
is one.
In this case the two situations than can be distinguished are the high
density phase (exit probability smaller than $1/2$) and the
maximum current phase (exit probability larger than $1/2$).
Interestingly the difference relates to an occupation density that
is larger or less than the critical point of  percolation.

In the lower density phase the average density is $1/2$
and current is maximum, at the density higher than the
percolation threshold it decreases with the exit rate.
\cite{BE}.
We derive an equation for the Mean Field concentration profile
which, in our setup, depends only on the vertical spatial
coordinate. The profile depends on the reservoir boundary
conditions and on the vertical drift, namely,
the ratio between down and up jump probabilities divided by their sum,
that, we recall, we do not considered necessarily equal
to one (as in the total asymmetric case discussed above).
In the large drift case, we recover density profiles consistent with
the phase diagram described above which has been proven
to be correct in the total asymmetric case \cite{DEHP}.

In this scenario, as already mentioned above, we setup two different
analytic computations of the residence time: a
semi--microscopic approach based on the analogy with a 1D
birth and death model with not homogeneous background and a purely
macroscopic Mean Field approximation. In the last approach the
residence time is finally computed in terms of the stationary
current. It is important to stress that, for zero drift, these
two different approach seem to explain the behavior of the model
in two different regimes, namely, low and high horizontal
displacement probability. From our point of view this result is
of absolutely high interest, indeed, the main issue we arise in this
paper is that of understanding the influence of lateral
displacements on the typical time
needed by a particle to cross a pipe.
Future investigation will be needed to understand deviations
from the average particle behavior. In this perspective the
residence time seems to be a very useful observable and its
distribution will be the key object of our future study.

The paper is structured in the following fashion:
we describe the dynamics of our stochastic lattice model in
Section~\ref{s:modello}.
Section~\ref{s:mf} is concerned primarily with the derivation
of the mean field equations governing the macroscopic evolution of the
density.
In the same section we study the stationary mean field
behavior, the thermalization time of the system (i.e. the time that the system takes to reach the steady state) as well as the numerical
testing of the accuracy of the mean field prediction of the stationary
density profile.
In Section~\ref{s:bd} we make the direct analogy between our
scenario and a biased birth and death model for which we can calculate
explicitly the unique invariant measure and the use this object to obtain
analytical lower and upper bounds on the residence time for three distinct
physically relevant scenarios, viz. (i) a homogeneous case, (ii) a linear
case, (iii) a totally asymmetric case.
The core of the paper is
Section~\ref{s:res} and
Section~\ref{s:res-num}.
Herein we use inspiration from the handling of the biased birth and death
model to get approximate analytical bounds on the residence
time for our problem for selected parameter regimes.
We also setup a purely Mean Field macroscopic
computation yielding a prediction of the residence time
which is fully discussed in the last two sections
Finally, we explore
numerically the residence time for all parameter regimes and compare the
results with the derived analytical bounds.

\begin{figure}[h]
\begin{picture}(400,330)(-30,0)
\thinlines
\multiput(0,0)(20,0){8}{\line(0,1){300}}
\multiput(0,0)(0,20){16}{\line(1,0){140}}
\put(60,-45){\vector(-1,0){60}}
\put(80,-45){\vector(1,0){60}}
\put(64,-49){${L_1}$}
\put(160,140){\vector(0,-1){140}}
\put(160,160){\vector(0,1){140}}
\put(154,145){${L_2}$}
\put(0,40){\circle*{5}}
\put(0,120){\circle*{5}}
\put(0,200){\circle*{5}}
\put(20,0){\circle*{5}}
\put(20,100){\circle*{5}}
\put(20,180){\circle*{5}}
\put(20,300){\circle*{5}}
\put(40,80){\circle*{5}}
\put(40,280){\circle*{5}}
\put(60,200){\circle*{5}}
\put(60,220){\circle*{5}}
\put(80,20){\circle*{5}}
\put(80,140){\circle*{5}}
\put(80,220){\circle*{5}}
\put(80,260){\circle*{5}}
\put(100,20){\circle*{5}}
\put(100,60){\circle*{5}}
\put(100,80){\circle*{5}}
\put(100,160){\circle*{5}}
\put(100,240){\circle*{5}}
\put(120,40){\circle*{5}}
\put(120,80){\circle*{5}}
\put(120,140){\circle*{5}}
\put(120,240){\circle*{5}}
\put(140,60){\circle*{5}}
\put(140,140){\circle*{5}}
\put(140,180){\circle*{5}}
\thinlines
\put(20,305){\vector(0,1){20}}
\put(25,310){${1-\varrho_\textrm{u}}$}
\put(120,325){\vector(0,-1){20}}
\put(125,310){${\varrho_\textrm{u}}$}
\put(120,330){\circle*{5}}
\put(20,-5){\vector(0,-1){20}}
\put(25,-15){${1-\varrho_\textrm{d}}$}
\put(120,-25){\vector(0,1){20}}
\put(125,-15){${\varrho_\textrm{d}}$}
\put(120,-30){\circle*{5}}
\qbezier(60,220)(70,240)(80,220)
\put(65.5,226.5){${\times}$}
\qbezier(40,220)(50,240)(60,220)
\put(53,229.5){\vector(-1,0){5}}
\qbezier(100,240)(110,260)(120,240)
\put(105.5,246.5){${\times}$}

\put(330,120){\line(0,1){60}}
\put(300,150){\line(1,0){60}}
\put(330,150){\circle*{5}}
\put(330,150){\vector(0,1){20}}
\put(335,185){$(1-h)(1-\delta)/2$}
\put(330,150){\vector(0,-1){20}}
\put(335,110){$(1-h)(1+\delta)/2$}
\put(330,150){\vector(1,0){20}}
\put(370,146){$h/2$}
\put(330,150){\vector(-1,0){20}}
\put(273,146){$h/2$}
\end{picture}
\vskip 2. cm
\caption{Schematic representation of the model: the lattice on the
left and the jumping probabilites on the right.}
\label{f:reticolo}
\end{figure}

\section{The model}
\label{s:modello}
\par\noindent
Let $L_1,L_2$ be two positive integers. Let $\Lambda\subset\bb{Z}^2$
denote the \emph{strip} $\{1,\dots,L_1\}\times\{1,\dots,L_2\}$.
We say that the coordinate directions $1$ and $2$ of the
strip are respectively the \definisco{horizontal} and the \definisco{vertical}
direction. We shall accordingly use the words \definisco{top},
\definisco{bottom}, \definisco{left}, and \definisco{right}.
On $\Lambda$ we define a discrete time stochastic process controlled by the
parameters
\begin{displaymath}
\varrho_\rr{u},\varrho_\rr{d}\in[0,1]
\;\;\;\textrm{ and }\;\;\;
h,u,d\in[0,1]\;\;\textrm{ such that }h+u+d=1
\end{displaymath}
whose meaning will be explained below.

The \emph{configuration} of the system at time $t\in\bb{Z}_+$
is given by the positive integer $n(t)$ denoting
the number of particles in the system at time $t$ and by the two
collections of integers
$x_1(1,t),\dots,x_1(n(t),t)\in\{1,\dots,L_1\}$ and
$x_2(1,t),\dots,x_2(n(t),t)\in\{1,\dots,L_2\}$ denoting, respectively,
the horizontal and the vertical coordinates of the $n(t)$ particles
in the strip $\Lambda$ at time $t$.
The $i$--th particle, with $i=1,\dots,n(t)$,
is then associated with the site $(x_1(i,t),x_2(i,t))\in\Lambda$ which
is called \definisco{position} of the particle at time $t$.
A site associated with a particle a time $t$ will be said to be
\definisco{occupied} at time $t$, otherwise we shall say that it is
\definisco{free} or \definisco{empty} at time $t$.
We let $n(0)=0$.

At each time $t\ge1$ we first set
$n(t)=n(t-1)$ and then
repeat the following algorithm $n(t-1)$ times.
One of the three actions
\emph{insert a particle through the top boundary},
\emph{insert a particle through the bottom boundary},
and
\emph{move a particle in the bulk} is performed with
the corresponding probabilities
$\varrho_\rr{u}L_1/
(\varrho_\rr{u}L_1+\varrho_\rr{d}L_1+n(t))$,
$\varrho_\rr{d}L_1/
(\varrho_\rr{u}L_1+\varrho_\rr{d}L_1+n(t))$,
and
$n(t)/
(\varrho_\rr{u}L_1+\varrho_\rr{d}L_1+n(t))$.

\emph{Insert a particle through the top boundary.\/}
Chose at random with uniform probability the integer $i\in\{1,\dots,L_1\}$
and, if the site $(1,i)$ is empty, with probability $d$
set $n(t)=n(t)+1$ and add a particle to site $(1,i)$.

\emph{Insert a particle through the bottom boundary.\/}
Chose at random with uniform probability the integer $i\in\{1,\dots,L_1\}$
and, if the site $(L_2,i)$ is empty, with probability $u$
set $n(t)=n(t)+1$ and add a particle to site $(L_2,i)$.

\emph{Move a particle in the bulk.\/}
Chose at random with uniform probability one of the $n(t)$ particles
in the bulk.
The chosen particle is moved according to the following rule:
one of the four
neighboring sites of the one occupied by the particle is chosen
at random with probability $h/2$ (left), $u$ (up), $h/2$ (right), and
$d$ (down). If the chosen site is in the strip (not on
the boundary) and it is free, the particle is moved there
leaving empty the site occupied at time $t$.
If the chosen site is on the boundary of the strip the dynamics is
defined as follows:
the left boundary $\{(0,z_2),\;z_2=1,\dots,L_2\}$ and
the right boundary $\{(L_1+1,z_2),\;z_2=1,\dots,L_2\}$
are \definisco{reflecting} (homogeneous Neumann boundary conditions)
in the sense that a particle trying to jump there is not moved.
The bottom and the top
boundary conditions are stochastic in the sense
that when a particle tries to jump to a site $(z_1,0)$, with
$z_1=1,\dots,L_1$, such a site has to be considered occupied
with probability $\varrho_\rr{u}$ and free with probability
$1-\varrho_\rr{u}$,
whereas
when a particle tries to jump to a site $(z_1,L_2+1)$, with
$z_1=1,\dots,L_1$, such a site has to be considered occupied
with probability $\varrho_\rr{d}$ and free with probability
$1-\varrho_\rr{d}$.
If the arrival site is considered free
the particle trying to jump there is removed
by the strip $\Lambda$ (it is said to \definisco{exit} the system)
and the number of particles is reduced by one, namely, $n(t)=n(t)-1$.
If the arrival site is occupied the particle is not moved.

We gave the definition of the model in an algorithmic way, but
note that the model is a Markov chain $\omega_0,\omega_1,\dots,\omega_t,\dots$
on the
\emph{state} or \emph{configuration space}
$\Omega:=\{0,1\}^\Lambda$ with transition probability
that can be deduced by the algorithmic definition.

