# Equilibrium Perturbations for Asymmetric Zero Range Process under Diffusive
Scaling in Dimensions $d\geq 2$

Linjie Zhao

###### Abstract

We consider the asymmetric zero range process in dimensions $d\geq 2$. Assume
the initial density profile is a perturbation of the constant density, which
has order $N^{-\alpha}$, $\alpha\in(0,1)$, and is constant along the drift
direction. Here, $N$ is the scaling parameter. We show that under some
constraints on the jump rate of the zero range process, the perturbed quantity
macroscopically obeys the heat equation under diffusive scaling.

_Keywords:_ asymmetric zero range process; diffusive scaling; spectral gap
estimate; logarithmic Sobolev inequality.

##  Introduction

It is well known that for asymmetric interacting particle systems with only
one conservation law, such as asymmetric zero range or exclusion processes,
the macroscopic density profile obeys the hyperbolic equation under hyperbolic
scaling, _i.e._ time sped up by $N$ and space divided by $N$ [13]. In order to
understand Navier-Stokes equations from a microscopic point of view,
asymmetric interacting particle systems have also been considered under
diffusive scaling, _i.e._ time sped up by $N^{2}$ and space divided by $N$. In
the seminal paper [4], Esposito, Marra and Yau prove that for asymmetric
exclusion processes in dimensions $d\geq 3$, if the initial density profile is
a perturbation of order $N^{-1}$ with respect to the constant density, then
the perturbed quantity evolves according to a parabolic equation under
diffusive scaling. In the literature this is called the incompressible limit,
which has also been extended to boundary driven asymmetric exclusion process
in [2]. Another interpretation of the Navier-Stokes equation is to describe
the evolution of system in the hyperplane orthogonal to the drift. Precisely
speaking, in [3, 9], it has been proven that for asymmetric zero range and
exclusion processes, if the initial density profile is constant along the
drift direction, then the macroscopic behavior is described by a parabolic
equation under diffusive scaling. The third interpretation is to consider the
first order correction to the hydrodynamic equation, which however is under
hyperbolic scaling. We refer to [7, P. 185] for more background and references
on understanding of Navier-Stokes equations.

In this note, we consider equilibrium perturbations for the asymmetric zero
range process under diffusive scaling in dimensions $d\geq 2$. We first recall
the result in [3]. Consider the zero range process $(\eta_{t},t\geq 0)$ with
jump rate $N^{2}g(\cdot)$ and transition probability $p(\cdot)$ such that
$m:=\sum_{x\in\mathbb{T}_{N}^{d}}xp(x)\neq 0$. Here, $N$ is the scaling
parameter. Assume the initial distribution of the process is associated to
some density profile $\varrho_{0}:\mathbb{T}^{d}\rightarrow\mathbb{R}_{+}$
such that $m\cdot\nabla\varrho_{0}(u)=0$ for any $u\in\mathbb{T}^{d}$. Then,
it is proved in [3] that for any continuous function
$F:\mathbb{T}^{d}\rightarrow\mathbb{R}$,

$\lim_{N\rightarrow\infty}\frac{1}{N^{d}}\sum_{x\in\mathbb{T}_{N}^{d}}\eta_{t}(x)F(\tfrac{x}{N})=\int_{\mathbb{T}^{d}}\varrho(t,u)F(u)du\quad\text{in
probability},$

where $\varrho(t,u)$ is the solution to the following parabolic equation

$\begin{cases}\partial_{t}\varrho(t,u)=\frac{1}{2}\sum_{i,j=1}^{d}\sigma_{i,j}\partial^{2}_{u_{i},u_{j}}\Phi(\varrho(t,u)),\\\
\varrho(0,u)=\varrho_{0}(u).\end{cases}$ (1.1)

Above, $\sigma_{i,j}=\sum_{x\in\mathbb{T}_{N}^{d}}x_{i}x_{j}p(x)$ for $1\leq
i,j\leq d$ and $\Phi(\varrho)$ is the expectation of the jump rate $g(\cdot)$
with respect to the invariant measure of the process with density $\varrho$.

We assume the initial density profile has the following form: fix a positive
constant $\rho_{*}$ and for $\alpha\in(0,1)$, let

$\varrho_{0}(u)=\rho_{*}+N^{-\alpha}\rho_{0}(u).$

Assume further that $m\cdot\nabla\rho_{0}(u)=0$ for any $u\in\mathbb{T}^{d}$.
Then, under some constraints on the jump rate $g(\cdot)$, we prove that for
any continuous function $F:\mathbb{T}^{d}\rightarrow\mathbb{R}$,

$\lim_{N\rightarrow\infty}\frac{1}{N^{d-\alpha}}\sum_{x\in\mathbb{T}_{N}^{d}}\big{(}\eta_{t}(x)-\rho_{*}\big{)}F(\tfrac{x}{N})=\int_{\mathbb{T}^{d}}\rho(t,u)F(u)du\quad\text{in
probability},$

where $\rho(t,u)$ is the solution to the following heat equation

$\begin{cases}\partial_{t}\rho(t,u)=\frac{1}{2}\sum_{i,j=1}^{d}\sigma_{i,j}\Phi^{\prime}(\rho_{*})\partial^{2}_{u_{i},u_{j}}\rho(t,u),\\\
\rho(0,u)=\rho_{0}(u).\end{cases}$ (1.2)

Note that if we assume a priori that the density profile of the process at
time $t$ is given by

$\varrho(t,u):=\rho_{*}+N^{-\alpha}\rho(t,u)$

and substitute $\varrho(t,u)$ into the parabolic equation (1.1), then we get
the above heat equation by using Taylor’s expansion and by letting
$N\rightarrow\infty$.

The proof is based on relative entropy method introduced by Yau [19].
Precisely speaking, let $\nu_{N,t}$ be the product measure with macroscopic
density profile $\rho_{*}+N^{-\alpha}\rho(t,u)$, and let $\mu_{N,t}$ be the
distribution of the process at time $t$. We prove in Theorem 2.3 that per
volume of the relative entropy of $\mu_{N,t}$ with respect to $\nu_{N,t}$ is
of order $o(N^{-2\alpha})$. From this and entropy inequality, it is easy to
obtain law of large numbers for the perturbed quantities. The main step to
bound the relative entropy is to prove a quantitative version of the so-called
one block estimate, for which we present two different proofs: one uses
spectral gap estimate and the other uses logarithmic Sobolev inequality. The
proof based on spectral gap estimate follows the steps in [15]. Logarithmic
Sobolev inequality has also been used to quantify block estimates in the
theory of hydrodynamic limit, see [5] for example.

Equilibrium perturbations from equilibrium have also been considered in other
contexts. For example, in dimension one, Seppäläinen [14] considers the
Hammersley’s model, adds a perturbation of order $N^{-\alpha}$ to the
equilibrium, and shows that the perturbation macroscopically obeys the invisid
Burgers equation in the time scale $N^{1+\alpha}t$ if $0<\alpha<1/2$, which is
extended by Tóth and Valkó [15] to a large class of one-dimensional
interacting particle systems based on relative entropy method, but only for
$0<\alpha<1/5$ and only in the smooth regime of the solution. For systems with
two conservation laws, Tóth and Valkó [16] obtain a two-by-two system for a
very rich class of systems. In [17], Valkó shows that small perturbations
around a hyperbolic equilibrium point evolve according to two decoupled
Burgers equations. For very rencent results, we refer to [6] for equilibrium
perturbations in weakly asymmetric exclusion processes, and to [18] for the
generalized exclusion process and anharmonic chains.

The rest of the note is organized as follows. In Section 2, we state the model
and main results. The proof of Theorem 2.3 is presented in Section 3 by
assuming the one block estimate holds, whose proof is postponed to Section 4.

##  Notation and Results

The state space of the zero range process is
$\Omega_{N}^{d}=(\mathbb{T}_{N}^{d})^{\mathbb{N}}$, where
$\mathbb{N}=\\{0,1,2,\ldots\\}$ and
$\mathbb{T}_{N}^{d}=\mathbb{Z}^{d}/(N\mathbb{Z}^{d})$ is the $d$ dimensional
discrete torus with $d\geq 2$. For a configuration $\eta\in\Omega_{N}^{d}$,
$\eta(x)$ is the number of particles at site $x$. Let $p(\cdot)$ be a
probability measure on $\mathbb{Z}^{d}$. We assume that

  1. (i)

$p(\cdot)$ is of finite range: there exists $R>0$ such that $p(x)=0$ for all
$|x|>R$;

  2. (ii)

$p(\cdot)$ is asymmetric: $m:=\sum_{x\in\mathbb{T}_{N}^{d}}xp(x)\neq 0$.

Above, $|x|=\max_{1\leq i\leq d}|x_{i}|$ for $x\in\mathbb{Z}^{d}$. Let
$g:\mathbb{N}\rightarrow\mathbb{R}_{+}$ be the jump rate of the zero range
process. To avoid degeneracy, assume $g(k)=0$ if and only if $k=0$. The
generator of the zero range process acting on functions
$f:\Omega_{N}^{d}\rightarrow\mathbb{R}$ is given by

$\mathscr{L}_{N}f(\eta)=\sum_{x,y\in\mathbb{T}_{N}^{d}}g(\eta(x))p(y)\big{[}f(\eta^{x,x+y})-f(\eta)\big{]}.$

Here, $\eta^{x,y}$ is the configuration obtained from $\eta$ after a particle
jumps from $x$ to $y$,

$\eta^{x,y}(z)=\begin{cases}\eta(x)-1,\quad&z=x,\\\ \eta(y)+1,\quad&z=y,\\\
\eta(z),\quad&z\neq x,y.\\\ \end{cases}$

Throughout the paper, we need the following assumptions on the rate function
$g(\cdot)$.

######  Assumption 2.1.

(i) There exists a constant $a_{0}$ such that for any $k\geq 1$,
$|g(k+1)-g(k)|\leq a_{0}$;
(ii) there exists $k_{0}>0$ and $a_{1}>0$ such that $g(k)-g(j)>a_{1}$ for any
$k\geq j+k_{0}$.

######  Remark 2.1.

Condition $(i)$ is needed in order the process to be well defined in the
infinite volume (_cf._ [1]). Condition $(ii)$ ensures spectral gap estimates
and logarithmic Sobolev inequality for the zero range process (_cf._ [8, 12]),
which are main tools of this note.

