# On Stable Constant Mean Curvature Spheres of
$\mathbb{H}^{n}\times\mathbb{R}$ and $\mathbb{S}^{n}\times\mathbb{R}$ and
their Uniqueness as Isoperimetric Hypersurfaces

Ronaldo F. de Lima, Maria F. Elbert and Barbara Nelli  Depto. de Matemática -
Univ. Federal do Rio Grande do Norte - RN, Brazil<EMAIL_ADDRESS>Instituto de Matemática, Universidade Federal do Rio de Janeiro - RJ, Brazil.
<EMAIL_ADDRESS>Dipartimento di Matematica Pura e Applicata, Universitá di
L’Aquila, Italy<EMAIL_ADDRESS>
###### Abstract.

We consider the rotational constant mean curvature spheres of
$\mathbb{H}^{n}\times\mathbb{R}$ obtained by Hsiang and Hsiang and show that
they are nested. We apply this property to prove the uniqueness of these
spheres as isoperimetric hypersurfaces of $\mathbb{H}^{n}\times\mathbb{R},$
filling in a gap in the original proof given by Hsiang and Hsiang. We also
give a direct proof of their stability, extending the result by Souam, who
considered the case $n=2.$ Analogous results are obtained for the class of the
rotational constant large mean curvature spheres of
$\mathbb{S}^{n}\times\mathbb{R}$ constructed by Pedrosa.

###### Key words and phrases:

mean curvature – stability – isoperimetric problem.

###### 2020 Mathematics Subject Classification:

53A10 (primary), 53B20 (secondary).

##  1\. Introduction

The isoperimetric problem is unarguably one of the most classical subjects in
differential geometry. In its more general setting, it can be stated as
follows: given an $(n+1)$-dimensional Riemannian manifold $\mathbb{M}^{n+1}$
and a positive real number $\mathcal{V},$ find a compact set $\varOmega$ in
$\mathbb{M}^{n+1}$ of volume $\mathcal{V}$ whose boundary $\partial\varOmega$
has least area. By _volume_ and _area_ we mean the $(n+1)$-dimensional and
$n$-dimensional Riemannian measures, respectively. Such a compact set
$\varOmega$ and its boundary $\partial\varOmega$ are then called an
_isoperimetric region_ and an _isoperimetric hypersurface_ , respectively.

The fundamental results of Almgren [1] ensure the existence of isoperimetric
regions of any prescribed volume $\mathcal{V}<{\rm
vol}\,\mathbb{M}\leq+\infty$ for a wide range of Riemannian manifolds
$\mathbb{M},$ including all the compact and all the homogeneous ones. In
addition, any connected component of the boundary of an isoperimetric region
has constant mean curvature at all of its regular points. Consequently,
present investigations on the isoperimetric problem are mainly focused on
regularity and uniqueness of isoperimetric regions (see [9] and the references
therein for a detailed account on the isoperimetric problem).

Compact hypersurfaces of constant mean curvature (CMC, for short) in
Riemannian manifolds are known to be critical points of the area functional
$\mathcal{A}$ for volume-preserving variations. The CMC hypersurface is called
_stable_ if it minimizes $\mathcal{A}$ up to second order for such variations.

In particular, any regular compact isoperimetric hypersurface is necessarily
stable. On the other hand, there exist stable CMC hypersurfaces in certain
Riemannian manifolds which are not isoperimetric (see, e.g., [2, 10]).

In [6], Hsiang and Hsiang showed that, for any $H>n-1\geq 1,$ there exists an
embedded rotational isoperimetric sphere $\varSigma_{H}$ of constant (non
normalized) mean curvature $H$ in $\mathbb{H}^{n}\times\mathbb{R}.$
Furthermore, any compact embedded hypersurface
$\mathbb{H}^{n}\times\mathbb{R}$ of constant mean curvature $H>0$ is
necessarily congruent to a rotational sphere $\varSigma_{H}.$ In particular,
$H$ must be larger than $n-1.$ They also showed that the isoperimetric region
$\varOmega_{H}$ determined by $\varSigma_{H}$ is the unique solution to the
isoperimetric problem for $\mathcal{V}={\rm vol}\,\varSigma_{H}.$ Lamentably,
their argument in the proof of this latter property is incorrect (cf. Remark 2
in Section 4).

Similar results on CMC hypersurfaces of $\mathbb{S}^{n}\times\mathbb{R}$ and
their relation with the isoperimetric problem were obtained by Pedrosa [8].
More specifically, he proved that there exists a rotational sphere of constant
mean curvature $H$ in $\mathbb{S}^{n}\times\mathbb{R}$ for any given $H>0.$
However, the region determined by such a sphere is a unique solution to the
isoperimetric problem only if $H$ is sufficiently large.

In this paper, we provide new proofs for the uniqueness of the regions
determined by the rotational spheres of $\mathbb{H}^{n}\times\mathbb{R}$ and
$\mathbb{S}^{n}\times\mathbb{R}$ as solutions to the isoperimetric problem.
Our proofs are different in nature because they are grounded on the fact that
all rotational CMC spheres of $\mathbb{H}^{n}\times\mathbb{R}$ and those of
$\mathbb{S}^{n}\times\mathbb{R}$ with sufficiently large mean curvature are
nested.

To be more precise, let us introduce some notation. First, denote by
$\mathbb{Q}_{\epsilon}^{n}$ the simply connected $n(\geq 2)$-space form of
constant sectional curvature $\epsilon\in\\{-1,1\\}$ (i.e.,
$\mathbb{H}^{n}=\mathbb{Q}_{-1}^{n}$ and $\mathbb{S}^{n}=\mathbb{Q}_{1}^{n}$).
Set $C_{\epsilon}:=n-1$ for $\epsilon=-1$ and $C_{\epsilon}:=0$ for
$\epsilon=1,$ and write $\mathscr{F}_{\epsilon}$ for the family of all
rotational spheres $\varSigma_{H}$ in
$\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R}$ of constant mean curvature
$H>C_{\epsilon}.$ Assume further that all spheres of $\mathscr{F}_{\epsilon}$
have the same rotational axis
$\\{o\\}\times\mathbb{R}\subset\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R}$ and
the same point $o\in\mathbb{Q}_{\epsilon}^{n}$ of minimal height, and denote
by $\varOmega_{H}$ the compact region determined by $\varSigma_{H}.$ In this
setting, the nesting property of the rotational CMC spheres $\varSigma_{H}$ of
$\mathscr{F}_{\epsilon}$ can be stated as follows.

######  Theorem 1.

There exists a nonnegative constant $\alpha\geq C_{\epsilon}$ such that,
whenever $\varSigma_{H},\varSigma_{H^{*}}\in\mathscr{F}_{\epsilon}$ satisfy
$H>H^{*}>\alpha,$ one has $\varOmega_{H}\subset\varOmega_{H^{*}}.$ Moreover,
the equality $\alpha=C_{\epsilon}$ holds for $\epsilon=-1,$ so that the whole
family $\mathscr{F}_{\epsilon}$ is nested in this case.

We point out that the constant $\alpha$ in the above theorem is strictly
positive for $n=2$ and $\epsilon=1,$ that is, the family of rotational CMC
spheres of $\mathbb{S}^{2}\times\mathbb{R}$ is not entirely nested (see Remark
1 in Section 4). We believe that this property holds for the rotational CMC
spheres of $\mathbb{S}^{n}\times\mathbb{R},$ $n>2,$ although we have no proof
for it.

As a direct consequence of Theorem 1 and the main results in [6] and [8], we
have the following

######  Corollary 1.

Given $\mathcal{V}>0$ (resp. a sufficiently small $\mathcal{V}>0$), there
exists a unique embedded rotational CMC sphere of $\mathscr{F}_{\epsilon}$ in
$\mathbb{H}^{n}\times\mathbb{R}$ (resp. $\mathbb{S}^{n}\times\mathbb{R}$)
which bounds a region $\varOmega$ of volume $\mathcal{V}.$ Consequently,
$\varOmega$ is the unique solution to the isoperimetric problem for prescribed
volume $\mathcal{V}.$

Souam [10] considered the rotational CMC spheres of
$\mathbb{H}^{2}\times\mathbb{R}$ and provided an ingenious direct proof of
their stability. He also proved that the rotational CMC spheres
$\varSigma_{H}$ of $\mathbb{S}^{2}\times\mathbb{R}$ are stable if and only if
$H\geq H_{0},$ where $H_{0}\approx 0.18.$ This result should be compared with
the one by Pedrosa in [8], where he shows that $\varSigma_{H}$ is
isoperimetric if and only if $H\geq H_{1}\approx 0.33>H_{0}.$ Indeed, from
these inequalities we conclude that all rotational spheres $\varSigma_{H}$ in
$\mathbb{S}^{2}\times\mathbb{R}$ such that $H_{0}\leq H<H_{1}$ are stable and
non isoperimetric.

A suitable application of Theorem 1 allows us to obtain the following
extension of Souam’s result (up to the statement on $H_{0}$):

######  Corollary 2.

For all $H>n-1$ (resp. sufficiently large $H>0$), the rotational sphere
$\varSigma_{H}$ of constant mean curvature $H$ of
$\mathbb{H}^{n}\times\mathbb{R}$ (resp. $\mathbb{S}^{n}\times\mathbb{R}$) is
stable.

The paper is organized as follows. In Section 2, we briefly discuss stability
of CMC hypersurfaces of $\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R}.$ In
Section 3, we establish two key lemmas to be applied in the proofs of our main
results. Finally, in Section 4, we prove Theorem 1 and Corollaries 1 and 2.

##  2\. Stability of CMC hypersurfaces of
$\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R}.$

Let $\varSigma$ be a compact oriented hypersurface of
$\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R},$ $n\geq 2,$ and recall that the
(non normalized) mean curvature function $H$ of $\varSigma$ is the trace of
its shape operator, that is, the sum of its principal curvatures. As is well
known, if $\varSigma$ has constant mean curvature, its stability is equivalent
to the non negativeness of the quadratic form $Q$ defined by

$Q(u,u)=-\int_{\varSigma}uLu,\,\,\,u\in C^{\infty}(\varSigma),$

for functions $u\in C^{\infty}(\varSigma)$ which satisfy
$\int_{\varSigma}u=0.$ Here, $L$ is the (self-adjoint) _Jacobi operator_ on
$C^{\infty}(\varSigma),$ which is defined as

$Lu:=\|\nabla u\|^{2}-({\rm Ric}(N)+\|\sigma\|^{2}),$

being $\nabla u$ the gradient of $u$, $\sigma$ the second fundamental form of
$\varSigma,$ and ${\rm Ric}(N)$ the Ricci curvature of $\varSigma$ evaluated
on its unit normal field $N.$

Suppose that $\varSigma=\varphi_{0}(M),$ where $\varphi_{0}\colon
M\to\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R}$ is an immersion of a compact
$n$-dimensional manifold $M$ into $\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R}.$
Given a smooth variation $\varphi(t,.)$ of $\varphi_{0}=\varphi(0,.),$ we have
that the Jacobi operator $L$ satisfies the following identity:

(1) $\frac{\partial H}{\partial t}(t,.)|_{t=0}=L\langle\xi,N\rangle,$

where $\xi$ is the _variational field_ on $\varSigma$ determined by
$\varphi(t,.),$ that is,

$\xi:=\frac{\partial\varphi}{\partial t}(t,.)|_{t=0}.$

In [7], Koiso established a fundamental result on stability of CMC surfaces
with boundary in Euclidean space $\mathbb{R}^{3}.$ As pointed out by Souam
[10], her result is easily extended to compact oriented CMC surfaces with
empty boundary in any Riemannian $3$-manifold. In fact, as seen in [3], where
the authors established a broad extension of Koiso’s Theorem, her arguments
hold in arbitrary dimension as well, giving the following result.