\section{Mean Field estimates}
\label{s:mf}
\par\noindent
Let the \emph{occupation number} of the site
$(z_1,z_2)\in\Lambda$ at time $t$
be equal to $1$ if such a site is occupied by a particle at time $t$
and to $0$ otherwise.
Let, also,
$m_t(z_1,z_2)$ be
the expectation of the occupation number at site $(z_1,z_2)\in\Lambda$
at time $t\ge0$. This quantity is  well defined in a stochastic context. But if one wants a
more intuitive idea of what such a quantity is, one can think of
running the dynamics many times independently and then
computing equal time averages with respect to
these different realizations of the process.
The mean over those independent realizations of the occupation
number at site $(z_1,z_2)$ at time $t$ will be
$m_t(z_1,z_2)$.

Now, we set up a Mean Field
computation for such a mean occupation time $m_t$. We shall follow
\cite{SLH}, but it is worth noting that,
in 1D and on the infinite volume $\bb{Z}$,
the equation we shall obtain
is proven rigorously to be the macroscopic limit
of the discrete space random process \cite{dMPS}.

We need to reproduce and slightly generalize the
approach in \cite{SLH} since there
a particular choice for the horizontal probability has been
considered, whereas we need a more general result.
Let the \emph{drift} be
\begin{equation}
\label{againmf00-1}
\delta=\frac{d-u}{d+u}
\;\;\;\;.
\end{equation}
This is indeed the physically meaningful definition of drift, since it
is the ratio between the difference of the probabilities to move
down and up and the total probability to perform a vertical displacement.
Note that, since $d+u=1-h$, a simple computation yields
\begin{equation}
\label{againmf00-2}
d=\frac{1-h}{2}(1+\delta)
\;\;\;\textrm{ and }\;\;\;
u=\frac{1-h}{2}(1-\delta)
\;\;\;\;.
\end{equation}

First of all, note that
in our dynamics
the probability that at time $t$ a specific particle is updated
is of order one,
since at each time we update at random
$n(t-1)$ particles,
where, we
recall, $n(t)$ is the number of particles in the system at time $t$.
Thus,
the Mean Field approximation consists in
letting
\begin{equation}
\label{newmf000}
\varepsilon=\frac{1}{L_2+1}
\;\;\;\textrm{ and }\;\;\;
\tau=\varepsilon^2
\;\;,
\end{equation}
considering
the macroscopic variables
\begin{equation}
\label{newmf010}
z'_1=\varepsilon z_1,\;\;
z'_2=\varepsilon z_2,\;\;
\;\;\;\textrm{ and }\;\;\;
t'=\tau t = \varepsilon^2 t
\;\;,
\end{equation}
and
writing, for an
arbitrary point in the bulk, the following
balance equation:
\begin{displaymath}
\begin{array}{l}
m_{t'+\tau}(z'_1,z'_2)-m_{t'}(z'_1,z'_2)
\\
\phantom{mm}
=(h/2) m_{t'}(z'_1-\varepsilon,z'_2)[1-m_{t'}(z'_1,z'_2)]
 +d m_{t'}(z'_1,z'_2-\varepsilon)[1-m_{t'}(z'_1,z'_2)]
\\
\phantom{mm=}
 +(h/2) m_{t'}(z'_1+\varepsilon,z'_2)[1-m_{t'}(z'_1,z'_2)]
 +u m_{t'}(z'_1,z'_2+\varepsilon)[1-m_{t'}(z'_1,z'_2)]
\\
\phantom{mm=}
 -(h/2) m_{t'}(z'_1,z'_2)[1-m_{t'}(z'_1+\varepsilon,z'_2)]
 -d m_{t'}(z'_1,z'_2)[1-m_{t'}(z'_1,z'_2+\varepsilon)]
\\
\phantom{mm=}
 -(h/2) m_{t'}(z'_1,z'_2)[1-m_{t'}(z'_1-\varepsilon,z'_2)]
 -u m_{t'}(z'_1,z'_2)[1-m_{t'}(z'_1,z'_2-\varepsilon)]
\phantom{mm=}
\\
\phantom{mm}
=(h/2)[m_{t'}(z'_1+\varepsilon,z'_2)-2m_{t'}(z'_1,z'_2)
 +m_{t'}(z'_1-\varepsilon,z'_2)]
\\
\phantom{mm=}
 +((1-h)/2)[m_{t'}(z'_1,z'_2+\varepsilon)
  -2m_{t'}(z'_1,z'_2)+m_{t'}(z'_1,z'_2-\varepsilon)]
\\
\phantom{mm=}
 -\delta((1-h)/2)\{
    [1-m_{t'}(z'_1,z'_2)][m_{t'}(z'_1,z'_2+\varepsilon)
                          -m_{t'}(z'_1,z'_2-\varepsilon)]
\\
\phantom{mm=-\delta((1-h)/2)\{}
    +m_{t'}(z'_1,z'_2)[(1-m_{t'}(z'_1,z'_2+\varepsilon))
                       -(1-m_{t'}(z'_1,z'_2-\varepsilon))]
                  \}
\;\;.
\end{array}
\end{displaymath}
Now, we multiply and divide suitably the space and time units
$\varepsilon$ and $\tau$ to obtain
\begin{displaymath}
\begin{array}{l}
[m_{t'+\tau}(z'_1,z'_2)-m_{t'}(z'_1,z'_2)]/\tau
\\
\phantom{mm}
=(h/2)\,[m_{t'}(z'_1+\varepsilon,z'_2)
         -2m_{t'}(z'_1,z'_2)+m_{t'}(z'_1-\varepsilon,z'_2)]/\varepsilon^2
\\
\phantom{mm=}
 +[(1-h)/2]\,[m_{t'}(z'_1,z'_2+\varepsilon)
              -2m_{t'}(z'_1,z'_2)+m_{t'}(z'_1,z'_2-\varepsilon)]
                            /\varepsilon^2
\\
\phantom{mm=}
 -[\delta(1-h)/\varepsilon]\,\{
            [1-m_{t'}(z'_1,z'_2)]
            [m_{t'}(z'_1,z'_2+\varepsilon)-m_{t'}(z'_1,z'_2-\varepsilon)]
\\
\phantom{mm=-[\delta(1-h)/\varepsilon]\{}
                    +m_{t'}(z'_1,z'_2)[(1-m_{t'}(z'_1,z'_2+\varepsilon))
                    -(1-m_{t'}(z'_1,z'_2-\varepsilon))]
                  \}/(2\varepsilon)
\;\;.
\end{array}
\end{displaymath}
Finally, if we assume that in the limit $\varepsilon\to0$, namely,
$L_2\to\infty$, the drift scales to zero as
\begin{displaymath}
\delta=\varepsilon\bar\delta
\;\;,
\end{displaymath}
we find the Mean Field limiting equation
\begin{equation}
\label{againmf000pre}
\frac{\partial m_{t'}}{\partial t'}
=
\frac{1}{2}h
\frac{\partial^2 m_{t'}}{\partial {z'_1}^2}
+
\frac{1}{2}
(1-h)
\frac{\partial^2 m_{t'}}{\partial {z'_2}^2}
-\bar\delta(1-h)
\frac{\partial }{\partial z'_2}[m_{t'}(1-m_{t'})]
\;\;.
\end{equation}

It is worth noting that the above equation is a diffusion--like
equation with a nonlinear anisotropic flux.
From the physical point of view the most interesting remark is that
the diffusion part of the equation is linear. The effect of the drift
is captured in nonlinear transport term.
This term vanishes when $\bar\delta=0$, so that linearity is
approximatively restored at very small $\bar\delta$. It is worth noting that, by the choice of scaling, there is no inbuilt bias towards diffusion- or drift-alone.

To compare the solution of the Mean Field equation to the numerical
simulations, we abuse the notation (recall that $t$, $z_1$, and $z_2$
denoted above integer numbers) and write
$t=t'/\tau$,
$z_1=z'_1/\varepsilon$,
and
$z_2=z'_2/\varepsilon$.
The above limiting equation then reads
\begin{equation}
\label{againmf000}
\frac{\partial m_t}{\partial t}
=
\frac{1}{2}
h
\frac{\partial^2 m_t}{\partial z_1^2}
+
\frac{1}{2}
(1-h)
\frac{\partial^2 m_t}{\partial z_2^2}
-\delta(1-h)
\frac{\partial }{\partial z_2}[m_t(1-m_t)]
\;\;.
\end{equation}

Since in
the top and bottom boundary
densities are constant in space (along the border),
the stationary solutions to \eqref{againmf000}
do not depend on the space variable $z_1$. We call
$\varrho(z_2)$ a
\emph{density profile}
of the stationary Mean Field equation
\begin{equation}
\label{againmf040}
\frac{d^2}{dz_2^2}\varrho-b\frac{\rr{d}}{\rr{d}z_2}\varrho(1-\varrho)=0
\;\;\;\textrm{ with }\;\;\;
b=\frac{2\delta(1-h)}{1-h}=2\delta
\end{equation}
with the Dirichlet boundary conditions
\begin{equation}
\label{againmf020}
\varrho(0)=\varrho_\textrm{u}
\;\;\;\textrm{ and }\;\;\;
\varrho(L_2+1)=\varrho_\textrm{d}
\;\;.
\end{equation}

\subsection{Stationary Mean Field behavior}
\label{s:stazMF}
\par\noindent
Finding the stationary profile $\varrho(z_2)$ means
solving the ordinary differential equation
\eqref{againmf040} with the Dirichlet boundary
conditions \eqref{againmf020}.
In the case $\delta=0$ the solution is trivially linear.
We now discuss the case $\delta >0$.
We integrate equation \eqref{againmf040} once with respect to the space
variable from $0$ to $z_2$ to get
\begin{equation}
\label{amf020}
\varrho'=b\varrho(1-\varrho)+c
\;\;\;\textrm{ where }\;\;\;
c=
\varrho'(0)-b\varrho_\textrm{u}(1-\varrho_\textrm{u})
\;\;.
\end{equation}

The structure of the
solutions of this equation, namely the phase space trajectories,
can be studied via a simple qualitative
analysis (see, for instance, \cite[Chapter 1]{Arnold}).
Let $f(\varrho)=b\varrho(1-\varrho)+c$ be
the right hand side of \eqref{amf020}. Five different
situations have to be considered:
$c>0$, $c=0$, $c<0$ and $b/4+c>0$, $b/4+c=0$, and $b/4+c<0$.
In Figure~\ref{f:mf} the phase diagram in the extended
phase space is shown in the two cases
$c<0$ and $b/4+c>0$, $b/4+c<0$.