######  Remark 2.2.

We assume condition $(ii)$ for simplicity. The results in this note should
also hold as long as the spectral gap of the zero range process shrinks at
rate at least $\ell^{-2}$ (not necessarily uniformly in the particle density),
where $\ell$ is the size of the underlying lattice. For example, the spectral
gap estimates have also been proven in the following cases: (a)
$g(k)=k^{\alpha}$, $\alpha\in(0,1)$, see [11]; (b) $g(k)=\mathbf{1}_{\\{k\geq
1\\}}$, see [10].

It is well known that the zero range process has a family of product invariant
measures indexed by the particle density. Precisely speaking, for each
$\varphi\geq 0$, let $\bar{\nu}^{N}_{\varphi}$ be the product measure on
$\Omega_{N}^{d}$ with marginals given by

$\bar{\nu}^{N}_{\varphi}(\eta(x)=k)=\frac{1}{Z(\varphi)}\frac{\varphi^{k}}{g(k)!},\quad
k\geq 0,\;x\in\mathbb{T}_{N}^{d}.$

Here, $Z(\varphi)$ is the normalizing constant and $g(k)!=\prod_{j=1}^{k}g(j)$
with the convention that $g(0)!=1$. Under Assumption 2.1, $Z(\varphi)<\infty$
for any $\varphi\geq 0$. In particular,

$E_{\bar{\nu}^{N}_{\varphi}}[e^{\lambda\eta(0)}]<\infty,\quad\forall\lambda\geq
0.$

For $\varphi\geq 0$, the particle density under $\bar{\nu}^{N}_{\varphi}$ is

$R(\varphi)=E_{\bar{\nu}^{N}_{\varphi}}[\eta(x)].$

It is easy to see that $R(\varphi)$ is strictly increasing in $\varphi$, hence
has an inverse denoted by $\Phi:=R^{-1}$. To index the invariant measures by
particle density $\rho\geq 0$, denote
$\nu^{N}_{\rho}:=\bar{\nu}^{N}_{\Phi(\rho)}$.

Fix $\rho_{*}>0$. Let $\rho_{0}:\mathbb{T}^{d}\rightarrow\mathbb{R}_{+}$ be
the initial density profile of the perturbed quantity. We assume $\rho_{0}$ is
continuously differentiable and is constant along the drift direction

$m\cdot\nabla\rho_{0}(u)=0,\quad\forall u\in\mathbb{T}^{d}.$ (2.1)

Let $\alpha>0$ denote the strength of the perturbed quantity. The initial
distribution of the process is

$\mu_{N,0}(d\eta)=\bigotimes_{x\in\mathbb{T}_{N}^{d}}\nu^{1}_{\rho_{*}+N^{-\alpha}\rho_{0}(x/N)}(d\eta(x)).$
(2.2)

Denote by $\mu_{N,t}$ the distribution of the process with generator
$N^{2}\mathscr{L}_{N}$ at time $t$ starting from $\mu_{N,0}$. The
corresponding accelerated process is denoted by $(\eta_{t},t\geq 0)$.

Let $\rho(t,u)$ be the solution to the following heat equation

$\begin{cases}\partial_{t}\rho(t,u)=\frac{1}{2}\sum_{i,j=1}^{d}\Phi^{\prime}(\rho_{*})\sigma_{i,j}\partial^{2}_{u_{i},u_{j}}\rho(t,u),\\\
\rho(0,u)=\rho_{0}(u).\end{cases}$ (2.3)

Above, $\sigma_{i,j}=\sum_{x\in\mathbb{T}_{N}^{d}}x_{i}x_{j}p(x)$ for $1\leq
i,j\leq d$. Define the reference measure as

$\nu_{N,t}(d\eta)=\bigotimes_{x\in\mathbb{T}_{N}^{d}}\nu_{\rho_{*}+N^{-\alpha}\rho(t,x/N)}^{1}(d\eta(x)).$

For two probability measures $\mu,\nu$ on $\Omega_{N}^{d}$ such that $\mu$ is
absolutely continuous with respect to $\nu$, recall the relative entropy of
$\mu$ with respect to $\nu$ is defined as

$H(\mu|\nu)=\int\log(d\mu/d\nu)d\mu.$

To make notation simple, denote

$h_{N}(t)=N^{-d}H(\mu_{N,t}|\nu_{N,t}).$

The following is the main result of this note.

######  Theorem 2.3.

Let $d\geq 2$. For any $\alpha\in(0,1)$, we have $h_{N}(t)=o(N^{-2\alpha})$.

######  Remark 2.4.

We do not actually need the initial measure $\mu_{N,0}$ to be product as given
in (2.2). But we need it to satisfy: $(i)\;h_{N}(0)=o(N^{-2\alpha})$,
$(ii)\;N^{-d}H(\mu_{N,0}|\nu^{N}_{\rho_{*}})\leq CN^{-2\alpha}$ for some
constant $C$ independent of $N$.

For any probability measure $\mu$ on $\Omega_{N}^{d}$, denote by
$\mathbb{P}_{\mu}$ the distribution of the process $(\eta_{t},t\geq 0)$ with
initial distribution $\mu$, and by $\mathbb{E}_{\mu}$ the corresponding
expectation. As a direct consequence of the above theorem, we have the
following law of large numbers, whose proof uses entropy inequality and is
very standard (_cf._ [7, Corollary 6.1.3] for example). For this reason, we
omit the proof.

######  Corollary 2.5.

Under the assumptions of Theorem 2.3, for any continuous function
$F:\mathbb{T}^{d}\rightarrow\mathbb{R}$, for any $t>0$ and for any
$\varepsilon>0$,

$\lim_{N\rightarrow\infty}\mathbb{P}_{\mu_{N,0}}\Big{(}\Big{|}\frac{1}{N^{d-\alpha}}\sum_{x\in\mathbb{T}_{N}^{d}}\big{(}\eta_{t}(x)-\rho_{*}\big{)}F\big{(}\tfrac{x}{N}\big{)}-\int_{\mathbb{T}^{d}}\rho(t,u)F(u)du\Big{|}>\varepsilon\Big{)}=0.$

Above, $\rho(t,u)$ is the solution to the heat equation (2.3).

##  Relative Entropy

In this section, we first bound the entropy production of the process in
Subsection 3.1, then calculate the relative entropy in Subsection 3.2, and
finally prove Theorem 2.3 in the last subsection. To make notations short, we
denote $\rho_{N}(t,u):=\rho_{*}+N^{-\alpha}\rho(t,u)$. Recall $\rho(t,u)$ is
the solution to the heat equation (2.3). We also underline that in the
following proof the constant $C$ may be different from line to line, but does
not depend on the scaling parameter $N$.

### 3.1 Entropy production.

For $s\geq 0$, denote

$f_{s}=f_{N,s}=\frac{d\mu_{N,s}}{d\nu_{\rho_{*}}^{N}}.$

For any $\nu_{\rho_{*}}^{N}$–density $f$, the Dirichlet form of $f$ with
respect to $\nu_{\rho_{*}}^{N}$ is defined as

$D_{N}(f;\nu_{\rho_{*}}^{N}):=\left\langle\sqrt{f}(-\mathscr{L}_{N})\sqrt{f}\right\rangle_{\nu_{\rho_{*}}^{N}}.$

Here, for any function $f$ and any distribution $\mu$ on $\Omega_{N}^{d}$,
$\langle f\rangle_{\mu}:=\int fdu$. Since $\nu_{\rho_{*}}^{N}$ is invariant
for the generator $\mathscr{L}_{N}$, direct calculations show that

$D_{N}(f;\nu_{\rho_{*}}^{N})=\sum_{x,y\in\mathbb{T}_{N}^{d}}D_{x,y}(f;\nu_{\rho_{*}}^{N}),$

where $D_{x,y}$ is the piece of Dirichlet form associated to the bond $(x,y)$,

$D_{x,y}(f;\nu_{\rho_{*}}^{N})=\frac{1}{2}\int
g(\eta(x))s(y-x)\big{[}\sqrt{f(\eta^{x,y})}-\sqrt{f(\eta)}\big{]}^{2}d\nu_{\rho_{*}}^{N}.$

Above, $s(x)=[p(x)+p(-x)]/2$.

######  Lemma 3.1.

There exists a constant $C>0$ independent of $N$ such that for any $t>0$,

$H(\mu_{N,t}|\nu_{\rho_{*}}^{N})+N^{2}\int_{0}^{t}D_{N}(f_{s};\nu_{\rho_{*}}^{N})ds\leq
CN^{d-2\alpha}.$

###### Proof.

Since $\nu_{\rho_{*}}^{N}$ is invariant for the zero range process, following
the proof in [7, Subsection 5.2] line by line, we have

$H(\mu_{N,t}|\nu_{\rho_{*}}^{N})+N^{2}\int_{0}^{t}D_{N}(f_{s};\nu_{\rho_{*}}^{N})ds\leq
H(\mu_{N,0}|\nu_{\rho_{*}}^{N}).$

Therefore, to conclude the proof, we only need to show

$H(\mu_{N,0}|\nu_{\rho_{*}}^{N})\leq CN^{d-2\alpha}.$

By direct calculations,

$H(\mu_{N,0}|\nu_{\rho_{*}}^{N})=\int\log\frac{\mu_{N,0}(\eta)}{\nu_{\rho_{*}}^{N}(\eta)}\mu_{N,0}(d\eta)\\\
=\sum_{x\in\mathbb{T}_{N}^{d}}\Big{\\{}\log\frac{Z(\Phi(\rho_{*}))}{Z(\Phi(\rho_{N}(0,\tfrac{x}{N}))}+\rho_{N}(0,\tfrac{x}{N})\log\frac{\Phi(\rho_{N}(0,\tfrac{x}{N}))}{\Phi(\rho_{*})}\Big{\\}}.$

Using the basic inequality $\log(1+x)\leq x$, we bound the last line by

$\sum_{x\in\mathbb{T}_{N}^{d}}\Big{\\{}\frac{Z(\Phi(\rho_{*}))-Z(\Phi(\rho_{N}(0,\tfrac{x}{N}))}{Z(\Phi(\rho_{N}(0,\tfrac{x}{N}))}+\rho_{N}(0,\tfrac{x}{N})\frac{\Phi(\rho_{N}(0,\tfrac{x}{N}))-\Phi(\rho_{*})}{\Phi(\rho_{*})}\Big{\\}}.$
(3.1)

