# Kähler families of Green’s functions

Vincent Guedj & Tat Dat Tô  Institut Universitaire de France et Institut de
Mathématiques de Toulouse
Université de Toulouse
118 route de Narbonne
31400 Toulouse, France
<EMAIL_ADDRESS>https://www.math.univ-toulouse.fr/ guedj/
Sorbonne Université, Institut de mathématiques de Jussieu – Paris Rive Gauche,
4, place Jussieu, 75252 Paris Cedex 05 France<EMAIL_ADDRESS>https://sites.google.com/site/totatdatmath/home

###### Abstract.

In a remarkable series of works, Guo, Phong, Song, and Sturm have obtained key
uniform estimates for the Green’s functions associated with certain Kähler
metrics. In this note, we broaden the scope of their techniques by removing
one of their assumptions and allowing the complex structure to vary. We apply
our results to various families of canonical Kähler metrics.

###### Key words and phrases:

Green’s function, Monge-Ampère equation, a priori estimates

###### 2010 Mathematics Subject Classification:

32W20, 32U05, 32Q15, 35A23

###### Contents

  1. 1 Uniform estimates for Monge-Ampère potentials
  2. 2 Green’s functions
  3. 3 Geometric applications

## Introduction

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$. We set
$V_{\omega}=\int_{X}\omega^{n}$ and let $G_{x}^{\omega}$ denote the Green’s
function of $\omega$ at the point $x$. This is the unique quasi-subharmonic
function such that $\int_{X}G_{x}^{\omega}\omega^{n}=0$ and

$\frac{1}{V_{\omega}}(\omega+dd^{c}G_{x}^{\omega})\wedge\omega^{n-1}=\delta_{x}.$

Here $\delta_{x}$ denotes the Dirac mass at point $x$. Equivalently
$\Delta_{\omega}G_{x}^{\omega}=n\left(V_{\omega}\delta_{x}-\omega^{n}\right).$

In a remarkable series of articles [GPS22, GPSS22, GPSS23], Guo, Phong, Song
and Sturm have obtained several key estimates for $G_{x}^{\omega}$ that are
uniform when $\omega$ varies in a large family ${\mathcal{F}}$ of Kähler forms
(our normalization for $G_{x}^{\omega}$ slightly differs from theirs, see
Definition 1.2 below). Although $G_{x}^{\omega}$ is a solution of the Laplace
equation, its dependence on $\omega$ is highly non-linear. Nonetheless, these
authors succeeded in obtaining spectacular estimates by comparing the problem
at hand with an auxiliary complex Monge-Ampère equation (an idea originating
from [CC21]), for which fine uniform estimates are available (see [Yau78,
Kol98, EGZ09, EGZ08, DP10, BEGZ10, GL21, GPT23]).

Fix $\beta$ a reference Kähler form on $X$, $A,B>0$ positive constants, and
$\gamma\geq 0$ a continuous function such that $(\gamma=0)$ has small
Hausdorff dimension. The family ${\mathcal{F}}$ consists in those Kähler forms
$\omega$ which satisfy the following three assumptions:

  1. (1)

an upper-bound on the cohomology class $\int_{X}\omega\wedge\beta^{n-1}\leq
A$;

  2. (2)

a uniform pointwise lower bound $f_{\omega}\geq\gamma$, where
$f_{\omega}:=V_{\omega}^{-1}\omega^{n}/\beta^{n}$;

  3. (3)

a uniform upper bound $\int_{X}f_{\omega}(\log f_{\omega})^{p}\beta^{n}\leq
B$, where $p>n$.

The first assumption can be restated as the cohomology class of $\omega$
remaining within a bounded subset of $H^{1,1}(X,\mathbb{R})$. It implies a
uniform upper bound on the volume $V_{\omega}\leq C(A)$, but it is a strictly
stronger assumption (see Example 1.7).

The third assumption ensures that $\omega$ admits uniformly bounded potentials
(as shown in [Kol98, EGZ08, DP10]), that moreover have good continuity
properties (see [Kol08, DDGHKZ14, GPTW21, GGZ23]). It can be slightly
generalized to

$\int_{X}f_{\omega}(\log f)^{n}(\log\log[3+f_{\omega}])^{p}\beta^{n}\leq
B\;\;\text{ with }\;\;p>2n,$

as shown in [GGZ23]. In this article we rather stick to the more classical
assumption $\int_{X}f_{\omega}^{p}\beta^{n}\leq B$ with $p>1$, to simplify the
exposition.

Our goal in this note is twofold. First, we eliminate the second assumption by
establishing a key new estimate (Proposition 2.2). Second, we demonstrate that
this new estimate, as well as all the estimates obtained in [GPS22, GPSS22,
GPSS23], remain uniform as the complex structure varies. This significantly
broadens the scope of geometric applications for these results.

The precise setting is as follows. Let $\mathcal{X}$ be an irreducible and
reduced complex space. We let $\pi:{\mathcal{X}}\rightarrow 2\mathbb{D}$
denote a proper, surjective holomorphic map with connected fibers such that
each fiber $X_{t}=\pi^{-1}(t)$ is a $n$-dimensional compact Kähler manifold,
for $t\in 2\mathbb{D}^{*}$. We allow the central fiber $X_{0}$ to be a
singular, irreducible and reduced complex space, as this is important for
geometric applications. Here $\mathbb{D}$ is the unit disk in $\mathbb{C}$; we
assume that the family is defined over a slightly larger disk $2\mathbb{D}$,
and will establish estimates that are uniform with respect to the parameter
$t\in\mathbb{D}^{*}$.

We fix $\beta$ a relative Kähler form on $\mathcal{X}$, i.e $\beta$ is a
smooth form on $\mathcal{X}$ whose restrictions $\beta_{t}=\beta_{|X_{t}}$ are
Kähler forms. We assume that the volumes
$V_{\beta_{t}}:=\int_{X_{t}}\beta_{t}^{n}$ are uniformly bounded away from 0
and $+\infty$, and let $dV_{X_{t}}=\beta_{t}^{n}/V_{\beta_{t}}$ denote the
corresponding probability volume form on $X_{t}$.

Fix $p>1$ and $A,B>0$. We let ${\mathcal{K}}(\mathcal{X},p,A,B)$ denote the
set of all relative Kähler forms $\omega$ on $\mathcal{X}\setminus X_{0}$ such
that for all $t\in\mathbb{D}^{*}$, $\\{\omega_{t}\\}\leq A\\{\beta_{t}\\}$ in
$H^{1,1}(X_{t},\mathbb{R})$, and

$\int_{X_{t}}f_{t}^{p}dV_{X_{t}}\leq B,\;\;\text{ where
}\;\;\frac{1}{V_{\omega_{t}}}\omega_{t}^{n}=f_{t}dV_{X_{t}}.$

Our main result is the following uniform control on Green’s functions
$G_{x}^{\omega_{t}}$.

######  Theorem A.

Fix $0<r<\frac{n}{n-1}$ and $0<s<\frac{2n}{2n-1}$. For all
$t\in\mathbb{D}^{*},x\in X_{t}$ and
$\omega\in{\mathcal{K}}(\mathcal{X},p,A,B)$ we have

  1. (1)

$\sup_{X_{t}}G_{x}^{\omega_{t}}\leq C_{0}=C_{0}(n,p,A,B)$;

  2. (2)

$\frac{1}{V_{\omega_{t}}}\int_{X_{t}}|G_{x}^{\omega_{t}}|^{r}\omega_{t}^{n}\leq
C_{1}=C_{1}(n,p,r,A,B)$;

  3. (3)

$\frac{1}{V_{\omega_{t}}}\int_{X_{t}}|\nabla
G_{x}^{\omega_{t}}|_{\omega_{t}}^{s}\omega_{t}^{n}\leq
C_{2}=C_{2}(n,p,s,A,B)$.

These estimates extend [GPS22, Theorem 1 and Theorem 2] and [GPSS22, Theorem
1.1]. The uniformity with respect to the complex structure crucially depends
on the uniform estimates obtained in [DnGG23].

As in [GPSS22] they easily imply a uniform control on diameters, as well as a
uniform non-collapsing estimate. Together with the uniform upper bound on
volumes, this yields in particular that the metric spaces $(X_{t},\omega_{t})$
are relatively compact in the Gromov-Hausdorff topology.

######  Theorem B.

Fix $0<\delta<1$, and $\omega\in{\mathcal{K}}(\mathcal{X},p,A,B)$. There
exists $C=C(n,p,\delta,A,B)>0$ such that for all $t\in\mathbb{D}^{*},x\in
X_{t}$ and $r>0$,

$C\min(1,r^{2n+\delta})\leq\frac{1}{V_{\omega_{t}}}{\rm
Vol}_{\omega_{t}}(B_{\omega_{t}}(x,r))\;\;\text{ and }\;\;{\rm
diam}(X_{t},\omega_{t})\leq C.$

Thus the family of compact metric spaces
$\\{(X_{t},d_{\omega_{t}}),\;\omega\in{\mathcal{K}}(\mathcal{X},p,A,B),\,t\in\mathbb{D}^{*}\\}$
is pre-compact in the Gromov-Hausdorff topology.

Following [GPSS23], we further show in Theorem 2.6 that the Sobolev constants
of the metrics $\omega_{t}$ are also uniformly bounded.

Theorems A and B have many geometric applications. We explain in Section 3.2
how they provide uniform estimates for families of constant scalar curvature
Kähler metrics (see Corollary 3.2), following [CC21] and [PTT23].

As in [GPSS22], we then apply our estimates to the study of the Kähler-Ricci
flow. Assume that $X$ is a a smooth minimal model ($K_{X}$ nef), and consider

$\frac{\partial\omega_{t}}{\partial t}=-{\rm Ric}(\omega_{t})-\omega_{t}$

the normalized Kähler-Ricci flow starting from an initial Kähler metric
$\omega_{0}$. It follows from [TZ06] that this flow exists for all times
$t>0$. We obtain here the following striking result which -in particular-
solves [Tos18, Conjecture 6.2]:

######  Theorem C.

Fix $0<\delta<1$. There exist $C=C(\omega_{0})>0$ and
$c=c(\omega_{0},\delta)>0$ such that for all $t>0$ and $x\in X$,

${\rm diam}(X,\omega_{t})\leq C\;\;\text{ and }\;\;{\rm
Vol}_{\omega_{t}}(B_{\omega_{t}}(x,r))\geq cr^{2n+\delta}V_{\omega_{t}},$

whenever $0<r<{\rm diam}(X,\omega_{t})$.

This result has been established by Guo-Phong-Song-Sturm under the extra
assumption that the Kodaira dimension ${\rm kod}(X)$ of $X$ is non-negative,
see [GPSS22, Theorem 2.3]. It is an old conjecture of Mumford that $K_{X}$ nef
implies ${\rm kod}(X)\geq 0$. The latter is a weak form of the abundance
conjecture, one of the most important open problem in birational geometry. We
refer the reader to [Tos18, Tos24] for recent overviews of the theory of the
Kähler-Ricci flow.