\begin{figure}[ht]
\begin{picture}(100,80)(55,10)
\thinlines
\put(140,0){\vector(0,1){90}}
\put(150,10){\vector(-1,0){70}}
\thicklines
\qbezier(140,15)(100,35)(140,55)
\qbezier(140,55)(180,75)(150,60)
\qbezier(140,15)(145,12.5)(150,10)
\put(75,0){{\scriptsize{$f(u)$}}}
\put(148,2){{\scriptsize{$c$}}}
\put(133,60){{\scriptsize{$1$}}}
\put(143,90){{\scriptsize{$u$}}}
\thinlines
\qbezier[21](140,60)(160,60)(180,60)
\qbezier[30](150,5)(150,30)(150,60)
\qbezier[21](140,55)(160,55)(180,55)
\qbezier[21](140,15)(160,15)(180,15)
\put(145,70){\vector(0,-1){10}}
\put(145,30){\vector(0,1){10}}
\thinlines
\put(180,0){\vector(0,1){90}}
\put(170,10){\vector(1,0){100}}
\put(183,2){{\scriptsize{$0$}}}
\put(270,2){{\scriptsize{$z_2$}}}
\put(172,90){{\scriptsize{$u$}}}
\thicklines
\put(180,55){\line(1,0){90}}
\put(180,15){\line(1,0){90}}
\qbezier(180,20)(190,53)(270,53)
\qbezier(180,60)(220,57)(270,57)
\thinlines
\put(335,0){\vector(0,1){90}}
\put(370,10){\vector(-1,0){70}}
\thicklines
\qbezier(360,15)(320,35)(360,55)
\qbezier(360,55)(400,75)(370,60)
\qbezier(360,15)(365,12.5)(370,10)
\put(295,0){{\scriptsize{$f(u)$}}}
\put(368,2){{\scriptsize{$c$}}}
\put(328,60){{\scriptsize{$1$}}}
\put(338,90){{\scriptsize{$u$}}}
\thinlines
\qbezier[35](335,60)(365,60)(400,60)
\qbezier[30](370,5)(370,30)(370,60)
\put(355,40){\vector(0,-1){10}}
\thinlines
\put(400,0){\vector(0,1){90}}
\put(390,10){\vector(1,0){100}}
\put(403,2){{\scriptsize{$0$}}}
\put(490,2){{\scriptsize{$z_2$}}}
\put(392,90){{\scriptsize{$u$}}}
\thicklines
\qbezier(400,60)(410,40)(440,35)
\qbezier(440,35)(470,30)(480,10)
\end{picture}
\vskip 0.5 cm
\caption{Phase diagram corresponding to \eqref{againmf040}.
The case
$c<0$ and $b/4+c>0$ is depicted on the left. The case $b/4+c<0$
is depicted on the right.}
\label{f:mf}
\end{figure}

Now, we find the solution of the stationary equation in the cases
of interest.
From the picture it is clear that:
\begin{itemize}
\item[--]
if $1\ge\varrho_\textrm{u}>1/2>\varrho_\textrm{d}\ge0$ the constant $c$
has to be such that $b/4+c<0$;
\item[--]
if $1\ge\varrho_\textrm{u}>\varrho_\textrm{d}>1/2$ the constant $c$
has to be such that either $b/4+c<0$ or $0>c>-b/4$.
\end{itemize}

It is important to remark that in the first case the constant
$c$ has to be necessarily larger that $-b/4$, while in the second
case there are two different possibilities, so that we will have to decide
which is the correct one.

\par\noindent
\textit{Case $1\ge\varrho_\textrm{u}>1/2>\varrho_\textrm{d}\ge0$.\/}
Assume $b/4+c<0$,
by performing a standard computation
we find the solution
\begin{equation}
\label{amf050}
\arctan\frac{\varrho(z_2)-1/2}{\sqrt{-c/b-1/4}}
=
\arctan\frac{\varrho_\rr{u}-1/2}{\sqrt{-c/b-1/4}}
-b\sqrt{-\frac{c}{b}-\frac{1}{4}} z_2
\end{equation}
with the constant $c$ given by
\begin{equation}
\label{amf060}
\arctan\frac{\varrho_\rr{u}-1/2}{\sqrt{-c/b-1/4}}
-
\arctan\frac{\varrho_\rr{d}-1/2}{\sqrt{-c/b-1/4}}
=
b\sqrt{-\frac{c}{b}-\frac{1}{4}} (L_2+1)
\;\;.
\end{equation}
It is not difficult to verify the following facts:
as a function of $c\in[-\infty,-b/4]$ the left hand side
of \eqref{amf060} is a monotonic function
increasing from $0$ to $-\pi$. Hence, \eqref{amf060}
admits a unique solution in this case.

We can then conclude that in such a case the stationary Mean Field
equation has the unique solution given by
\eqref{amf050} with the constant $c$ as in \eqref{amf060}.

\par\noindent
\textit{Case $1\ge\varrho_\textrm{u}>\varrho_\textrm{d}>1/2$.\/}
In this case,
the left hand side of
\eqref{amf060} tends to zero for $c\to-b/4$,
so that \eqref{amf060} has a solution
provided
\begin{displaymath}
\lim_{c\to-b/4}
\frac{{\displaystyle
       \arctan\frac{\varrho_\rr{u}-1/2}{\sqrt{-c/b-1/4}}
       -
       \arctan\frac{\varrho_\rr{d}-1/2}{\sqrt{-c/b-1/4}}
     }}
     {{\displaystyle
       b\sqrt{-\frac{c}{b}-\frac{1}{4}} (L_2+1)
     }}
>1
\;\;.
\end{displaymath}
By computing the limit above, we get the condition
\begin{equation}
\label{amf070}
\frac{\varrho_\textrm{u}-\varrho_\textrm{d}}
     {b(L_2+1)(\varrho_\textrm{u}-1/2)(\varrho_\textrm{d}-1/2)}
>1
\;\;.
\end{equation}

We recall, now, that in this case it is also possible to find a solution
of the Mean Field equation \eqref{againmf040} with $0>c>-b/4$.
By a standard computation we find the solution
\begin{equation}
\label{amf090}
\varrho(z_2)=\frac{{\displaystyle
                  u_2-u_1\exp\{-bz_2(u_2-u_1)
                    -\log[(\varrho_\textrm{u}-u_1)/(\varrho_\textrm{u}-u_2]\}
                }}
                {{\displaystyle
                  1-\exp\{-bz_2(u_2-u_1)
                    -\log[(\varrho_\textrm{u}-u_1)/(\varrho_\textrm{u}-u_2]\}
                }}
\;\;,
\end{equation}
where
\begin{equation}
\label{amf100}
u_1=\frac{1-\sqrt{1+4c/b}}{2}
<
u_2=\frac{1+\sqrt{1+4c/b}}{2}
\;\;,
\end{equation}
where
the constant $c$, hidden in the expressions of $u_1$ and $u_2$,
can be obtained by requiring $u(L_2+1)=\varrho_\textrm{d}$,
namely,
\begin{equation}
\label{amf110}
\varrho_\textrm{d}=\frac{{\displaystyle
                  u_2-u_1\exp\{-b(L_2+1)(u_2-u_1)
                    -\log[(\varrho_\textrm{u}-u_1)/(\varrho_\textrm{u}-u_2]\}
                }}
                {{\displaystyle
                  1-\exp\{-b(L_2+1)(u_2-u_1)
                    -\log[(\varrho_\textrm{u}-u_1)/(\varrho_\textrm{u}-u_2]\}
                }}
\;\;.
\end{equation}

By computing the limit, for $c\to-b/4$ of the ratio on the right
hand side of \eqref{amf110}, it is possible to show that
such a limit is either larger or smaller than $\varrho_\textrm{d}$
if and only if the equality \eqref{amf070} is satisfied.
This occurrence is related to the existence of solutions to
\eqref{amf110}.
Hence, we have that
the solution of the stationary Mean Field equation is given by
\eqref{amf050} provided \eqref{amf070} is satisfied,
otherwise it is given by \eqref{amf090}.

\begin{figure}[t]
\begin{picture}(200,290)(-10,0)
\put(0,0)
{
\resizebox{14cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-den03.pdf}}}
}
\put(235,0)
{
\resizebox{14cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-den04.pdf}}}
}
\put(0,150)
{
\resizebox{14cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-den01.pdf}}}
}
\put(235,150)
{
\resizebox{14cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-den02.pdf}}}
}
\end{picture}
\caption{Density profiles. Comparison with numerical results (dots) and
the Mean Field solution (solid lines). The cases described in
Section~\ref{s:MFn} are considered:
(i) top left, (ii) top right, (iii) bottom left, and
(iv) bottom right.
}
\label{f:dens-mf}
\end{figure}

\subsection{Numerical verification of the Mean Field prediction}
\label{s:MFn}
\par\noindent
We now test numerically how accurate is the Mean Field prediction of the
stationary density profile.
To measure experimentally such a profile we proceed as follows.
We chose two numbers $1\ll t_\textrm{term}\ll t_\textrm{max}$ that
are called, respectively, \emph{thermalization} and \emph{maximal}
time.
As an estimator for the density $\varrho(z_2)$ we use
\begin{equation}
\label{mis030}
\frac{1}{L_1}
\frac{1}{t_\textrm{max}-t_\textrm{term}+1}
\sum_{s=t_\textrm{term}+1}^{t_\textrm{max}}
\sum_{i=1}^{n(s)}
  \bb{I}_{\{x_2(i,s)=z_2\}}
\;\;,
\end{equation}
where $\bb{I}_{\{\textrm{condition}\}}$ is equal to one if the condition
is true and to zero otherwise.

The thermalization and maximal time are chosen \emph{ad hoc} so that
the measure is performed when the system is
in the stationary state and so that
the measure is sufficiently stable.

We choose the geometrical parameters $L_1=100$ and $L_2=200$
and the probabilistic one $h=1/2$. Then we vary the remaining
ones according to the following four cases:
\begin{itemize}
\item[(i)]
$\varrho_\textrm{u}=1$, $\varrho_\textrm{d}=0$, and $\delta=0.008$:
the mean field solution is given by \eqref{amf050} with
$c=-0.007887$;
\item[(ii)]
$\varrho_\textrm{u}=1$, $\varrho_\textrm{d}=0$, and $\delta=0.8$:
the mean field solution is given by \eqref{amf050} with
$c=-0.400149$;
\item[(iii)]
$\varrho_\textrm{u}=0.8$, $\varrho_\textrm{d}=0.55$, and $\delta=0.008$:
since the first term of inequality \eqref{amf070} is equal to $5.18242$,
the mean field solution is given by \eqref{amf050} with
$c=-0.00478572$;
\item[(iv)]
$\varrho_\textrm{u}=0.8$, $\varrho_\textrm{d}=0.55$, and $\delta=0.8$:
since the first term of inequality \eqref{amf070} is equal to $0.0518242$,
the mean field solution is given by \eqref{amf090} with
$c=-0.396$.
\end{itemize}

The numerical simulations are performed with $t_\textrm{term}=10^5$
and $t_\textrm{max}=5\times10^5$ and the corresponding
results are depicted in
Figure~\ref{f:dens-mf}. In all the considered cases the match between the
numerical data and the Mean Field prediction is strikingly good.

\begin{figure}[t]
\begin{picture}(200,290)(-10,0)
\put(0,0)
{
\resizebox{14cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-flu03.pdf}}}
}
\put(235,0)
{
\resizebox{14cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-flu04.pdf}}}
}
\put(0,150)
{
\resizebox{14cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-flu01.pdf}}}
}
\put(235,150)
{
\resizebox{14cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-flu02.pdf}}}
}
\end{picture}
\caption{Numerical estimate of the ingoing ($+$) and outgoing ($\times$)
flux for the four cases described in
Section~\ref{s:MFn}:
(i) top left, (ii) top right, (iii) bottom left, and
(iv) bottom right.
}
\label{f:flusso}
\end{figure}

\subsection{Thermalization time}
\label{s:term}
\par\noindent
Since the measuring of the density profile has to be performed
in the stationary states, the
choice of the thermalization time has to be done with care.
One possibility to check if the system has reached stationarity is
to compare the typical number of particles entering the system
through the top boundary to that of the particles
exiting from the bottom.