Note that for any $\rho>0$,

$\frac{Z^{\prime}(\Phi(\rho))}{Z(\Phi(\rho))}=\frac{\rho}{\Phi(\rho)}.$

Then, by Taylor’s expansion, the first term inside the brace in (3.1) equals

$\frac{\rho_{N}(0,\tfrac{x}{N})}{\Phi(\rho_{N}(0,\tfrac{x}{N}))}\big{[}\Phi(\rho_{*})-\Phi(\rho_{N}(0,\tfrac{x}{N}))\big{]}+\mathcal{O}(N^{-2\alpha}).$

Therefore, we may bound the term (3.1) by

$\sum_{x\in\mathbb{T}^{d}_{N}}\rho_{N}(0,\tfrac{x}{N})\big{(}\Phi(\rho_{N}(0,\tfrac{x}{N}))-\Phi(\rho_{*})\big{)}\Big{[}\frac{1}{\Phi(\rho_{*})}-\frac{1}{\Phi(\rho_{N}(0,\tfrac{x}{N}))}\Big{]}+\mathcal{O}(N^{d-2\alpha}).$

We conclude the proof by noting that
$|\Phi(\rho_{N}(0,\tfrac{x}{N}))-\Phi(\rho_{*})|\leq CN^{-\alpha}$. ∎

### 3.2 Calculations.

Let

$\psi_{N,t}:=\frac{d\nu_{N,t}}{d\nu_{\rho_{*}}^{N}}.$

Since $\nu_{N,t}$ and $\nu_{\rho_{*}}^{N}$ are both product measures,
$\psi_{N,t}$ is explicitly given by

$\psi_{N,t}(\eta)=\prod_{y\in\mathbb{T}_{N}^{d}}\frac{Z(\Phi(\rho_{*}))}{Z(\Phi(\rho_{N}(t,\tfrac{y}{N})))}\Big{[}\frac{\Phi(\rho_{N}(t,\tfrac{y}{N}))}{\Phi(\rho_{*})}\Big{]}^{\eta(y)}.$

Using Yau’s relative entropy inequality (_cf._ [7, Lemma 6.1.4] for example),

$\frac{dh_{N}(t)}{dt}\leq
N^{-d}\int\big{\\{}\psi_{N,t}^{-1}N^{2}\mathscr{L}_{N}^{*}\psi_{N,t}-\partial_{t}\log\psi_{N,t}\big{\\}}d\mu_{N,t}.$
(3.2)

Above, $\mathscr{L}_{N}^{*}$ is the adjoint generator of $\mathscr{L}_{N}$ in
$L^{2}(\nu_{\rho_{*}}^{N})$ and corresponds to the zero range process with
jump rate $p(-\cdot)$. Precisely speaking, for any
$f:\Omega_{N}^{d}\rightarrow\mathbb{R}$,

$\mathscr{L}_{N}^{*}f(\eta)=\sum_{x,y\in\mathbb{T}_{N}^{d}}g(\eta(x))p(-y)\big{[}f(\eta^{x,x+y})-f(\eta)\big{]}.$

Now we calculate the right hand side of (3.2). By direct calculations,

$\psi_{N,t}^{-1}N^{2-d}\mathscr{L}_{N}^{*}\psi_{N,t}=N^{2-d}\sum_{x,y\in\mathbb{T}_{N}^{d}}g(\eta(y))p(y-x)\bigg{[}\frac{\Phi(\rho_{N}(t,\tfrac{x}{N}))}{\Phi(\rho_{N}(t,\tfrac{y}{N}))}-1\bigg{]}\\\
=-N^{1-d}\sum_{y\in\mathbb{T}_{N}^{d}}\sum_{i=1}^{d}\frac{m_{i}\partial_{u_{i}}\Phi(\rho_{N}(t,\tfrac{y}{N}))}{\Phi(\rho_{N}(t,\tfrac{y}{N}))}\big{[}g(\eta(y))-\Phi(\rho_{N}(t,\tfrac{y}{N}))\big{]}\\\
+\frac{1}{2N^{d}}\sum_{y\in\mathbb{T}_{N}^{d}}\sum_{i,j=1}^{d}\frac{\sigma_{i,j}\partial^{2}_{u_{i},u_{j}}\Phi(\rho_{N}(t,\tfrac{y}{N}))}{\Phi(\rho_{N}(t,\tfrac{y}{N}))}\big{[}g(\eta(y))-\Phi(\rho_{N}(t,\tfrac{y}{N}))\big{]}+\mathcal{E}_{N,1}+\mathcal{O}(N^{-1-\alpha}).$
(3.3)

Above, the term $\mathcal{O}(N^{-1-\alpha})$ comes from the errors by Taylor’s
expansion since there exists some constant $C$ independent of $N$ such that

$\sup_{u\in\mathbb{T}^{d}}\big{|}\partial^{3}_{u_{i},u_{j},u_{k}}\Phi(\rho_{N}(t,u))\big{|}\leq
CN^{-\alpha},\quad\forall 1\leq i,j,k\leq d,$

and the other error term is given by

$\mathcal{E}_{N,1}=-N^{1-d}\sum_{y\in\mathbb{T}_{N}^{d}}\sum_{i=1}^{d}m_{i}\partial_{u_{i}}\Phi(\rho_{N}(t,\tfrac{y}{N}))+\frac{1}{2N^{d}}\sum_{y\in\mathbb{T}_{N}^{d}}\sum_{i,j=1}^{d}\sigma_{i,j}\partial^{2}_{u_{i},u_{j}}\Phi(\rho_{N}(t,\tfrac{y}{N})).$
(3.4)

######  Lemma 3.2.

There exists a constant $C$ independent of $N$ such that

$|\mathcal{E}_{N,1}|\leq C\big{(}N^{-1-\alpha}+N^{-3\alpha}\big{)}.$

In particular, since $\alpha\in(0,1)$, $\mathcal{E}_{N,1}=o(N^{-2\alpha})$.

###### Proof.

By direct calculations, for $1\leq i,j\leq d$,

$\displaystyle\partial_{u_{i}}\Phi(\rho_{N}(t,u))$
$\displaystyle=N^{-\alpha}\Phi^{\prime}(\rho_{N}(t,u))\partial_{u_{i}}\rho(t,u),$
(3.5) $\displaystyle\partial^{2}_{u_{i},u_{j}}\Phi(\rho_{N}(t,u))$
$\displaystyle=N^{-\alpha}\Phi^{\prime}(\rho_{N}(t,u))\partial^{2}_{u_{i},u_{j}}\rho(t,u)+N^{-2\alpha}\Phi^{\prime\prime}(\rho_{N}(t,u))\partial_{u_{i}}\rho(t,u)\partial_{u_{j}}\rho(t,u).$
(3.6)

By assumption (2.1),

$m\cdot\nabla\rho(t,u)=0$ (3.7)

for all $t\geq 0$ and for all $u\in\mathbb{T}^{d}$. Therefore, the first term
on the right hand side of (3.4) is equal to zero. Since

$\Phi^{\prime}(\rho_{N}(t,u))=\Phi^{\prime}(\rho_{*})+N^{-\alpha}\Phi^{\prime\prime}(\rho_{*})\rho(t,u)+\mathcal{O}(N^{-2\alpha}),\quad\Phi^{\prime\prime}(\rho_{N}(t,u))=\Phi^{\prime\prime}(\rho_{*})+\mathcal{O}(N^{-\alpha}),$

by (3.6), the second term in (3.4) equals

$\frac{1}{2N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}\sum_{i,j=1}^{d}\Phi^{\prime}(\rho_{*})\sigma_{i,j}\partial_{u_{i},u_{j}}^{2}\rho(t,\tfrac{y}{N})\\\
+\frac{1}{2N^{d+2\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}\sum_{i,j=1}^{d}\Phi^{\prime\prime}(\rho_{*})\sigma_{i,j}\big{[}\rho(t,\tfrac{y}{N})\partial_{u_{i},u_{j}}^{2}\rho(t,\tfrac{y}{N})+\partial_{u_{i}}\rho(t,\tfrac{y}{N})\partial_{u_{j}}\rho(t,\tfrac{y}{N})\big{]}\\\
+\mathcal{O}(N^{-3\alpha}).$ (3.8)

Since

$\Big{|}\frac{1}{N^{d}}\sum_{y\in\mathbb{T}_{N}^{d}}\sum_{i,j=1}^{d}\Phi^{\prime}(\rho_{*})\sigma_{i,j}\partial_{u_{i},u_{j}}^{2}\rho(t,\tfrac{y}{N})-\int_{\mathbb{T}^{d}}\sum_{i,j=1}^{d}\Phi^{\prime}(\rho_{*})\sigma_{i,j}\partial_{u_{i},u_{j}}^{2}\rho(t,u)du\Big{|}\leq\frac{C}{N},$

and by the conservation of particle numbers,

$\int_{\mathbb{T}^{d}}\frac{1}{2}\sum_{i,j=1}^{d}\Phi^{\prime}(\rho_{*})\sigma_{i,j}\partial_{u_{i},u_{j}}^{2}\rho(t,u)du=\partial_{t}\int_{\mathbb{T}^{d}}\rho(t,u)du=0,$

the first term in (3.8) is bounded by $CN^{-1-\alpha}$. Similarly, the second
term in (3.8) is bounded by $CN^{-1-2\alpha}$ since

$\int_{\mathbb{T}^{d}}\sum_{i,j=1}^{d}\sigma_{i,j}\big{[}\rho(t,u)\partial_{u_{i},u_{j}}^{2}\rho(t,u)+\partial_{u_{i}}\rho(t,u)\partial_{u_{j}}\rho(t,u)\big{]}du=0.$

This concludes the proof. ∎

By (3.5) and (3.7), the first term on the right hand side of (3.3) equals
zero. By (3.6) and Lemma 3.2, we rewrite
$\psi_{N,t}^{-1}N^{2-d}\mathscr{L}_{N}^{*}\psi_{N,t}$ as

$\psi_{N,t}^{-1}N^{2-d}\mathscr{L}_{N}^{*}\psi_{N,t}\\\
=\frac{1}{2N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}\sum_{i,j=1}^{d}\frac{\sigma_{i,j}\Phi^{\prime}(\rho_{N}(t,\tfrac{y}{N}))\partial^{2}_{u_{i},u_{j}}\rho(t,\tfrac{y}{N})}{\Phi(\rho_{N}(t,\tfrac{y}{N}))}\big{[}g(\eta(y))-\Phi(\rho_{N}(t,\tfrac{y}{N}))\big{]}\\\
+\frac{1}{2N^{d+2\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}\sum_{i,j=1}^{d}\frac{\sigma_{i,j}\Phi^{\prime\prime}(\rho_{N}(t,\tfrac{y}{N}))\partial_{u_{i}}\rho(t,\tfrac{y}{N})\partial_{u_{j}}\rho(t,\tfrac{y}{N})}{\Phi(\rho_{N}(t,\tfrac{y}{N}))}\big{[}g(\eta(y))-\Phi(\rho_{N}(t,\tfrac{y}{N}))\big{]}+o(N^{-2\alpha}).$
(3.9)