In Section 3.4 we show how our estimates provide an alternative proof of an
important diameter bound of Y.Li [Li23, Theorem 1.4]. We anticipate many more
applications of Theorems A and B (see [CGNPPW23, GP24] for recent
developments). In [GPSS23], the authors have partially extended their
techniques to the case of mildly singular Kähler varieties, showing in
particular that Kähler-Einstein currents have bounded diameter. Our key
Proposition 2.2 can be extended to this context as well, as it relies solely
on uniform estimates for solutions to complex Monge-Ampère equations, which
are applicable here according to [DnGG23, Theorem 1.9]. Therefore, we expect
that most of the results presented in this note can be extended to families of
Kähler varieties (see [GP24, Section 3] for some first steps). Very recently
Vu [Vu24] announced a proof of Theorem B for a fixed variety and an
independent proof of Theorem C by using analysis on Sobolev spaces associated
to currents; shortly afterwards Guo-Phong-Song-Sturm [GPSS24] have provided
and alternative proof of Proposition 2.2.

######  Acknowledgements.

We thank D.H.Phong for enlightning discussions about his joint works with Guo,
Song and Sturm on Green’s functions. We are grateful to H.C.Lu for several
useful comments on a preliminary draft. T.D.T. is partially supported by the
project MARGE ANR-21-CE40-0011. V.G. is partially supported by the Institut
Universitaire de France and fondation Charles Defforey.

##  1\. Uniform estimates for Monge-Ampère potentials

###  1.1. Quasi-(pluri)subharmonic functions

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$.

####  1.1.1. $\omega$-subharmonic functions

######  Definition 1.1.

A function $v:X\rightarrow\mathbb{R}\cup\\{-\infty\\}$ is $\omega$-subharmonic
($\omega$-sh for short) if it is locally the sum of a smooth and a subharmonic
function, and $(\omega+dd^{c}u)\wedge\omega^{n-1}\geq 0$ in the sense of
distributions. Alternatively

$\Delta_{\omega}v=n\frac{dd^{c}v\wedge\omega^{n-1}}{\omega^{n}}\geq-n.$

The maximum principle ensures that two $\omega$-sh functions $u,v$ satisfy
$\Delta_{\omega}u=\Delta_{\omega}v$ if and only if they differ by a constant.
We can thus ensure that $u=v$ by normalizing them by
$\int_{X}u\omega^{n}=\int_{X}v\omega^{n}=0$.

######  Definition 1.2.

We let $G_{x}$ denote the Green function of $\omega$ at the point $x$. This is
the unique $\omega$-sh function such that $\int_{X}G_{x}\omega^{n}=0$ and

$\frac{1}{V_{\omega}}(\omega+dd^{c}G_{x})\wedge\omega^{n-1}=\delta_{x}.$

Here $\delta_{x}$ denotes the Dirac mass at point $x$.

Let us stress that our definition differs from that of [GPSS22] in two ways:
we use the opposite sign convention, as well as a different volume
normalization. If $\tilde{G}_{x}$ denotes the Green function from [GPSS22],
then $G_{x}=-\frac{V_{\omega}}{n}\tilde{G}_{x}$.

For any $\omega$-sh function $u$ such that $\int_{X}u\omega^{n}=0$, Stokes
theorem yields

$u(x)=\frac{1}{V_{\omega}}\int_{X}u(\omega+dd^{c}G_{x})\wedge\omega^{n-1}=\frac{1}{V_{\omega}}\int_{X}G_{x}(\omega+dd^{c}u)\wedge\omega^{n-1}.$

In particular for $u=G_{y}$ we obtain the symmetry relation
$G_{x}(y)=G_{y}(x)$. Green’s functions are classical objects of study in
Riemannian geometry. In particular it is known that
$G_{x}\in{\mathcal{C}}^{\infty}(X\setminus\\{x\\})$ and
$G_{x}(y)\rightarrow-\infty$ as $y\rightarrow x$ (either at a logarithmic
speed if $n=1$, or at a polynomial speed if $n\geq 2$).

While the Laplace operator $\Delta_{\omega}$ is linear, it depends on $\omega$
in a non linear way. Following [GPS22, GPSS22, GPSS23] we are going to
establish uniform estimates on normalized $\omega$-sh functions, by comparing
them with $\omega$-plurisubharmonic solutions to certain complex Monge-Ampère
equations.

####  1.1.2. $\omega$-plurisubharmonic functions

A function is quasi-plurisubharmonic (quasi-psh) if it is locally given as the
sum of a smooth and a psh function. Quasi-psh functions
$\varphi:X\rightarrow\mathbb{R}\cup\\{-\infty\\}$ satisfying
$\omega_{\varphi}:=\omega+dd^{c}\varphi\geq 0$ in the weak sense of currents
are called $\omega$-plurisubharmonic ($\omega$-psh for short).

######  Definition 1.3.

We let ${\rm PSH}(X,\omega)$ denote the set of all $\omega$-plurisubharmonic
functions which are not identically $-\infty$.

Note that constant functions are $\omega$-psh functions. A
${\mathcal{C}}^{2}$-smooth function $u$ has bounded Hessian, hence
$\varepsilon u$ is $\omega$-psh if $0<\varepsilon$ is small enough and
$\omega$ is positive.

The set ${\rm PSH}(X,\omega)$ is a closed subset of $L^{1}(X)$, for the
$L^{1}$-topology. Subsets of $\omega$-psh functions enjoy strong compactness
and integrability properties, we mention notably the following: for any fixed
$r\geq 1$,

  * •

${\rm PSH}(X,\omega)\subset L^{r}(X)$; the induced $L^{r}$-topologies are
equivalent;

  * •

${\rm PSH}_{A}(X,\omega):=\\{u\in{\rm PSH}(X,\omega),\,-A\leq\sup_{X}u\leq
0\\}$ is compact in $L^{r}$.

We refer the reader to [Dem, GZ] for further basic properties of $\omega$-psh
functions.

####  1.1.3. Laplace vs Monge-Ampère solutions

We shall regularly compare solutions to the Laplace equation and $\omega$-psh
solutions to an auxiliary Monge-Ampère equation. The following principle will
play an important role.

######  Proposition 1.4.

Fix $t>0$, $p>1$ and $0\leq f\in L^{p}(\omega^{n})$. Let $v$ (resp. $\varphi$)
be the unique bounded $\omega$-sh (resp. $\omega$-psh) function such that

$(\omega+dd^{c}v)\wedge\omega^{n-1}=e^{tv}f\omega^{n}\;\;\text{ and
}\;\;(\omega+dd^{c}\varphi)^{n}=e^{nt\varphi}f^{n}\omega^{n}.$

Then $\varphi\leq v$.

The existence of $v$ is classical. For the existence and uniqueness of
$\varphi$, see [GZ].

###### Proof.

It follows from the maximum principle that $v$ is the envelope of bounded
$\omega$-sh subsolutions to this twisted Laplace equation, i.e. for all $x\in
X$,

$v(x)=\sup\left\\{u(x);\;u\in SH(X,\omega)\cap L^{\infty}(X)\;\text{ and
}\;(\omega+dd^{c}u)\wedge\omega^{n-1}\geq e^{tu}f\omega^{n}\right\\}.$

Since the function $\varphi_{2}=0$ is the trivial solution to the twisted
Monge-Ampère equation
$(\omega+dd^{c}\varphi_{2})^{n}=e^{nt\varphi_{2}}1^{n}\omega^{n}$, the AM-GM
inequality (see [DK14, Theorem 2.12]) ensures that

$(\omega+dd^{c}\varphi)\wedge\omega^{n-1}\geq e^{t\varphi}f\omega^{n}.$

The conclusion follows as $PSH(X,\omega)\subset SH(X,\omega)$ and $\varphi$ is
a bounded subsolution of the twisted Laplace equation. ∎

Uniform a priori estimates for solutions to complex Monge-Ampère equations
have been intensively studied in the past decades. The previous proposition
will allow one to use them and gain uniform $L^{\infty}$ a priori estimates
for solutions of the Laplace equation, as the reference form $\omega$ varies
(see Section 2).

###  1.2. Kähler families

####  1.2.1. Assumptions

We shall consider the following set of assumptions.

######  Setting 1.5.

Let $\mathcal{X}$ be an irreducible and reduced complex space. We let
$\pi:{\mathcal{X}}\rightarrow 2\mathbb{D}$ denote a proper, surjective
holomorphic map with connected fibers such that each fiber $X_{t}=\pi^{-1}(t)$
is a $n$-dimensional compact Kähler manifold, for $t\in 2\mathbb{D}^{*}$ and
$X_{0}$ is an irreducible and reduced complex space. We fix $\beta$ a relative
Kähler form on $\mathcal{X}$, set $\beta_{t}=\beta_{|X_{t}}$ and assume that
the volumes $V_{\beta_{t}}:=\int_{X_{t}}\beta_{t}^{n}$ are uniformly bounded
away from 0 and $+\infty$. Let $dV_{X_{t}}=\beta_{t}^{n}/V_{\beta_{t}}$ denote
the corresponding probability volume form on $X_{t}$.

Here $\mathbb{D}$ is the unit disk in $\mathbb{C}$; we assume that the family
is defined over a slightly larger disk (e.g. $2\mathbb{D}$), and will
establish estimates that are uniform with respect to the parameter
$t\in\mathbb{D}^{*}$.

######  Definition 1.6.

Fix $p>1$ and $A,B>0$. We let ${\mathcal{K}}(\mathcal{X},p,A,B)$ denote the
set of all relative Kähler forms $\omega$ on $\mathcal{X}\setminus X_{0}$ such
that for all $t\in\mathbb{D}^{*}$, $\\{\omega_{t}\\}\leq A\\{\beta_{t}\\}$ in
$H^{1,1}(X_{t},\mathbb{R})$, and

$\int_{X_{t}}f_{t}^{p}dV_{X_{t}}\leq B,\;\;\text{ where
}\;\;\frac{1}{V_{\omega_{t}}}\omega_{t}^{n}=f_{t}dV_{X_{t}}.$

The cohomological assumption ensures that the volumes
$V_{t}=\int_{X_{t}}\omega_{t}^{n}\leq C(A)$ are uniformly bounded fom above,
but it is a strictly stronger assumption as we observe in the following
example.

######  Example 1.7.

Assume ${X}=\mathbb{P}^{1}\times\mathbb{P}^{1}$ is the product of two Riemann
spheres, endowed with the Kähler form
$\omega_{\lambda}(x,y)=\lambda\omega_{\mathbb{P}^{1}}(x)+\lambda^{-1}\omega_{\mathbb{P}^{1}}(y)$,
where $\lambda>0$. Observe that the volume
$V_{\omega_{\lambda}}=\int_{X}\omega_{\lambda}^{2}=\int_{X}2\omega_{\mathbb{P}^{1}}(x)\wedge\omega_{\mathbb{P}^{1}}(y)=2$
is constant, while the diameter ${\rm
diam}(X,\omega_{\lambda})\sim\lambda\rightarrow\infty$ as
$\lambda\rightarrow\infty$.