We proceed as following:
to account for the number of particle crossing the top and
bottom boundaries, we let
$I(t)$ be the difference between the number of particles
that entered at time $t$ through the
top boundary and that of the particles that exited through the same
boundary.
On the other hand,
let
$O(t)$ be the difference between the number of particles
that exited at time $t$ through the
bottom boundary and that of the particles that entered through the same
boundary.
Both $I(t)$ and $O(t)$ are stochastic variables that can be
positive or negative. We can expect that, if a
stationary state is observed, in such a state $I(t)$ and $O(t)$
will both
fluctuate around the same value.

We also define two additional quantities:
given the positive integer $T$, we let
the \emph{local ingoing and outgoing} fluxes at time $t$
be as
\begin{equation}
\label{mis010}
F^\rr{i}_T(t)=\frac{1}{T}\sum_{s=\max\{0,t-T\}}^t I(s)
\;\;\;\textrm{ and }\;\;\;
F^\rr{o}_T(t)=\frac{1}{T}\sum_{s=\max\{0,t-T\}}^t O(s)
\;\;,
\end{equation}
respectively.

In Figure~\ref{f:flusso} we plot the local ingoing and outgoing fluxes
measured on the interval $T=100$ for the four cases (i)--(iv)
considered above.
The data shown in the figures indicate the choice
$10^5$ for the thermalization time.

\section{A biased birth and death model}
\label{s:bd}
\par\noindent
The main physical problem we discuss in this paper
is that of the typical time that (at stationarity)
a particle spends
in the strip before finding its way to go out.
We refer to this time as the
\emph{residence time} and we
will
discuss its definition in detail in the next section.

Since from this point of view
only vertical displacements are relevant, we can think
to the ``vertical" history of the particle as to the
evolution of a non--homogeneous birth and death process for the
vertical coordinate with a rule depending on the
stationary density profile. In this section we recall
general results on this birth and death model and in the next section we
discuss how to apply such results to our 2D model.
We mainly follow \cite{BBF} for the general discussion. We
derive explicit formulas in two specific simple cases, that
will turn out to be very important from the physical point of view.
For the sake of completeness we outline these
computations in Subsections~\ref{s:cost} and \ref{s:lin}.

Let $L$ be a positive integer,
$[[0,L]]:=\{0,1,\dots,L\}$, and
consider a random walk on $[[0,L]]$
with transition probabilities
$p(x,y)$ with $x,y\in[[0,L]]$.
We denote by $X_a(t)$, with $t=0,1,\dots$ the position of the
walker at time $t$ with initial position $a$.
We assume that at each time the walker either does not move
or moves to one of the neighboring sites, i.e.,
we assume $p(x,y)=0$ for $x,y\in[[0,L]]$ such that $|x-y|\ge2$.
Moreover,
we let
$p_x=p(x,x+1)$ with $x=0,\dots,L-1$ be the probabilities
to go to the right and
$q_x=p(x,x-1)$ with $x=1,\dots,L$ be the probabilities
to go to the left.
We assume $0\le p_{x-1}\le q_x<1$ for
$x=1,\dots,L$, namely, at each site the walker has
the chance to go both to the left and to the right and
the walk is left--biased.
Furthermore,
we assume that at least one of the ``rest probabilities"
$p(x,x)=1-(p_x+q_x)$ is different from zero so that the
chain is aperiodic.
We note that the more general situation in which the
bias condition is removed can be treated as well, but
for the sake of clarity, we consider the biased case.

In the case $0< p_{x-1}\le q_x<1$ for $x=1,\dots,L$,
since the birth and death process is aperiodic and
positive recurrent, it has a unique invariant measure that can be
written as
\begin{equation}
\label{art040}
\pi(x)=\pi(0)\prod_{i=1}^x\frac{p_{i-1}}{q_i}
\;\;\textrm{ for all }x=1,\dots,L
\;\;,
\end{equation}
see \cite[equation~(4.1)]{BBF}. Note that the proof of the above statement
is immediate, since one just has to verify that the above measure
satisfies the detailed balance equation $\pi(x)p(x,x+1)=\pi(x+1)p(x+1,x)$
for all $x=1,\dots,L-1$.

In \cite{BBF} the authors study in detail the properties of the
first hitting time of the chain to any point of the lattice $[[0,L]]$
with any initial condition (initial position of the walker).
We are interested
only to the hitting time
\begin{equation}
\label{hitting}
T:=\inf\{t\ge1,\,X_L(t)=0\}
\;\;,
\end{equation}
i.e.,
the random time that
the walker started at $L$ needs to reach the origin for the first
time.
The expectation of such a hitting time
is given by
\begin{equation}
\label{art050}
\bb{E}[T]
=
\sum_{i=1}^{L}\frac{1}{q_i\pi(i)}\,\sum_{j=i}^{L}\pi(j)
=
\frac{1}{q_L}
+
\sum_{i=1}^{L-1}\frac{1}{q_i}\,
      \Big(1+\sum_{j=i+1}^{L}\prod_{k=i+1}^j\frac{p_{k-1}}{q_k}
      \Big)
\;\;,
\end{equation}
see \cite[equation~(4.3)]{BBF}.

The first very simple remark is that,
since the velocity of the
particle is bounded by one, the mean hitting time to $0$ cannot be
smaller than $L$. More precisely, by using \eqref{art050}
and recalling that $q_x<1$ for $x=1,\dots,L$
we get the ballistic lower bound
\begin{equation}
\label{hitlow}
\bb{E}[T]
\ge
L
\;\;.
\end{equation}

A natural question is under which assumptions on the bias there
exists a ballistic upper bound to the first hitting time to zero.
We prove this bound in a very simple case, namely,
when we assume that each bond is left--biased. More precisely,
we show that if there exists $0<\eta<1$ such that
$p_{x-1}/q_x\le\eta$ for $x=1,\dots,L$, then
\begin{equation}
\label{hitup}
\bb{E}[T]
\le
\frac{L}{\underline{q}(1-\eta)}
\end{equation}
where we have let $\underline{q}=\min\{q_1,\dots,q_L\}$.
Indeed, from \eqref{art050} we have that
\begin{displaymath}
\bb{E}[T]
\le
\frac{1}{\underline{q}}
+
\sum_{i=1}^{L-1}\frac{1}{\underline{q}}\,
      \Big(1+\sum_{j=i+1}^{L}\prod_{k=i+1}^j\eta
      \Big)
=
\frac{1}{\underline{q}}
+
\frac{1}{\underline{q}}
\sum_{i=1}^{L-1}
      \sum_{k=0}^{L-i}\eta^k
\le
\frac{1}{\underline{q}}
+
\frac{1}{\underline{q}}
\sum_{i=1}^{L-1}
      \sum_{k=0}^\infty\eta^k
\end{displaymath}
where we have omitted few simple steps. The statement \eqref{hitup}
follows recalling that $\eta<1$.

Finally, we remark that using equations \eqref{art040}
and \eqref{art050} we can
compute the expected value of the first hitting time $T$. In practice
this is explicitly feasible only for some particularly easy choices
of the probabilities $p_x$ and $q_x$. In the next subsections
we discuss two physically relevant cases.

\subsection{Homogeneous case}
\label{s:cost}
\par\noindent
Let $0<p\le q<1$ be two real numbers and assume
$p_x=p$ and $q_x=q$ for all $x=0,\dots,L$.
Note that this choice satisfies all the basic assumptions on
the birth and death chain.

To compute the expected value of the first hitting time $T$
it is convenient to set $\lambda=p/q$, so that, from
\eqref{art040}, we have
$\pi(x)=\pi(0)\lambda^x$.
Equation
\eqref{art050} yields
\begin{displaymath}
\bb{E}[T]
=
\frac{1}{q}
\sum_{i=1}^{L} \frac{1}{\lambda^i} \,\sum_{j=i}^{L} \lambda^j
=
\frac{1}{q}
\sum_{i=1}^{L} \,\sum_{j=i}^{L} \lambda^{j-i}
=
\frac{1}{q}
\sum_{i=1}^{L} \,\sum_{k=0}^{L-i} \lambda^{k}
=
\frac{1}{q(1-\lambda)}
\sum_{i=1}^{L} (1-\lambda^{L-i+1})
\;\;.
\end{displaymath}
By reordering the sum, we obtain
\begin{displaymath}
\bb{E}[T]
=
\frac{1}{q(1-\lambda)}
\Big(L-\sum_{k=1}^{L} \lambda^{k}\Big)
=
\frac{1}{q(1-\lambda)}
\Big(L+1-\frac{1-\lambda^{L+1}}{1-\lambda}\Big)
=
\frac{L}{q(1-\lambda)}
-
\frac{\lambda(1-\lambda^L)}{q(1-\lambda)^2}
\;\;.
\end{displaymath}
Recalling $\lambda=p/q$, we finally have
the expression
\begin{equation}
\label{art110}
\bb{E}[T]
=
\frac{L}{q-p}-\frac{p}{(q-p)^2}\Big[1-\Big(\frac{p}{q}\Big)^L\Big]
\;\;.
\end{equation}

A physical comment is useful:
if $p/q<1$, the behavior of the mean first hitting time
on $L$ is ballistic.
On the other hand, we can prove that
\begin{displaymath}
\lim_{p\to q} \bb{E}[T]=\frac{L(L+1)}{2q}
\end{displaymath}
Hence, in the symmetric limiting case the
diffusive dependence
of the mean hitting time on the length $L$ is found.
Note that the totally asymmetric limit $p\to0$ will be considered in
Section~\ref{s:tot} and the ballistic scaling will be found.

\subsection{A linear case}
\label{s:lin}
\par\noindent
We now consider a case in which the transition probabilities
$q_x$ and $p_x$ decrease linearly in the interval $[[0,L]]$.
The physical interest of the peculiar choice we shall do will
be clear in the next section.
Let $A>0$ and take
\begin{equation}
\label{lin00}
q_x=2A+A(L-x)
\;\;\;\textrm{ and }\;\;\;
p_x=A(L-x)
\;\;.
\end{equation}

By using \eqref{art040} for the stationary measure,
we find that
\begin{displaymath}
\pi(x)=
       \pi(0)\,
       \frac{L}{L+1}\,
       \frac{L-1}{L}\,
       \frac{L-2}{L-1}\,\cdots\cdots
       \frac{L-x+1}{L-x+2}
=
 \pi(0)\,
 \frac{L-x+1}{L+1}
\;\;.
\end{displaymath}
By \eqref{art050}, we get
\begin{displaymath}
\bb{E}[T]
=
\frac{1}{A}
\sum_{i=1}^{L}\frac{1}{L+2-i}\,\frac{1}{L+1-i}
\sum_{j=i}^{L}(L+1-j)
\;\;.
\end{displaymath}
Since, it holds that
\begin{displaymath}
\sum_{j=i}^{L}(L+1-j)
=
\frac{1}{2}(L+2-i)(L-i+1)
\;\;,
\end{displaymath}
we finally get
\begin{equation}
\label{lin040}
\bb{E}[T]
=
\frac{L}{2A}
\;\;.
\end{equation}
It is worth noting that,
if $A$ is a constant then the scaling is ballistic. But
if $A$ is small with $L$, then we can possibly expect to have
a diffusive scaling.