Direct calculations yield

$N^{-d}\partial_{t}\log\psi_{N,t}=\frac{1}{N^{d}}\sum_{y\in\mathbb{T}_{N}^{d}}\Big{\\{}\frac{\partial_{t}\Phi(\rho_{N}(t,\tfrac{y}{N}))}{\Phi(\rho_{N}(t,\tfrac{y}{N}))}\eta(y)-\partial_{t}\log
Z\big{(}\Phi(\rho_{N}(t,\tfrac{y}{N}))\big{)}\Big{\\}}.$

Since
$E_{\nu_{N,t}}[\partial_{t}\log\psi_{N,t}]=E_{\nu_{\rho_{*}}^{N}}[\partial_{t}\psi_{N,t}]=0$
and $\rho_{N}(t,\frac{y}{N})=E_{\nu_{N,t}}[\eta(y)]$, we rewrite the last line
as

$N^{-d}\partial_{t}\log\psi_{N,t}=\frac{1}{N^{d}}\sum_{y\in\mathbb{T}_{N}^{d}}\frac{\partial_{t}\Phi(\rho_{N}(t,\tfrac{y}{N}))}{\Phi(\rho_{N}(t,\tfrac{y}{N}))}\big{(}\eta(y)-\rho_{N}(t,\tfrac{y}{N})\big{)}.$

Note that

$\partial_{t}\Phi(\rho_{N}(t,u))=N^{-\alpha}\Phi^{\prime}(\rho_{N}(t,u))\partial_{t}\rho(t,u)=\frac{\Phi^{\prime}(\rho_{N}(t,u))}{2N^{\alpha}}\sum_{i,j=1}^{d}\sigma_{i,j}\Phi^{\prime}(\rho_{*})\partial^{2}_{u_{i},u_{j}}\rho(t,u).$

Together with (3.9) and (3.2), we have

$\frac{dh_{N}(t)}{dt}\leq\int\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}a_{N,t}(y)\big{[}g(\eta(y))-\Phi(\rho_{N}(t,\tfrac{y}{N}))-\Phi^{\prime}(\rho_{*})\big{(}\eta(y)-\rho_{N}(t,\tfrac{y}{N})\big{)}\big{]}d\mu_{N,t}\\\
+\int\frac{1}{N^{d+2\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}b_{N,t}(y)\big{[}g(\eta(y))-\Phi(\rho_{N}(t,\tfrac{y}{N}))\big{]}d\mu_{N,t}+o(N^{-2\alpha}).$
(3.10)

Above,

$\displaystyle a_{N,t}(y)$
$\displaystyle=\frac{1}{2}\sum_{i,j=1}^{d}\frac{\sigma_{i,j}\Phi^{\prime}(\rho_{N}(t,\tfrac{y}{N}))\partial^{2}_{u_{i},u_{j}}\rho(t,\tfrac{y}{N})}{\Phi(\rho_{N}(t,\tfrac{y}{N}))},$
$\displaystyle b_{N,t}(y)$
$\displaystyle=\frac{1}{2}\sum_{i,j=1}^{d}\frac{\sigma_{i,j}\Phi^{\prime\prime}(\rho_{N}(t,\tfrac{y}{N}))\partial_{u_{i}}\rho(t,\tfrac{y}{N})\partial_{u_{j}}\rho(t,\tfrac{y}{N})}{\Phi(\rho_{N}(t,\tfrac{y}{N}))}.$

### 3.3 Proof of Theorem 2.3.

In this subsection, we deal with the terms on the right hand side of (3.10)
respectively. We first deal with the second one since it is simpler.

######  Lemma 3.3.

There exists a constant $C$ independent of $N$ such that

$\int\frac{1}{N^{d+2\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}b_{N,t}(y)\big{[}g(\eta(y))-\Phi(\rho_{N}(t,\tfrac{y}{N}))\big{]}d\mu_{N,t}\leq
CN^{-3\alpha}.$

###### Proof.

Since $|\Phi(\rho_{N}(t,\tfrac{y}{N}))-\Phi(\rho_{*})|\leq CN^{-\alpha}$, we
only need to prove

$\int\frac{1}{N^{d+2\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}b_{N,t}(y)\big{[}g(\eta(y))-\Phi(\rho_{*})\big{]}d\mu_{N,t}\leq
CN^{-3\alpha}.$

By entropy inequality (_cf._ [7, Appendix 1.8] for example), the left hand
side above is bounded by

$\frac{H(\mu_{N,t}|\nu_{\rho_{*}}^{N})}{N^{d+\alpha}}+\frac{1}{N^{d+\alpha}}\log
E_{\nu_{\rho_{*}}^{N}}\Big{[}\exp\Big{\\{}\frac{1}{N^{\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}b_{N,t}(y)\big{[}g(\eta(y))-\Phi(\rho_{*})\big{]}\Big{\\}}\Big{]}.$

By Lemma 3.1, $H(\mu_{N,t}|\nu_{\rho_{*}}^{N})\leq CN^{d-2\alpha}$, hence the
first term above is bounded by $CN^{-3\alpha}$. Since $\nu_{\rho_{*}}^{N}$ is
product measure, the second term in the last expression equals

$\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}\log
E_{\nu_{\rho_{*}}^{N}}\Big{[}\exp\Big{\\{}\frac{1}{N^{\alpha}}b_{N,t}(y)\big{[}g(\eta(y))-\Phi(\rho_{*})\big{]}\Big{\\}}\Big{]}.$
(3.11)

For a random variable $X$ such that $E[X]=0$ and $E[e^{\lambda X}]<\infty$ for
all $\lambda\in\mathbb{R}$, we claim that for any $\lambda_{0}>0$, there
exists a constant $C=C(\lambda_{0})$ such that for all
$0<\lambda<\lambda_{0}$,

$\log E[e^{\lambda X}]\leq CE[X^{4}]^{1/2}\lambda^{2}.$ (3.12)

Indeed, since $e^{x}\leq 1+x+(x^{2}/2)e^{|x|}$ and $\log(1+x)\leq x$, for any
$0<\lambda<\lambda_{0}$,

$\log E[e^{\lambda
X}]\leq\frac{\lambda^{2}}{2}E\big{[}X^{2}e^{\lambda|X|}\big{]}\leq\frac{\lambda^{2}}{2}E\big{[}X^{4}]^{1/2}E\big{[}e^{2\lambda_{0}|X|}\big{]}^{1/2}=:C(\lambda_{0})E[X^{4}]^{1/2}\lambda^{2}.$

Using the above claim, we bound (3.11) by $CN^{-3\alpha}$ for large $N$. This
completes the proof. ∎

Using the same argument as in Lemma 3.3, we could replace
$\Phi^{\prime}(\rho_{*})$ by $\Phi^{\prime}(\rho_{N}(t,\tfrac{y}{N}))$ in the
first line in (3.10) plus an error of order $\mathcal{O}(N^{-3\alpha})$. Up to
now, we have shown

$\frac{dh_{N}(t)}{dt}\leq\int\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}a_{N,t}(y)\big{[}g(\eta(y))-\Phi(\rho_{N}(t,\tfrac{y}{N}))\\\
-\Phi^{\prime}(\rho_{N}(t,\tfrac{y}{N}))\big{(}\eta(y)-\rho_{N}(t,\tfrac{y}{N})\big{)}\big{]}d\mu_{N,t}+o(N^{-2\alpha}).$
(3.13)

For a positive integer $\ell=\ell(N)$ and for any sequence
$\\{\phi(x)\\}_{x\in\mathbb{Z}^{d}}$, define

$\bar{\phi}^{\ell}(x)=\frac{1}{(2\ell+1)^{d}}\sum_{|y-x|\leq\ell}\phi(y).$

In the following, we shall take

  * •

either $\ell=cN^{\frac{\alpha+d/2}{d+1}}$ for some sufficiently small constant
$c>0$ when applying the spectral gap estimate (_cf._ Lemma 4.2),

  * •

or $\ell=N^{\frac{\alpha+1}{d+1}}$ when applying the logarithmic Sobolev
inequality (_cf._ Lemma 4.4).

######  Lemma 3.4.

There exists a constant $C$ independent of $N$ such that

$\int\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}\big{[}a_{N,t}(y)-\bar{a}_{N,t}^{\ell}(y)\big{]}\big{[}g(\eta(y))-\Phi(\rho_{N}(t,\tfrac{y}{N}))\\\
-\Phi^{\prime}(\rho_{N}(t,\tfrac{y}{N}))\big{(}\eta(y)-\rho_{N}(t,\tfrac{y}{N})\big{)}\big{]}d\mu_{N,t}\leq\frac{C\ell^{2}}{N^{2+2\alpha}}.$
(3.14)

In particular, with the above choice of $\ell$, the above term is of order
$o(N^{-2\alpha})$.

######  Remark 3.5.

By Taylor’s expansion, the above formula is bounded by
$C\ell^{2}/N^{2+\alpha}$. However, the bound is of order $o(N^{-2\alpha})$
only for $\alpha<(d+2)/(d+3)$ when $\ell=cN^{\frac{\alpha+d/2}{d+1}}$, and
only for $\alpha<2d/(d+3)$ when $\ell=N^{\frac{\alpha+1}{d+1}}$, which is not
optimal.