######  Generalized Koiso’s Theorem.

Let $\varSigma$ be a smooth compact immersed oriented CMC hypersurface in a
complete $(n+1)$-dimensional Riemannian manifold $\mathbb{M}^{n+1}.$ Denoting
by $\lambda_{1}$ and $\lambda_{2}$ the first and second eigenvalues of the
Jacobi operator $L$ on $\varSigma,$ respectively, suppose that the following
conditions are satisfied:

  * i)

$\lambda_{1}<0=\lambda_{2}.$

  * ii)

$\int_{\varSigma}g=0$ for any eigenfunction $g$ associated to $\lambda_{2}=0.$

Then, there exists a unique smooth function $u\in(\ker L)^{\perp}$ (orthogonal
with respect to the $L^{2}$ norm) which satisfies $Lu=1$ and determines the
stability condition of $\varSigma;$ namely, $\varSigma$ is stable if and only
if $\int_{\varSigma}u\geq 0.$

##  3\. Two Key Lemmas

The proof of Theorem 1 will be based on the fundamental properties of the
solutions of the first order linear ODE:

(2) $y^{\prime}=-(n-1)\cot_{\epsilon}(s)y+1,\,\,\,s\in(0,s_{0}),$

where $n>1$ is an integer, $\cot_{\epsilon}:=\cos_{\epsilon}/\sin_{\epsilon},$
and $\cos_{\epsilon},\,\sin_{\epsilon}$ are as in Table 1.

Function | $\epsilon=1$ | $\epsilon=-1$
---|---|---
$\cos_{\epsilon}$ | $\cos$ | $\cosh$
$\sin_{\epsilon}$ | $\sin$ | $\sinh$
Table 1. Definition of $\cos_{\epsilon}$ and $\sin_{\epsilon}.$

We will assume that $s_{0}\in(0,\mathcal{R}_{\epsilon}),$ where

(3)
$\mathcal{R}_{\epsilon}:=\left\\{\begin{array}[]{ccl}+\infty&\text{if}&\epsilon=-1,\\\
\pi&\text{if}&\epsilon=1.\end{array}\right.$

In this setting, it is easily checked that the function

(4)
$\varphi_{\epsilon}(s):=\frac{1}{\sin_{\epsilon}^{n-1}(s)}\int_{0}^{s}{\sin_{\epsilon}^{n-1}(u)}du,\,\,\,s\in(0,s_{0}),$

is a solution to (2) which extends smoothly to both $s=0$ and $s=s_{0},$ and
satisfies $\varphi_{\epsilon}(0)=0.$ Hence, $\varphi_{\epsilon}$ is defined in
the closed interval $[0,s_{0}].$

######  Lemma 1.

The function $\varphi_{\epsilon}\colon[0,s_{0}]\rightarrow\mathbb{R}$ has the
following properties:

  * i)

$\varphi_{\epsilon}^{\prime}(s)>0$ for all $s\in(0,s_{0}].$

  * ii)

$\varphi_{\epsilon}^{\prime}(0)=1/n.$

  * iii)

$\epsilon\varphi_{\epsilon}^{\prime\prime}(s)>0\,\forall s\in(0,s_{0}].$

###### Proof.

To prove (i), we first observe that $\varphi_{\epsilon}$ is increasing near
$0,$ since it vanishes at $0$ and is positive in $(0,s_{0}).$ Assume that
$\varphi_{\epsilon}$ is not increasing in $(0,s_{0}).$ In this case,
$\varphi_{\epsilon}$ has a first critical point $s_{1}$ in $(0,s_{0})$ which
is necessarily a local maximum.

Since $\varphi_{\epsilon}$ is a solution to (2), one has

(5)
$\varphi_{\epsilon}^{\prime}=-(n-1)\cot_{\epsilon}(s)\varphi_{\epsilon}+1,\,\,\,s\in(0,s_{0}).$

So,
$\varphi_{\epsilon}^{\prime\prime}(s_{1})=(n-1)\csc_{\epsilon}^{2}(s_{1})\varphi_{\epsilon}(s_{1})>0,$
which gives that $s_{1}$ is a local minimum — a contradiction. This proves
(i).

Regarding (ii), we have from (4) and (5) that

(6)
$\varphi_{\epsilon}^{\prime}(s)=-(n-1)\frac{\cos_{\epsilon}(s)I(s)}{\sin_{\epsilon}^{n}(s)}+1,$

where $I(s):=\int_{0}^{s}{\sin_{\epsilon}^{n-1}(u)}du.$ In addition, we have

$\displaystyle\lim_{s\rightarrow
0}\frac{\cos_{\epsilon}(s)I(s)}{\sin_{\epsilon}^{n}(s)}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\lim_{s\rightarrow 0}\left(\frac{-\epsilon
I(s)}{\sin_{\epsilon}^{n-2}(s)}+\cos(s)\right)$ $\displaystyle=$
$\displaystyle\frac{1}{n}\lim_{s\rightarrow
0}\left(-\epsilon\varphi_{\epsilon}(s)\sin_{\epsilon}(s)+\cos_{\epsilon}(s)\right)=1/n,$

which proves (ii).

To prove (iii), we first observe that, for $n=2,$ one has from equality (4)
that
$\varphi_{\epsilon}(s)=\epsilon(1-\cos_{\epsilon}(s))/\sin_{\epsilon}(s).$ In
this case, it can be easily checked that
$\epsilon\varphi_{\epsilon}^{\prime\prime}>0$ in $(0,s_{0}].$ So, we can
assume $n>2.$ Under this hypothesis, let us show that

(7)
$\omega_{\epsilon}(s):=\sin_{\epsilon}^{n}(s)-(n-1)\cos_{\epsilon}(s)I(s)>0\,\,\forall
s\in(0,s_{0}].$

Since $\omega_{\epsilon}(0)=0,$ it suffices to prove that
$\omega_{\epsilon}^{\prime}>0$ in $(0,s_{0}].$ From a direct computation, we
get

$\omega_{\epsilon}^{\prime}(s)=\sin_{\epsilon}(s)(\epsilon(n-1)I(s)+\cos_{\epsilon}(s)\sin_{\epsilon}^{n-2}(s)).$

Setting
$\mu_{\epsilon}(s):=\epsilon(n-1)I(s)+\cos_{\epsilon}(s)\sin_{\epsilon}^{n-2}(s),$
one has $\omega_{\epsilon}^{\prime}(s)=\sin_{\epsilon}(s)\mu_{\epsilon}(s)$.
Besides, $\mu_{\epsilon}(0)=0$ and

$\mu_{\epsilon}^{\prime}(s)=(n-2)\sin_{\epsilon}^{n-3}(s)(\cos_{\epsilon}^{2}(s)+\epsilon\sin_{\epsilon}^{2}(s))=(n-2)\sin_{\epsilon}^{n-3}(s)>0\,\,\,\forall
s\in(0,s_{0}].$

Therefore, $\mu_{\epsilon},$ and so $\omega_{\epsilon}^{\prime},$ is positive
in $(0,s_{0}].$ This finishes the proof of (7).

Now, we have from (6) that

(8)
$\varphi_{\epsilon}^{\prime\prime}(s)=(n-1)\frac{\varLambda(s)}{\sin_{\epsilon}^{n+1}(s)},$

where $\varLambda$ is the function

$\varLambda(s):=\epsilon\sin_{\epsilon}^{2}(s)I(s)-\cos_{\epsilon}(s)(\sin_{\epsilon}^{n}(s)-n\cos_{\epsilon}(s)I(s)).$

Again, we have $\varLambda(0)=0.$ In addition, it can be easily checked that

$\varLambda^{\prime}(s)=2\epsilon\sin_{\epsilon}(s)\omega_{\epsilon}(s),$

which yields $\epsilon\varLambda^{\prime}>0$ in $(0,s_{0}].$ Hence,
$\epsilon\varLambda(s)$ is positive in $(0,s_{0}].$ From this and (8), we
finally have $\epsilon\varphi_{\epsilon}^{\prime\prime}>0$ in $(0,s_{0}],$ as
we wished to prove. ∎

######  Lemma 2.

Given $s_{0}>0,$ let $F_{\epsilon}:[0,s_{0}]\rightarrow\mathbb{R}$ be the
function defined by

(9)
$F_{\epsilon}(s):=\varphi_{\epsilon}(s_{0})\varphi_{\epsilon}^{\prime}(s)-\varphi_{\epsilon}(s)\varphi_{\epsilon}^{\prime}(s_{0}).$

Then, $F_{\epsilon}$ is positive for $\epsilon=-1,$ and is positive for
$\epsilon=1$ provided that $s_{0}$ is sufficiently small.

###### Proof.

We have from Lemma 1-(ii) that
$F_{\epsilon}(0)=\varphi_{\epsilon}(s_{0})\varphi_{\epsilon}^{\prime}(0)=\varphi_{\epsilon}(s_{0})/n>0.$
Since $F_{\epsilon}(s_{0})=0,$ to conclude that $F_{\epsilon}$ is nonnegative,
it suffices to prove that $F_{\epsilon}$ is decreasing.

For the derivative of $F_{\epsilon},$ we have

$F_{\epsilon}^{\prime}(s)=\varphi_{\epsilon}(s_{0})\varphi_{\epsilon}^{\prime\prime}(s)-\varphi_{\epsilon}^{\prime}(s)\varphi_{\epsilon}^{\prime}(s_{0}).$

Thus, if $\epsilon=-1,$ we have from Lemma 1-(iii) that
$F_{\epsilon}^{\prime}<0$ in $(0,s_{0}],$ which implies that $F_{\epsilon}$ is
decreasing in this case.

Assume now that $\epsilon=1.$ Then, $\varphi_{\epsilon}$ satisfies:

$\varphi_{\epsilon}^{\prime}(s)=a(s)\varphi_{\epsilon}(s)+1,\,\,\,a(s)=-(n-1)\cot
s.$

From Lemma 1-(iii), $\varphi_{\epsilon}^{\prime\prime}>0.$ Thus,
$\varphi_{\epsilon}^{\prime}(s)>\varphi_{\epsilon}^{\prime}(0)=1/n\,\forall
s\in(0,s_{0}].$ In particular,

$n\varphi_{\epsilon}^{\prime}>1=\varphi_{\epsilon}^{\prime}-a\varphi_{\epsilon},$

which implies that

(10)
$\varphi_{\epsilon}^{\prime}(s)>-\frac{a(s)\varphi_{\epsilon}(s)}{n-1}=\cot(s)\varphi_{\epsilon}(s).$

Suppose that $(0,s_{0}]\subset(0,\pi/2).$ In this case, it follows from (10)
that, for all $s\in(0,s_{0}],$ the following inequalities hold:

(11)
$a(s)\varphi_{\epsilon}^{\prime}(s)<-\frac{a^{2}(s)\varphi_{\epsilon}(s)}{n-1}\phantom{-}\quad\text{and}\quad-\varphi_{\epsilon}^{\prime}(s_{0})\varphi_{\epsilon}^{\prime}(s)<\frac{a(s)\varphi_{\epsilon}(s)\varphi_{\epsilon}^{\prime}(s_{0})}{n-1}\cdot$

Now, we can estimate the derivative of $F$ as follows:

(12) $\displaystyle F_{\epsilon}^{\prime}$ $\displaystyle=$
$\displaystyle\varphi_{\epsilon}(s_{0})\varphi_{\epsilon}^{\prime\prime}-\varphi_{\epsilon}^{\prime}(s_{0})\varphi_{\epsilon}^{\prime},$
$\displaystyle=$
$\displaystyle\varphi_{\epsilon}(s_{0})(a\varphi_{\epsilon}+1)^{\prime}-\varphi_{\epsilon}^{\prime}(s_{0})\varphi_{\epsilon}^{\prime}$
$\displaystyle=$
$\displaystyle\varphi_{\epsilon}(s_{0})(a^{\prime}\varphi_{\epsilon}+a\varphi_{\epsilon}^{\prime})-\varphi_{\epsilon}^{\prime}(s_{0})\varphi_{\epsilon}^{\prime}$
$\displaystyle<$
$\displaystyle\varphi_{\epsilon}(s_{0})\varphi_{\epsilon}\left(a^{\prime}-\frac{a^{2}}{n-1}+\frac{a}{n-1}\varphi_{\epsilon}^{\prime}(s_{0})\right).$

From (10),
$\varphi_{\epsilon}^{\prime}(s_{0})>\cot(s_{0})\varphi_{\epsilon}(s_{0}).$ In
addition, we have $a/(n-1)=-\cot(s)$ and

$a^{\prime}(s)-\frac{a^{2}(s)}{n-1}=(n-1)\left(\frac{1}{\sin^{2}(s)}-\frac{\cos^{2}(s)}{\sin^{2}(s)}\right)=n-1.$

These facts, together with (12), yield

(13)
$F_{\epsilon}^{\prime}(s)<\varphi_{\epsilon}(s_{0})\varphi_{\epsilon}(s)\left(n-1-\varphi_{\epsilon}(s_{0})\cot(s_{0})\cot(s)\right).$

For the last summand in the brackets, we have

(14)
$\varphi_{\epsilon}(s_{0})\cot(s_{0})\cot(s)>\varphi_{\epsilon}(s_{0})\cot^{2}(s_{0})=\frac{\varphi_{\epsilon}(s_{0})}{\tan^{2}(s_{0})}\cdot$

By taking the limit of this fraction as $s_{0}\to 0,$ we get

(15) $\lim_{s_{0}\to
0}\frac{\varphi_{\epsilon}(s_{0})}{\tan^{2}(s_{0})}=\lim_{s_{0}\to
0}\frac{\varphi_{\epsilon}^{\prime}(s_{0})}{{2\tan(s_{0})\sec^{2}(s_{0})}}=\lim_{s_{0}\to
0}\frac{1}{2n\tan(s_{0})\sec^{2}(s_{0})}=+\infty,$

where, in the second equality, we used Lemma 1-(ii).