####  1.2.2. Uniform a priori estimates

We shall need the following uniform integrability result in families, that
generalizes previous results of Skoda and Zeriahi:

######  Theorem 1.8.

Fix $p>1$, $A,B>0$. There exists $\alpha=\alpha(n,p,A,B)>0$ such that for all
$\omega\in{\mathcal{K}}(\mathcal{X},p,A,B)$, $t\in\mathbb{D}^{*}$ and
$\varphi_{t}\in PSH(X_{t},\omega_{t})$ with $\sup_{X_{t}}\varphi_{t}=0$,

$\int_{X_{t}}\exp(-\alpha\varphi_{t})dV_{X_{t}}\leq C_{\alpha},$

where $C=C(\alpha,n,p,A,B)>0$ is independent of $\omega,t,\varphi_{t}$.

###### Proof.

The cohomological assumption means that there exists a smooth closed
$(1,1)$-form $\theta_{t}$ on $X_{t}$, cohomologous to $\omega_{t}$, such that
$\theta_{t}\leq A\beta_{t}$. The $\partial\overline{\partial}$-lemma ensures
that $\omega_{t}=\theta_{t}+dd^{c}u_{t}$, where $\int_{X}u_{t}dV_{X_{t}}=0$
and $u_{t}\in PSH(X_{t},\theta_{t})$. Since we also have $u_{t}\in
PSH(X_{t},A\beta_{t})$, it follows from [DnGG23, Conjecture 3.1] and [Ou22,
Corollary 4.8] that $0\leq\sup_{X_{t}}u_{t}\leq M_{0}$ for some uniform
constant $M_{0}$.

Set $\tilde{\varphi}_{t}=\varphi_{t}-\int_{X_{t}}\varphi_{t}dV_{X_{t}}$ so
that $\int_{X_{t}}\tilde{\varphi}_{t}dV_{X_{t}}=0$. Observe similarly that
$\psi_{t}=\tilde{\varphi}_{t}+u_{t}\in PSH(X_{t},\theta_{t})\subset
PSH(X_{t},A\beta_{t})$ is normalized by $\int_{X}\psi_{t}dV_{X_{t}}=0$, hence
$0\leq\sup_{X_{t}}\psi_{t}\leq M_{0}$. It follows therefore from [DnGG23,
Theorem 2.9] that

$\int_{X_{t}}e^{2\alpha|\psi|}dV_{X_{t}}+\int_{X_{t}}e^{2\alpha|u|}dV_{X_{t}}\leq
C_{\alpha},$

for some uniform constants $\alpha,C_{\alpha}>0$. Cauchy-Schwarz inequality
now yields

$\int_{X_{t}}e^{\alpha|\tilde{\varphi}_{t}|}dV_{X_{t}}\leq\int_{X_{t}}e^{\alpha|\psi_{t}|}e^{\alpha|u_{t}|}dV_{X_{t}}\leq
C_{\alpha}.$

Now [DnGG23, Theorem 1.9] ensures that $||u_{t}-V_{\theta_{t}}||_{\infty}\leq
M_{1}$, where $V_{\theta_{t}}=\sup\\{v,\;v\in PSH(X,\theta_{t})\;\text{ with
}\;v\leq 0\\}$. Thus $\psi_{t}\leq V_{\theta_{t}}+\sup_{X_{t}}\psi_{t}\leq
u_{t}+M_{0}+M_{1}$ and $\sup_{X_{t}}\tilde{\varphi}_{t}\leq M_{0}+M_{1}$. The
conclusion follows as $-\varphi_{t}\leq|\tilde{\varphi}_{t}|+M_{0}+M_{1}$. ∎

The following uniform estimate is the key tool for all results to follow.

######  Theorem 1.9.

Fix $p>1$, $A,B>0$ and $\omega\in{\mathcal{K}}(\mathcal{X},p,A,B)$. Assume
that there exists $\varphi_{t}\in PSH(X,\omega_{t})\cap L^{\infty}(X_{t})$,
$p^{\prime}>1$ and $B^{\prime}>0$ independent of $t$ such that

$\frac{1}{V_{\omega_{t}}}(\omega_{t}+dd^{c}\varphi_{t})^{n}=g_{t}dV_{X_{t}},$

with $\int_{X_{t}}g_{t}^{p^{\prime}}dV_{X_{t}}\leq B^{\prime}$. Then ${\rm
Osc}_{X}(\varphi_{t})\leq C=C(p,p^{\prime},A,B,B^{\prime})$.

These uniform estimates have a long history. For a fixed cohomology class they
have been established by Kolodziej [Kol98], generalizing a celebrated a priori
estimate of Yau [Yau78]. These have been further generalized in [EGZ09, EGZ08,
DP10, BEGZ10] in order to deal with less positive or collapsing families of
cohomology classes on Kähler manifolds. Alternative proofs have been provided
in [GL21, GPT23], while the family version needed here is obtained in
[DnGG23].

###### Proof.

The result follows from [DnGG23, Theorem A] and Theorem 1.8. ∎

These uniform estimates remain valid under less restrictive integrability
assumptions on the densities. To gain clarity in the proofs to follow, we have
chosen to restrict to this setting which is the most useful one in geometric
applications. We refer the reader to [GGZ23, Section 5] for a discussion of
the optimality of the integrability assumptions that ensure finiteness of
diameters of Kähler metrics.

##  2\. Green’s functions

In this section we prove Theorem A. In the setting 1.5 we fix $p>1$, $A,B>0$
and $\omega\in{\mathcal{K}}(\mathcal{X},p,A,B)$. For $t\in{\mathbb{D}^{*}}$ we
consider the Green function $G^{\omega_{t}}_{x}$ of the Kähler form
$\omega_{t}$ at the point $x\in X_{t}$ and establish integrability estimates
for the latter that are uniform in $t\in{\mathbb{D}^{*}}$ and $x\in X_{t}$. To
lighten the notation, we get rid of the $t$-subscript and write $X$ instead of
$X_{t}$ and $G_{x}$ instead of $G^{\omega_{t}}_{x}$.

###  2.1. Bounding $\omega$-subharmonic functions from above

######  Lemma 2.1.

Fix $a>0$ and let $v$ be a quasi-subharmonic function on $X$ such that
$\Delta_{\omega}v\geq-a$ and $\int_{X}v\omega^{n}=0$. Then

$\sup_{X}v\leq C\left[a+\frac{1}{V_{\omega}}\int_{X}|v|\omega^{n}\right],$

where $C=C(n,p,A,B)>0$ only depends on $n,p,A,B$.

This result is a family version of [GPSS22, Lemma 5.1].

###### Proof.

Both the statement and the assumptions are homogeneous of degree $1$. Changing
$v$ in $\frac{n}{a}v$ we thus reduce to the case $a=n$. Observe that
$\Delta_{\omega}v\geq-n$ is equivalent to
$(\omega+dd^{c}v)\wedge\omega^{n-1}\geq 0$.

Regularizing $v$ we can assume that it is smooth. We set
$v_{+}=\widetilde{\max}(v,0)$, where $\widetilde{\max}$ denotes a convex
regularized maximum such that $0\leq\widetilde{\max}\leq 1+\max$ . Since
$\sup_{X}v\leq\sup_{X}v_{+}$, it suffices to bound $v_{+}$ from above. Let
$\varphi\in PSH(X,\omega)$ be the unique smooth function such that

$(\omega+dd^{c}\varphi)^{n}=\frac{1+v_{+}}{1+M}\omega^{n}$

and $\sup_{X}\varphi=-1$, where
$M=\int_{X}v_{+}\frac{\omega^{n}}{V_{\omega}}\leq
1+\frac{1}{2}\int_{X}|v|\frac{\omega^{n}}{V_{\omega}}$.

Set $H=1+v_{+}-\varepsilon(-\varphi)^{\alpha}$, where $\alpha=\frac{n}{n+1}$
and $\varepsilon>0$ is chosen so that
$\frac{\varepsilon^{n+1}\alpha^{n}}{(1+\alpha\varepsilon)^{n}}=1+M$. We are
going to show that $H\leq 0$. Observe that

$-dd^{c}(-\varphi)^{\alpha}=\alpha(1-\alpha)(-\varphi)^{\alpha-2}d\varphi\wedge
d^{c}\varphi+\alpha(-\varphi)^{\alpha-1}dd^{c}\varphi,$

hence
$\Delta_{\omega}(-\varepsilon(-\varphi)^{\alpha})\geq\alpha\varepsilon(-\varphi)^{\alpha-1}\Delta_{\omega}\varphi\geq
n\alpha\varepsilon(-\varphi)^{\alpha-1}\left[\left(\frac{1+v_{+}}{1+M}\right)^{\frac{1}{n}}-1\right]$.
We infer

$\Delta_{\omega}H\geq-n+n\alpha\varepsilon(-\varphi)^{\alpha-1}\left[\left(\frac{1+v_{+}}{1+M}\right)^{\frac{1}{n}}-1\right].$

At the point $x_{0}$ where $H$ reaches its maximum, we have
$0\geq\Delta_{\omega}H$ hence

$(1+\alpha\varepsilon)(-\varphi)^{1-\alpha}\geq(-\varphi)^{1-\alpha}+\alpha\varepsilon\geq\alpha\varepsilon\left(\frac{1+v_{+}}{1+M}\right)^{\frac{1}{n}},$

using that $(-\varphi)^{1-\alpha}\geq 1$. Thus

$\varepsilon(-\varphi)^{\alpha}=\varepsilon(-\varphi)^{n(1-\alpha)}\geq\frac{\alpha^{n}\varepsilon^{n+1}}{(1+\alpha\varepsilon)^{n}}\frac{1+v_{+}}{1+M}=1+v_{+},$

by our choice of $\varepsilon$ and $\alpha$. This shows that $H\leq 0$ hence
$1+v_{+}\leq\varepsilon(-\varphi)^{\alpha}$.