###### Proof.

Since the proof is exactly the same as in Lemma 3.3, we only sketch it. Note
that $\big{|}a_{N,t}(y)-\bar{a}_{N,t}^{\ell}(y)\big{|}\leq C\ell^{2}/N^{2}$.
Let

$A_{N,t}^{\ell}(\eta)=\frac{N^{2}}{\ell^{2}}\big{[}a_{N,t}(0)-\bar{a}_{N,t}^{\ell}(0)\big{]}\big{[}g(\eta(0))-\Phi(\rho_{*})-\Phi^{\prime}(\rho_{*})\big{(}\eta(0)-\rho_{*}\big{)}\big{]}.$

We pay a price of order $\mathcal{O}(\ell^{2}N^{-2-2\alpha})$ by replacing
$\rho_{N}(t,\frac{y}{N})$ with $\rho_{*}$ in (3.14). By entropy inequality,
Lemma 3.1 and inequality (3.12),

$\int\frac{\ell^{2}}{N^{d+\alpha+2}}\sum_{y\in\mathbb{T}_{N}^{d}}\tau_{y}A_{N,t}^{\ell}(\eta)d\mu_{N,t}\leq\frac{H(\mu_{N,t}|\nu_{\rho_{*}}^{N})}{N^{d+2}\ell^{-2}}\\\
+\frac{1}{N^{d+2}\ell^{-2}}\sum_{y\in\mathbb{T}_{N}^{d}}\log\int\exp\Big{\\{}\frac{1}{N^{\alpha}}\tau_{y}A^{\ell}_{N,t}(\eta)\Big{\\}}d\nu_{\rho_{*}}^{N}\leq\frac{C\ell^{2}}{N^{2+2\alpha}}.$

This concludes the proof. ∎

Using summation by parts formula,

$\int\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}\bar{a}_{N,t}^{\ell}(y)\big{[}g(\eta(y))-\Phi(\rho_{N}(t,\tfrac{y}{N}))-\Phi^{\prime}(\rho_{N}(t,\tfrac{y}{N}))\big{(}\eta(y)-\rho_{N}(t,\tfrac{y}{N})\big{)}\big{]}d\mu_{N,t}\\\
=\int\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}a_{N,t}(y)\big{[}\bar{g}^{\ell}(\eta(y))-\Phi(\bar{\rho}^{\ell}_{N}(t,\tfrac{y}{N}))-\Phi^{\prime}(\bar{\rho}^{\ell}_{N}(t,\tfrac{y}{N}))\big{(}\bar{\eta}^{\ell}(y)-\bar{\rho}^{\ell}_{N}(t,\tfrac{y}{N})\big{)}\big{]}d\mu_{N,t}\\\
+\sum_{j=2}^{3}\mathcal{E}_{N,j},$ (3.15)

where

$\displaystyle\mathcal{E}_{N,2}=$
$\displaystyle\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}a_{N,t}(y)\Big{[}\Phi(\bar{\rho}^{\ell}_{N}(t,\tfrac{y}{N}))-\bar{\Phi}^{\ell}(\rho_{N}(t,\tfrac{y}{N}))\Big{]},$
$\displaystyle\mathcal{E}_{N,3}=$
$\displaystyle\int\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}a_{N,t}(y)\Big{[}\Phi^{\prime}(\bar{\rho}^{\ell}_{N}(t,\tfrac{y}{N}))\big{(}\bar{\eta}^{\ell}(y)-\bar{\rho}^{\ell}_{N}(t,\tfrac{y}{N})\big{)}$
$\displaystyle-\frac{1}{(2\ell+1)^{d}}\sum_{|x-y|\leq\ell}\Phi^{\prime}(\rho_{N}(t,\tfrac{x}{N}))\big{(}\eta(x)-\rho_{N}(t,\tfrac{x}{N})\big{)}\Big{]}d\mu_{N,t}.$

Since for any $|x-y|\leq\ell$,
$|\rho_{N}(t,\tfrac{x}{N})-\rho_{N}(t,\tfrac{y}{N})|\leq C\ell N^{-1-\alpha}$,
it is easy to see

$\big{|}\mathcal{E}_{N,j}\big{|}\leq C\ell
N^{-1-2\alpha}=o(N^{-2\alpha}),\quad j=2,3.$

In order to deal with the first term on the right hand side of (3.15), we need
the following one-block estimate, whose proof is postponed to Section 4.

######  Lemma 3.6 (One-block estimate).

With the choice of $\ell(N)$ as above, for any continuous function
$F:\mathbb{R}_{+}\times\mathbb{T}^{d}\rightarrow\mathbb{R}$,

$\int_{0}^{t}E_{\mu_{N,s}}\Big{[}\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}F\big{(}s,\tfrac{y}{N}\big{)}\tau_{y}V_{g}^{\ell}(\eta)\Big{]}ds=o(N^{-2\alpha}),$

where

$V_{g}^{\ell}(\eta):=\bar{g}^{\ell}(\eta(0))-\Phi\big{(}\bar{\eta}^{\ell}(0)\big{)}.$

By Lemma 3.6 and (3.15), we have shown that

$h_{N}(t)\leq
h_{N}(0)+\int^{t}_{0}E_{\mu_{N,s}}\Big{[}\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}a_{N,s}(y)W^{\ell}_{y}(s,\eta)\Big{]}ds+o(N^{-2\alpha}),$
(3.16)

where

$W^{\ell}_{y}(t,\eta):=\Phi(\bar{\eta}^{\ell}(y))-\Phi(\bar{\rho}^{\ell}_{N}(t,\tfrac{y}{N}))-\Phi^{\prime}(\bar{\rho}^{\ell}_{N}(t,\tfrac{y}{N}))\big{(}\bar{\eta}^{\ell}(y)-\bar{\rho}^{\ell}_{N}(t,\tfrac{y}{N})\big{)}.$

In order to deal with the integral in (3.16), we first state a lemma and refer
the readers to [15, P.193] for its proof.

######  Lemma 3.7 ([15, Lemma 2]).

Let $\zeta_{i},\,i\geq 1,$ be independent random variables with mean zero. For
any $\lambda>0$, assume

$\Lambda_{i}(\lambda)=\log E[e^{\lambda\zeta_{i}}]<\infty.$

Assume further that there exist constants $\lambda_{0}>0$ and $C_{0}>0$ such
that $\Lambda_{i}(\lambda)\leq C_{0}\lambda^{2}$ for all
$0<\lambda<\lambda_{0}$ and for all $i\geq 1$. Let
$G:\mathbb{R}\rightarrow\mathbb{R}_{+}$ be smooth and satisfy $G(u)\leq
C_{1}(|u|\wedge u^{2})$ for some constant $C_{1}>0$. Denote
$S_{\ell}=\sum_{i=1}^{\ell}\zeta_{i}$. Then, there exist constants
$\gamma_{0}>0$ and $C_{2}<\infty$ such that for any $\ell>\gamma_{0}^{-1}$,
for any $0<\gamma<\gamma_{0}$,

$\log E\big{[}\exp\big{\\{}\gamma\ell G(S_{\ell}/\ell)\big{\\}}\big{]}<C_{2}.$

Now we continue to treat (3.16). By entropy inequality, for any $\gamma>0$,

$E_{\mu_{N,s}}\Big{[}\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}a_{N,s}(y)W^{\ell}_{y}(s,\eta)\Big{]}\leq\frac{h_{N}(s)}{\gamma}+\frac{1}{\gamma
N^{d}}\log
E_{\nu_{N,s}}\Big{[}\exp\Big{\\{}\frac{\gamma}{N^{\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}a_{N,s}(y)W^{\ell}_{y}(s,\eta)\Big{\\}}\Big{]}.$

Since $W^{\ell}_{y}(s,\eta)$ and $W^{\ell}_{y^{\prime}}(s,\eta)$ are
independent under $\nu_{N,s}$ if $|y-y^{\prime}|>2\ell$, by Hölder’s
inequality, the second term in the last line is bounded by

$\frac{1}{\gamma N^{d}(2\ell+1)^{d}}\sum_{y\in\mathbb{T}_{N}^{d}}\log
E_{\nu_{N,s}}\Big{[}\exp\Big{\\{}\frac{\gamma(2\ell+1)^{d}}{N^{\alpha}}a_{N,s}(y)W^{\ell}_{y}(s,\eta)\Big{\\}}\Big{]}.$

Take $\gamma=\gamma_{0}N^{\alpha}$ for some fixed but small enough
$\gamma_{0}$ and take

$G(u)=\Phi(u+\bar{\rho}_{N}^{\ell}(t,y))-\Phi(\bar{\rho}_{N}^{\ell}(t,\tfrac{y}{N}))-\Phi^{\prime}(\bar{\rho}_{N}^{\ell}(t,\tfrac{y}{N}))u,\quad
u=\bar{\eta}^{\ell}(y)-\bar{\rho}_{N}^{\ell}(t,\tfrac{y}{N})$

in Lemma 3.7, then the above term is bounded by $C\ell^{-d}N^{-\alpha}$ for
some constant $C=C(\gamma_{0})$. Therefore,

$h_{N}(t)\leq
h_{N}(0)+\frac{C}{N^{\alpha}}\int_{0}^{t}h_{N}(s)ds+\frac{C}{\ell^{d}N^{\alpha}}+o(N^{-2\alpha}).$
(3.17)

With of choices of $\ell$ above,

$\frac{1}{\ell^{d}N^{\alpha}}=\mathcal{O}(N^{-\frac{(4d+2)\alpha+d^{2}}{2(d+1)}})\quad\text{or}\quad\frac{1}{\ell^{d}N^{\alpha}}=\mathcal{O}(N^{-\frac{(2d+1)\alpha+d}{d+1}}).$

In both cases, the third term in (3.17) is of order $o(N^{-2\alpha})$ if
$d\geq 2$ and $\alpha\in(0,1)$. We conclude the proof of Theorem 2.3 by using
Grönwall’s inequality.

##  One-block Estimate

In this section, we prove Lemma 3.6. In order to cut off large densities, for
each $M>0$, let

$V_{g,M}^{\ell}(\eta)=V_{g}^{\ell}(\eta)\mathbf{1}_{\\{\bar{\eta}^{\ell}(0)\leq
M\\}}.$

Then, the expression in Lemma 3.6 equals

$\int_{0}^{t}E_{\mu_{N,s}}\Big{[}\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}F\big{(}s,\tfrac{y}{N}\big{)}\tau_{y}\big{(}V_{g}^{\ell}(\eta)\mathbf{1}_{\\{\bar{\eta}^{\ell}(0)>M\\}}\big{)}\Big{]}ds\\\
+\int_{0}^{t}E_{\mu_{N,s}}\Big{[}\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}F\big{(}s,\tfrac{y}{N}\big{)}\tau_{y}V_{g,M}^{\ell}(\eta)\Big{]}ds.$
(4.1)

### 4.1 Cut off large densities.

In this subsection, we bound the first term in (4.1).

######  Lemma 4.1.

There exists some constant $C=C(M,F,t)$ such that for $N$ large enough,

$\int_{0}^{t}E_{\mu_{N,s}}\Big{[}\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}F\big{(}s,\tfrac{y}{N}\big{)}\tau_{y}\big{(}V_{g}^{\ell}(\eta)\mathbf{1}_{\\{\bar{\eta}^{\ell}(0)>M\\}}\big{)}\Big{]}ds\leq
C\Big{(}\frac{\log N}{N^{3\alpha}}+\frac{\log
N}{\ell^{d}N^{\alpha}}\exp\\{-C\ell^{d}\\}\Big{)}.$

In particular, with the choices of $\ell=\ell(N)$ given in Subsection 3.3, the
above term has order $o(N^{-2\alpha})$.