From (13), (14), and (15), we conclude that, for a sufficiently small
$s_{0}\in(0,\pi/2),$ the inequality $F_{\epsilon}^{\prime}(s)<0$ holds for all
$s\in(0,s_{0}],$ as we wished to prove. ∎

##  4\. Proofs of Theorem 1 and Corollaries 1 and 2

In this section, we shall approach the rotational CMC spheres $\varSigma_{H}$
of $\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R}$ as in [4], where the more
general case of hypersurfaces of constant higher order mean curvature was
considered (see also [5]).

Firstly, we point out that each sphere $\varSigma_{H}$ is symmetric with
respect to a horizontal hyperplane $P$ of
$\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R}.$ In addition, the lower hemisphere
of $\varSigma_{H}$ determined by $P$ is a radial graph $\varSigma_{H}^{-}$
over an open geodesic ball
$B(o,s_{0})\subset\mathbb{Q}_{\epsilon}^{n}\times\\{0\\}$ centered at a fixed
point $o\in\mathbb{Q}_{\epsilon}^{n}$ and of radius
$s_{0}\in(0,\mathcal{R}_{\epsilon}),$ where $\mathcal{R}_{\epsilon}$ is as in
(3) (Fig. 1).

Figure 1. A rotational CMC hemisphere in
$\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R}.$

The (radial) height function of $\varSigma_{H}$ is

(16)
$\phi(s)=\int_{0}^{s}\frac{\varrho(u)}{\sqrt{1-\varrho^{2}(u)}}du,\,\,\,\,s\in[0,s_{0}),$

where $\varrho\colon[0,s_{0})\to\mathbb{R}$ is defined as

(17) $\varrho(s):=H\varphi_{\epsilon}(s),\,\,\,s\in[0,s_{0}),$

being $\varphi_{\epsilon}$ the function defined in (4) (cf. [4, Section 4]).

Writing $N$ for the unit normal to $\varSigma_{H}^{-}$, and $\partial_{t}$ for
the gradient of the height function of
$\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R},$ it can be proved that the _angle
function_ $\varTheta:=\langle N,\partial_{t}\rangle$ of $\varSigma_{H}^{-}$
relates with the function $\varrho$ as

$\varTheta^{2}=1-\varrho^{2}.$

Since $\varphi_{\epsilon}$ is a solution to (2), we have that $\varrho$
satisfies:

(18)
$\varrho^{\prime}(s)=-(n-1)\cot_{\epsilon}(s)\varrho(s)+H,\,\,\,s\in[0,s_{0}).$

Also, $\varrho$ extends to $s_{0}$ and satisfies:

$0=\varrho(0)<\varrho(s)<1=\varrho(s_{0})\,\,\,\forall s\in(0,s_{0}).$

In this setting, the horizontal hyperplane $P$ of symmetry of $\varSigma_{H}$
is given by

$P=\mathbb{Q}_{\epsilon}^{n}\times\\{\phi(s_{0})\\}.$

This follows from the fact that $\varrho(s_{0})=1.$ It should also be
mentioned that any CMC sphere $\varSigma_{H}$ is strictly convex, except for
the case $\epsilon=1$ and $0<H\leq 1/S(n),$ where
$S(n)=\int_{0}^{\pi/2}\sin^{n-1}(s)ds\leq 1.$

###### Proof of Theorem 1.

Keeping the above notation, assume $H>H^{*}$ and denote the height and
$\varrho$ functions of $\varSigma_{H^{*}}^{-}$ by $\phi^{*}$ and
$\varrho^{*},$ respectively. Let $s_{0}^{*}>0$ be such that
$\varrho^{*}(s_{0}^{*})=1$ and set $s_{1}=\min\\{s_{0},s_{0}^{*}\\}.$ Under
these hypotheses, we have from (17) that $\varrho(s)>\varrho^{*}(s)\,\forall
s\in(0,s_{1}),$ which, together with (16), yields

(19) $\phi(s)>\phi^{*}(s)\,\,\,\forall s\in(0,s_{1}).$

Since the $\varrho$ and $\varTheta$ functions relate as
$\varTheta^{2}=1-\varrho^{2},$ we also have

$\varTheta(s)<\varTheta^{*}(s)\,\,\,\forall s\in(0,s_{1}),$

which implies that $s_{0}<s_{0}^{*}$ (i.e., $s_{1}=s_{0}$), for
$\varTheta(0)=\varTheta^{*}(0)=1$ and
$\varTheta(s_{0})=\varTheta^{*}(s_{0}^{*})=0.$ So, it remains to prove that
(see Fig. 2)

(20) $\phi(s_{0})\leq\phi^{*}(s_{0}^{*}).$

Figure 2. Nested CMC hemispheres in
$\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R}.$

To verify (20), let us consider $s_{0}$ as a variable and show that the height
function $\phi$ is increasing with respect to $s_{0}.$ In this setting, we
have from (17) that

(21) $\varrho(s,s_{0}):=H(s_{0})\varphi_{\epsilon}(s)\quad\text{and}\quad
H(s_{0})=\frac{1}{\varphi_{\epsilon}(s_{0})}\cdot$

These equalities, together with (16), yield

$\phi(s_{0})=\int_{0}^{s_{0}}\frac{\varrho(s,s_{0})}{\sqrt{1-\varrho^{2}(s,s_{0})}}ds.$

Calling $J(s,s_{0})$ the integrand of this integral, one has from the Leibniz
rule that

(22) $\displaystyle\frac{\partial\phi}{\partial s_{0}}(s_{0})$
$\displaystyle=$ $\displaystyle\int_{0}^{s_{0}}\left(\frac{\partial
J}{\partial s}(s,s_{0})+\frac{\partial J}{\partial s_{0}}(s,s_{0})\right)ds$
$\displaystyle=$
$\displaystyle\int_{0}^{s_{0}}\left(\frac{\partial\varrho}{\partial
s}(s,s_{0})+\frac{\partial\varrho}{\partial
s_{0}}(s,s_{0})\right)\Psi(s,s_{0})ds,$

where $\Psi$ is a positive function.

From the second equality in (21), one has

$\frac{\partial H}{\partial
s_{0}}=-\frac{\varphi_{\epsilon}^{\prime}(s_{0})}{\varphi_{\epsilon}^{2}(s_{0})}\cdot$

Therefore,

(23) $\displaystyle\frac{\partial\varrho}{\partial
s}(s,s_{0})+\frac{\partial\varrho}{\partial s_{0}}(s,s_{0})$ $\displaystyle=$
$\displaystyle\frac{\varphi_{\epsilon}^{\prime}(s)}{\varphi_{\epsilon}(s_{0})}-\frac{\varphi_{\epsilon}^{\prime}(s_{0})}{\varphi_{\epsilon}^{2}(s_{0})}\varphi_{\epsilon}(s)$
$\displaystyle=$
$\displaystyle\frac{1}{\varphi_{\epsilon}^{2}(s_{0})}\left(\varphi_{\epsilon}^{\prime}(s)\varphi_{\epsilon}(s_{0})-\varphi_{\epsilon}^{\prime}(s_{0})\varphi_{\epsilon}(s)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\varphi_{\epsilon}^{2}(s_{0})}F_{\epsilon}(s),$

where $F_{\epsilon}$ is the function defined in (9).

It follows from the above and Lemma 2 that

$\varLambda(s,s_{0}):=\frac{\partial\varrho}{\partial
s}(s,s_{0})+\frac{\partial\varrho}{\partial s_{0}}(s,s_{0})$

is positive for all $s\in(0,s_{0})$, provided that, in the case $\epsilon=1,$
$s_{0}$ is sufficiently small. This means that there is a constant
$\alpha=\alpha(n)\geq C_{\epsilon}$ such that, whenever $H>\alpha$ and
$s\in(0,s_{0}),$ one has $\varLambda(s,s_{0})>0.$ For $\epsilon=-1,$
$\alpha=C_{\epsilon},$ since there are no restrictions on $s_{0}$ in this
case. Thus, we have ${\partial\varrho}/{\partial
s}(s,s_{0})+{\partial\varrho}/{\partial s_{0}}(s,s_{0})>0$ for such values of
$H$ and for all $s\in(0,s_{0}).$ This last inequality and (22) then give
$\partial\phi/\partial s_{0}>0,$ so that $\phi$ is increasing with respect to
$s_{0}.$ This shows (20) and concludes our proof. ∎

######  Remark 1.

For $n=2$ and $\epsilon=1,$ one can verify that the family
$\mathscr{F}_{\epsilon}$ is not nested. Indeed, in this case, one has
$\varphi_{\epsilon}(s)=(1-\cos s)/\sin(s).$ Proceeding as in the proof of
Theorem 1, we obtain:

$\varLambda(s,s_{0}):=\frac{\partial\varrho}{\partial
s}(s,s_{0})+\frac{\partial\varrho}{\partial
s_{0}}(s,s_{0})=\varphi(s)\varphi(s_{0})\left(\frac{1}{\sin s}-\frac{1}{\sin
s_{0}}\right).$

Therefore, $\varLambda(s,s_{0})<0$ if $s,s_{0}\in(\pi/2,\pi).$ From (22), we
have

$\frac{\partial\phi}{\partial s_{0}}(s_{0})=\int_{0}^{\pi-
s_{0}}\varLambda(s,s_{0})\Psi(s,s_{0})ds+\int_{\pi-
s_{0}}^{s_{0}}\varLambda(s,s_{0})\Psi(s,s_{0})ds.$

So, if $\pi<s_{0}<\pi/2,$ the first integral in the above equality is positive
and converges to zero as $s_{0}\rightarrow\pi,$ whereas the second integral is
negative and stays bounded away from zero as $s_{0}\rightarrow\pi.$ Therefore,
for $s_{0}$ sufficiently close to $\pi,$ one has ${\partial\phi}/{\partial
s_{0}}<0.$ This implies that, for $H$ sufficiently small (so that the
corresponding value $s_{0}=s_{0}(H)$ is sufficiently close to $\pi$), the
height function $\phi$ of the CMC spheres of $\mathscr{F}_{1}$ is decreasing
with respect to $s_{0},$ which gives that such spheres are not nested.

######  Remark 2.

The case $\epsilon=-1$ of Theorem 1 was considered in [6] as part of the proof
of the uniqueness of solutions to the isoperimetric problem in
$\mathbb{H}^{n}\times\mathbb{R}.$ However, the proof of the nesting property
of the rotational CMC spheres the authors provided there is incorrect. More
precisely, on page 48 of [6], the factor $\partial y/\partial h$ is missing in
the last equality of the first line of equations in (11)(i)(i)(i)We are indebt
to R. Souam for pointing this out to us.. Therefore, the reasoning that
follows is flawed. Furthermore, if that proof in [6] were correct, one could
easily adapt it to show that, in $\mathbb{S}^{2}\times\mathbb{R},$ the entire
family $\mathscr{F}_{1}$ of rotational CMC spheres is nested, which is not
true, as seen in the preceding remark. As a matter of fact, this inconsistency
gave us a tip of the incorrectness of the proof in [6].

We conclude with the proofs of Corollaries 1 and 2.