\subsection{The totally asymmetric case}
\label{s:tot}
\par\noindent
A situation that will be useful in our discussion and that is not
included in the results discussed above is the case
$0= p_{x-1}<q_x<1$ for $x=1,\dots,L$, which we refer as
the \textit{totally asymmetric} case. In such a case, by using the
same strategy of proof as in \cite{BBF}, it is easy to show that
\begin{equation}
\label{hitting-tot}
\bb{E}[T]
=
\sum_{i=1}^{L}\frac{1}{q_i}
\;\;.
\end{equation}

The dependence of the mean hitting time to zero on the length
of the system is, in this case, trivially ballistic. Indeed,
from \eqref{hitting-tot} we get the bounds
\begin{equation}
\label{hittot}
\bb{E}[T]\ge\frac{L}{\bar{q}}
\;\;\textrm{ and }\;\;
\bb{E}[T]\le\frac{L}{\underline{q}}
\;\;,
\end{equation}
where we have set
$\underline{q}=\min\{q_1,\dots,q_L\}$ and
$\bar{q}=\max\{q_1,\dots,q_L\}$.
Note that in the totally asymmetric homogeneous case,
that is to say, $0<q_x=q<1$ for $x=1,\dots,L$, we get $\bb{E}[T]=L/q$.

\section{Residence time at stationarity}
\label{s:res}
\par\noindent
The main question we pose in
this paper is to find estimates for
the typical time spent by a particle
in the strip before exiting through the bottom boundary.

To give a precise definition of such a time for the
simple exclusion model on the strip defined
in Section~\ref{s:modello} we
set $\varrho_\rr{u}=1$ so that particles cannot exit the
strip through the top boundary.
Once the stationary state is reached, we look at the
particles that enter from the top boundary and exit from the
bottom one.
We count after how many steps of the dynamics the particle exits
from the bottom boundary and call \textit{residence time} the average
of such a time over all the particles entered in the system
after the stationary state is reached.
Note that, at each step of the dynamics, a number of particles
equal to the number of particles in the system at the end
of the preceding time step is tentatively moved.

Despite its evident physical interest, the residence time
is quite a difficult object to treat mathematically.
In this section we discuss a microscopic approach in which we follow
the motion of a single particle in a stationary density profile
that is treated as a fixed background. In Section~\ref{s:res-mf}
we shall treat the problem macroscopically, by using the Mean Field
theory.

Recall that, at stationarity, the average density
profile is given by a function that we have denoted by
$\varrho(z_2)$. Making a thought experiment,
imagine that a new particle is injected into the stationary system
through the top boundary.
To estimate the typical time this particle needs to find its way
out through the bottom boundary we note that
only vertical displacements are relevant.
Moreover,  we can think
of the ``vertical" history of the particle as to the
evolution of the not homogeneous birth and death process
defined in Section~\ref{s:bd}
with the peculiar choice of the jump probabilities $p_x$ and $q_x$
that will be discussed below.
We let $L=L_2$ and
imagine that
the value $x=0$ of the birth and death process represents the particle
at the row $L_2+1$ of the lattice (bottom boundary),
the value $x=L_2$ represents the particle
at the row $1$ of the lattice (the row close to the top boundary),
and
the generic value $x$ represents the particle
at the row $L_2+1-x$ of the lattice.
The only not zero transition probabilities of the birth and death
process are chosen as
\begin{equation}
\label{art010}
\left\{
\begin{array}{ll}
{\displaystyle
 q_x
 =\frac{1-h}{2}(1+\delta)[1-\varrho(L_2+1-x+1)]
}
&
\;\;\;\textrm{ for }
x=1,\dots,L_2
\vphantom{\Bigg\{_\}}
\\
{\displaystyle
 p_x
 =\frac{1-h}{2}(1-\delta)[1-\varrho(L_2+1-x-1)]
}
&
\;\;\;\textrm{ for }
x=0,\dots,L_2-1
\\
\end{array}
\right.
\;\;.
\end{equation}
Indeed,
recalling the expressions \eqref{againmf00-2} for
the probabilities $u$ and $d$,
the prefactor $(1-h)(1+\delta)/2$
in the expression of $q_x$
is nothing but the probability $d$ to move the particle in the
real space downwards; while the second factor is nothing
but the probability that, at stationarity, the site where
the particle tries to jump is empty.
The expression for $p_x$ can be justified similarly. The rest probability $1-(p_x+q_x)$ is nonzero as the particle can stay at the same site or perform a unit step in the horizontal direction.

We now propose a conjecture for the residence
time of the exclusion model and we will test it numerically
in the next section.

\begin{guess}
The residence time\footnote{Since in the real model a particle at the
fictitious row $L_2+1$ cannot jump back to the real row $L_2$,
one should set the probability $p(0,1)=0$. This would be a problem
for the birth and death process, where all the jumping probabilities
has to be assumed strictly positive, but for our purpose it is
not necessary, since we are only interested to the first hitting time
to $0$, so that when the 1D birth and death process
reaches such a state  it is stopped. Hence all our results will not
depend on the choice of the probability $p(0,1)$.}
of the model introduced in Section~\ref{s:modello} is equal to
the mean value of the first hitting time to $0$ for the
birth and death process started at $L_2$.
\end{guess}

The main properties of birth and death processes have been
recalled in Section~\ref{s:bd}.
Those results and the conjecture based on the above analogy
suggest that, for $\delta<1$,
the residence time is given by equation \eqref{art050} where
the stationary measure $\pi$ is given by \eqref{art040} with
the jump probabilities $p_i$ and $q_i$ defined in \eqref{art010}.
Since the stationary density profile is given with great accuracy by
the stationary Mean Field equation, we can use
these equations to give an estimate of the residence time of the model.
It is worth noting that, since the expression of the
stationary density profile is rather complicated, see \eqref{amf050}
and \eqref{amf090}, it will not be possible to derive in general explicit
formulas for the residence time. On the other hand, since
in \eqref{art050} and \eqref{art040} only finite product and
sums are involved, we will be able to
compute estimates for the residence time
numerically for any values of the parameters of the model.
In the case $\delta=1$ the expression \eqref{hitting-tot} should
be used.

Let us now discuss three simple cases in which explicit expressions
of the residence time can be derived explicitly.

\subsection{Totally vertically asymmetric model}
\label{s:td}
\par\noindent
Recall we assumed $\varrho_\rr{u}=1$.
Consider, more, the case $\delta=1$.
From the profiles in Figure~\ref{f:dens-mf} (graphs on the right)
it is rather clear
that in this situation the density profile is with very good approximation
constant throughout the strip.

Denote by $\bar\varrho$ such a constant value of the
density profile. By \eqref{art010}, it follows that
the birth and death model has the jump probabilities
\begin{displaymath}
q_x=(1-h)(1-\bar\varrho)
\;\;x=1,\dots,L_2
\;\;\textrm{ and }\;\;
p_x=0
\;\;x=0,\dots,L_2-1
\end{displaymath}
Hence, by the results in Section~\ref{s:tot},
we have that the residence time is given by
\begin{equation}
\label{td010}
R
=
\frac{L_2}{(1-h)(1-\bar\varrho)}
\;\;.
\end{equation}
In particular,
the large $L_2$ behavior of the residence time given by \eqref{td010}
is ballistic with a
slope depending on the parameters $h$ and $\bar\varrho$.

\subsection{Large drift case}
\label{s:ld}
\par\noindent
Recall we assumed $\varrho_\rr{u}=1$.
Consider $\delta$ close to one.
From the profiles in Figure~\ref{f:dens-mf} (graphs on the right)
it is rather clear
that in this situation the density profile is with very good approximation
equal to a constant, denoted again by $\bar\varrho$.
By \eqref{art010} it follows that
the birth and death model
has the jump probabilities
\begin{displaymath}
q_x=\frac{1-h}{2}(1+\delta)(1-\bar\varrho)
\;\;x=1,\dots,L_2
\;\;\textrm{ and }\;\;
p_x=\frac{1-h}{2}(1-\delta)(1-\bar\varrho)
\;\;x=0,\dots,L_2-1
\end{displaymath}
Hence, by \eqref{art110}, we have that the residence time is given by
\begin{equation}
\label{ld010}
R
=
\frac{L_2}{(1-h)(1-\bar\varrho)\delta}
-
\frac{1-\delta}{2(1-h)(1-\bar\varrho)\delta^2}
\Big[
     1-\Big(\frac{1-\delta}{1+\delta}\Big)^{L_2}
\Big]
\;\;.
\end{equation}
The formula \eqref{ld010} suggests that in this case
the dependence of the residence time on $L_2$ is
ballistic, but this statement needs more care. Indeed,
the equation above is based on the assumption that
the density profile is constant with good approximation
in this regime and such an assumption is based on the
results obtained via the Mean Field equation. But we have to recall
that the Mean Field equation has been (not rigorously)
derived in the large
volume limit with the drift parameter scaling to zero
with $L_2$. For this reason it is not correct, in principle,
to fix $\delta$ and let $L_2\to\infty$ in the above formula.

We remark, that in this case we shall as well be able to deduce
the ballistic scaling of the residence time with $L_2$
in Section~\ref{s:scaling} via a different argument.

\subsection{Zero drift}
\label{s:zd}
\par\noindent
Consider, now, the case $\delta=0$ (zero drift)
again for $\varrho_\rr{u}=1$.
In the absence of drift the stationary density
profile is linear, hence
\begin{equation}
\label{zd000}
\varrho(z_2)=1-\frac{1-\varrho_\textrm{d}}{L_2+1}z_2
\end{equation}
for $z_2=0,\dots,L_2+1$.
By \eqref{art010}, we get also that
\begin{displaymath}
q_x=\frac{1-h}{2}\frac{1-\varrho_\textrm{d}}{L_2+1}(L_2+2-x)
\;\;\;\textrm{ and }\;\;\;
p_x=\frac{1-h}{2}\frac{1-\varrho_\textrm{d}}{L_2+1}(L_2-x)
\;\;.
\end{displaymath}
Thus, by using the result \eqref{lin040} from Section~\ref{s:lin}, we find
the residence time
\begin{equation}
\label{zd020}
R=
\frac{L_2(L_2+1)}{(1-h)(1-\varrho_\textrm{d})}
\;\;.
\end{equation}
As expected,
the large volume ($L_2\to\infty)$ behavior of the residence
time is quadratic in
the purely diffusive zero drift case.