###### Proof.

By Assumption 2.1 $(i)$, $\big{|}V_{g}^{\ell}(\eta)\big{|}\leq
C(a_{0})\bar{\eta}^{\ell}(0)$. Therefore, the term on the right hand side in
the lemma is bounded by

$C\|F\|_{\infty}\int_{0}^{t}E_{\mu_{N,s}}\Big{[}\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}\bar{\eta}^{\ell}(y)\mathbf{1}_{\\{\bar{\eta}^{\ell}(y)>M\\}}\Big{]}ds.$

By entropy inequality [7, Appendix 1.8], for any $\gamma>0$, the integral
above is bounded by

$\int_{0}^{t}\frac{H(\mu_{N,s}|\nu_{\rho_{*}}^{N})}{\gamma}ds+\frac{t}{\gamma}\log
E_{\nu_{\rho_{*}}^{N}}\Big{[}\exp\Big{\\{}\frac{\gamma}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}\bar{\eta}^{\ell}(y)\mathbf{1}_{\\{\bar{\eta}^{\ell}(y)>M\\}}\Big{\\}}\Big{]}$
(4.2)

By Lemma 3.1, for any $0\leq s\leq t$,

$H(\mu_{N,s}|\nu_{\rho_{*}}^{N})\leq H(\mu_{N,0}|\nu_{\rho_{*}}^{N})\leq
CN^{d-2\alpha}.$

For the second term in (4.2), since $\bar{\eta}^{\ell}(y)$ and
$\bar{\eta}^{\ell}(z)$ are independent if $|y-z|>2\ell$, by Hölder’s
inequality,

$\log
E_{\nu_{\rho_{*}}^{N}}\Big{[}\exp\Big{\\{}\frac{\gamma}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}\bar{\eta}^{\ell}(y)\mathbf{1}_{\\{\bar{\eta}^{\ell}(y)>M\\}}\Big{\\}}\Big{]}\\\
\leq\frac{N^{d}}{(2\ell+1)^{d}}\log
E_{\nu_{\rho_{*}}^{N}}\Big{[}\exp\Big{\\{}\frac{\gamma(2\ell+1)^{d}}{N^{d+\alpha}}\bar{\eta}^{\ell}(0)\mathbf{1}_{\\{\bar{\eta}^{\ell}(0)>M\\}}\Big{\\}}\Big{]}$

By using the basic inequality $\log(1+x)\leq x$, we bound the last line by

$\frac{N^{d}}{(2\ell+1)^{d}}\log
E_{\nu_{\rho_{*}}^{N}}\Big{[}1+\exp\Big{\\{}\frac{\gamma(2\ell+1)^{d}}{N^{d+\alpha}}\bar{\eta}^{\ell}(0)\Big{\\}}\mathbf{1}_{\\{\bar{\eta}^{\ell}(0)>M\\}}\Big{]}\\\
\leq\frac{N^{d}}{(2\ell+1)^{d}}E_{\nu_{\rho_{*}}^{N}}\Big{[}\exp\Big{\\{}\frac{\gamma(2\ell+1)^{d}}{N^{d+\alpha}}\bar{\eta}^{\ell}(0)\Big{\\}}\mathbf{1}_{\\{\bar{\eta}^{\ell}(0)>M\\}}\Big{]}.$

By Cauchy-Schwarz inequality, the expectation in the last line is bounded by

$E_{\nu_{\rho_{*}}^{N}}\Big{[}\exp\Big{\\{}\frac{2\gamma(2\ell+1)^{d}}{N^{d+\alpha}}\bar{\eta}^{\ell}(0)\Big{\\}}\Big{]}^{1/2}P_{\nu_{\rho_{*}}^{N}}\Big{(}\bar{\eta}^{\ell}(0)>M\Big{)}^{1/2}.$
(4.3)

For $\lambda\geq 0$ and $\theta\in\mathbb{R}$, define

$\Lambda(\lambda):=\log
E_{\nu_{\rho_{*}}^{N}}\big{[}e^{\lambda\eta(0)}\big{]},\quad
I(\theta):=\sup_{\lambda}\\{\lambda\theta-\Lambda(\lambda)\\}.$

Recall under Assumption 2.1, $\Lambda(\lambda)<\infty$ for any $\lambda\geq
0$. By standard large deviation estimates, we may bound (4.3) by

$\exp\Big{\\{}C(2\ell+1)^{d}\big{[}\Lambda(2\gamma/N^{d+\alpha})-I(M)\big{]}\Big{\\}}.$

To sum up, we bound (4.2) by

$\frac{C\|F\|_{\infty}tN^{d}}{\gamma
N^{2\alpha}}+\frac{C\|F\|_{\infty}tN^{d}}{\gamma(2\ell+1)^{d}}\exp\Big{\\{}C(2\ell+1)^{d}\big{[}\Lambda(2\gamma/N^{d+\alpha})-I(M)\big{]}\Big{\\}}.$
(4.4)

Now, take $\gamma=N^{d+\alpha}/\log N$. Since $\lim_{\lambda\rightarrow
0}\Lambda(\lambda)=0$, for $N$ large enough,

$\Lambda(2\gamma/N^{d+\alpha})=\Lambda(2/\log N)<I(M)/2.$

Hence, (4.4) is bounded by

$C\|F\|_{\infty}t\Big{(}\frac{\log N}{N^{3\alpha}}+\frac{\log
N}{\ell^{d}N^{\alpha}}\exp\\{-CI(M)\ell^{d}\\}\Big{)}.$

This concludes the proof of the lemma. ∎

### 4.2 Estimates by spectral gap inequality.

In this subsection, we bound the second term in (4.1) by using the spectral
gap estimates for the zero range process.

######  Lemma 4.2.

Let $F:\mathbb{R}_{+}\times\mathbb{T}^{d}\rightarrow\mathbb{R}$ be continuous.
There exists $\varepsilon_{0}>0$ such that for any $\ell$ satisfying
$M\ell^{d+1}<\varepsilon_{0}N^{d/2+\alpha}$, there exists $C=C(M,F,t)$ such
that

$\int_{0}^{t}E_{\mu_{N,s}}\Big{[}\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}F\big{(}s,\tfrac{y}{N}\big{)}\tau_{y}V_{g,M}^{\ell}(\eta)\Big{]}ds\leq
C\Big{(}\frac{1}{N^{\alpha}\ell^{d}}+\frac{\ell}{N^{2\alpha+d/2}}\Big{)}.$
(4.5)

In particular, by taking

$\ell=\Big{(}\frac{\varepsilon_{0}}{2M}\Big{)}^{1/(d+1)}N^{\frac{\alpha+d/2}{d+1}},$

the left hand side in (4.5) is bounded by
$CN^{-\frac{(4d+2)\alpha+d^{2}}{2(d+1)}}=o(N^{-2\alpha})$.

###### Proof.

By entropy inequality, for any $\gamma>0$,

$\int_{0}^{t}E_{\mu_{N,s}}\Big{[}\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}F\big{(}s,\tfrac{y}{N}\big{)}\tau_{y}V_{g,M}^{\ell}\Big{]}ds\leq\frac{H(\mu_{N,0}|\nu_{\rho_{*}}^{N})}{\gamma
N^{d}}\\\ +\frac{1}{\gamma
N^{d}}\log\mathbb{E}_{\nu_{\rho_{*}}^{N}}\Big{[}\exp\Big{\\{}\int_{0}^{t}\frac{\gamma\|F\|_{\infty}}{N^{\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}\big{|}\tau_{y}V_{g,M}^{\ell}(\eta_{s})\big{|}ds\Big{\\}}\Big{]}.$
(4.6)

By Lemma 3.1, $H(\mu_{N,0}|\nu_{\rho_{*}}^{N})\leq CN^{d-2\alpha}$. Using the
basic inequality

$e^{|x|}\leq e^{x}+e^{-x},\forall x,\quad\text{and}\quad\log(a+b)\leq\log
2+\max\\{\log a,\log b\\},\forall a,b>0,$

we could remove the absolute value inside the exponential in the second term
above. By Feynman-Kac formula [7, Lemma A1.7.2], the second term in (4.6) is
bounded by $t\Gamma_{N}/(\gamma N^{d})$, where $\Gamma_{N}$ is the largest
eigenvalue of the operator
$N^{2}\mathscr{L}_{N}^{s}+\frac{\gamma\|F\|_{\infty}}{N^{\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}\tau_{y}V_{g,M}^{\ell}(\eta)$
with $\mathscr{L}_{N}^{s}:=(\mathscr{L}_{N}+\mathscr{L}_{N}^{*})/2$ being the
symmetric part of $\mathscr{L}_{N}$ in $L^{2}(\nu_{\rho_{*}}^{N})$. Moreover,
$t\Gamma_{N}/(\gamma N^{d})$ may be written as the following variational form

$t\,\sup_{f:\nu_{\rho_{*}}^{N}-density}\Big{\\{}\frac{\|F\|_{\infty}}{N^{d+\alpha}}E_{\nu_{\rho_{*}}^{N}}\Big{[}\sum_{y\in\mathbb{T}_{N}^{d}}\tau_{y}V_{g,M}^{\ell}(\eta)f(\eta)\Big{]}-\frac{N^{2}}{\gamma
N^{d}}D_{N}(f;\nu_{\rho_{*}}^{N})\Big{\\}}.$ (4.7)

Above, recall $D_{N}(f;\nu_{\rho_{*}}^{N})$ is the Dirichlet form of the
density $f$ with respect to the measure $\nu_{\rho_{*}}^{N}$,

$D_{N}(f;\nu_{\rho_{*}}^{N}):=\left\langle\sqrt{f}(-\mathscr{L}_{N})\sqrt{f}\right\rangle_{\nu_{\rho_{*}}^{N}}=\left\langle\sqrt{f}(-\mathscr{L}_{N}^{s})\sqrt{f}\right\rangle_{\nu_{\rho_{*}}^{N}}.$

For any $\nu_{\rho_{*}}^{N}$–density $f$, denote

$\bar{f}=\frac{1}{N^{d}}\sum_{y\in\mathbb{T}_{N}^{d}}\tau_{y}f.$

By translation invariance and convexity of the Dirichlet form,

$D_{N}(f;\nu_{\rho_{*}}^{N})\geq D_{N}(\bar{f};\nu_{\rho_{*}}^{N}).$

This permits us to bound (4.7) by

$t\,\sup_{f}\Big{\\{}\frac{\|F\|_{\infty}}{N^{\alpha}}E_{\nu_{\rho_{*}}^{N}}\Big{[}V_{g,M}^{\ell}(\eta)f(\eta)\Big{]}-\frac{N^{2}}{\gamma
N^{d}}D_{N}(f;\nu_{\rho_{*}}^{N})\Big{\\}}.$ (4.8)

Above, the supremum is over all translation invariant
$\nu_{\rho_{*}}^{N}$–densities $f$.