Note that $\varepsilon\leq c_{n}(1+M)$ since
$\frac{\varepsilon^{n+1}\alpha^{n}}{(1+\alpha\varepsilon)^{n}}=1+M$. Thus
$\frac{(\omega+dd^{c}\varphi)^{n}}{V_{\omega}}=FdV_{X}$ with

$F=\frac{1+v_{+}}{1+M}f\leq\frac{\varepsilon(-\varphi)^{\alpha}}{1+M}f\leq
c_{n}(-\varphi)^{\alpha}f.$

Since $\int_{X}f^{p}dV_{X}\leq A$, we can fix $1<r<p$ and use Hölder
inequality to obtain

$\int_{X}F^{r}dV_{X}\leq\varepsilon^{r}\int_{X}(-\varphi)^{r\alpha}f^{r}dV_{X}\leq\left(\int_{X}f^{p}dV_{X}\right)^{\frac{r}{p}}\left(\int_{X}(-\varphi)^{\frac{rp\alpha}{p-r}}dV_{X}\right)^{\frac{p-r}{p}}.$

The first integral is controlled by $A$ by assumption, the second one is
uniformly bounded by Theorem 1.8. Thus $\varphi$ is uniformly bounded by
Theorem 1.9, hence

$\sup_{X}v\leq\sup_{X}v_{+}\leq\varepsilon(-\varphi)^{\alpha}\leq
c_{n}[1+M]C_{0},$

and the conclusion follows as
$M\leq\frac{1}{2V_{\omega}}\int_{X}|v|\omega^{n}+1$. ∎

The following result is a key improvement on the results obtained in [GPSS22].

######  Proposition 2.2.

Let $u$ be a continuous function such that $\int_{X}u\omega^{n}=0$ and
$|\Delta_{\omega}u|\leq 1$. Then

$||u||_{L^{\infty}(X)}\leq C,$

where $C=C(n,p,A,B)>0$ only depends on $n,p,A,B$.

###### Proof.

We let $C_{1}>0$ denote the uniform constant from Lemma 2.1 and we set

$\delta=\frac{1}{4(1+4n^{2}C_{1}^{2})^{2}}.$

Observe that the statement to be proved is homogeneous of degree $1$, so it
suffices to show that if $u$ is a continuous function such that
$\int_{X}u\omega^{n}=0$ and $|\Delta_{\omega}u|\leq\delta$, then
$M=\frac{1}{V_{\omega}}\int_{X}|u|\omega^{n}\leq C$ is uniformly bounded from
above independently of $u$. We assume that $M\geq 1$ (otherwise we are done),
and we set

$v:=\frac{u}{M}=\varepsilon u\;\;\text{ where
}\;\;0<\varepsilon:=\frac{1}{M}\leq 1.$

Since $\|\Delta_{\omega}v\|_{L^{\infty}(X)}\leq
1,\frac{1}{V_{\omega}}\int_{X}|v|\omega^{n}=1$ and $\int_{X}v\omega^{n}=0$,
Lemma 2.1 yields

(2.1) $\|v\|_{L^{\infty}(X)}\leq 2C_{1}.$

Set

$H:=\frac{1}{n}\Delta_{\omega}u=\frac{dd^{c}u\wedge\omega^{n-1}}{\omega^{n}}\;\;\text{
so that }\;\;\|H\|_{L^{\infty}(X)}\leq\delta.$

Set $t=\sqrt{\delta}\in(0,1)$ and observe that the function $v$ is
$\varepsilon\omega$-sh and satisfies

$(\varepsilon\omega+dd^{c}v)\wedge(\varepsilon\omega)^{n-1}=\varepsilon^{n}(1+H)\omega^{n}=e^{tv}e^{-tv}(1+H)(\varepsilon\omega)^{n}.$

We let $\varphi\in PSH(X,\varepsilon\omega)\cap L^{\infty}(X)$ be the unique
bounded $\varepsilon\omega$-psh solution of the complex Monge-Ampère equation

$(\varepsilon\omega+dd^{c}\varphi)^{n}=e^{nt\varphi}e^{-ntv}(1+H)^{n}(\varepsilon\omega)^{n}.$

It follows from Proposition 1.4 (applied to $\varepsilon\omega$ and
$f=e^{-tv}(1+H)$) that

(2.2) $\varphi\leq v.$

Setting $\psi=\varphi/\varepsilon\in PSH(X,\omega)$, the equation can be
rewritten as

(2.3)
$(\omega+dd^{c}\psi)^{n}=e^{nt(\varepsilon\psi-v)}(1+H)^{n}\omega^{n}\leq
2^{n}\omega^{n},$

since $\varepsilon\psi-v=\varphi-v\leq 0$ and
$\|H\|_{L^{\infty}}\leq\delta\leq 1$. It follows from Theorem 1.9 that

${\rm Osc}_{X}(\psi)=\|\tilde{\psi}\|_{L^{\infty}(X)}\leq
C_{0}=C_{0}(n,p,A,B),$

where $\tilde{\psi}:=\psi-\sup_{X}\psi\leq 0$. Integrating (2.3) we obtain

$1=e^{nt\varepsilon\sup_{X}\psi}\int_{X}e^{nt\varepsilon\tilde{\psi}}e^{-ntv}(1+H)^{n}\frac{\omega^{n}}{V_{\omega}}\leq
e^{nt\varepsilon\sup_{X}\psi}(1+\delta)^{n}\int_{X}e^{-ntv}\frac{\omega^{n}}{V_{\omega}}.$

We let the reader check that $e^{x}\leq 1+x+x^{2}$ for $|x|\leq 1$. Since
$t<n^{-1}C_{1}^{-1}$ we infer

$\displaystyle\int_{X}e^{-ntv}\frac{\omega^{n}}{V_{\omega}}$
$\displaystyle\leq
1-nt\int_{X}v\frac{\omega^{n}}{V_{\omega}}+n^{2}t^{2}\int_{X}v^{2}\frac{\omega^{n}}{V_{\omega}}$
$\displaystyle=1+n^{2}t^{2}\int_{X}v^{2}\frac{\omega^{n}}{V_{\omega}}\quad(\text{
since }\int_{X}v\omega^{n}=0)$ $\displaystyle\leq 1+4n^{2}C_{1}^{2}t^{2},$

since $||v||_{L^{\infty}(X)}\leq 2C_{1}$. Using that $\log(1+x)\leq x$ we
therefore obtain

$\displaystyle nt\varepsilon\sup_{X}\psi$
$\displaystyle\geq-n\log(1+\delta)-\log\left(1+4n^{2}C_{1}^{2}t^{2}\right)$
$\displaystyle\geq-n\delta-4n^{2}C_{1}^{2}t^{2}.$

Using that $t=\sqrt{\delta}=\frac{1}{2[1+4n^{2}C_{1}^{2}]}$ we conclude that

$\varepsilon\sup_{X}\psi\geq-\frac{1}{2}.$

It follows that
$\varphi=(\varphi-\sup_{X}\varphi)+\sup_{X}\varphi=\varepsilon\tilde{\psi}+\varepsilon\sup_{X}\psi\geq-
C_{0}\varepsilon-\frac{1}{2}$. Together with (2.2), this shows that

$v\geq-C_{0}\varepsilon-\frac{1}{2}.$

Repeating the same argument for $-v$, we get $-v\geq-
C_{0}\varepsilon-\frac{1}{2}$ hence

$\displaystyle\|v\|_{L^{\infty}(X)}\leq C_{0}\varepsilon+\frac{1}{2}.$

Therefore

$1=\frac{1}{V_{\omega}}\int_{X}|v|\omega^{n}\leq\frac{1}{V_{\omega}}\int_{X}\|v\|_{L^{\infty}(X)}\omega^{n}\leq
C_{0}\varepsilon+\frac{1}{2},$

which yields

$\frac{1}{V_{\omega}}\int_{X}|u|\omega^{n}\leq 2C_{0}.$

Unravelling the normalizations we have made, we see that the constant $C$ from
the statement can be chosen as $C=8C_{0}[1+4n^{2}C_{1}^{2}]^{2}$, where
$C_{0}$ is the uniform constant provided by Theorem 1.9 (taking
$g_{t}=2^{n}\omega_{t}^{n}/dV_{X_{t}}$) and $C_{1}$ is the uniform constant
from Lemma 2.1. ∎

###  2.2. Green’s functions

######  Theorem 2.3.

Fix $0<r<\frac{n}{n-1}$ and $0<s<\frac{2n}{2n-1}$. For all
$t\in\mathbb{D}^{*},x\in X_{t}$ and
$\omega\in{\mathcal{K}}(\mathcal{X},p,A,B)$ we have

  1. (1)

$\sup_{X_{t}}G^{\omega_{t}}_{x}\leq C_{0}=C_{0}(n,p,A,B)$;

  2. (2)

$\frac{1}{V_{\omega_{t}}}\int_{X_{t}}|G^{\omega_{t}}_{x}|^{r}\omega_{t}^{n}\leq
C_{1}=C_{1}(n,p,r,A,B)$;

  3. (3)

$\frac{1}{V_{\omega_{t}}}\int_{X_{t}}|\nabla
G^{\omega_{t}}_{x}|_{\omega_{t}}^{s}\omega_{t}^{n}\leq
C_{2}=C_{2}(n,p,s,A,B)$.

Given Proposition 2.2 above, the proof is a combination of the main results of
[GPS22, GPSS22] with the uniform estimates provided by Theorem 1.9.

###### Proof.

Step 1. Consider $h=-{\bf 1}_{\\{G_{x}\leq 0\\}}+\int_{\\{G_{x}\leq
0\\}}\frac{\omega^{n}}{V_{\omega}}$. Observe that $-1\leq h\leq 1$ and
$\int_{X}h\omega^{n}=0$. We let $v$ denote the unique solution
$\Delta_{\omega}v=h$ with $\int_{X}v\omega^{n}=0$. It follows from Proposition
2.2 that $||v||_{L^{\infty}(X)}\leq C$. Thus

$\displaystyle C\geq v(x)$ $\displaystyle=$
$\displaystyle\frac{1}{V_{\omega}}\int_{X}v(\omega+dd^{c}G_{x})\wedge\omega^{n-1}$
$\displaystyle=$
$\displaystyle\frac{1}{V_{\omega}}\int_{X}G_{x}dd^{c}v\wedge\omega^{n-1}=n\int_{\\{G_{x}\leq
0\\}}(-G_{x})\frac{\omega^{n}}{V_{\omega}}.$

Since $\int_{X}G_{x}\omega^{n}=0$, we infer

$\int_{X}|G_{x}|\frac{\omega^{n}}{V_{\omega}}=2\int_{\\{G_{x}\leq
0\\}}(-G_{x})\frac{\omega^{n}}{V_{\omega}}\leq\frac{2C}{n}.$

It therefore follows from Lemma 2.1 that $\sup_{X}G_{x}\leq C_{0}$, proving
(1).