\subsection{Dependence of the residence time on the length of the strip}
\label{s:scaling}
\par\noindent
In this section we focus on
the dependence of the residence time
on the length $L_2$ of the strip.
We expect that for $\delta>0$, since there is a
preferred direction in the movement, the particles
will have a not zero average vertical velocity. As a
consequence, we expect a ballistic behavior and a
linear dependence of the residence time on $L_2$.

We have already shown in Section~\ref{s:td} that this is indeed the
case for $\delta=1$. Assume, now, that the drift $\delta$ is fixed and
$0<\delta<1$. From \eqref{art010}, we see
\begin{displaymath}
\frac{p_{x-1}}{q_x}
=
\frac{1-\delta}{1+\delta}
\frac{1-\varrho(L_2+1-x)}{1-\varrho(L_2+1-x+1)}
\end{displaymath}
for any $x=1,\dots,L_2$.
Since, $\varrho_\rr{u}=1>\varrho_\rr{d}\ge0$
we can reasonably assume that the density profile is a decreasing
function of the vertical spatial coordinate. Thus,
$\varrho(L_2+1-x)>\varrho(L_2+1-x+1)$ implies that
there exists $\eta<1$ such that
$p_{x-1}/q_x\le\eta$.
This remark and \eqref{hitup} ensure that, for any positive
finite $\delta$, the residence
time has a ballistic dependence on the length of the strip.

Finally, for $\delta=0$ we have shown in Section~\ref{s:zd}, cf.\
\eqref{zd020}, that the residence time is quadratic
in $L_2$ (diffusive scaling). We stress that this
result is not trivial at all. Indeed, even in the case
$\delta=0$ the birth and death model that we use to
investigate the property of the residence time is not
symmetric. The lack of symmetry is due to
\begin{displaymath}
q_x-p_{x-1}
=
\frac{1-h}{2}\,
\frac{1-\varrho_\rr{d}}{L_2+1}
\end{displaymath}
for any $x=1,\dots,L_2$. The diffusive scaling is a consequence of
the fact that this difference vanishes as $1/(L_2+1)$.

\subsection{Single file regime}
\label{s:single}
\par\noindent
The model we study is two-dimensional. Particles move
in a strip from the top boundary towards the bottom one due
to the presence of a drift $\delta$ or just because
of a vertical biased diffusion due to the difference between
the boundary densities on the top and on the bottom
end of the strip. The particles are subjected
also to horizontal displacements whose probability
is controlled by the parameter $h$.

In the particular case $h=0$, our model becomes 1D
and particles move, each on its vertical line, as in a single
file system. In other words in this case our model
reduces to the 1D simple exclusion
model with open boundaries. The specification
``open boundaries" means that the system is finite
and at the boundaries there is a rule prescribing the
rate at which particles can either enter or leave the system.

This model has been widely studied, see for instance \cite{DEHP}
for the seminal paper where the model was solved exactly
in the totally asymmetric case. See, also,
\cite{BE} for a general review.
Our results can be compared easily to those in \cite{DEHP} which
refer to the totally asymmetric case, namely, for our case $\delta=1$.
In \cite{DEHP} one computes the stationary current
$J$, namely, the number of particles that for unit of time
crosses one bond at stationarity. With our notation
one finds
\begin{equation}
\label{derrida}
J=
\left\{
\begin{array}{ll}
1/4
&
\;\;\;\;\;\textrm{ for } \varrho_\rr{d}\le 1/2\\
\varrho_\rr{d}(1-\varrho_\rr{d})
&
\;\;\;\;\;\textrm{ for } \varrho_\rr{d}> 1/2\\
\end{array}
\right.
\end{equation}
in the case $\varrho_\rr{u}=1$ (see \cite[equations (58) and (60)]{DEHP}).

Now, since in the totally asymmetric case the stationary density
throughout the system is equal to a constant $\bar\varrho$,
we can write
$J\approx\bar\varrho L_2/R$. Hence, we have that
\begin{displaymath}
R\approx\bar\varrho\,\frac{L_2}{J}
\end{displaymath}
which reduces to our result \eqref{td010} with $h=0$
once we use \eqref{derrida} and notice that
$\bar\varrho=1/2$ for $\varrho_\rr{d}\le1/2$ and
$\bar\varrho=\varrho_\rr{d}$ for $\varrho_\rr{d}>1/2$.

Additionally,
we mention here an interesting result which is valid for the symmetric
simple exclusion model in 1D on the whole line (no boundary).
In both \cite{fedders} and \cite{DF}, the authors compute
the variance of the position of a tagged particle and they
prove that it is proportional to the square root of time, meaning that
the typical distance spanned by the (symmetric)
walker is proportional to $t^{1/4}$.
In our problem we do not find any $L_2^4$ scaling for the
residence time even in the single--file regime. Indeed,
we think that there is no direct connection between the two problems.
One important point is that the results in \cite{fedders,DF}
deal with a simple exclusion model without drift (symmetric)
and without boundaries. In our problem, even in the zero drift
($\delta=0$) case, a net flux is present due to the boundary conditions.

\subsection{Mean Field approximation for the residence time}
\label{s:res-mf}
\par\noindent
In this section we approach the residence time computation
from a macroscopic point of view. In the Mean Field approximation
the evolution of the systems can be described in terms
of the density profile $m_t(z_1,z_2)$ evolving according
to the Partial Differential Equation \eqref{againmf000}.
Such an equation can be tought of as a continuity equation
for the current two--dimensional vector
\begin{displaymath}
\vec{J}_t=-\frac{1}{2}h\frac{\partial m_t}{\partial z_1}\vec{e}_1
        +\Big(
           -\frac{1}{2}(1-h)\frac{\partial m_t}{\partial z_2}
           +\delta(1-h)m_t(1-m_t)
         \Big)\vec{e}_2
\end{displaymath}
where $\vec{e}_1$ and $\vec{e}_2$ are the unitary vectors of the
lattice.

At stationarity, the density profile $\varrho(z_2)$ does not depend on
the horizontal coordinate, so that the current is
parallel to the vertical direction and its intensity is given by
\begin{equation}
\label{res-mf000}
J= -\frac{1}{2}(1-h)\frac{\partial \varrho}{\partial z_2}(z_2)
           +\delta(1-h)\varrho(z_2)[1-\varrho(z_2)]
      =-\frac{1}{2}(1-h)\varrho'(0)
\end{equation}
where we have used \eqref{againmf040}, \eqref{amf020}, and the fact that
$\varrho_\rr{u}=1$.

As we have already done in Section~\ref{s:single}, we can
relate the stationary current to the residence time.
Indeed, it is quite reasonable to assume that, at stationarity,
the average velocity of a particle $v(z_2)$ is such that
$\varrho(z_2)v(z_2)=J_2$. Then, an easy integration gives
\begin{equation}
\label{res-mf010}
R=
\int_0^{L_2+1}\frac{\varrho(z_2)}{J}\,\rr{d}z_2
=
-\frac{2}{(1-h)\varrho'(0)}
\int_0^{L_2+1}\varrho(z_2)\,\rr{d}z_2
\end{equation}

In general, we cannot use the above equation to write an explicit
expression of the residence time in terms of the parameter of the model,
indeed the quantity $\varrho'(0)$ is nothing but the constant
$c$ in equation \eqref{amf020} (recall $\varrho_\rr{u}=1$) which
is, in turn, the solution of either the equation
\eqref{amf060} ot \eqref{amf110}.

But in the zero drift case, i.e.,
$\delta=0$, the density profile is the linear function
$\varrho(z_2)=1-(1-\varrho_\rr{d})z_2/(L_2+1)$ for $z_2\in[0,L_2+1]$.
Hence an easy computation yields
\begin{equation}
\label{res-mf020}
R=\frac{1}{(1-h)}\frac{1+\varrho_\rr{d}}{1-\varrho_\rr{d}}(L_2+1)^2.
\end{equation}
This formula has to be compared to the prediction \eqref{zd020}
for the residence time in the zero drift case obtained in the
framework of the birth and death model approximation.

The mean field residence time expression (\ref{res-mf020}) can be considered the residence time according to the birth and death model corrected by  a term proportional to $\frac{\varrho_\rr{d}}{1-\varrho_\rr{d}}$. When $\varrho_\rr{d}$ is zero both expressions are the same. The correction can be understood is due to the correlated motion  in the one dimensional bouncing back  case. Fedders \cite{fedders} found that the diffusion constant  has to be corrected by a similar term in the stationary case.
We will see that expression  (\ref{res-mf020}) only agrees well the Monte Carlo  simulations  as long as $h=0$. On the other hand, the expression (\ref{zd020}) applies rigorously as long as $\delta=0$.

\section{Numerical estimates of the residence time}
\label{s:res-num}
\par\noindent

In this section we test numerically the validity of the analytic
computations developed in
Section~\ref{s:res}. We choose the
parameters of the model as follows: take $\varrho_\rr{u}=1$,
$L_1=100$, and consider the cases
\begin{displaymath}
L_2=100,200,\;\;\;
\delta=0,0.2,0.4,0.6,0.8,1,\;\;
\varrho_\rr{d}=0,0.4,0.8,\;\;
h=0,0.4,0.8.
\end{displaymath}

\begin{figure}[h]
\begin{picture}(200,125)(0,10)
\put(0,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-el100.pdf}}}
}
\put(235,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-el200.pdf}}}
}
\end{picture}
\caption{Residence time versus drift in the cases
$\varrho_\rr{d}=0$ and $L_2=100$ (left)
and $L_2=200$ (right).
The symbols $\bullet$, $\blacktriangle$, and $\blacksquare$
refer, respectively, to the cases $h=0,0.4,0.8$.
The inset in the left picture is just a zoom
of part of the data of the same picture.
Solid lines are the birth and death theoretical prediction.
Open circles are the Mean Field theoretical prediction \eqref{res-mf010}.
}
\label{f:res-dr-100}
\end{figure}

\begin{figure}[h]
\begin{picture}(200,125)(0,10)
\put(0,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-ph100.pdf}}}
}
\put(235,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-ph200.pdf}}}
}
\end{picture}
\caption{Residence time versus the horizontal displacement probability
$h$ in the cases $\varrho_\rr{d}=0$ and $L_2=100$ (left)
and $L_2=200$ (right).
The symbols $\blacksquare$, $\bullet$, and $\blacktriangle$
refer, respectively, to the cases $\delta=0.6,0.8,1$.
Solid lines are the birth and death theoretical prediction.
Open squares are the approximated theoretical prediction
\eqref{ld010} with $\bar\varrho=1/2$, which is
valid in the large drift regime.
Open circles are the Mean Field theoretical prediction \eqref{res-mf010}.
}
\label{f:res-ph-100}
\end{figure}