###### Proof of Corollary 1.

Since $\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R}$ is homogeneous, there exists
a solution to the isoperimetric problem for any preassigned volume
$\mathcal{V}>0.$ In addition, from the main results in [6, 8]; namely [6,
Theorems 1 and 2 ] and [8, Theorems 1 and 2], these solutions are the
rotational regions $\varOmega_{H}$ for all $\mathcal{V}>0,$ if $\epsilon=-1,$
and for all sufficiently small $\mathcal{V}>0,$ if $\epsilon=1$ (in the latter
case, a slab $\mathbb{S}^{n}\times[a,b]$ could be a solution for large
$\mathcal{V}$). However, by Theorem 1, the family of regions $\varOmega_{H}$
is nested. Hence, for each such value $\mathcal{V}>0,$ there exists one and
only one $H>0$ such that $\varOmega_{H}$ has volume $\mathcal{V}.$ ∎

###### Proof of Corollary 2.

Except for the concluding argument, the reasoning employed by Souam in his
proof for the case $n=2$ in [10] applies to the letter to the general
$n$-dimensional case. Consequently, denoting by $\lambda_{1}$ and
$\lambda_{2}$ the first and second eigenvalues of the Jacobi operator $L$ on
$\varSigma_{H},$ one has $\lambda_{1}<0=\lambda_{2}$ (cf. proof of [10,
Theorem 2.2]). So, the condition (i) in Koiso’s Theorem is fulfilled (cf.
Section 2).

Still following Souam, we fix $H^{*}>C_{\epsilon}$ and consider
$\mathscr{F}_{\epsilon}$ as a one-parameter family $\varphi_{H}$ of CMC
immersions of $\varSigma_{H^{*}}$ into
$\mathbb{Q}_{\epsilon}^{n}\times\mathbb{R}.$ Then, denoting by $\xi$ the
corresponding variational field on $\varSigma_{H^{*}},$ we have from the
fundamental equality (1) that the function $f:=\langle\xi,N\rangle$ satisfies
$Lf=1$ on $\varSigma_{H^{*}},$ since the parameter $t$ of this variation is
$H$ itself. In particular, for all $g\in\ker L,$ one has

$\int_{\varSigma_{H^{*}}}g=\int_{\varSigma_{H^{*}}}gLf=\int_{\varSigma_{H^{*}}}fLg=0,$

for $L$ is self-adjoint. Hence, condition (ii) of Koiso’s Theorem holds for
$\varSigma_{H^{*}}.$

Finally, assuming $H^{*}$ sufficiently large in the case $\epsilon=1,$ we have
from Theorem 1 that $f$ is nonnegative and non identically zero on
$\varSigma_{H^{*}}.$ Hence, writing $f=g+u,$ where $g\in\ker L$ and $u\in(\ker
L)^{\perp},$ we have that $Lu=Lf=1$ and

$\int_{\varSigma_{H^{*}}}u=\int_{\varSigma_{H^{*}}}f>0.$

The result, then, follows from Koiso’s Theorem. ∎

## References

  * [1] F. J. Almgren: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. AMS 165 (1976).
  * [2] J. L. Barbosa, M. do Carmo, and J. Escchenburg: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds Math. Z. 197, 123–138 (1988).
  * [3] L. Damasceno, M.F. Elbert: Stable free boundary hypersurfaces with constant higher order mean curvature. Preprint available at https://arxiv.org/abs/2212.14130.
  * [4] R. F. de Lima, F. Manfio, J. P. dos Santos: Hypersurfaces of constant higher order mean curvature in $M\times\mathbb{R}.$ Ann. Mat. Pura Appl. 201, 2979–3028 (2022)
  * [5] M. F. Elbert, R. Sa Earp: Constructions of $H_{r}$-hypersurfaces, barriers and Alexandrov theorem in $\mathbb{H}^{n}\times\mathbb{R}$. Ann. Mat. Pura Appl. 194, 1809-1834 (2015).
  * [6] W-T Hsiang, W-y Hsiang: On the uniqueness of isoperimetric solutions and imbedded soap bubbles in non-compact symmetric spaces, I, Invent. math. 98, 39-58 (1989).
  * [7] M. Koiso.: Deformation and stability of surfaces with constant mean curvature. Tohoku Math. J. Second Series, 54, 1, 145-159 (2002).
  * [8] R. Pedrosa: The isoperimetric problem in spherical cylinders, Ann. Glob. Anal. and Geom. 26, 333-354 (2004).
  * [9] A. Ros. The isoperimetric problem. Global theory of minimal surfaces, 175–209, Clay Math. Proc., 2. Amer. Math. Soc., Providence, RI, 2005.
  * [10] R. Souam: On stable constant mean curvature surfaces in $\mathbb{S}^{2}\times\mathbb{R}$ and $\mathbb{H}^{2}\times\mathbb{R}.$ Trans. Am. Math. Soc. 362, 6, 2845-2857 (2010).


# Macroscopic fluctuations of a driven tracer in the symmetric exclusion
process

Rahul Dandekar and Kirone Mallick  Institut de Physique Theorique, CEA, CNRS,
Universite Paris–Saclay, F–91191 Gif-sur-Yvette cedex, France

###### Abstract

The dynamics of an asymmetric tracer in the symmetric simple exclusion process
is mapped, in the continuous scaling limit, to the local current through the
origin in the zero-range process (ZRP) with a biased bond. This allows us to
study the hydrodynamics of the SEP with an asymmetric tracer with a step
initial condition, leading to the average displacement as a function of the
bias and the densities on both sides. We then derive the cumulant generating
function of the process in the high-density limit, by using the Macroscopic
Fluctuation Theory and obtain agreement with the microscopic results of Poncet
et al (2021). For more general initial conditions, we show that the tracer
variance in the high-density limit depends only on the generalized
susceptibility in the initial condition.

##  1 Introduction

Stochastic lattice gases of interacting particles are invaluable models to
explore the dynamics of many-body systems: powerful theoretical tools have
been developed to study them in the physics and the mathematical literature
(some general references are [1, 2, 3, 4, 5]). Many of these processes
appeared in biophysics to represent molecular motors or biopolymerization on
nucleic acid templates [6, 7]. Among numerous models, the one-dimensional
simple exclusion process – in which particles perform random walks subject to
the constraint that a site can be occupied by at most one particle at a given
time – has acquired a paradigmatic status, thanks to its rich combinatorial
structure that has lead to many exact analytical results [8, 9]. In
particular, the macroscopic hydrodynamics of the simple exclusion process is
by now well understood [10, 1] and the analysis of the microscopic process
allows one to probe fluctuations beyond the average hydrodynamic behavior.
Indeed, noise at the lattice size scale can affect the coarse-grained
evolution of the system and generate rare events (or large deviations) with
drastic impact [11, 12, 9].

Another manner to explore the interplay between the microscopic and
macroscopic scales is to consider a probe particle (also called a tracer) and
follow its motion in the surrounding fluid of particles. If the tracer follows
the same dynamical rules as the other particles, it behaves as a passive
scalar and does not affect the hydrodynamics. However, if the probe behaves in
a different manner (it is sensitive to an external drive, or it is active in
some way), then the overall collective dynamics can be altered. This is
precisely the type of effect we wish to investigate in the present work.

We shall consider the one-dimensional symmetric simple exclusion process (SEP)
on the infinite one-dimensional lattice with a single biased tracer (or tagged
particle). This biased tracer hops with rate $r_{+}$ to its right neighbor and
$r_{-}$ to its left, while all other particles hop at rate $1$ in either
direction. All particles, including the tracer, are subject to the exclusion
condition and their order remains unchanged: this is a single file-dynamics.
It is known [13, 14, 15, 16] that the average displacement of the biased
tracer in an infinite system grows, for large times, as

$\langle X_{t}\rangle=\sqrt{2ct}$ (1)

where $X_{t}$ is the displacement of the tracer at time $t$, and $c$ is a
number that depends on the initial density profile and on the tracer bias
through the ratio $\frac{r_{+}}{r_{+}+r_{-}}$. If the tracer particle is
symmetric and has the same dynamics as all other particles, the variance of
the tracer displacement for large times grows as

$\langle X_{t}^{2}\rangle-\langle X_{t}\rangle^{2}=\sigma_{\rm{ic}}\sqrt{t}$
(2)

where $\sigma_{\rm{ic}}$ is a constant that depends on the initial densities
on either side of the tracer and on the initial set-up of the system [17, 18,
19, 20, 21, 22, 23]. More generally, for reflecting Brownian motions [22, 24]
and for the SEP [25], it is possible to calculate all the cumulants of the
position of an unbiased tracer, that all grow with time as $\sqrt{t}$.

For a biased tracer, the behavior of the variance is not known in most cases
even for the SEP. In the high-density limit, $\rho\rightarrow 1$, it has been
shown that equation (2) is valid for the driven tracer as well, and the
coefficient $\sigma_{\rm{ic}}$ can be computed microscopically [15, 26]. In
fact, in the high-density limit, the full cumulant generating function of the
tracer position can be calculated from microscopic dynamics [27]. However,
many questions remain unanswered and a more general framework for studying the
driven tracer would be useful.

In the present work, we study the asymmetric tracer in the framework of
Macroscopic Fluctuation Theory (MFT), by extending the hydrodynamic approach
to asymmetric tracers developed by Landim, Olla and Volchan [28, 29]. For
symmetric single-file systems, the MFT [30, 9] allows one to derive
statistical results for quantities like the current fluctuations [31] and the
tracer position [23] in terms of the diffusivity and conductivity coefficients
of the hydrodynamic description. However, the presence of the biased tracer
generates a moving boundary condition at the macroscopic level. A suitable way
to derive this condition is to use a mapping between single-file systems and
mass transfer processes. In particular, the SEP in 1D is mapped to the 1D
Zero-range process (ZRP) [32, 33] and the biased particle in the SEP becomes a
biased bond in the ZRP. It can be shown (see section 3) that, for any density,
the dynamics depends on $r_{-}$ and $r_{+}$ only through the parameter
$\frac{r_{+}}{r_{+}+r_{-}}$. In the high-density limit of the SEP, the MFT
equations can be solved and the full cumulant generating function of the
tracer position can be calculated. Our results agree with the microscopic
derivations in [15, 26]. Recently, the dependence of the variance of the
tracer position on the initial state has been investigated in a broader class
of initial states and shown to depend on the initial state only through a
specific quantity, the generalized compressibility [34]. The MFT allows us to
study these more general initial states and we show that

$\langle X_{t}^{2}\rangle-\langle
X_{t}\rangle^{2}=(\alpha_{\rm{ic}}\sigma_{\rm{annealed}}+(1-\alpha_{\rm{ic}})\sigma_{\rm{quenched}})\sqrt{t}$
(3)

where $\alpha_{\rm{ic}}$ is the generalized susceptibility of the initial
ensemble, and $\sigma_{\rm{annealed}}$ and $\sigma_{\rm{quenched}}$ are the
coefficients of the tracer variance for quenched (deterministic) and annealed
(fluctuating) initial conditions respectively.

The outline of this work is as follows. In Section 2, we describe the general
mapping between single-file systems and a dual mass-transfer process at the
hydrodynamic level, that generalizes a known microscopic mapping between the
SEP and the ZRP. In Section 3, we first describe the hydrodynamic approach to
calculating the average displacement of a driven tracer in the SEP for any
density and then, we derive the complete MFT boundary conditions at the tracer
using the mapping to a mass transfer process. In section 4, we solve the case
of the SEP in the high-density limit, $\rho\rightarrow 1$ and calculate the
full cumulant generating function for the tracer position using the MFT,
recovering the microscopic results of [15, 26]. We also calculate the
dependence of the variance on the initial condition and show that it depends
only on the generalized compressibility. Section 5 is devoted to concluding
remarks.

##  2 Single-file and mass-transfer processes

In a single-file process on a one-dimensional lattice the particle order is
conserved by the dynamical rules, because overtaking is forbidden. The
Symmetric Simple Exclusion Process (SEP) is an archetypical single-file model
and its well-known mapping to the Zero Range Process (ZRP) [32, 33] goes as
follows (see Fig. 1):

  1. 1.

Label the SEP particles at the initial time ($t=0$), starting from the origin,
with $i=0$ being the particle with the smallest non-negative position, $i=1$
being the next particle to the right, and so on. Call $z_{i}$ the number of
empty sites between particles $i$ and $i+1$.

  2. 2.

Consider a ZRP with masses $z_{i}$ on site $i$ and rate $r=1$ of mass transfer
from each occupied site to each of its neighbors, irrespective of the mass $z$
on the site.