Step 2. We have shown (2) for $r=1$ in the previous step. We now show (2) for
$r<1+\frac{1}{n}$ Set ${\mathcal{G}}_{x}=G_{x}-C_{0}-1\leq-1$ and consider $u$
the $\omega$-sh solution of

$\frac{1}{V_{\omega}}(\omega+dd^{c}u)\wedge\omega^{n-1}=\frac{(-{\mathcal{G}}_{x})^{\beta}\omega^{n}}{\int_{X}(-{\mathcal{G}}_{x})^{\beta}\omega^{n}},$

with $\int_{X}u\omega^{n}=0$, where $0<\beta<\frac{1}{n}$. We are going to
show that $u\geq-C$ is uniformly bounded below. It will follow that

$\displaystyle-C\leq u(x)$ $\displaystyle=$
$\displaystyle\int_{X}u\frac{(\omega+dd^{c}{\mathcal{G}}_{x})\wedge\omega^{n-1}}{V_{\omega}}$
$\displaystyle=$
$\displaystyle\int_{X}{\mathcal{G}}_{x}\frac{(\omega+dd^{c}u)\wedge\omega^{n-1}}{V_{\omega}}=-\frac{\int_{X}(-{\mathcal{G}}_{x})^{1+\beta}\frac{\omega^{n}}{V_{\omega}}}{\int_{X}(-{\mathcal{G}}_{x})^{\beta}\frac{\omega^{n}}{V_{\omega}}}.$

Since $1\leq-{\mathcal{G}}_{x}$ we obtain
$\int_{X}(-{\mathcal{G}}_{x})^{\beta}\frac{\omega^{n}}{V_{\omega}}\leq\int_{X}(-{\mathcal{G}}_{x})\frac{\omega^{n}}{V_{\omega}}=1+C_{0}$,
hence

$\int_{X}(-{\mathcal{G}}_{x})^{1+\beta}\frac{\omega^{n}}{V_{\omega}}\leq
C[1+C_{0}],$

proving (2) for $r=1+\beta$, as ${\mathcal{G}}_{x}$ differs from $G_{x}$ by a
uniform additive constant.

To prove that $u$ is uniformly bounded below, we consider the normalized
solution $\varphi\in PSH(X,\omega)\cap L^{\infty}(X)$, $\sup_{X}\varphi=0$, of
the Monge-Ampère equation

$\frac{1}{V_{\omega}}(\omega+dd^{c}\varphi)^{n}=\frac{(-{\mathcal{G}}_{x})^{n\beta}\omega^{n}}{\int_{X}(-{\mathcal{G}}_{x})^{n\beta}\omega^{n}}.$

Since $-{\mathcal{G}}_{x}\geq 1$, the density of the RHS is bounded from above
by $(-{\mathcal{G}}_{x})^{n\beta}f_{\omega}$. It follows from Hölder
inequality that the latter belongs to $L^{p^{\prime}}(dV_{X})$,
$p^{\prime}<p$, since

$\displaystyle\int_{X}(-{\mathcal{G}}_{x})^{n\beta
p^{\prime}}f_{\omega}^{p^{\prime}}dV_{X}$ $\displaystyle=$
$\displaystyle\int_{X}(-{\mathcal{G}}_{x})^{n\beta
p^{\prime}}f_{\omega}^{p^{\prime}-1}\frac{\omega^{n}}{V_{\omega}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{X}f_{\omega}^{p}dV_{X}\right)^{\frac{p^{\prime}-1}{p-1}}\left(\int_{X}(-{\mathcal{G}}_{x})^{n\beta
p^{\prime}s^{\prime}}\frac{\omega^{n}}{V_{\omega}}\right)^{\frac{1}{s^{\prime}}}\leq
C^{\prime},$

where $s^{\prime}=\frac{p-1}{p-p^{\prime}}$ is the conjugate exponent of
$s=\frac{p-1}{p^{\prime}-1}$: we choose $p^{\prime}>1$ very close to $1$ (thus
$s^{\prime}>1$ is very close to $1$ as well) so that $n\beta
p^{\prime}s^{\prime}<1$, and the last integral is under control by the first
step. Theorem 1.9 now ensures that

$-M_{0}\leq\varphi\leq 0$

for some uniform $M_{0}$. Since
$\int_{X}(-{\mathcal{G}}_{x})^{n\beta}\frac{\omega^{n}}{V_{\omega}}\leq\int_{X}(-{\mathcal{G}}_{x})\frac{\omega^{n}}{V_{\omega}}\leq
1+C_{0}=:C_{1}^{n}$, it follows from the AM-GM inequality that
$(\omega+dd^{c}\varphi)\wedge\omega^{n-1}\geq\frac{(-{\mathcal{G}}_{x})^{\beta}}{C_{1}}\omega^{n}$,
while
$(-{\mathcal{G}}_{x})^{\beta}\omega^{n}\geq(\omega+dd^{c}u)\wedge\omega^{n-1}$
since $-{\mathcal{G}}_{x}\geq 1$. We infer that $\varphi-u/C_{1}$ is a
normalized $\omega$-sh function; it is bounded from above by Lemma 2.1, hence

$-M_{0}C_{1}-C_{2}\leq C_{1}\varphi-C_{2}\leq u\leq C_{2}^{\prime},$

showing that $u$ is uniformly bounded, as claimed.

Step 3. We now establish (2) for the optimal values of $r$ by a recursive
argument. Indeed the reasoning from Step 2 shows that if
$\int_{X}(-{\mathcal{G}}_{x})^{n\beta}\frac{\omega^{n}}{V_{\omega}}\leq C$ for
$n\beta\leq\beta^{\prime}$, then
$\int_{X}(-{\mathcal{G}}_{x})^{r}\frac{\omega^{n}}{V_{\omega}}\leq C^{\prime}$
for $r<1+\frac{\beta^{\prime}}{n}$. By induction this yields, for all
$k\in\mathbb{N}$,

$\int_{X}(-{\mathcal{G}}_{x})^{r}\frac{\omega^{n}}{V_{\omega}}\leq
C_{r}\;\;\text{ for }r<1+\frac{1}{n}+\frac{1}{n^{2}}+\cdots+\frac{1}{n^{k}}.$

Thus a uniform control can be obtained for all $r<\frac{n}{n-1}$.

Step 4. It follows from Step 3, Lemma 2.5 below and Hölder inequality that (3)
holds for $s<\frac{2n}{2n-1}$. Indeed set $r=\frac{s}{2-s}(1+\beta)$, and
observe that by choosing $0<\beta$ very small and $r$ arbitrarily close to
$\frac{n}{n-1}$, we obtain $s$ arbitrarily close to $\frac{2n}{2n-1}$. Setting
$2\alpha=s(1+\beta)$, we thus get

$\displaystyle\int_{X}|\nabla{G}_{x}|^{s}{\omega^{n}}=\int_{X}|\nabla{\mathcal{G}}_{x}|^{s}{\omega^{n}}$
$\displaystyle=$
$\displaystyle\int_{X}\frac{|\nabla{\mathcal{G}}_{x}|^{s}}{|{\mathcal{G}}_{x}|^{\alpha}}|{\mathcal{G}}_{x}|^{\alpha}{\omega^{n}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{X}\frac{|\nabla{\mathcal{G}}_{x}|^{2}}{{|{\mathcal{G}}_{x}}|^{\frac{2\alpha}{s}}}{\omega^{n}}\right)^{\frac{s}{2}}\left(\int_{X}|{\mathcal{G}}_{x}|^{\frac{2\alpha}{2-s}}{\omega^{n}}\right)^{\frac{2-s}{2}}$
$\displaystyle=$
$\displaystyle\left(\int_{X}\frac{|\nabla{\mathcal{G}}_{x}|^{2}}{{|{\mathcal{G}}_{x}}|^{1+\beta}}{\omega^{n}}\right)^{\frac{s}{2}}\left(\int_{X}|{\mathcal{G}}_{x}|^{r}{\omega^{n}}\right)^{\frac{2-s}{2}}\leq
C(s).$

∎

######  Remark 2.4.

All these estimates are valid, more generally, for any $\omega$-psh function
$v$ which is normalized by $\int_{X}v\omega^{n}=0$. Indeed using Stokes
theorem we obtain

$v(x)=\frac{1}{V_{\omega}}\int_{X}G_{x}(\omega+dd^{c}v)\wedge\omega^{n-1}\leq\sup_{X}G_{x}\leq
C_{0}.$

The other estimates follow by using the symetry $G_{x}(y)=G_{y}(x)$.

We have used the following observation [GPSS22, Lemma 5.6].

######  Lemma 2.5.

Fix $\beta>0$. Then

$\frac{1}{V_{\omega}}\int_{X}\frac{dG_{x}\wedge
d^{c}G_{x}\wedge\omega^{n-1}}{(-G_{x}+C_{0}+1)^{1+\beta}}\leq\frac{1}{\beta}.$

Here $C_{0}$ denotes the uniform constant from Theorem 2.3.1.

###### Proof.

The function $u(y)=(-G_{x}(y)+C_{0}+1)^{-\beta}$ takes values in $[0,1]$ with
$u(x)=0$. Since $du=\frac{\beta dG_{x}}{(-G_{x}+C_{0}+1)^{1+\beta}}$, we infer

$0=\frac{1}{V_{\omega}}\int_{X}u(\omega+dd^{c}G_{x})\wedge\omega^{n-1}=\frac{1}{V_{\omega}}\int_{X}u\omega^{n}-\frac{\beta}{V_{\omega}}\int_{X}\frac{dG_{x}\wedge
d^{c}G_{x}\wedge\omega^{n-1}}{(-G_{x}+C_{0}+1)^{\beta+1}}.$

The result follows. ∎

###  2.3. Sobolev estimates

The following is our improved and family version of [GPSS23, Theorem 2.1 and
Lemma 6.2].

######  Theorem 2.6.

Fix $1<r<\frac{2n}{n-1}$, $t\in\mathbb{D}^{*}$ and
$\omega\in{\mathcal{K}}(\mathcal{X},p,A,B)$.

1) For all $u\in W^{1,2}(X_{t})$, we have

$\left(\frac{1}{V_{\omega_{t}}}\int_{X_{t}}|u-\overline{u}|^{2r}\omega_{t}^{n}\right)^{1/r}\leq
C_{1}\frac{1}{V_{\omega_{t}}}\int_{X_{t}}|\nabla
u|^{2}_{\omega_{t}}\omega_{t}^{n},$

where $\overline{u}=\frac{1}{V_{\omega_{t}}}\int_{X_{t}}u\omega_{t}^{n}$ and
$C_{1}=C_{1}(n,p,r,A,B)>0$.

2) If $\Omega\subset X_{t}$ is a domain and $u\in W^{1,2}(\Omega)$ has compact
support in $\Omega$, then

$\left(\frac{1}{V_{\omega_{t}}}\int_{\Omega}|u|^{2r}\omega_{t}^{n}\right)^{1/r}\leq
C_{2}\left[1+\frac{V_{\omega_{t}}(\Omega)}{V_{\omega_{t}}(X_{t}\setminus\Omega)}\right]\frac{1}{V_{\omega_{t}}}\int_{\Omega}|\nabla
u|^{2}_{\omega_{t}}\omega_{t}^{n},$

where $C_{2}=C_{2}(n,p,r,A,B)>0$.