\begin{figure}[h]
\begin{picture}(200,125)(0,10)
\put(0,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-el01.pdf}}}
}
\put(235,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-el02.pdf}}}
}
\end{picture}
\caption{Residence time versus length of the strip $L_2$
in the case $\varrho_\rr{d}=0$ and $h=0.4$.
The symbol $\blacksquare$ refers to the
case $\delta=0$ (left picture) while
the symbols $\bullet$ and $\blacktriangle$
refer, respectively, to the cases $\delta=0.6,1$.
Solid lines are the theoretical prediction based on the birth and death
model;
in the picture in the left the explicit expression \eqref{zd020} is used,
Open squares are the approximated theoretical prediction
\eqref{ld010} with $\bar\varrho=1/2$, which is
valid in the large drift regime.
Open circles are the Mean Field theoretical prediction \eqref{res-mf010}.
}
\label{f:res-el}
\end{figure}

\begin{figure}[h]
\begin{picture}(200,125)(0,10)
\put(0,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-ph100-dr1.pdf}}}
}
\put(235,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-ph200-dr1.pdf}}}
}
\end{picture}
\caption{Residence time versus the horizontal displacement probability
$h$ in the cases $\delta=1$ and $L_2=100$ (left)
and $L_2=200$ (right).
The symbols $\bullet$, $\blacktriangle$, and $\blacksquare$
refer, respectively, to the cases $\varrho_\rr{d}=0,0.4,0.8$.
Note that bullets and black triangle are perfectly coinciding,
this is due to the fact that in this regime the residence time
does not depend that much on the bottom boundary density, provided
it is smaller that $1/2$. In such a case, indeed,
the density profile inside the strip is almost perfectly constant and equal
to $1/2$.
Solid lines are the theoretical prediction based on the birth and
death model.
Open squares are the approximated theoretical prediction
\eqref{ld010} valid in the large drift regime
with $\bar\varrho=1/2$ for the cases $\varrho_\rr{d}=0,0.4$
and
with $\bar\varrho=8/10$ in the case $\varrho_\rr{d}=0.8$.
Indeed, in the first two cases the density profile is almost constantly
equal to $1/2$, while in the last case it is almost constantly equal
to $8/10$.
Open circles are the Mean Field theoretical prediction \eqref{res-mf010}.
}
\label{f:res-ph-100-dr1}
\end{figure}

\begin{figure}[h]
\begin{picture}(200,125)(0,10)
\put(0,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-rd100-dr1.pdf}}}
}
\put(235,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-rd200-dr1.pdf}}}
}
\end{picture}
\caption{Residence time versus the bottom boundary density
$\varrho_\rr{d}$
in the cases $\delta=1$ and
$L_2=100$ (left)
and $L_2=200$ (right).
The symbols $\bullet$, $\blacktriangle$, and $\blacksquare$
refer, respectively, to the cases $h=0,0.4,0.8$.
Solid lines are the theoretical prediction based on the birth and
death model.
Open squares are the approximated theoretical prediction
\eqref{ld010} valid in the large drift regime
with $\bar\varrho=1/2$ for $\varrho_\rr{d}\le 1/2$
and
with $\bar\varrho=\varrho_\rr{d}$ for $\varrho_\rr{d}>1/2$.
Indeed, when $\varrho_\rr{d}<1/2$ the density profile is almost constantly
equal to $1/2$, while for $\varrho_\rr{d}>1/2$
it is almost constantly equal to $\varrho_\rr{d}$.
Open circles are the Mean Field theoretical prediction \eqref{res-mf010}.
}
\label{f:res-rd-100-dr1}
\end{figure}

\begin{figure}[h]
\begin{picture}(200,140)(0,10)
\put(80,0)
{
\resizebox{12cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-el-dr1.pdf}}}
}
\end{picture}
\caption{Residence time versus the vertical length of the strip $L_2$
in the case $\delta=1$
and $h=0.4$.
The symbols $\bullet$, $\blacktriangle$, and $\blacksquare$
refer, respectively, to the cases $\varrho_\rr{d}=0,0.4,0.8$.
Note that bullets and black triangle are perfectly coinciding,
this is due to the fact that in this regime the residence time
does not depend that much on the bottom boundary density, provided
it is smaller that $1/2$. In such a case, indeed,
the density profile inside the strip is almost perfectly constant and equal
to $1/2$.
Solid lines are the theoretical prediction based on the birth and death
model.
Open squares are the approximated theoretical prediction
\eqref{ld010} valid in the large drift regime
with $\bar\varrho=1/2$ for the cases $\varrho_\rr{d}=0,0.4$
and
with $\bar\varrho=8/10$ in the case $\varrho_\rr{d}=0.8$.
Indeed, in the first two cases the density profile is almost constantly
equal to $1/2$, while in the last case it is almost constantly equal
to $8/10$.
Open circles are the Mean Field theoretical prediction \eqref{res-mf010}.
}
\label{f:res-el-dr1}
\end{figure}

\begin{figure}[h]
\begin{picture}(200,125)(0,10)
\put(0,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-ph100-dr0.pdf}}}
}
\put(235,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-ph200-dr0.pdf}}}
}
\end{picture}
\caption{Residence time versus the horizontal displacement probability
$h$ in the cases $\delta=0$ and $L_2=100$ (left)
and $L_2=200$ (right).
The symbols $\bullet$, $\blacktriangle$, and $\blacksquare$
refer, respectively, to the cases $\varrho_\rr{d}=0,0.4,0.8$.
Solid lines are the theoretical prediction \eqref{zd020}
from the bottom to the top for the cases $\varrho_\rr{d}=0,0.4,0.8$.
Dotted lines are the Mean Field  prediction \eqref{res-mf020}
from the bottom to the top for the cases $\varrho_\rr{d}=0,0.4,0.8$.
The dotted and the solid lines corresponding to the case
$\varrho_\rr{d}=0$ cannot be distinguished in the picture.
The insets are just a zoom
of part of the data of the same picture.
}
\label{f:res-ph-100-dr0}
\end{figure}

\begin{figure}[h]
\begin{picture}(200,125)(0,10)
\put(0,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-rd100-dr0.pdf}}}
}
\put(235,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-rd200-dr0.pdf}}}
}
\end{picture}
\caption{Residence time versus the bottom boundary density
$\varrho_\rr{d}$
in the cases $\delta=0$ and
$L_2=100$ (left)
and $L_2=200$ (right).
The symbols $\bullet$, $\blacktriangle$, and $\blacksquare$
refer, respectively, to the cases $h=0,0.4,0.8$.
Solid lines are the theoretical prediction \eqref{zd020}
from the bottom to the top for the cases $h=0,0.4,0.8$.
Dotted lines are the Mean Field  prediction \eqref{res-mf020}
from the bottom to the top for the cases $h=0,0.4,0.8$.
The insets are just a zoom
of part of the data of the same picture.
}
\label{f:res-rd-100-dr0}
\end{figure}

\begin{figure}[h]
\begin{picture}(200,375)(0,10)
\put(0,280)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-el-dr0-h0-left.pdf}}}
}
\put(235,280)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-el-dr0-h0-right.pdf}}}
}
\put(0,140)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-el-dr0-h04-left.pdf}}}
}
\put(235,140)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-el-dr0-h04-right.pdf}}}
}
\put(0,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-el-dr0-h08-left.pdf}}}
}
\put(235,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-el-dr0-h08-right.pdf}}}
}
\end{picture}
\caption{Residence time versus the vertical length of the strip $L_2$
in the case $\delta=0$
and $h=0,0.4,0.8$ from the top to the bottom.
The symbols $\bullet$, $\blacktriangle$, and $\blacksquare$
refer, respectively, to the cases $\varrho_\rr{d}=0,0.4,0.8$.
Solid thin lines are the theoretical prediction \eqref{zd020}.
Dotted lines are the Mean Field prediction \eqref{res-mf020}.
The solid thick lines are quadratic fitting of the experimental data:
$3.16\times L_2^2$ in the case $h=0.4$ and $\varrho_\rr{d}=0.4$ and
$10.53\times L_2^2$ in the case $h=0.4$ and $\varrho_\rr{d}=0.8$,
The dotted and the solid lines corresponding to the cases
$\varrho_\rr{d}=0$ cannot be distinguished in the picture.
}
\label{f:res-el-dr0}
\end{figure}

\begin{figure}[h]
\begin{picture}(200,125)(0,10)
\put(0,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-ph-nuova-left.pdf}}}
}
\put(235,0)
{
\resizebox{10cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-ph-nuova-right.pdf}}}
}
\end{picture}
\caption{Residence time versus the horizontal displacement
probability in the case
$L_2=200$, $\delta=0$, and
and $\varrho_\rr{d}=0.1$ (left) and $\varrho_\rr{d}=0.4$ (right).
Solid thin lines are the theoretical prediction \eqref{zd020}.
Dotted lines are the Mean Field prediction \eqref{res-mf020}.
}
\label{f:res-ph-nuova}
\end{figure}

In all these cases, the residence time has been evaluated
by averaging over all the particles entered in the system through
the top boundary after the time $t_\rr{term}$, namely, at
stationarity, and exited through the bottom one at a time
smaller than $t_\rr{max}$.
The simulations have been performed with
$t_\rr{term}=5\times10^5$
and
$t_\rr{max}=5\times10^6$.

To check the dependence of the residence time on the length
of the strip we had to consider a few more cases with larger $L_2$,
viz.\
$L_2=300,400$.
In these cases, due to the length of the strip, we had to use
$t_\rr{term}=10^6$
and increase $t_\rr{max}$ up to $4\times10^7$ in the more
delicate situations ($h=0.8$).
It is important to note that in the cases $L_2=300,400$
and $\varrho_\rr{d}=0.8$,
the initial configuration of the system has been chosen by
populating the system by putting particles
with probability $0.95$. Indeed, by starting the system with all
the sites empty, the dynamics was trapped in a sort of ``meta--stationary"
state with density approximatively constant and slightly larger than
$0.8$.

In all the pictures the vertical bar representing the
statistical error is not visible since it is smaller than
the symbol representing the measured value of the residence time.

We compare the numerical results with the theoretical predictions based
both on the
birth and death analogy and on the Mean Field approach.
In general the birth and death theoretical prediction of the residence time is
the value given by \eqref{art050} with
the stationary measure $\pi$ given by \eqref{art040} where
the jump probabilities $p_i$ and $q_i$ are defined in \eqref{art010}.
Since we could not derive explicit expressions in terms of the
model parameters,
the birth and death theoretical prediction has been computed by performing
sums and products numerically.
In the case $\delta=1$ (see, the discussion in
Section~\ref{s:caso010})
the birth and death theoretical prediction is
the value given by \eqref{hitting-tot} with
the $q_i$'s defined in \eqref{art010}.
In the case $\delta=0$ (see for instance the discussion in
Section~\ref{s:caso020})
the theoretical prediction is given explicitly by \eqref{zd020}.

We find that the match between the theoretical predictions
based both on the birth and death analogy and on the
Mean Field approach and the numerical data
is perfect for any choice of the parameters
of the model provided either $\varrho_\rr{d}=0$
or $\delta=1$ (or both).
To illustrate this fact we first discuss
our data at $\varrho_\rr{d}=0$ and let
$h=0,0.4,0.8$ and $\delta=0,0.2,0.4,0.6,0.8,1$.
Then, we consider $\delta=1$ and let
$h=0,0.4,0.8$ and $\varrho_\rr{d}=0,0.4,0.8$.