Denote by $\Lambda_{\ell}^{d}:=\\{-\ell,-\ell+1,\ldots,\ell\\}^{d}$ the
$d$-dimensional cube of length $2\ell+1$ and denote
$\Omega_{\ell}^{d}:=(\Lambda_{\ell}^{d})^{\mathbb{N}}$. Let
$\nu_{\rho_{*}}^{\ell}$ be the product measure on $\Omega_{\ell}^{d}$ with
particle density $\rho_{*}$. Denote
$f_{\ell}:=E_{\nu_{\rho_{*}}^{N}}[f|\mathcal{F}_{\ell}]$ with
$\mathcal{F}_{\ell}$ being the $\sigma$-algebra generated by
$(\eta(x),x\in\Lambda_{\ell}^{d})$. Let

$\displaystyle D_{\ell}(f;\nu_{\rho_{*}}^{N})$
$\displaystyle=\sum_{x,y\in\Lambda_{\ell}^{d}}D_{x,y}(f;\nu_{\rho_{*}}^{N}),$
$\displaystyle D_{\ell}(f_{\ell};\nu_{\rho_{*}}^{\ell})$
$\displaystyle=\left\langle\sqrt{f}(-\mathscr{L}_{\ell}^{s})\sqrt{f}\right\rangle_{\nu_{\rho_{*}}^{\ell}}$
$\displaystyle=\frac{1}{2}\sum_{x,y\in\Lambda_{\ell}^{d}}\int_{\Omega_{\ell}^{d}}g(\eta(x))s(y-x)\big{[}\sqrt{f(\eta^{x,y})}-\sqrt{f(\eta)}\big{]}^{2}d\nu_{\rho_{*}}^{\ell},$

where the symmetric generator $\mathscr{L}_{\ell}^{s}$ acts on functions
$f:\Omega_{\ell}^{d}\rightarrow\mathbb{R}$ as

$\mathscr{L}_{\ell}^{s}f(\eta)=\sum_{x,y\in\Lambda_{\ell}^{d}}g(\eta(x))s(y-x)\big{[}f(\eta^{x,y})-f(\eta)\big{]}.$

Since $f$ is translation invariant, by convexity of the Dirichlet form,

$D_{N}(f;\nu_{\rho_{*}}^{N})\geq\frac{N^{d}}{(2\ell+1)^{d}}D_{\ell}(f;\nu_{\rho_{*}}^{N})\geq\frac{N^{d}}{(2\ell+1)^{d}}D_{\ell}(f_{\ell};\nu_{\rho_{*}}^{\ell}).$

Therefore, (4.8) is bounded by

$t\,\sup_{f}\Big{\\{}\frac{\|F\|_{\infty}}{N^{\alpha}}E_{\nu_{\rho_{*}}^{\ell}}\Big{[}V_{g,M}^{\ell}(\eta)f_{\ell}(\eta)\Big{]}-\frac{N^{2}}{\gamma(2\ell+1)^{d}}D_{\ell}(f_{\ell};\nu_{\rho_{*}}^{\ell})\Big{\\}}.$
(4.9)

To decompose the above term on hyperplanes with fixed number of particles, we
first introduce the following notations. For $j\geq 0$, let

$\Omega_{\ell,j}^{d}=\big{\\{}\eta\in\Omega_{\ell}^{d}:\sum_{x\in\Lambda_{\ell}^{d}}\eta(x)=j\big{\\}},$

and let $\nu_{\ell,j}$ be the conditional measure of $\nu_{\rho_{*}}^{\ell}$
on $\Omega_{\ell,j}^{d}$, _i.e._
$\nu_{\ell,j}(\cdot)=\nu_{\rho_{*}}^{\ell}(\cdot|\Omega_{\ell,j}^{d})$. Note
that $\nu_{\ell,j}$ is independent of the density $\rho_{*}$. Define

$m_{\ell,j}(f)=E_{\nu_{\rho_{*}}^{\ell}}[f_{\ell}\mathbf{1}_{\\{\bar{\eta}^{\ell}(0)=j/(2\ell+1)^{d}\\}}],\quad
f_{\ell,j}=\frac{f_{\ell}}{E_{\nu_{\ell,j}}[f_{\ell}]}.$

Then,

$E_{\nu_{\rho_{*}}^{\ell}}\Big{[}V_{g,M}^{\ell}(\eta)f_{\ell}(\eta)\Big{]}=\sum_{j=0}^{M(2\ell+1)^{d}}m_{\ell,j}(f)E_{\nu_{\ell,j}}[V_{g,M}^{\ell}f_{\ell,j}].$

Moreover,

$D_{\ell}(f_{\ell};\nu_{\rho_{*}}^{\ell})=\sum_{j=0}^{\infty}m_{\ell,j}(f)D_{\ell}(f_{\ell,j};\nu_{\ell,j})\geq\sum_{j=0}^{M(2\ell+1)^{d}}m_{\ell,j}(f)D_{\ell}(f_{\ell,j};\nu_{\ell,j}).$

Since $f_{\ell}$ is density with respect to $\nu_{\rho_{*}}^{\ell}$,

$\sum_{j=0}^{M(2\ell+1)^{d}}m_{\ell,j}(f)\leq
E_{\nu^{\ell}_{\rho_{*}}}[f_{\ell}]=1.$

Then, we may bound (4.9) by

$t\sup_{f}\sup_{0\leq
j\leq(2\ell+1)^{d}M}\Big{\\{}\frac{\|F\|_{\infty}}{N^{\alpha}}E_{\nu_{\ell,j}}[V_{g,M}^{\ell}f_{\ell,j}]-\frac{N^{d}}{\gamma(2\ell+1)^{d}}D_{\ell}(f_{\ell,j};\nu_{\ell,j})\Big{\\}}$
(4.10)

Note that $V_{g,M}^{\ell}$ is not mean zero with respect to the measure
$\nu_{\ell,j}$. In order to make it centralized, define

$\displaystyle U_{g,M}^{\ell,j}:=$
$\displaystyle\;\bar{g}^{\ell}(\eta(0))\mathbf{1}_{\\{\bar{\eta}^{\ell}(0)\leq
M\\}}-E_{\nu_{\ell,j}}\big{[}\bar{g}^{\ell}(\eta(0))\mathbf{1}_{\\{\bar{\eta}^{\ell}(0)\leq
M\\}}\big{]}$ $\displaystyle=$
$\displaystyle\;\bar{g}^{\ell}(\eta(0))\mathbf{1}_{\\{\bar{\eta}^{\ell}(0)\leq
M\\}}-E_{\nu_{\ell,j}}\big{[}g(\eta(0))\big{]}.$

The last identity holds because $j\leq M(2\ell+1)^{d}$. We may rewrite
$V_{g,M}^{\ell}$ as

$V_{g,M}^{\ell}=U_{g,M}^{\ell,j}+E_{\nu_{\ell,j}}\big{[}g(\eta(0))\big{]}-\Phi(\bar{\eta}^{\ell}(0))\mathbf{1}_{\\{\bar{\eta}^{\ell}(0)\leq
M\\}}.$

Let $\lambda_{g}^{\ell,j}(-\mathscr{L}_{\ell}^{s})$ be the largest eigenvalue
associated to the generator
$-\mathscr{L}_{\ell}^{s}+\gamma(2\ell+1)^{d}\|F\|_{\infty}U_{g,M}^{\ell,j}/N^{d+\alpha}$,
Then, (4.10) is bounded by

$\frac{t\|F\|_{\infty}}{N^{\alpha}}\sup_{0\leq
j\leq(2\ell+1)^{d}M}\big{\\{}E_{\nu_{\ell,j}}[g(\eta(0))]-\Phi\big{(}j/(2\ell+1)^{d}\big{)}\big{\\}}+\frac{tN^{d}}{\gamma(2\ell+1)^{d}}\;\sup_{0\leq
j\leq(2\ell+1)^{d}M}\lambda_{g}^{\ell,j}(-\mathscr{L}_{\ell}^{s}).$ (4.11)

By the standard equivalence of ensembles [7, Corollary A2.1.7],

$\sup_{0\leq
j\leq(2\ell+1)^{d}M}\big{\\{}E_{\nu_{\ell,j}}[g(\eta(0))]-\Phi\big{(}j/(2\ell+1)^{d}\big{)}\big{\\}}\leq
C\ell^{-d}.$

Hence, the first term in (4.11) is bounded by $CN^{-\alpha}\ell^{-d}$. In
order to bound the second term in (4.11), we need the following lemma to bound
the largest eigenvalue of the small perturbation of the reversible generator
$-\mathscr{L}_{\ell}^{s}$.