###### Proof.

The proof is very similar to that of [GPSS23, Theorem 2.1 and Lemma 6.2], so
we only sketch it. We fix $\beta\in(0,1)$ such that
$(1+\beta)r<\frac{n}{n-1}$. Green’s formula and Hölder inequality yield

$\displaystyle|u(x)-\bar{u}|$
$\displaystyle=\left|\frac{1}{V_{\omega}}\int_{X}du\wedge
d^{c}{\mathcal{G}_{x}}\wedge\omega^{n-1}\right|$
$\displaystyle\leq\left(\frac{1}{V_{\omega}}\int_{X}\frac{d\mathcal{G}_{x}\wedge
d^{c}\mathcal{G}_{x}\wedge\omega^{n-1}}{(-\mathcal{G}_{x})^{1+\beta}}\right)^{1/2}\left(\frac{1}{V_{\omega}}\int_{X}(-\mathcal{G}_{x})^{1+\beta}|\nabla
u|^{2}_{\omega}\omega^{n}\right)^{1/2}$
$\displaystyle\leq\frac{1}{\beta^{1/2}}\left(\frac{1}{V_{\omega}}\int_{X}(-\mathcal{G}_{x})^{1+\beta}|\nabla
u|^{2}_{\omega}\omega^{n}\right)^{1/2},$

hence

(2.4)
$\left(\int_{X}|u-\overline{u}|^{2r}\omega^{n}\right)^{1/r}\leq\frac{1}{\beta}\|\frac{1}{V_{\omega}}\int_{X}(-\mathcal{G}_{x})^{1+\beta}|\nabla
u|^{2}_{\omega}\omega^{n}\|_{L^{r}(X,\omega)}.$

It follows from Minskowski’s inequality for integrals that

$\displaystyle\|\frac{1}{V_{\omega}}\int_{X}(-\mathcal{G}_{x})^{1+\beta}|\nabla
u|^{2}_{\omega}\omega^{n}\|_{L^{r}(X,\omega)}$
$\displaystyle\leq\frac{1}{V_{\omega}}\int_{X}\left(\int_{X}(-\mathcal{G}_{x})^{r(1+\beta)}\omega^{n}(x)\right)^{\frac{1}{r}}|\nabla
u|^{2}_{\omega}(y)\omega^{n}(y)$ $\displaystyle\leq
C_{1}\frac{V_{\omega}^{1/r}}{V_{\omega}}\int_{X}|\nabla
u|^{2}_{\omega}\omega^{n},$

using Theorem 2.3.2. Together with (2.4) we obtain

$\left(\frac{1}{V_{\omega}}\int_{X}|u-\overline{u}|^{2r}\omega^{n}\right)^{1/r}\leq
C_{1}\frac{1}{V_{\omega}}\int_{X}|\nabla u|^{2}_{\omega}\omega^{n}.$

For the second inequality, as in [GPSS23, Lemma 6.2], using Green formula, one
has for any $x\in\Omega$ and $\beta>0$,

$|u(x)|^{2}\leq\frac{C}{V_{\omega}(\Omega^{c})}\int_{X}|\nabla
u|^{2}_{\omega}\omega^{2}+\frac{1}{\beta
V_{\omega}}\int_{X}(-\mathcal{G}(x,y))^{1+\beta}|\nabla
u|^{2}_{\omega}\omega^{n}.$

Raising to the power $r$ on both sides and arguing similarly to what we have
done above, we obtain the required inequality. ∎

##  3\. Geometric applications

###  3.1. Diameter bounds and non-collapsing

Among the various applications of Theorem A, we stress the following diameter
and non-collapsing estimates.

######  Theorem 3.1.

Fix $0<\delta<1$ and $\omega\in{\mathcal{K}}(\mathcal{X},p,A,B)$. There exists
$C=C(n,p,\delta,A,B)>0$ such that for all $t\in\mathbb{D}^{*},x\in X_{t},$ and
$r>0$,

$C\min(1,r^{2n+\delta})\leq\frac{1}{V_{\omega_{t}}}{\rm
Vol}_{\omega_{t}}(B_{\omega_{t}}(x,r))\;\;\text{ and }\;\;{\rm
diam}(X_{t},\omega_{t})\leq C.$

Thus the family of compact metric spaces
$\\{(X_{t},d_{\omega_{t}}),\;\omega\in{\mathcal{K}}(\mathcal{X},p,A,B),\,t\in\mathbb{D}^{*}\\}$
is pre-compact in the Gromov-Hausdorff topology.

Due to its prominent role in geometric analysis, there has been an intensive
search for such uniform diameter estimates in the past decade. We simply list
the most recent contributions which require as an extra assumption

  * •

either a Ricci lower bound [LTZ17, FGS20, GS22];

  * •

or that $X$ be of general type [Bam18, Wang18, JS22];

  * •

to control the continuity of Monge-Ampère potentials [Li21, GGZ23, Vu23];

  * •

orelse a uniform lower bound on $\omega^{n}/dV_{X}$ [GPS22, GPSS22, GPSS23].

The proof of Theorem 3.1 is similar to that of [GPSS22, Theorem 1.1].

###### Proof.

We fix $(x_{0},y_{0})\in X^{2}$ such that $d_{\omega}(x_{0},y_{0})={\rm
diam}(X,\omega)$. The function $\rho:x\in X\mapsto
d_{\omega}(x_{0},x)\in\mathbb{R}^{+}$ is $1$-Lipschitz with $\rho(x_{0})=0$.
Green’s formula applied to the function $\rho$ at the point $x_{0}$ yields

$\int_{X}\rho\omega^{n}=\int_{X}d\rho\wedge
d^{c}G_{x_{0}}\wedge\omega^{n-1}\leq\int_{X}|\nabla
G_{x_{0}}|_{\omega}\omega^{n}.$

Using again Green’s formula at $y_{0}$ and the previous inequality, we obtain

$\displaystyle{\rm diam(X,\omega)}$ $\displaystyle=$
$\displaystyle\frac{1}{V_{\omega}}\int_{X}\rho\omega^{n}-\frac{1}{V_{\omega}}\int_{X}d\rho\wedge
d^{c}G_{y_{0}}\wedge\omega^{n-1}$ $\displaystyle\leq$
$\displaystyle\frac{1}{V_{\omega}}\int_{X}|\nabla
G_{x_{0}}|_{\omega}\omega^{n}+\frac{1}{V_{\omega}}\int_{X}|\nabla
G_{y_{0}}|_{\omega}\omega^{n}\leq C,$

where the last inequality follows from Theorem 2.3.3.

Next we prove the non-collapsing estimate. We fix $x\in X$ and consider the
uniformly bounded function $\rho:y\in X\mapsto
d_{\omega}(x,y)\in\mathbb{R}^{+}$. Fix $r\in(0,1]$ and let $\chi$ be a non-
negative smooth cut-off function with support in $B_{\omega}(x,r)$ such that
$\chi\equiv 1$ on $\overline{B}_{\omega}(x,\frac{r}{2})$ and
$\sup_{X}|\nabla\chi|_{\omega}\leq\frac{C}{r}$. Thus $\rho\chi$ is a
$C$-Lipschitz function.

We pick $s\in(1,\frac{2n}{2n-1})$ and denote by
$s^{*}=\frac{s}{s-1}\in(2n,\infty)$ the conjugate exponent of $s$. Applying
Green’s formula to $\rho\chi$ at $y\in\overline{B_{\omega}(x,r)}^{c}$, we
obtain

$\displaystyle\int_{X}\rho\chi\omega^{n}$
$\displaystyle=\frac{1}{V_{\omega}}\int_{X}d(\chi\rho)\wedge
d^{c}G_{y}\wedge\omega^{n-1}$ $\displaystyle\leq C\left(\int_{X}|\nabla
G_{y}|^{s}_{\omega}\omega^{n}\right)^{1/s}\left({\rm
Vol}_{\omega}(B_{\omega}(x,r))\right)^{1/s^{*}}$ $\displaystyle\leq
CV_{\omega}^{1/s}\left({\rm Vol}_{\omega}(B_{\omega}(x,r))\right)^{1/s^{*}},$

by Theorem 2.3.3. Applying Green’s formula again with $z\in\partial
B_{\omega}(x,r/2)$, we infer

$\displaystyle\frac{r}{2}$
$\displaystyle=\frac{1}{V_{\omega}}\int_{X}\rho\chi\omega^{n}+\frac{1}{V_{\omega}}\int_{X}d(\chi\rho)\wedge
d^{c}G_{z}\wedge\omega^{n-1}$ $\displaystyle\leq
2CV_{\omega}^{-1+1/s}\left({\rm
Vol}_{\omega}(B(x,r))\right)^{1/s^{*}}=2C\left(\frac{{\rm
Vol}_{\omega}(B(x,r))}{V_{\omega}}\right)^{1/s^{*}}.$

Since $s^{*}=\frac{s}{s-1}\in(2n,\infty)$, this implies the non-collapsing
estimate.

Set
${\mathcal{F}}:=\\{(X_{t},d_{\omega_{t}}),\;\omega\in{\mathcal{K}}(\mathcal{X},p,A,B),\,t\in\mathbb{D}^{*}\\}$.
Gromov’s theorem [BBI01, Theorem 7.4.15] ensures that this family is pre-
compact in the Gromov-Hausdorff topology iff there is a uniform bound on the
diameters and for each $\varepsilon>0$ one can find in each
$X\in{\mathcal{F}}$ an $\varepsilon$-net consisting of no more than
$N=N(\varepsilon)$-points. This follows easily from the uniform non-collapsing
estimate, together with the uniform upper bound on the global volumes ${\rm
Vol}_{\omega_{t}}(X_{t})\leq V_{0}$. ∎

###  3.2. Diameters of smoothable cscK metrics

In this section we consider $(X,\beta)$ a compact $n$-dimensional Kähler
variety with Kawamata log terminal (klt) singularities which admits a
$\mathbb{Q}$-Gorenstein smoothing $\pi:{\mathcal{X}}\rightarrow\mathbb{D}$,
i.e.

  * •

${\mathcal{X}}$ is a $\mathbb{Q}$-Gorenstein complex space of complex
dimension $n+1$,

  * •

$\pi$ is a proper surjective holomorphic map such that
${\mathcal{X}}_{|\pi^{-1}(0)}\sim X_{0}$,

  * •

$X_{t}={\mathcal{X}}_{|\pi^{-1}(t)}$ is smooth for all $t\in\mathbb{D}^{*}$,

  * •

there is a smooth form $\beta_{{\mathcal{X}}}$ such that
$\beta_{t}={\beta_{{\mathcal{X}}}}_{|X_{t}}$ is Kähler with $\beta_{0}=\beta$.

When the Mabuchi functional $M_{\beta}$ is coercive, it has been shown in
[PTT23, Theorem C] that so are the Mabuchi functionals $M_{\beta_{t}}$ , hence
there exists a unique constant scalar curvature Kähler metric $\omega_{t}$
cohomologous to $\beta_{t}$.