  3. 3.

A mass transfer to the right results in $z_{i}$ decreasing by one, and
$z_{i+1}$ increasing by one. Thus, in the SEP, it corresponds to particle
$i+1$ hopping to the left. Thus, the mapping between the SEP and ZRP dynamics
is that the current (or mass transfer) in the ZRP along the bond $[i-1,i]$ to
the left (right) corresponds to the particle with label $i$ in the SEP hopping
to the right (left).

Figure 1: A configuration in the SEP (left) and its corresponding ZRP
configuration (right). The particle with label $i$ in SEP becomes the site $i$
in ZRP and the number of marbles over site $i$ in ZRP corresponds to the
number of holes between the successive particles $i$ and $i+1$ in SEP.

The particle with label $i=0$, having the smallest non-negative initial
position will be hereafter called the tracer. Its position with time will be
denoted as $X(t)$. In the ZRP language, the tracer is mapped to the site 0 and
tracer movement to mass transfer along the bond $[-1,0]$.

###  2.1 Mapping a single-file to a mass-transfer process

In this subsection we present a general mapping of the continuum limit of an
arbitrary single-file diffusion system to a mass-transfer process,
generalizing the mapping of the previous section. Our aim is to relate
precisely the statistical fluctuations of these two types of models. From now
on, observables that refer to a single-file process will be written with a
subscript $s$ and the spatial coordinate in the single-file process is denoted
by $x$. Quantities related to a mass-transfer model have a subscript $m$ and
the spatial coordinate is denoted by $z$. Hence, $\rho_{s}(x)$ and
$\rho_{m}(z)$ represent the local density in the single-file process and in
the mass-transfer process, respectively.

Using the same underlying idea as that the one explained in the previous
subsection, the transformation laws for the coordinates and the densities can
be written as follows:

$\displaystyle z=\int_{X(t)}^{x}\rho_{s}(u,t)\,du$ (4)
$\displaystyle\int_{0}^{z}\rho_{m}(u,t)\,\rm{d}u=\int_{X(t)}^{x}(1-\rho_{s}(u,t))\,du$
(5)

The first equation says that a distance $z$ in the mass-transfer process
corresponds to the total number of particles between the tracer (located at
$X(t)$) and the position $x$ of the $z^{\rm{th}}$ particle in the single-file.
The second equation expresses the fact that the total mass between 0 and $z$
in the mass-transfer process is equal to the total number of holes between
$X(t)$ and $x$ in the single-file. Finally, the position of the tracer $X(t)$
is given by minus the mass-transfer through the origin

$\displaystyle
X(t)=-J(t)=\int_{0}^{\infty}\left(\rho_{m}(u,0)-\rho_{m}(u,t)\right)\,du$ (6)

Equations (4), (5) and (6) define a complete macroscopic one-to-one mapping
between the single-file and the mass-transfer processes. In particular,
differentiating eqns. (4) and (5) with respect to $x$, we deduce that the
local densities in the two processes are related through

$\displaystyle\rho_{s}(x,t)=\frac{1}{\rho_{m}(z,t)+1}$ (7)

The transformation (4) and (5) can be inverted as

$\displaystyle x(z,t)$ $\displaystyle=$ $\displaystyle
X(t)+z+\int_{0}^{z}\rho_{m}(u,t)\,du,$ (8)

and hence

$\displaystyle\frac{dx}{dz}=1+\rho_{m}(z,t)=\frac{1}{\rho_{s}(x,t)}.$ (9)

We now relate the diffusivities of the two models to each other. Consider the
current in a coarse-grained single-file system in the presence of a uniform
density gradient,

$j_{s}=-{D_{s}}(\rho_{s})\frac{d{\rho_{s}}}{dx}$

This would also cause a current in the mass transfer process. Since transfer
across a bond in the mass transfer process corresponds to the motion of a
particle (in the opposite direction) in the single-file system, we have, for
the current in the mass transfer process,

$j_{s}=-\rho_{s}j_{m},\mbox{ that is, }j_{m}=-\frac{{j_{s}}}{{\rho_{s}}}.$
(10)

Using (7) and (8), we obtain (note that the density gradient is of opposite
sign after the transformation)

$j_{m}=-\frac{1}{(1+\rho_{m})^{2}}{D_{s}}\left(\frac{1}{1+\rho_{m}}\right)\frac{d\rho_{m}}{dz}=-D_{m}(\rho_{m})\frac{d\rho_{m}}{dz}$

Hence, the diffusivity for the mass transfer process is related to the
diffusivity of the single-file system through the relation

$D_{m}(\rho_{m})=\frac{1}{(1+\rho_{m})^{2}}{D_{s}}\left(\frac{1}{1+\rho_{m}}\right),\mbox{
or
}D_{s}(\rho_{s})=\frac{1}{\rho_{s}^{2}}D_{m}\left(\frac{1}{\rho_{s}}-1\right)$
(11)

###  2.2 MFT equations and the cumulant generating function for the current

The Macroscopic Fluctuation Theory is defined in terms of two hydrodynamic
coefficients, the diffusivity $D(\rho)$ and the conductivity $\sigma(\rho)$.
The Macroscopic Fluctuating Theory equations for a stochastic process with
diffusivity $D$ and conductivity $\sigma$ are given, in the bulk, by

$\displaystyle\partial_{t}q$ $\displaystyle=$
$\displaystyle\partial_{x}(D(q)\partial_{x}q)-\partial_{x}(\sigma(q)\partial_{x}p)$
(12a) $\displaystyle\partial_{t}p$ $\displaystyle=$
$\displaystyle-D(q)\partial_{xx}p-\sigma^{\prime}(q)(\partial_{x}p)^{2}$ (12b)

We will consider the calculation of the cumulant generating function for the
current across the origin in time $T$, $J(T)$

$\mu(\lambda)=\log{\big{\langle}e^{\lambda J(T)}\big{\rangle}}$ (13)

where the angular brackets denote the average over all paths, weighted by the
path action $S$. $\mu(\lambda)$ can be calculated from the current through the
origin in the MFT. (See also [35]). Let the notation $J[\mathcal{P}]$ denote
the value of $J$ calculated along a particular trajectory $\mathcal{P}$ of the
system. Differentiating, we have

$\mu^{\prime}(\lambda)=\frac{\langle Je^{\lambda J}\rangle}{\langle e^{\lambda
J}\rangle}=\langle J\rangle_{\lambda}$

where the final average is over the tilted action, $S+\lambda J$.

Let us assume that, in the hydrodynamic limit, the tilted action is dominated
by a single trajectory, $\mathcal{P}_{S}(\lambda)$, the saddle-point
trajectory. Then,

$\mu^{\prime}(\lambda)=J[\mathcal{P}_{S}(\lambda)]=J_{MFT}$ (14)

since the MFT equations provide the saddle-point solution to the tilted
action. One can thus calculate the CGF from the current in the MFT solution,
along with the condition that $\mu(\lambda=0)=0$.

The MFT equations (12a) and (12b) are supplemented by boundary conditions,
which depend on the quantity being calculated. For the CGF of the current, the
appropriate boundary condition for $p$ at the final time is [21]

$p(x,T)=\lambda\theta(x)$ (15)

We start with a step initial density, with uniform density $\rho_{-\infty}$ to
the left of the origin and $\rho_{+\infty}$ to the right of the origin, and
consider two kinds of initial conditions, annealed and quenched [21, 9, 30,
31]. In the quenched case, the step initial condition is deterministic:

$q(x,0)=\rho_{-\infty}\theta(-x)+\rho_{+\infty}\theta(x)$ (16)

In the annealed case, the initial profile is allowed to fluctuate, leading to
an implicit equation for $q(x,0)$ in terms of $p(x,0)$:

$p(x,0)=\lambda\theta(x)+\theta(-x)\int_{\rho_{-\infty}}^{q(x,0)}dq\frac{2D(q)}{\sigma(q)}+\theta(x)\int_{\rho_{+\infty}}^{q(x,0)}dq\frac{2D(q)}{\sigma(q)}$
(17)

###  2.3 Mapping the MFT equations

In this section, we show how the conductivity $\sigma(q)$ and the field $p$
transform in the mapping from a single-file process to a mass-transport model.
We use the Hamiltonian formulation [30] of the MFT equations of motion (12a)
and (12b) in terms of the fields $p_{m}$ and $q_{m}$ for the mass-transfer
model

$H=\int
dz\left(-D_{m}(q_{m})(\partial_{z}q_{m})(\partial_{z}p_{m})+\frac{\sigma_{m}(q_{m})}{2}(\partial_{z}p_{m})^{2}\right)$
(18)

We transform this Hamiltonian to the single-file system by using equations
(7), (8) and (11):

$\displaystyle H$ $\displaystyle=$ $\displaystyle\int
dx\left(\frac{dz}{dx}\right)\bigg{(}-q_{s}^{2}{D_{s}}(q_{s})(\partial_{x}q_{s}^{-1})(\partial_{x}p_{m})\left(\frac{dx}{dz}\right)^{2}$
$\displaystyle+\frac{1}{2}{\sigma_{m}}\left(\frac{1}{q_{s}}-1\right)(\partial_{x}p_{m})^{2}\left(\frac{dx}{dz}\right)^{2}\bigg{)}$
$\displaystyle=$ $\displaystyle\int
dx\left({D_{s}}(q_{s})(\partial_{x}q_{s})\left(\frac{\partial_{x}p_{m}}{q_{s}}\right)+\frac{q_{s}}{2}{\sigma_{m}}\left(\frac{1}{q_{s}}-1\right)\left(\frac{\partial_{x}p_{m}}{q_{s}}\right)^{2}\right)$

We demand that this transformation preserves the Hamiltonian structure of the
MFT equations: hence, the above Hamiltonian must take the form

$\displaystyle H$ $\displaystyle=$ $\displaystyle\int
dx\left(-{D_{s}}(q_{s})(\partial_{x}q_{s})(\partial_{x}p_{s})+\frac{\sigma_{s}(q_{s})}{2}(\partial_{x}p_{s})^{2}\right)$

We deduce the following transformations rules for $(\partial_{x}p)$ and
$\sigma$:

$\displaystyle\partial_{x}{p_{s}}(x)$ $\displaystyle=$
$\displaystyle-\frac{\partial_{x}p_{m}}{q_{s}}=-\partial_{z}p_{m}(z)$ (19)
$\displaystyle\sigma_{s}(\rho_{s})$ $\displaystyle=$
$\displaystyle\rho_{s}{\sigma_{m}}\left(\frac{1}{\rho_{s}}-1\right),\mbox{ or,
}{\sigma_{m}}(\rho_{m})=(1+\rho_{m})\,\sigma_{s}\left(\frac{1}{1+\rho_{m}}\right)$
(20)

This completes the mapping at the hydrodynamic level between a single-file
system and the corresponding mass transfer process. We can verify that the
hydrodynamic coefficients of SEP [30] and its corresponding ZRP are related by
the transformations found above:

$\displaystyle D_{m}(\rho_{m})$ $\displaystyle=$
$\displaystyle\frac{1}{(1+\rho_{m})^{2}},~{}~{}~{}\sigma(\rho_{m})=\frac{2\rho_{m}}{1+\rho_{m}}$
(21) $\displaystyle D_{s}(\rho_{s})$ $\displaystyle=$ $\displaystyle
1,\quad\quad\quad\quad\quad\sigma(\rho_{s})=2\rho_{s}(1-\rho_{s})$ (22)

##  3 Driven tracer in SEP

We now consider a biased (or driven) tracer in SEP, initially located at the
origin, that hops with rate $r_{+}$ to the right, and rate $r_{-}$ to the
left. The initial density is a step profile with density $\rho_{-\infty}$ for
$x<0$ and density $\rho_{+\infty}$ for $x>0$. The particular dynamics of the
tracer will affect the density profile in its neighborhood. We start by
rederiving the expression of the density profile in a spirit close to that of
[13] (see [36] for a mathematically similar problem).Then, we shall state the
problem in the framework of the MFT and solve it in the high density regime.