Finally, for the region where the match between the theoretical
predictions and the numerical results is not good,
we consider
the worst case from the point of view of drift, namely,
we choose $\delta=0$
and let, again,
$h=0,0.4,0.8$ and $\varrho_\rr{d}=0,0.4,0.8$.
We shall notice that the match between theory and experiment, even
if qualitatively correct, it becomes progressively worse with increasing $\varrho_\rr{d}$.

\subsection{Case $\varrho_\rr{d}=0$}
\label{s:caso000}
\par\noindent
We discuss first the case $\varrho_\rr{d}=0$ and show that here
the theoretical predictions of the residence time
based on both the birth and death analogy and the
Mean Field computation discussed in Section~\ref{s:res}
agree perfectly with the numerical results.

In Figure~\ref{f:res-dr-100} it is shown the dependence
of the residence time on the drift $\delta$ for
$\varrho_\rr{d}=0$ and for different values
of the horizontal displacement probability $h$.
The dependence
of the residence time on the horizontal displacement
probability for different values of the drift
is shown in Figure~\ref{f:res-ph-100}.

The fact that the residence time decreases with the drift and increases
with the horizontal displacement probability is completely
reasonable.
The match between the simulation data and the theoretical predictions is
perfect.
It is remarkable the fact that even the approximated
expression \eqref{ld010} (which in the case $\delta=1$
reduces to \eqref{td010})
gives a perfect estimate of the residence time. This is due to the
fact that for the values of the parameter that we have chosen
the stationary density profile throughout the system
is constantly equal to $1/2$  with very a good approximation.

In Figure~\ref{f:res-el},
the residence time in the case $\varrho_\rr{d}$ has been plotted as
a function of the
length $L_2$ of the strip for different values of the drift and
for $h=0.4$.
It is remarkable to note the striking match between theory and
simulation. In particular the fact that the
behavior is linear (ballistic) for $\delta>0$ and
quadratic (diffusive) for $\delta=0$, see the theoretical
discussion in Section~\ref{s:scaling}, is confirmed by the
numerical experiment.

\subsection{Case $\delta=1$}
\label{s:caso010}
\par\noindent
This is
the case of totally asymmetric simple exclusion rule
along the vertical direction. Here,
particles can never jump upwards.
We show that, for this scenario,
the theoretical predictions of the residence time
based on both the birth and death analogy and the
Mean Field computation discussed in Section~\ref{s:res}
agree perfectly with the numerical results.

We show three pictures:
the dependence of the residence time
on the horizontal displacement probability $h$ (Figure~\ref{f:res-ph-100-dr1}),
the dependence on the bottom boundary density $\varrho_\rr{d}$
(Figure~\ref{f:res-rd-100-dr1}),
and, finally,
the dependence on the length of the strip $L_2$
(Figure~\ref{f:res-el-dr1}).

We do not repeat the discussion in detail. We just mention
that the linearity of the residence time with respect to
the length of the strip
is confirmed by the experimental data and refer the
reader to the caption of the pictures for more specific
comments.

\subsection{Case $\delta=0$}
\label{s:caso020}
\par\noindent
We discuss now the case of a
symmetric simple exclusion rule
along the vertical direction.
As we already mentioned at the beginning of this
section, in this case the match between the theoretical
prediction and the numerical data is only qualitatively good and
quantitavely worst and worst
when $\varrho_\rr{d}$ is increased.

We also remark that, fixed $\varrho_\rr{d}$, the match with the
prediction based on the birth and death analogy is better
at larger values of the horizontal displacement probability $h$.
On the other hand,
fixed $\varrho_\rr{d}$, the match with the
Mean Field prediction is better
at smaller values of the horizontal displacement probability $h$ and
perfect at $h=0$.

The physical interpretation of this fact is that at large
$\varrho_\rr{d}$ particles accumulate
at the bottom exit and, due to bouncing back of particles,
fluctuations are not more negligible. For this reason our theoretical
predictions, which are based only on the stationary shape of the
density profile, are not anymore always efficient.
We shall discuss this point also in Section~\ref{s:caso030}.

We show three pictures:
the dependence of the residence time
on the horizontal displacement probability $h$ (Figure~\ref{f:res-ph-100-dr0}),
the dependence on the bottom boundary density $\varrho_\rr{d}$
(Figure~\ref{f:res-rd-100-dr0}),
and, finally,
the dependence on the length of the strip $L_2$
(Figure~\ref{f:res-el-dr0}).

Note that in this section, since
$\delta=0$, the theoretical prediction based on the birth and death analogy is
given by the explicit formula \eqref{zd020}, whereas the Mean Field
prediction is \eqref{res-mf020}.

The relevant comment now, see Figures~\ref{f:res-ph-100-dr0},
\ref{f:res-el-dr0}, and \ref{f:res-ph-nuova},
is that the match between the Mean Field prediction and
the experiment is perfect at $h=0$, whereas it gets worst and
worst when $h$ is increased.
On the other hand, the birth and death analogy poorly predicts
the residence time behavior
$h=0$, whereas it captures the phenomenon better and better when
$h$ is increased.
This phenomenon can be explained via two mechanisms: bouncing back and
the possibility to bypass clusters of blocking particles.
The former suggests that correlations become important in this
regime so that a model based only on the average stationary
density profile cannot explain the behavior of the system.
On the other hand, the latter phenomenon suggests that the effect
of bouncing back is milder when the horizontal displacement probability is
larger, since particles have a good chance to avoid blocking
clusters.

In is interesting to remark that
the Mean Field expression \eqref{res-mf020} for the case $\delta=0$,
and $L_2$ large,
is given by the expression \eqref{zd020} predicted by the
birth and death analogy plus a term that can be ascribed
the the two point correlations, see \cite{fedders}.
We could then guess that the correct expression of the residence time
could be the birth and death prediction plus the two point
correlation contribution weighted by a function depending on $h$
tending to $1$ for $h\to0$ and to $0$ for $h\to1$.

\subsection{Non--monotonic behavior in the bouncing back regime}
\label{s:caso030}
\par\noindent
As it has been seen in the previous section
the residence time is typically an increasing function of the
horizontal displacement probability.
This is an obvious fact. Indeed, when $h$ is increased
particles spend a lot of time in performing horizontal
jumps which are a waste of time in the run towards the bottom
exit.

In this section we show that in the regime in which
the bouncing back phenomenon is present a small not zero
horizontal probability displacement can favour the
exit of the particles.

We have performed the following simulations:
$\varrho_\rr{u}=1$ (as always),
$L_1=100$,
$L_2=200$,
$\delta=0$,
\begin{displaymath}
\varrho_\rr{d}=0,0.05,\dots,0.5,0.6,0.7,0.8,\;\;
\;\;\;\textrm{ and }\;\;\;
h=0,0.005,0.010,\dots,0.125\;\;.
\end{displaymath}
As before,
in all these cases, the residence time has been evaluated
by averaging over all the particles entered in the system through
the top boundary after the time $t_\rr{term}$, namely, at
stationarity, and exited through the bottom one at a time
smaller than $t_\rr{max}$.
The simulations have been performed with
$t_\rr{term}=1\times10^6$
and
$t_\rr{max}=1\times10^7$.

\begin{figure}[htbp]
\begin{picture}(200,450)(0,10)
\put(0,450)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono000.pdf}}}
}
\put(160,450)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono005.pdf}}}
}
\put(320,450)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono010.pdf}}}
}
\put(0,350)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono015.pdf}}}
}
\put(160,350)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono020.pdf}}}
}
\put(320,350)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono025.pdf}}}
}
\put(0,250)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono030.pdf}}}
}
\put(160,250)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono035.pdf}}}
}
\put(320,250)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono040.pdf}}}
}
\put(0,150)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono045.pdf}}}
}
\put(160,150)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono050.pdf}}}
}
\put(320,150)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono060.pdf}}}
}
\put(0,50)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono070.pdf}}}
}
\put(160,50)
{
\resizebox{9cm}{!}{\rotatebox{0}{\includegraphics{fig-cms001-res-notmono080.pdf}}}
}
\end{picture}
\vskip -0.5 cm
\caption{Residence time versus the horizontal displacement
probability $h$ in the case $\delta=0$
and $L_2=200$.
From the left to the right and from the top to the bottom
the cases
$\varrho_\rr{d}=0,0.05,0.01,\dots,0.045,0.5,0.6,0.7,0.8$
are depicted.
}
\label{f:res-notmono}
\end{figure}

In Figure~\ref{f:res-notmono}, the residence time is plotted
as function of the horizontal displacement probability. It is
remarkable the presence of a minimum at small values of $h$.
This fact can be interpreted as follows: in the bouncing
back regime, namely, when $\varrho_\rr{d}$ is high and $\delta$ low,
a particle
can find a blocking cluster of particles
in its way out through the bottom boundary.
In such a situation, then,
having a larger horizontal displacement probability can help
the particle to bypass the obstacle. Obviously, if $h$ is increased
too much this effect disappears due to the time wasted by the
particles in horizontal movements.

\section{Conclusions}
\label{s:conclusioni}
\par\noindent
We focused our attention on the study of the simplest
2D model that mimics the flow of particles
in a straight pipe under the effect of different driving boundary
conditions and external fields.
We studied the residence time, i.e., the typical time a particle
entering the strip at stationarity from the
top boundary needs to exit through the bottom one,
under the assumption that
particles in the strip interact only via hard--core exclusion and
vertical boundaries are reflecting.

We explored the
dependence of the residence time on the
external driving force,
length of the strip,
horizontal diffusion, and
boundary conditions.
We have shown that, in almost all the considered regimes,
the mean residence time is equal to the average time needed to cross the
strip by a particle performing a random walk in a background
prescribed by the stationary density profile of the original model.
In this way, in particular,
we recover the structure of the fluxes as well as the residence times
proven mathematically in 1D in \cite{DEHP}.

A completely macroscopic point of view has also been alternatively adopted,
i.e. a Mean Field computation of the residence time has been performed
by connecting such a quantity to the stationary current in the
system. It has been shown that also this point of view
porvides very satisfactory results.

This picture fails to be correct in the case in which there is
a particle accumulation close to the bottom boundary (exit).
In this regime, we  discover new effects
that are consequence of the two--dimensionality of the system.
The most relevant is the non--monotonic
behavior in changes in the horizontal displacement
probability in the bouncing back regime.

A second very interesting result is the fact that in this
bouncing back regime the Mean Field approach gives a very good prediction
of the residence time in the single file (zero horizontal
displacement probability) case. On the other hand,
if the horizontal displacement probability is large (close to one)
the Mean Field prediction is poor and the one based on the analogy
with the birth and death model on the background of the stationary
density profile yield a very good prediction.

This particle accumulation
situation can be realized artificially by inserting
obstacles in the strip. We then expect interesting non--linear
phenomena to show up. We shall study this regime in a
follow--up paper.

\newpage
\newpage
\newpage

redpajama_set_name: RedPajamaArXiv