######  Corollary 3.2.

In the setting above, there exists $D>0$ such that for all
$t\in\mathbb{D}^{*}$,

${\rm diam}(X_{t},\omega_{t})\leq D.$

###### Proof.

For $t\in\mathbb{D}^{*}$, the unique constant scalar curvature Kähler metric
$\omega_{t}\in[\beta_{t}]$ satisfies the following coupled equations:

$\left\\{\begin{matrix}(\beta_{t}+dd^{c}\varphi_{t})^{n}=e^{F_{t}}\beta_{t}^{n}\\\
\Delta_{\omega_{t}}F_{t}=-\bar{s}_{t}+{\rm Tr}_{\omega_{t}}{\rm
Ric}(\beta_{t}).\end{matrix}\right.$

The volumes $\int_{X}\beta_{t}^{n}$ are uniformly bounded away from zero and
infinity, while it follows from [PTT23, Theorem 5.3] that the smooth densities
$f_{t}=e^{F_{t}}$ satisfy $\|f_{t}\|_{L^{p}(X_{t},\beta_{t}^{n})}\leq B$, for
some $p>1$ and for all $t\in\mathbb{D}^{*}$. Thus
$\omega\in{\mathcal{K}}(\mathcal{X},p,1,B)$, and the uniform bound for ${\rm
diam}(X_{t},\omega_{t})$ follows from Theorem 3.1. ∎

###  3.3. Estimates along the Kähler-Ricci flow

We assume here that $X$ is a compact Kähler manifold with $K_{X}$ nef (a
smooth minimal model). We consider

$\frac{\partial\omega_{t}}{\partial t}=-{\rm Ric}(\omega_{t})-\omega_{t}$

the normalized Kähler-Ricci flow starting from an initial Kähler metric
$\omega_{0}$. It follows from [TZ06] that this flow exists for all times
$t>0$. We refer the reader to [Tos18, Tos24] for recent overviews of the
theory of the Kähler-Ricci flow.

It has been a challenging open problem up to now to obtain uniform geometric
bounds along the flow as $t\rightarrow+\infty$. We obtain here the following
striking result which -in particular- solves [Tos18, Conjecture 6.2]:

######  Theorem 3.3.

Fix $0<\delta<1$. There exist $C=C(\omega_{0})>0$ and
$c=c(\omega_{0},\delta)>0$ such that for all $t>0$ and $x\in X$,

${\rm diam}(X,\omega_{t})\leq C\;\;\text{ and }\;\;{\rm
Vol}_{\omega_{t}}(B_{\omega_{t}}(x,r))\geq cr^{2n+\delta}V_{\omega_{t}},$

whenever $0<r<{\rm diam}(X,\omega_{t})$.

This result has been established by Guo-Phong-Song-Sturm under the extra
assumption that the Kodaira dimension ${\rm kod}(X)$ of $X$ is non-negative
(see [GPSS22, Theorem 2.3]). Recall that

${\rm kod}(X)=\limsup_{m\rightarrow+\infty}\left[\frac{\log\dim
H^{0}(X,mK_{X})}{\log m}\right]$

measures the asymptotic growth of the number of holomorphic pluricanonical
forms, while the numerical dimension $\nu=\nu(X)=\sup\\{k\geq
0,c_{1}(K_{X})^{k}\neq 0\\}.$ measures the asymptotic growth of volumes under
the NKRF,

(3.1) ${\rm Vol}_{\omega_{t}}(X)=\left(\begin{array}[]{c}n\\\
\nu\end{array}\right)c_{1}(K_{X})^{\nu}\\{\omega_{0}\\}^{n-\nu}e^{-(n-\nu)t}[1+o(1)].$

It is known that $\nu(X)\geq{\rm kod}(X)$ and the equality turns out to be
equivalent to the abundance conjecture (see [Tos18, Conjecture 6.3]). The
extra assumption made in [GPSS22, Theorem 2.3] requires that ${\rm
kod}(X)\neq-\infty$; it is equivalent to

$\nu(X)\geq 0\stackrel{{\scriptstyle?}}{{\Longrightarrow}}{\rm kod}(X)\geq 0,$

which has been an open problem for the last fifty years.

###### Proof.

Fix $\chi$ a smooth closed $(1,1)$-form representing $c_{1}(K_{X})$. It
follows from the $\partial\overline{\partial}$-lemma that
$\omega_{t}=e^{-t}\omega_{0}+(1-e^{-t})\chi+dd^{c}\varphi_{t}$ for some
$\varphi_{t}\in{\mathcal{C}}^{\infty}(X)$. One can normalize the latter so
that the NKRF is equivalent to the parabolic equation

$(e^{-t}\omega_{0}+(1-e^{-t})\chi+dd^{c}\varphi_{t})^{n}=e^{\partial_{t}\varphi_{t}+\varphi_{t}-(n-\nu)t}\omega_{0}^{n}$

on $X\times\mathbb{R}^{+}$, with $\varphi_{0}=0$. Here $\nu$ denotes the
numerical dimension of $K_{X}$.

We claim that there exists $C_{0},C_{1}>0$ such that for all $t>0$ and $x\in
X$,

$\varphi_{t}(x)\leq C_{0}\;\;\text{ and }\;\;\partial_{t}\varphi_{t}(x)\leq
C_{1}.$

It will then follow from (3.1) that
$V_{\omega_{t}}^{-1}\omega_{t}^{n}=f_{t}\omega_{0}^{n}$ with
$||f_{t}||_{\infty}\leq C_{2}$. The uniform diameter and non-collapsing
estimates are thus consequences of Theorem A.

Upper bound on $\varphi_{t}$. We set $V_{t}=\int_{X}\omega_{t}^{n}$ and
$I(t)=\int_{X}\varphi_{t}\frac{\omega_{0}^{n}}{V_{0}}$. It follows from [GZ,
Proposition 8.5] that $\sup_{X}\varphi_{t}\leq I(t)+C$ for some uniform
constant $C>0$, hence it suffices to bound $I(t)$ from above. The concavity of
the logarithm ensures that

$I^{\prime}(t)=\int_{X}\partial_{t}\varphi_{t}\frac{\omega_{0}^{n}}{V_{0}}\leq-I(t)+[\log
V_{t}+(n-\nu)t-\log V_{0}]\leq-I(t)-C^{\prime}.$

Using that $I(0)=0$ we conclude that $I(t)\leq C^{\prime}$, as desired.

Upper bound on $\partial\varphi_{t}$. Consider
$H(t,x)=(e^{t}-1)\partial_{t}\varphi_{t}(x)-\varphi_{t}(x)-h(t)$, where
$h(t)=\nu t+(n-\nu)e^{t}$. A direct computation shows that

$\left(\frac{\partial}{\partial t}-\Delta_{\omega_{t}}\right)(H)=-{\rm
Tr}_{\omega_{t}}(\omega_{0})+(n-\nu)[e^{t}-1]+n-h^{\prime}(t)\leq 0.$

The maximum principle thus ensures that $H(t,x)\leq\max_{x\in X}H(0,x)\leq 0$,
hence

$(e^{t}-1)\partial_{t}\varphi_{t}(x)\leq\varphi_{t}(x)+\nu t+(n-\nu)e^{t}\leq
C_{0}+ne^{t}.$

Thus $\partial_{t}\varphi_{t}(x)\leq C_{1}$ for all $t\geq 1$, while such an
upper bound is clear on $X\times[0,1]$ by compactness. The proof is complete.
∎

###  3.4. Fiberwise Calabi-Yau metrics

Our estimates can be applied in the study of adiabatic limits of Ricci-flat
Kähler metrics on a Calabi-Yau manifold under the degeneration of the Kähler
class, as initiated by Tosatti in [Tos10].

Let $(X,\beta_{X})$ be an $N$-dimensional Kähler manifold with nowhere
vanishing holomorphic volume form $\Omega$ normalized so that
$\int_{X}i^{N^{2}}\Omega\wedge\overline{\Omega}=1$. Let $\pi:X\rightarrow Y$
be a holomorphic fibration onto a Riemann surface $(Y,\beta_{Y})$, with
connected fibres by $X_{y}$, $y\in Y$. We assume without loss of generality
that $\int_{X_{y}}\beta_{X}^{N-1}=1$ and $\int_{Y}\beta_{Y}=1$, and that the
singular fibres have at worst canonical singularities. We let $\omega_{t}$
denote the Calabi–Yau metrics on X in the class of
$\beta_{t}=t\beta_{X}+\pi^{*}\beta_{Y}$, $t>0$.

Understanding the behavior of $(X,\omega_{t})$ as $t\rightarrow 0$ as been the
subject of intensive studies in the past decade, notably through
collaborations of Gross, Hein, Li, Tosatti, Weinkove, Yang and Zhang (see
[Tos20] and the references therein). Theorem A allows one to provide an
alternative proof of the main result of [Li23].

######  Theorem 3.4.

[Li23, Theorem 1.4] There exists $C>0$ such that

${\rm diam}\left(X_{y},\frac{\omega_{t}}{t}\right)\leq C$

for all $0<t\leq 1$ and for all $y\in Y\setminus S$.

Here $S\subset Y$ denotes the discriminant locus; $\pi$ is a submersion over
$Y\setminus S$ so that every fiber $X_{y},y\in Y\setminus S$ is smooth.

###### Proof.

Set $n=N-1$ and $\omega_{y}={{\frac{\omega_{t}}{t}}}_{|X_{y}}$. Observe that
$V_{\omega_{y}}=\int_{X_{y}}\omega_{y}^{n}=\int_{X_{y}}\beta_{X}^{N-1}=1$. It
follows from [Li23, Proposition 2.3] and [DnGG23, Lemma 4.4] that there exists
$p>1$ such that

$\omega_{y}^{n}=f({\beta_{X}}_{|X_{y}})^{n}\;\;\text{ with
}\;\;\int_{X_{y}}f^{p}({\beta_{X}}_{|X_{y}})^{n}\leq B,$

for some constant $B>0$ independent of $t,y$. The conclusion follows therefore
from Theorem 3.1 applied to the family $(X_{y},\omega_{y})$ with $A=1$. ∎

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# Finite-size excess-entropy scaling for simple liquids

Mauricio Sevilla  Max Planck Institute for Polymer Research, Ackermannweg 10,
55128, Mainz  Atreyee Banerjee  Max Planck Institute for Polymer Research,
Ackermannweg 10, 55128, Mainz  Robinson Cortes-Huerto  corteshu@mpip-
mainz.mpg.de Max Planck Institute for Polymer Research, Ackermannweg 10,
55128, Mainz

(August 27, 2024)

###### Abstract

We introduce and validate a finite-size two-body excess entropy integral
equation. By using analytical arguments and computer simulations of
prototypical simple liquids, we show that the excess entropy $s_{2}$ exhibits
a finite-size scaling with the inverse of the linear size of the simulation
box. Since the self-diffusivity coefficient $D^{*}$ displays a similar finite-
size effect, we show that the scaling entropy relation $D^{*}=A\exp(\alpha
s_{2})$ also depends on the simulation box size. By extrapolating to the
thermodynamic limit, we report values for the coefficients $A$ and $\alpha$
that agree well with values available in the literature. Finally, we find a
power law relation between the scaling coefficients for $D^{*}$ and $s_{2}$,
suggesting a constant viscosity to entropy ratio.