###  3.1 Density profile and mean displacement

First we notice that the average density $\rho(x,t)$ of the host particles
satisfies the diffusion equation

$\frac{\partial\rho}{\partial t}=\frac{\partial^{2}\rho}{\partial x^{2}}$ (23)

In the long-time limit, fluctuations are ignored and we treat $X(t)$ as a
deterministic variable. The profile $\rho(x,t)$ as seen from the tracer will
adopt a stationary scaling form. This profile may be discontinuous at the
position of the tracer: we denote the density profile for $x\geq X(t)$ by
$\rho_{R}(x,t)$, and the density profile for $x\leq X(t)$ by $\rho_{L}(x,t)$.
Similarly, $\rho_{+}$ and $\rho_{-}$ will denote the limiting values of the
density just to the right $(x=X(t)+0^{+})$ and just to the left
$(x=X(t)-0^{+})$ of the tracer respectively.

The diffusion equation (23) has to be supplemented by the following boundary
conditions:

$\displaystyle\rho_{+}\frac{dX}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\partial\rho_{R}(x,t)}{\partial x}\Big{|}_{x=X^{+}}$ (24)
$\displaystyle\rho_{-}\frac{dX}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\partial\rho_{L}(x,t)}{\partial x}\Big{|}_{x=X^{-}}$ (25)
$\displaystyle r_{+}(1-\rho_{+})$ $\displaystyle=$ $\displaystyle
r_{-}(1-\rho_{-})$ (26)

The Stefan-type boundary conditions Eq. (24) and Eq. (25), result from the
conservation of the number of particles. Eq. (26) expresses the fact that the
average speed of the tracer vanishes.

Hence, we must solve two diffusion equations (23) in the regions
$-\infty<x<X(t)$ and $X(t)<x<+\infty$ subject to the initial condition

$\rho(x,0)=\rho_{-\infty}\theta(-x)+\rho_{+\infty}\theta(x)$ (27)

with the boundary conditions (24) and (25). The two solutions are matched
using (26).

The Stefan problem (24) admits a scaling solution of the form

$\rho_{R}(x,t)=f(\xi),\quad\mathrm{where}\qquad\xi=\frac{x}{X}$ (28)

With this scaling form, the heat equation and the boundary condition (24)
reduce to the ordinary differential equations

$\displaystyle f^{\prime\prime}$ $\displaystyle=$ $\displaystyle-
c_{R}\,\xi\,f^{\prime}\,\,\hbox{ with
}\,\,c_{R}=-\frac{f^{\prime}(1)}{\rho_{+}}$ (29) $\displaystyle X\,\dot{X}$
$\displaystyle=$ $\displaystyle c_{R}$ (30)

where the prime (the dot) denotes the derivative with respect to the scaled
distance $\xi$ (the time). Solving with the condition $f(1)=\rho_{+}$ leads us
to

$\displaystyle\rho_{R}(x,t)$ $\displaystyle=$
$\displaystyle\rho_{+}\left(1-c_{R}\int_{1}^{\frac{x}{X}}e^{c_{R}(1-\xi^{2})/2}d\xi\right)$
(31) $\displaystyle X_{t}$ $\displaystyle=$ $\displaystyle\sqrt{2c_{R}t}$ (32)

The validity of the solution requires that $c_{R}>0$. Since $\rho_{R}$ is
monotonically decreasing, this implies that $\rho_{+}>\rho_{+\infty}.$ In this
case, the tracer moves in the positive direction.

$\displaystyle\rho_{L}(x,t)$ $\displaystyle=$
$\displaystyle\rho_{-}\left(1+c_{L}\int_{\frac{x}{X}}^{1}e^{c_{L}(1-\xi^{2})/2}d\xi\right)$
(33) $\displaystyle X_{t}$ $\displaystyle=$ $\displaystyle\sqrt{2c_{L}t}$ (34)

Comparing (32) with (34) gives us $c_{R}=c_{L}$: this common value will be
denoted by $c$. Recalling that $\rho_{R}(+\infty)=\rho_{+\infty}$ and
$\rho_{R}(-\infty)=\rho_{-\infty}$, we can also deduce the relations

$\displaystyle\frac{\rho_{+\infty}}{\rho_{+}}$ $\displaystyle=$ $\displaystyle
1-c_{R}\int_{1}^{+\infty}e^{c_{R}(1-\xi^{2})/2}d\xi=1-\sqrt{2\pi
c}\,\,e^{c/2}\,\,\frac{1-\hbox{erf}(\sqrt{\frac{c}{2}})}{2}$ (35)
$\displaystyle\frac{\rho_{-\infty}}{\rho_{-}}$ $\displaystyle=$ $\displaystyle
1+c_{L}\int_{-\infty}^{1}e^{c_{L}(1-\xi^{2})/2}d\xi=1+\sqrt{2\pi
c}\,\,e^{c/2}\,\,\frac{1+\hbox{erf}(\sqrt{\frac{c}{2}})}{2}$ (36)

Equations (35), (36) together with the matching condition
$r_{+}(1-\rho_{+})=r_{-}(1-\rho_{-})$ (see eqn. (26)), provide us with three
independent relations for the three unknowns $\rho_{+},\rho_{-}$ and $c$. Eqn.
(26) shows that the absolute values of $r_{+}$ and $r_{-}$ do not matter, only
the asymmetry ratio $-1<\alpha=\frac{r_{+}-r_{-}}{r_{+}+r_{-}}<1$ does. 111A
simple procedure to analyze these equations is as follows: for any given
values of $\rho_{\pm\infty}$, choose a $c>0$ and this provides us with values
of $\rho_{+}(c),\rho_{-}(c)$ and $\alpha(c)$.

Figure 2: (a) A plot of $\kappa$ and $c$ as a function of $r_{+}$, on the
hyperplane of parameters where $r_{+}+r_{-}=1$, for $\rho_{+\infty}=0.7$ and
$\rho_{-\infty}=0.3$. (b) A plot of the density profile for $t=10$ for three
values of $r_{+}$, with the same values of other parameters as in (a). It can
be seen that the particle remains stationary for $r_{+}=0.7$, and the density
profile remains constant with time.

So far, we have assumed that $c>0$, and this implies that
$\rho_{+}\geq\rho_{+\infty}$ and $\rho_{-}\leq\rho_{-\infty}$. Since eqn. (26)
holds, we define the quantity

$\kappa=r_{+}(1-\rho_{+\infty})-r_{-}(1-\rho_{-\infty})$ (37)

It can be seen that $\kappa>0$ when $c>0$ and $\kappa=0$ when $c=0$. Thus,
when

$\displaystyle r_{+}(1-\rho_{+\infty})=r_{-}(1-\rho_{-\infty}),\mbox{ we have
}X_{t}=0$

and the tracer does not move. Now, if $\kappa<0$, assuming for the moment that
$r_{+}>r_{-}$, the bias is not large enough to overcome the push of the
density gradient, and the particle moves in the negative direction. To obtain
the solution for the case $\kappa<0$, we consider the mirrored system, where
one exchanges $r_{+}$ and $r_{-}$, and exchanges $\rho_{+\infty}$ and
$\rho_{-\infty}$. For this mirrored system, we have $\kappa>0$ and thus can
apply the method above. Let $\rho_{\rm{mirror}}$ be the density profile for
this mirrored system, and $c_{\rm{mirror}}$ the value of $c$. The solution for
$\kappa<0$ is then

$\displaystyle\rho(x,t)$ $\displaystyle=$
$\displaystyle\rho_{\rm{mirror}}(-x,t)$ $\displaystyle X_{t}$ $\displaystyle=$
$\displaystyle-\sqrt{2c_{\rm{mirror}}t}.$

Thus, we have that $X=\mbox{Sgn}(\kappa)\sqrt{2ct}$. Fig. 2 shows a plot of
the density profile around the tracer for various values of $\kappa$.

One can look at the limit of a small asymmetry around the equilibrium state,
$r_{\pm}=(1\pm\epsilon)$, and $\rho_{+\infty}=\rho_{-\infty}=\rho$. In this
limit,

$\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\frac{(1-\rho)^{2}}{\rho^{2}}\epsilon^{2}+O(\epsilon^{3}),$
$\displaystyle X_{t}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{\pi}}\frac{(1-\rho)}{\rho}\epsilon\sqrt{t}+O(\epsilon^{2})$
(38)

The value of the drift under a small bias is thus the same as the (annealed)
variance without a bias, as expected from the Einstein relation [37, 28].

In the totally asymmetric case $r_{-}=0$, we have $\rho_{+}=1$ irrespective of
the value of $r_{+}$. Then, in the limiting cases of small and large density,
the average position of the tracer exhibits the following asymptotic
behaviors:

$X_{t}\simeq\sqrt{t}\times\begin{cases}\sqrt{2/\rho_{+\infty}}&\rho_{+\infty}\downarrow
0\\\ 2\pi^{-1/2}(1-\rho_{+\infty})&\rho_{+\infty}\uparrow 1\end{cases}$ (39)

###  3.2 Boundary conditions at the origin

In the SEP, a biased tracer attempts a hop to the right with probability
$r_{+}$, and to the left with probability $r_{-}$. If $r_{+}=r_{-}=1$, the
tracer is unbiased and performs the same motion as all other particles. We now
consider a driven tracer in a general single-file system and define the biased
dynamics carefully. If an unbiased tracer in a configuration $C$ has a
probability $p_{\rm{T}}(C)$ of hopping to the right or the left, we define the
driven tracer to hop to the right with rate $r_{+}p_{\rm{T}}(C)$ and to the
left with $r_{-}p_{\rm{T}}(C)$.

In the mass-transfer picture, the movement of the tracer corresponds to biased
mass transfer across the bond $[-1,0]$. The mass transfer across this bond to
the left is enhanced by a factor $r_{+}$ while transfer to the right is
enhanced (or suppressed) by a factor $r_{-}$. The total displacement of the
tracer is, in the ZRP picture, is minus the total current through the origin
from time $0$ to time $t$, using eqn. (6).

We will consider the MFT solutions on the two sides of the origin as separate
systems on the two half-planes [38], whose solutions are related through
certain boundary conditions at the origin. Assuming that the coarse-grained
hydrodynamic limit remains valid, the hydrodynamic local density field during
the evolution is then given by $q(x,t)$, the solution of the MFT equations.
This implies that local equilibrium holds, with the local density parameter
given by $q([\ell x],t)$ where $\ell$ is the coarse-graining length. We also
assume that the joint distribution on different sites factorizes [28, 29]. For
local equilibrium to hold on both sides,

$r_{-}R(\rho_{-1})=r_{+}R(\rho_{0})$

where $R(\rho)$ denotes the total rate of outward mass transfer from a site in
the steady state with density $\rho$ (we have assumed local equilibrium).
Thus, in the hydrodynamic limit,

$r_{-}R(q(0^{-},t))=r_{+}R(q(0^{+},t)),\mbox{ or }r_{-}R(q_{-})=r_{+}R(q_{+})$
(40)

where we have used the notation $g_{+}\equiv g(x=0^{+})$ and $g_{-}\equiv
g(x=0^{-})$ for the limits of a function $g$ on either sides of the origin.
The function $R$ is related to the diffusivity as, in the presence of a
uniform density gradient, the current in the continuum limit is given by

$j=-\frac{1}{2}\partial_{x}R(\rho),\mbox{ and hence,
}R^{\prime}(\rho)=2D(\rho)$ (41)

where $D(\rho)$ is the diffusivity. For the ZRP, we have
$R(q)=\frac{2q}{1+q}$, and the condition eqn. (40) becomes

$r_{-}\frac{q_{-}}{1+q_{-}}=r_{+}\frac{q_{+}}{1+q_{+}}$ (42)

It is seen that this is equivalent to the SEP condition for the density
profile, eqn. (26).

We now find the boundary conditions for $p$ across the origin by considering
the variation of the action. The action for a trajectory is given by [30]

$S=\int_{0}^{T}dt\int_{-\infty}^{\infty}dx\left(p\partial_{t}q-\frac{\sigma(q)}{2}(\partial_{x}p)^{2}+D(q)(\partial_{x}p)(\partial_{x}q)\right)$
(43)

To derive the boundary conditions on $p$ at the origin, we consider the
variation of the action with respect to $q$ and $p$. (See the A for the
details of the derivation.) First, we consider the variation with respect to
$q$, which gives boundary conditions relating the gradient of $p$ on the two
sides of the origin:

$r_{-}(\partial_{x}p)_{+}=r_{+}(\partial_{x}p)_{-}$ (44)

Similarly, we can consider the variation of the action with respect to $p$.
This gives boundary conditions that relate $p$ on the two sides of the origin,

$p_{+}=p_{-}$ (45)

Eqns. (40), (44) and (45) complete the description of boundary conditions at
the origin for a biased particle. It can be seen that the values of $r_{+}$
and $r_{-}$ come into the boundary conditions only upto an overall
multiplicative constant, and hence only the value of
$\frac{r_{+}}{r_{+}+r_{-}}$ matters to the dynamics.