######  Lemma 4.3 ([7, Theorem A3.1.1]).

Let $\mathscr{L}$ be a reversible irreducible generator on some countable
state space $\Omega$ with respect to some probability measure $\nu$ on
$\Omega$. Let $U$ be a mean zero bounded function with respect to $\nu$. For
$\varepsilon>0$, denote by $\lambda_{\varepsilon}$ the upper bound of the
spectrum of $\mathscr{L}+\varepsilon U$. Then,

$0\leq\lambda_{\varepsilon}\leq\frac{\varepsilon^{2}\kappa}{1-2\|U\|_{\infty}\varepsilon\kappa}{\rm
Var}(U;\nu),$

where $\kappa^{-1}$ is the spectral gap of the generator $\mathscr{L}$ in
$L^{2}(\nu)$,

$\kappa^{-1}=\inf_{f:E_{\nu}[f]=0}\frac{\left\langle
f(-\mathscr{L})f\right\rangle_{\nu}}{{\rm Var}(f;\nu)}$

We continue to bound the second term in (4.11). Note that
$E_{\nu_{\ell,j}}[U_{g,M}^{\ell,j}]=0$ and $|U_{g,M}^{\ell,j}|\leq CM$. By the
above lemma, the second term in (4.11) is bounded by

$\sup_{0\leq
j\leq(2\ell+1)^{d}M}\Big{\\{}\frac{C(F,t)N^{d}}{\gamma\ell^{d}}\frac{\gamma^{2}\ell^{2d}N^{-2d-2\alpha}\kappa}{1-C(F)M\gamma\ell^{d}N^{-d-\alpha}\kappa}{\rm
Var}\big{(}U_{g,M}^{\ell,j};\nu_{\ell,j}\big{)}\Big{\\}},$

where $\kappa=\kappa(\ell,j)$ is the reciprocal of the spectral gap of the
generator $\mathscr{L}_{\ell}^{s}$,

$\kappa^{-1}=\inf_{f}\frac{\left\langle
f(-\mathscr{L}_{\ell}^{s})f\right\rangle_{\nu_{\ell,j}}}{{\rm
Var}(f;\nu_{\ell,j})}.$

Above, the infimum is over all functions $f$ such that
$E_{\nu_{\ell,j}}[f]=0$. In [8], Landim _et al._ prove the following lower
bound for the spectral gap of the zero range process under Assumption 2.1,

$\kappa(\ell,j)^{-1}\geq C\ell^{-2}$

uniformly in $j$. By [15, Lemma 4],

$\sup_{0\leq j\leq(2\ell+1)^{d}M}{\rm
Var}\big{(}U_{g,M}^{\ell,j};\nu_{\ell,j}\big{)}\leq C\ell^{-d}.$

Therefore, the second term in (4.11) is bounded by
$C(M,F,t)\gamma\ell^{2}N^{-d-2\alpha}$ as long as $\gamma
M\ell^{d+2}<\varepsilon_{0}N^{d+\alpha}$ for some fixed but small
$\varepsilon_{0}$. Adding up the above estimates (4.6) and (4.11), we bound
the left hand side in (4.5) by

$C(M,F,t)\Big{(}\frac{1}{N^{\alpha}\ell^{d}}+\frac{\gamma\ell^{2}}{N^{d+2\alpha}}+\frac{1}{\gamma
N^{2\alpha}}\Big{)}.$

Take $\gamma=N^{d/2}\ell^{-1}$, we get the desired bound

$C(M,F,t)\Big{(}\frac{1}{N^{\alpha}\ell^{d}}+\frac{\ell}{N^{2\alpha+d/2}}\Big{)}$

as long as $M\ell^{d+1}<\varepsilon_{0}N^{d/2+\alpha}$, which concludes the
proof. ∎

### 4.3 Estimates by logarithmic Sobolev inequality.

In this subsection, we present a second proof to bound the second term in
(4.1) by using the logarithmic Sobolev inequality for the zero range process.

######  Lemma 4.4.

There exists $C=C(M,F,t)$ such that

$\int_{0}^{t}E_{\mu_{N,s}}\Big{[}\frac{1}{N^{d+\alpha}}\sum_{y\in\mathbb{T}_{N}^{d}}F\big{(}s,\tfrac{y}{N}\big{)}\tau_{y}V_{g,M}^{\ell}\Big{]}ds\leq
C\Big{(}\frac{1}{N^{\alpha}\ell^{d}}+\frac{\ell^{d+2}}{N^{3\alpha+2}}\Big{)}.$
(4.12)

In particular, by taking $\ell=N^{\frac{\alpha+1}{d+1}}$, the left hand side
in (4.12) is bounded by $CN^{-\frac{(2d+1)\alpha+d}{d+1}}$, which has order
$o(N^{-2\alpha})$.

######  Remark 4.5.

Compared the estimate in the above lemma with that in Lemma 4.2, we get a
better bound by using logarithmic Sobolev inequality only in dimension one.
Indeed,

$N^{-\frac{(2d+1)\alpha+d}{d+1}}<N^{-\frac{(4d+2)\alpha+d^{2}}{2(d+1)}}\quad\text{if
and only if}\quad d<2.$

###### Proof.

Denote

$f_{s}=f_{N,s}=\frac{d\mu_{N,s}}{d\nu_{\rho_{*}}^{N}},\quad\bar{f}_{s}=\frac{1}{N^{d}}\sum_{y\in\mathbb{T}_{N}^{d}}\tau_{y}f_{s},\quad\bar{f}_{s,\ell}=E_{\nu_{\rho_{*}}^{N}}[\bar{f}_{s}|\mathcal{F}_{\ell}].$

Then, we may bound the left hand side in (4.12) by

$\int_{0}^{t}E_{\nu_{\rho_{*}}^{N}}\Big{[}\frac{\|F\|_{\infty}}{N^{\alpha}}V_{g,M}^{\ell}\bar{f}_{s}\Big{]}ds=\int_{0}^{t}E_{\nu_{\rho_{*}}^{\ell}}\Big{[}\frac{\|F\|_{\infty}}{N^{\alpha}}V_{g,M}^{\ell}\bar{f}_{s,\ell}\Big{]}ds$

As in the proof of Lemma 4.2, we denote

$m_{\ell,j}(\bar{f}_{s,\ell})=E_{\nu_{\rho_{*}}^{\ell}}[\bar{f}_{s,\ell}\mathbf{1}_{\\{\bar{\eta}^{\ell}(0)=j/(2\ell+1)^{d}\\}}],\quad\bar{f}_{s,\ell,j}=\frac{\bar{f}_{s,\ell}}{E_{\nu_{\ell,j}}[\bar{f}_{s,\ell}]}.$

Then, we rewrite the last formula as

$\sum_{j=0}^{M(2\ell+1)^{d}}\int_{0}^{t}m_{\ell,j}(\bar{f}_{s,\ell})E_{\nu_{\ell,j}}\Big{[}\frac{\|F\|_{\infty}}{N^{\alpha}}V_{g,M}^{\ell}\bar{f}_{s,\ell,j}\Big{]}ds.$

By entropy inequality, for any $\gamma>0$, the above formula is bounded by

$\sum_{j=0}^{M(2\ell+1)^{d}}\frac{1}{\gamma}\int_{0}^{t}m_{\ell,j}(\bar{f}_{s,\ell})E_{\nu_{\ell,j}}\big{[}\bar{f}_{s,\ell,j}\log\bar{f}_{s,\ell,j}\big{]}ds\\\
+\sum_{j=0}^{M(2\ell+1)^{d}}\frac{1}{\gamma}\int_{0}^{t}m_{\ell,j}(\bar{f}_{s,\ell})\log
E_{\nu_{\ell,j}}\Big{[}\exp\Big{\\{}\frac{\gamma\|F\|_{\infty}}{N^{\alpha}}V_{g,M}^{\ell}\Big{\\}}\Big{]}ds.$
(4.13)

The logarithmic Sobolev inequality for the zero range process reads (_cf._
[12])

$E_{\nu_{\ell,j}}\big{[}\bar{f}_{s,\ell,j}\log\bar{f}_{s,\ell,j}\big{]}\leq
C\ell^{2}D_{\ell}\big{(}\bar{f}_{s,\ell,j};\nu_{\ell,j}\big{)}$

uniformly in $j$. By convexity and translation invariance of the Dirichlet
form,

$\sum_{j=0}^{M(2\ell+1)^{d}}m_{\ell,j}(\bar{f}_{s,\ell})D_{\ell}\big{(}\bar{f}_{s,\ell,j};\nu_{\ell,j}\big{)}\leq
D_{\ell}\big{(}\bar{f}_{s,\ell};\nu_{\rho_{*}}^{\ell}\big{)}\leq
D_{\ell}\big{(}\bar{f}_{s};\nu_{\rho_{*}}^{N}\big{)}\leq\frac{(2\ell+1)^{d}}{N^{d}}D_{N}\big{(}\bar{f}_{s};\nu_{\rho_{*}}^{N}\big{)}.$

Since by Lemma 3.1,

$\int_{0}^{t}D_{N}\big{(}\bar{f}_{s};\nu_{\rho_{*}}^{N}\big{)}ds\leq
CN^{d-2\alpha-2},$

we bound the first term in (4.13) by $C\ell^{d+2}/(\gamma N^{2\alpha+2})$.

Now we bound the second term in (4.13). Note that
$\big{|}V_{g,M}^{\ell}\big{|}\leq CM$. Using the basic inequality

$e^{x}\leq 1+x+\tfrac{x^{2}}{2}e^{|x|},\quad\log(1+x)\leq x,$

we bound the second term in (4.13) by

$\frac{C(M,F,t)}{\gamma}\sup_{0\leq j\leq
M(2\ell+1)^{d}}\Big{\\{}\frac{\gamma}{N^{\alpha}}E_{\nu_{\ell,j}}[V^{\ell}_{g,M}]+\frac{\gamma^{2}}{N^{2\alpha}}E_{\nu_{\ell,j}}[(V^{\ell}_{g,M})^{2}]\Big{\\}},\quad\forall\gamma\leq
N^{\alpha}.$

Using the standard equivalence of ensembles (_cf._ [7, Corollary A2.1.7]), for
any $j\leq M(2\ell+1)^{d}$,

$E_{\nu_{\ell,j}}[V^{\ell}_{g,M}]=E_{\nu_{\ell,j}}[g(\eta(0))]-\Phi(j/(2\ell+1)^{d})\leq
C\ell^{-d}.$

By Cauchy-Schwarz inequality and [15, Lemma 4],

$E_{\nu_{\ell,j}}[(V^{\ell}_{g,M})^{2}]\leq
2\big{(}E_{\nu_{\ell,j}}[g(\eta(0))]-\Phi(j/(2\ell+1)^{d})\big{)}^{2}+2{\rm
Var}\big{(}\bar{g}^{\ell}(0);\nu_{\ell,j}\big{)}\leq C\ell^{-d}.$

This permits us to bound the second term in (4.13) by
$C(M,F,t)\Big{(}N^{-\alpha}\ell^{-d}+\gamma N^{-2\alpha}\ell^{-d}\Big{)}$.

In conclusion, for any $\gamma\leq N^{\alpha}$, we bound the left hand side in
(4.12) by

$C(M,F,t)\Big{(}\frac{1}{N^{\alpha}\ell^{d}}+\frac{\ell^{d+2}}{\gamma
N^{2\alpha+2}}+\frac{\gamma}{N^{2\alpha}\ell^{d}}\Big{)}.$

We finish the proof by taking $\gamma=N^{\alpha}$. ∎

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Linjie Zhao

School of Mathematics and Statistics, Huazhong University of Science &
Technology, Wuhan, China.
<EMAIL_ADDRESS>