Since the current at the origin is related to the displacement of the tracer,
we have

$\frac{dX}{dt}=-j_{m}|_{x=0^{+}}=-j_{m}|_{x=0^{-}}$ (46)

Eqn. (42) becomes, in the SEP MFT,

$r_{+}(1-q_{s})|_{x=X^{+}}=r_{-}(1-q_{s})|_{x=X^{-}}$ (47)

The two equations above are the MFT generalization of the conditions in eqns.
(24), (25) and (26). Based on these MFT equations, we propose the following
fluctuating hydrodynamic equations to model the tracer hydrodynamics. In the
bulk on either side of the tracer we have the equations

$\displaystyle\partial_{t}\rho_{s}$ $\displaystyle=$
$\displaystyle-\partial_{x}j_{s},$ (48) $\displaystyle j_{s}$ $\displaystyle=$
$\displaystyle-D(\rho_{s})\partial_{x}\rho_{s}+\sqrt{\sigma_{s}(\rho_{s})}\eta$
(49)

where $\eta$ is a noise uncorrelated in space and time. These bulk equations
are supplemented by the boundary conditions at the tracer:

$\displaystyle\frac{dX}{dt}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{\rho_{s}}j_{s}\right)\bigg{|}_{x=X^{+}}=\left(\frac{1}{\rho_{s}}j_{s}\right)\bigg{|}_{x=X^{-}}$
(50) $\displaystyle r_{+}(1-\rho_{s})_{+}$ $\displaystyle=$ $\displaystyle
r_{-}(1-\rho_{s})_{-}$ (51)

where for the first equation we have used eqns. (10) and (46).

##  4 Fluctuations of the driven tracer in the high density limit

We now consider the low-density limit in the ZRP, which corresponds to the
high-density limit of the SEP. In this limit, the hydrodynamic coefficients
for the ZRP become

$D(q)=1+O(q),~{}~{}~{}\sigma(q)=2q+O(q^{2})$ (52)

Away from the tracer particle, the bulk MFT equations for the ZRP hold,

$\displaystyle\partial_{t}q$ $\displaystyle=$
$\displaystyle\partial_{x}((\partial_{x}q)-2q(\partial_{x}p))$ (53a)
$\displaystyle\partial_{t}p$ $\displaystyle=$
$\displaystyle-\partial_{x}^{2}p-(\partial_{x}p)^{2}$ (53b)

The Cole-Hopf transformation

$\displaystyle Q=qe^{-p},~{}~{}~{}~{}P=e^{p}$ (54)

simplifies the E-L equations into pure diffusion equations

$\partial_{t}Q=\partial_{x}^{2}Q,~{}~{}~{}~{}\partial_{t}P=-\partial_{x}^{2}P$
(55)

In the ZRP, in the high-density limit, the SEP initial condition (27), the
initial condition for the average density becomes

$\rho_{m}(x,0)=(1-\rho_{+\infty})\theta(x)+(1-\rho_{-\infty})\theta(-x)+O((1-\rho)^{2})$
(56)

The boundary condition on $p(x,T)$, eqn. (15) becomes

$P(x,T)=1+(e^{\lambda}-1)\theta(x)$ (57)

The quenched initial condition (16) gives

$Q(x,0)=\frac{1}{P(x,0)}\left((1-\rho_{-\infty})\theta(-x)+(1-\rho_{+\infty})\theta(x)\right)$
(58)

For the annealed initial condition, we have, from (17) with $D(q)=1$ and
$\sigma(q)=2q$,

$\displaystyle q(x,0)$ $\displaystyle=$
$\displaystyle(1-\rho_{-\infty})e^{p(x,0)}\theta(-x)+(1-\rho_{+\infty})e^{p(x,0)-\lambda}\theta(x),\mbox{
giving,}$ (59) $\displaystyle Q(x,0)$ $\displaystyle=$
$\displaystyle(1-\rho_{-\infty})\theta(-x)+(1-\rho_{+\infty})e^{-\lambda}\theta(x).$
(60)

###  4.1 Analytic solution for $P(x,t)$

$P(x,t)$ obeys the anti-diffusion equation

$\partial_{t}P=-\partial_{x}^{2}P$ (61)

with initial condition

$P(x,T)=1+(e^{\lambda}-1)\theta(x)$

and boundary conditions

$\displaystyle P(x,t)$ $\displaystyle\rightarrow$ $\displaystyle
e^{\lambda}\mbox{ as }x\rightarrow\infty,$ $\displaystyle P(x,t)$
$\displaystyle\rightarrow$ $\displaystyle 1\mbox{ as }x\rightarrow-\infty$

The general solution is thus

$\displaystyle
P(x,t)=\begin{cases}&e^{\lambda}+A\left(1-\mbox{erf}\left(\frac{x}{\sqrt{4(T-t)}}\right)\right)\mbox{
for }x>0\\\
&1+B\left(1+\mbox{erf}\left(\frac{x}{\sqrt{4(T-t)}}\right)\right)\mbox{ for
}x<0\end{cases}$ (62)

where $A$ and $B$ are constants to be determined from the boundary conditions.
We now use the two conditions (44) and (45)

$P_{+}=P_{-},\mbox{ and }r_{-}(\partial_{x}P)_{+}=r_{+}(\partial_{x}P)_{-}$

to solve for $A$ and $B$. This gives,

$\displaystyle A=\frac{r_{+}}{r_{+}+r_{-}}(1-e^{\lambda}),\mbox{ and
}B=\frac{r_{-}}{r_{+}+r_{-}}(e^{\lambda}-1)$

Thus, we have

$\displaystyle
P(x,t)=\begin{cases}&e^{\lambda}+\frac{r_{+}}{r_{+}+r_{-}}(1-e^{\lambda})\mbox{erfc}\left(\frac{x}{\sqrt{4(T-t)}}\right)\mbox{
for }x>0\\\
&1+\frac{r_{-}}{r_{+}+r_{-}}(e^{\lambda}-1)\mbox{erfc}\left(\frac{|x|}{\sqrt{4(T-t)}}\right)\mbox{
for }x<0\end{cases}$ (63)

We want to determine the cumulant generating function (CGF) of the position of
the tagged particle after time $T$,

$\mu^{p}(\lambda)=\log{\big{\langle}e^{\lambda X_{T}}\big{\rangle}}$ (64)

As the position of the tagged particle in the SEP is related to the current in
the ZRP through (6), we have the relation

$\mu^{p}(\lambda)=\log{\big{\langle}e^{\lambda
X_{T}}\big{\rangle}}=\log{\big{\langle}e^{-\lambda
J(T)}\big{\rangle}}=\mu(-\lambda)$ (65)

where $\mu(\lambda)$ is the CGF of the current in the ZRP, defined in (13).
Now, using (14) we can calculate $\mu(\lambda)$ through the MFT solution,

$\displaystyle\mu^{\prime}(\lambda)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx(q(x,T)-q(x,0))$ (66) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\left(Q(x,T)P(x,T)-Q(x,0)P(x,0)\right)$

$Q(x,t)$ evolves according to a pure diffusion equation with the boundary
condition at the origin, using eqn. (42),

$r_{+}Q(0^{+},t)=r_{-}Q(0^{-},t)$ (67)

Conservation of the current of $q$ across the origin gives

$(\partial_{x}Q)_{-}=(\partial_{x}Q)_{+}$ (68)

Consider a general initial condition

$Q(x,0)=Q_{1}(x)\Theta(x)+Q_{2}(x)\Theta(-x)$ (69)

The general solution for $x>0$ satisfying the boundary conditions is (see B
for details)

$\displaystyle Q(x,t)=\int_{0}^{\infty}\frac{dy}{\sqrt{4\pi t}}$
$\displaystyle\bigg{[}$ $\displaystyle
Q_{1}(y)\left(e^{-\frac{(x-y)^{2}}{4t}}-\frac{r_{+}-r_{-}}{r_{+}+r_{-}}e^{-\frac{(x+y)^{2}}{4t}}\right)$
$\displaystyle+\frac{2r_{-}}{r_{+}+r_{-}}Q_{2}(y)e^{-\frac{(x+y)^{2}}{4t}}\bigg{]}$

Inserting the solutions for $Q$ and $P$ into eqn. (66), and performing the
integral over $x$, we get

$\displaystyle\mu^{\prime}(\lambda)$ $\displaystyle=$
$\displaystyle-\int_{0}^{\infty}dyQ_{1}(y)\left(e^{\lambda}+\frac{r_{+}}{r_{+}+r_{-}}(1-e^{\lambda})\mbox{erfc}\left(\frac{y}{\sqrt{4T}}\right)\right)$
$\displaystyle+\int_{0}^{\infty}dyQ_{1}(y)e^{\lambda}\left(1-\frac{r_{+}}{r_{+}+r_{-}}\mbox{erfc}\left(\frac{y}{\sqrt{4T}}\right)\right)$
$\displaystyle+\int_{0}^{\infty}dyQ_{2}(y)e^{\lambda}\frac{r_{-}}{r_{+}+r_{-}}\mbox{erfc}\left(\frac{y}{\sqrt{4T}}\right)$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}dy\left(-\frac{r_{+}}{r_{+}+r_{-}}Q_{1}(y)\mbox{erfc}\left(\frac{y}{\sqrt{4T}}\right)+\frac{r_{-}}{r_{+}+r_{-}}Q_{2}(y)e^{\lambda}\mbox{erfc}\left(\frac{y}{\sqrt{4T}}\right)\right)$

###  4.2 Cumulants in the Annealed Case

For the annealed case, inserting the initial condition (60) into eqn.
(LABEL:eq:mup1),

$\displaystyle\mu_{a}^{\prime}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{4T}}{r_{+}+r_{-}}\left(-(1-\rho_{+\infty})r_{+}e^{-\lambda}+(1-\rho_{-\infty})r_{-}e^{\lambda}\right)\int_{0}^{\infty}dy~{}\mbox{erfc}(y)$
(71) $\displaystyle=$
$\displaystyle\sqrt{\frac{4T}{\pi}}\frac{1}{r_{+}+r_{-}}\left(-(1-\rho_{+\infty})r_{+}e^{-\lambda}+(1-\rho_{-\infty})r_{-}e^{\lambda}\right)$

We thus have, using the condition that $\mu_{a}(0)=0$,

$\frac{\mu^{p}_{a}(\lambda)}{\sqrt{4T}}=\frac{\mu_{a}(-\lambda)}{\sqrt{4T}}=\frac{(1-\rho_{+\infty})r_{+}(e^{\lambda}-1)+(1-\rho_{-\infty})r_{-}(e^{-\lambda}-1)}{\sqrt{\pi}(r_{+}+r_{-})}+O((1-\rho)^{2})$
(72)

where we have made explicit that the results here are derived in the high-
density limit of the SEP.

###  4.3 Cumulants in the Quenched Case

For the quenched case, we have the initial condition (58), which gives,

$\mu_{q}^{\prime}(\lambda)=\int_{0}^{\infty}dy\left(-\frac{(1-\rho_{+\infty})\frac{r_{+}}{r_{+}+r_{-}}e^{-\lambda}\mbox{erfc}\left(\frac{y}{\sqrt{4T}}\right)}{1+\frac{r_{+}}{r_{+}+r_{-}}(e^{-\lambda}-1)\mbox{erfc}\left(\frac{y}{\sqrt{4T}}\right)}+\frac{(1-\rho_{-\infty})\frac{r_{-}}{r_{+}+r_{-}}e^{\lambda}\mbox{erfc}\left(\frac{y}{\sqrt{4T}}\right)}{1+\frac{r_{-}}{r_{+}+r_{-}}(e^{\lambda}-1)\mbox{erfc}\left(\frac{y}{\sqrt{4T}}\right)}\right)$

Which then gives, for the CGF,

$\displaystyle\frac{\mu^{p}_{q}(\lambda)}{\sqrt{4T}}$ $\displaystyle=$
$\displaystyle\frac{\mu_{q}(-\lambda)}{\sqrt{4T}}=\int_{0}^{\infty}dy\bigg{[}(1-\rho_{+\infty})\log{\left(1+\frac{r_{+}}{r_{+}+r_{-}}(e^{\lambda}-1)\mbox{erfc}(y)\right)}$
$\displaystyle+(1-\rho_{-\infty})\log{\left(1+\frac{r_{-}}{r_{+}+r_{-}}(e^{-\lambda}-1)\mbox{erfc}(y)\right)}\bigg{]}+O((1-\rho)^{2})$

where we have made the high-density limit explicit. Equations (72) and
(LABEL:eq:muqfinal) are in agreement with the results of [15, 26], derived
using a microscopic approach.

###  4.4 The variance for a general initial condition

Following Banerjee et al [34], we now consider more general initial conditions
described by a generalized susceptibility, such that the log-probability of a
small density variation in the initial ensemble is given by

$F(\\{q(x,0)\\})\approx\frac{1}{2}\int_{-\infty}^{0}dx\frac{(q(x,0)-(1-\rho_{-\infty}))^{2}}{\alpha_{\rm{ic}}(1-\rho_{-\infty})}+\frac{1}{2}\int_{0}^{\infty}dx\frac{(q(x,0)-(1-\rho_{+\infty}))^{2}}{\alpha_{\rm{ic}}(1-\rho_{+\infty})}$
(74)

where we have kept terms only to quadratic order in the variation. The
quantity $\alpha_{\rm{ic}}$ defines a generalized susceptibility, and is
related to the Fano factor [34]. The generalized susceptibility
$\alpha_{\rm{ic}}$ describes the amount of fluctuations in the initial
ensemble. For the quenched or hyperuniform [39, 40] initial conditions,
$\alpha_{\rm{ic}}=0$, while for annealed initial conditions, because local
equilibrium is a Poisson state, $\alpha_{\rm{ic}}=1$.

We analyze the dependence of the variance of the biased tracer position on
$\alpha_{\rm{ic}}$. This involves terms to second order in the CGF
$\mu(\lambda)$, and hence to first order in $q(x,t)$. For an ensemble of
initial conditions described by the log-probability function $F$, the initial
condition for $q$ is given implicitly by [21]

$p(x,0)=\lambda\theta(x)+\frac{\delta F}{\delta q(x,0)},$ (75)

which gives

$q(x,0)\approx\theta(-x)(1-\rho_{-\infty})\big{(}1+\alpha_{\rm{ic}}p(x,0)\big{)}+\theta(x)(1-\rho_{+\infty})\big{(}1+\alpha_{\rm{ic}}(p(x,0)-\lambda)\big{)}$
(76)

to $O(\lambda^{2})$ and $O((1-\rho)^{2})$. For $\alpha_{\rm{ic}}=0$, we have
the quenched initial conditions, while the case $\alpha_{\rm{ic}}=1$ is
equivalent to eqn. (59) first order in $\lambda$. Since $q(x,t)$ is a linear
function of $\alpha_{\rm{ic}}$, and the MFT equations (12a) and (12b) are
linear to first order in $\lambda$ [31], it is expected that
$\mu^{\prime}(\lambda)$ to first order for general $\alpha_{\rm{ic}}$ will be
a linear combination of the quenched and annealed answers.

Using the Cole-Hopf transformation and eqn. (63), and keeping terms only to
$O(\lambda)$ and in the high-density limit,

$\displaystyle Q_{1}(x)$ $\displaystyle=$
$\displaystyle(1-\rho_{+\infty})-\lambda~{}(1-\rho_{+\infty})(1-\alpha_{\rm{ic}})\left(1-\frac{r_{+}}{r_{+}+r_{-}}\mbox{Erfc}\left(\frac{y}{\sqrt{4T}}\right)\right)$
$\displaystyle Q_{2}(x)$ $\displaystyle=$
$\displaystyle(1-\rho_{-\infty})-\lambda~{}\frac{r_{-}}{r_{+}+r_{-}}(1-\rho_{-\infty})(1-\alpha_{\rm{ic}})\mbox{Erfc}\left(\frac{y}{\sqrt{4T}}\right)$

Inserting into eqn. (LABEL:eq:mup1), we have

$\displaystyle\mu^{\prime}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{4T}}{\sqrt{\pi}(r_{+}+r_{-})}\bigg{\\{}(1-\rho_{-\infty})r_{-}-(1-\rho_{+\infty})r_{+}$
(77)
$\displaystyle+\lambda\bigg{[}(1-\rho_{+\infty})r_{+}\left(1-\left(\sqrt{2}-2\right)(\alpha_{\rm{ic}}-1)\frac{r_{+}}{r_{+}+r_{-}}\right)$
$\displaystyle+(1-\rho_{-\infty})r_{-}\left(1-\left(\sqrt{2}-2\right)(\alpha_{\rm{ic}}-1)\frac{r_{-}}{r_{+}+r_{-}}\right)\bigg{]}\bigg{\\}}$
$\displaystyle=$ $\displaystyle
J_{\rm{quenched}}+\alpha_{\rm{ic}}\left(J_{\rm{annealed}}-J_{\rm{quenched}}\right)+O(\lambda^{2})+O((1-\rho)^{2})$

The cases $\alpha_{\rm{ic}}=0$ and $\alpha_{\rm{ic}}=1$ give back the results,
to $O(\lambda)$, of eqns. (LABEL:eq:muqfinal) and (72). From eqn. (77), we get
that the second cumulant of the biased tracer position in the high-density
limit is

$\langle X^{2}\rangle_{c}=\langle
X^{2}\rangle_{c,\rm{quenched}}+\alpha_{\rm{ic}}\left(\langle
X^{2}\rangle_{c,\rm{annealed}}-\langle X^{2}\rangle_{c,\rm{quenched}}\right)$
(78)

Thus, the dependence of the second cumulant on $\alpha_{\rm{ic}}$ follows the
same pattern for a biased tracer as that found for unbiased tracers in [34].

##  5 Concluding remarks

In this paper, we have investigated the problem of a biased tracer in a
single-file system. Thanks to a mapping to a ZRP with a biased bond at the
origin, we could write exact boundary conditions at the origin within the MFT
framework, solve the high density limit, retrieve at the macroscopic scale the
results of microscopic calculations [15, 26] and study a whole family of
initial conditions, thus generalizing the results of [34] to biased tracers.

We emphasize that the boundary conditions derived here are valid for general
single-file systems and thus open up the MFT framework for investigating
biased tracers. The fluctuating hydrodynamic equations (48)- (51) could also
be useful for numerical investigation as well as for perturbative
calculations. We expect that this continuous approach, that takes fluctuations
into account, will be useful for investigating the effect of a local defect at
the global hydrodynamic scale.

While completing this manuscript, we became aware of a related work ‘Duality
in single-file diffusion’ by our collegues P. Rizkallah, A. Grabsch, P. Illien
and Olivier Bénichou from Sorbonne University (arXiv 2207.07549): they also
consider the mapping from a single-file to a mass-transfer process to find
relations between various models and to transfer exact solutions between them.
We were motivated by a different problem, the biased tracer and the mapping
was introduced to rewrite a (stochastic) moving Stefan problem as a localized
boundary condition.

## Acknowledgements

We would like to thank Paul Krapivsky for early collaboration and helpful
discussions and comments on the manuscript.

##  Appendix A Boundary conditions on $p$ at the biased particle

First we consider the variation in the MFT action $S$ (eqn. (43)) in terms of
the variation in $q$. We only keep the terms which are total derivatives in
space (the bulk variation gives the E-L equations (12a) and (12b) and the
total time-derivative terms give the boundary conditions [21]),

$\displaystyle\delta S$ $\displaystyle=$
$\displaystyle\int_{0}^{T}dt\bigg{(}\big{[}\delta
qD(q)\partial_{x}p\big{]}_{x=\infty}-\big{[}\delta
qD(q)\partial_{x}p\big{]}_{x=0^{+}}$ (79) $\displaystyle+\big{[}\delta
qD(q)\partial_{x}p\big{]}_{x=0^{-}}-\big{[}\delta
qD(q)\partial_{x}p\big{]}_{x=-\infty}\bigg{)}$

We now use the fact that $(\partial_{x}p)\rightarrow 0$ as
$x\rightarrow\pm\infty$, to get

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{0}^{T}dt\left(\big{[}\delta
q_{+}D(q_{+})\partial_{x}p\big{]}_{+}-\big{[}\delta
q_{-}D(q_{-})\partial_{x}p\big{]}_{-}\right)$ $\displaystyle=$
$\displaystyle\int_{0}^{T}dt\left(\delta R(q_{+})(\partial_{x}p)_{+}-\delta
R(q_{-})(\partial_{x}p)_{-}\right)$

where we have used eqn. (41). Now using eqn. (40), we get that

$r_{+}\delta R(q_{+})=r_{-}\delta R(q_{-})$

and hence,

$\delta S=\int_{0}^{T}dt~{}\delta
R(q_{+})\left(r_{-}(\partial_{x}p)_{+}-r_{+}(\partial_{x}p)_{-}\right)=0$

and hence,

$r_{-}(\partial_{x}p)_{+}=r_{+}(\partial_{x}p)_{-}$ (80)

The boundary terms in the variation in $p$ can be similarly computed to be

$\delta S=\int_{0}^{T}dt\left(\delta
p_{+}(\sigma(q)(\partial_{x}p)-D(q)(\partial_{x}q))_{+}-\delta
p_{-}(\sigma(q)(\partial_{x}p)-D(q)(\partial_{x}q))_{-}\right)$

Now, it is known that in the MFT, the density field $q$ is conserved, and
hence, the current flowing from the right to the origin is equal to the
current flowing to the left from the origin,

$(\sigma(q)(\partial_{x}p)-D(q)(\partial_{x}q))_{+}=(\sigma(q)(\partial_{x}p)-D(q)(\partial_{x}q))_{-}$
(81)

For this to happen, it must be that case that $\delta p_{+}$ and $\delta
p_{-}$ cannot vary independently, that is,

$\delta p_{-}=\delta p_{+},\mbox{ that is, }p_{+}=p_{-}+C$ (82)

The constant $C$ can be fixed by looking at the right hand side of (12b),
where $C\neq 0$ would lead to an unphysical delta-function-squared term.
Hence,

$p_{+}=p_{-}$ (83)

##  Appendix B General solution for $Q(x,t)$

In this Appendix we calculate a general formula for $Q(x,t)=q(x,t)e^{-p(x,t)}$
evolving according to the diffusion equation (55) and the boundary conditions

$r_{+}Q(0^{+},t)=r_{-}Q(0^{-},t)\mbox{ for $t>0$, and
}(\partial_{x}Q)_{+}=(\partial_{x}Q)_{+}$ (84)

The latter equation comes from the conservation of the current of $q$ across
the origin, (81) and using the boundary conditions (42), (44) and (45). We
consider the initial condition

$\displaystyle Q(x,0)=Q_{1}(x)\Theta(x)+Q_{2}(x)\Theta(-x)$

Since the values of $Q$ on either side of the origin are constrained to be
different, we consider a partial mirror-image solution on the two sides of the
origin,

$\displaystyle Q(x,0)=\begin{cases}\int_{0}^{\infty}\frac{dy}{\sqrt{4\pi
t}}\left(Q_{1}(y)\left(e^{-\frac{(x-y)^{2}}{4t}}-ae^{-\frac{(x+y)^{2}}{4t}}\right)+(1-a)Q_{2}(y)e^{-\frac{(x+y)^{2}}{4t}}\right)\\\
\mbox{ for $x>0$}\\\ \int_{-\infty}^{0}\frac{dy}{\sqrt{4\pi
t}}\left(Q_{2}(y)\left(e^{-\frac{(x-y)^{2}}{4t}}-be^{-\frac{(x+y)^{2}}{4t}}\right)+(1-b)Q_{1}(y)e^{-\frac{(x+y)^{2}}{4t}}\right)\\\
\mbox{ for $x<0$}\end{cases}$

where $a$ and $b$ are constants determined from the boundary conditions in
(84), which give

$r_{+}(1-a)=r_{-}(1-b)\mbox{ and }a=-b,$

Hence, we have

$a=-b=\frac{r_{+}-r_{-}}{r_{+}+r_{-}}$ (85)

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