##  I Introduction

Excess entropy ($s_{\rm exc}$), the difference between the entropy of a system
and its ideal gas counterpart at the same temperature and density, is
connected to the dynamical properties of simple liquids (See Ref.
dyre2018perspective  for a recent review). This observation was first reported
by Rosenfeld Rosenfeld1977 ; *Rosenfeld1999quasi, who showed that, for simple
model liquids, reduced transport properties such as diffusivity, viscosity and
thermal conductivity scale with the excess entropy as

$X^{*}=A\exp\left(\alpha s_{\rm exc}\right)\,,$ (1)

with $X^{*}$ a dimensionless transport property and $A$ and $\alpha$
parameters, independent of the thermodynamic state, determined by the
interparticle potential.
Following similar physical arguments and assuming that the major contribution
to $s_{\rm exc}$ comes from two-body terms, Dzugutov proposed a similar
scaling relation between self-diffusivity and a two-body approximation to the
excess entropy $s_{2}$, namely Dzugutov1996

$D^{*}=A\exp(\alpha s_{2})\,,$ (2)

with $D^{*}=\frac{D}{\Gamma\sigma_{r}^{2}}$ where $D$ is the self-diffusion
coefficient, $\sigma_{r}$ measures the linear size of the particles and
$\Gamma=4\sigma^{2}g(\sigma_{r})\rho\sqrt{\frac{\pi k_{\rm B}T}{m}}$ the
collision frequency given by the Enskog theory chapman1990  where
$g(\sigma_{r})$ is the value of the radial distribution function at a distance
$\sigma_{r}$. In this case, a large variety of simple liquids satisfy Eq. (2)
with the _universal_ choice of parameters $A=0.049$ and $\alpha=1$
Dzugutov1996 .
This excess entropy scaling has been widely validated for a large variety of
simple PhysRevLett.85.594 ; doi:10.1063/1.1516594 ; PhysRevE.68.031204 ;
Zhu2005 ; Jeff2011 ; Atreyee_JChemSci2017 ; widom2019first  and molecular
liquidsGoel_etal_JCP2008 ; Malvaldi_Chiappe_JCP2010 ; Chopra_etal_JCP2010 ;
Galliero_etal_JCP2011 , including specially water
doi:10.1080/00268970802378662 ; Agarwal_etal_JPCB2010 ; chopra2010use ;
Agarwal_JPCB2011 . We also highlight that experimental studies have tested
entropy scaling in somewhat challenging scenarios Tong2013 ;
spieckermann2022structure , and the fact that Rosenfeld and Dzugutov relations
are empirical but have been justified on theoretical grounds Alok_etal_PRL2001
; seki2015relationship . Furthermore, the structure–dynamics connection in Eq.
(2) has been proposed as a tool to investigate the relation between dynamical
properties of computational models at different resolutions, Jeff2012 , which
is now routinely considered in the context of coarse-grained models
rondina2020predicting ; jin2022understanding .
Transport properties exhibit implicit size effects due to the finite size of
the simulation box and the use of periodic boundary conditions (PBC)
Duenweg1993 ; Pascal_PRB2016 ; Zaoui_etal_PRB2016 . In the particular case of
the reduced self-diffusion coefficient $D^{*}$, given a cubic simulation box
of linear size $L$, $D^{*}\equiv D^{*}(L)$ takes the form Duenweg1993 ;
Hummer2004 ; Kikugawa_etal_JCP2015I ; Kikugawa_etal_JCP2015II ; Rotenberg2015
(See Figure 1)

Figure 1:  Reduced self-diffusion coefficient $D^{*}$ as a function of the
inverse of the box linear size $1/L$ for a Lennard-Jones liquid with density
$\rho\sigma_{\mathrm{LJ}}^{3}=0.864$ in the range of temperatures $k_{\rm
B}T=[0.7\epsilon,7\epsilon]$.

$D^{*}(L)=D^{*\infty}-\frac{\delta}{L}\,,$ (3)

with $\delta=\frac{k_{\rm B}T\zeta}{6\pi\eta\Gamma\sigma_{r}^{2}}$ with
$\zeta\approx 2.837297$ and $\eta$ the system’s viscosity. In the
thermodynamic limit (TL), namely, in the limit $L\to\infty$, the self-
diffusion coefficient takes the value $D^{*\infty}$.
Given the finite-size scaling of $D^{*}$, we expect that Eq. (2) also depends
on the size of the simulation box. Recent computational studies investigating
entropy scaling for liquid water using ab initio molecular dynamics
simulations Herrero_etal_PNAS2022  emphasise the relevance of this remark. In
this case, the systems under consideration are rather small, and finite-size
effects become increasingly important.
In this paper, we investigate the finite-size scaling of Eq. (2) by focusing
on implicit and explicit finite-size effects present on the two-body excess
entropy $s_{2}$. We find that $s_{2}$ obeys a finite-size scaling relation
similar to $D^{*}$, which implies that the _universal_ parameters $A$ and
$\alpha$ in Eq. (2) also depends on the size of the simulation box. Finally,
and perhaps more interestingly, our results indicate that a power law relates
the finite-size scaling coefficients of $D^{*}$ and $s_{2}$, suggesting a
constant viscosity/entropy ratio Kovtun_etal_PRL2005 ;
Angiella_etal_PhysLettA2009 ; Faussurier_etal_HEDP2014 ; Hohm_ChemPhys2014 .
The paper is organised as follows: In Section II we present the model and
computational details. We show that $s_{2}$ is ensemble invariant and that the
only relevant finite-size effect comes from using finite integration domains
in Section III. In Section IV, we introduce and validate a finite-size version
of $s_{2}$. We then present the finite-size scaling of the Dzugutov relation
(Eq. (2)) in Section V. Finally, we conclude and provide our outlook in
Section VI.

##  II Computational details

We investigate the excess entropy scaling for liquids whose potential energy
is described by a 12–6 Lennard–Jones potential truncated, with cutoff radius
$r_{c}/\sigma_{\mathrm{LJ}}=2.5$, and shifted. The parameters $\epsilon$,
$\sigma_{\mathrm{LJ}}$ and $m$, define the energy, length and mass units,
respectively. All the results are expressed in LJ units with time
$\sigma_{\mathrm{LJ}}(m/\epsilon)^{1/2}$, temperature $\epsilon/k_{\rm B}$ and
pressure $\epsilon/\sigma_{\mathrm{LJ}}^{3}$. In the following, we identify
$\sigma_{r}$ of Eq. (3) with $\sigma_{\mathrm{LJ}}$. We consider cubic
simulation boxes with linear sizes in the interval
$L/\sigma_{\mathrm{LJ}}=[5,50]$, with fixed density
$\rho\sigma_{\mathrm{LJ}}^{3}=0.864$. The systems are equilibrated at
temperatures in the interval $k_{\rm B}T=[0.7\epsilon,7.0\epsilon]$, enforced
with a Langevin thermostat with damping coefficient
$\gamma(\sigma(m/\epsilon)^{1/2})=1.0$. We equilibrate the samples for
$10\times 10^{6}$ molecular dynamics (MD) steps using a time step of $\delta
t/(\sigma_{\mathrm{LJ}}(m/\epsilon)^{1/2})=10^{-3}$, followed by additional
$10\times 10^{6}$ MD steps on the NVE ensemble to verify that the temperature
does not deviate substantially from the target value. Production runs span
$10\times 10^{6}$ MD steps. All the simulations have been performed with the
LAMMPS simulation package LAMMPS .

##  III Implicit and explicit finite-size effects

In this section, we identify which finite-size effects are expected to affect
the calculation of the excess entropy. We start with the definition of excess
entropy for an $N$–particle system with respect to the ideal gas:

$s_{\rm exc}=\frac{S-S_{\rm IG}}{Nk_{\rm B}}=\frac{S_{2}+S_{3}+\cdots}{Nk_{\rm
B}}\,,$ (4)

with $k_{\rm B}$ the Boltzmann constant. In the following, we focus on two-
body contributions, which mostly amount to 80–90$\%$ of the overall value of
the excess entropy for simple liquids. borzsak1992convergence ;
banerjee2014role  In particular, we have Raveche_JCP55_2242_1971 ;
Mountain_Raveche_JCP55_2250_1971

$s_{2}=-\frac{\rho}{2V}\int_{V}\int_{V}d\mathbf{r}_{1}\,d\mathbf{r}_{2}\,\left[g(\mathbf{r})\ln
g(\mathbf{r})-(g(\mathbf{r})-1)\right]\,,$ (5)

with $s_{2}=\frac{S_{2}}{Nk_{\rm B}}$ the two-body excess entropy per
particle. By taking the thermodynamic limit and assuming that the liquid is
homogeneous and isotropic, we obtain the familiar expression

$s_{2}^{\infty}=-2\pi\rho\int_{0}^{\infty}dr\,r^{2}\left[g(r)\ln
g(r)-(g(r)-1)\right]\,.$ (6)

When performing molecular dynamics simulations, we usually consider systems
with a finite number of particles, typically not large enough to reach the
thermodynamic limit. Furthermore, when evaluating the double integral in Eq.
(5) we need to consider that the volume $V$ is finite. For such a reason, and
following the strategy used to compute the compressibility equation Roman2008
; CortesHuerto_Entropy2018  and the Kirkwood-Buff integrals schnell-etal-
JPhysChemLett4-235-2013 ; CortesHuerto_Communication2016 ; Robin2018b  in
computer simulations, we define a finite–size two–body excess entropy
evaluated in a subvolume $V$ of a system with a total number of particles
$N_{0}$ and the volume $V_{0}$

$\begin{split}s_{2}(V;N_{0})=-\frac{\rho}{2V}\int_{V}\int_{V}&d\mathbf{r}_{1}\,d\mathbf{r}_{2}\,\left[g(\mathbf{r};N_{0})\ln
g(\mathbf{r};N_{0})\right.\\\
&\left.-(g(\mathbf{r};N_{0})-1)\right]\,,\end{split}$ (7)

with $g(\mathbf{r};N_{0})$ the finite-size RDF. The asymptotic correction to
the finite-size RDF, given by the difference in the thermodynamic ensemble,
gives Percus1961 ; Percus1961II ; Salacuse-etal-PRE53-2382-1996 ; Roman-etal-
JChemPhys107-4635-1997 ; Villamaina-Trizac-EurJPhys35-035011-2014 ; Roman-
etal-AmJPhys67-1149-1999

$g(\mathbf{r};N_{0})=g(\mathbf{r})-\frac{\chi_{T}^{\infty}}{N_{0}}$ (8)

with $\chi_{T}^{\infty}=\rho k_{\rm B}T\kappa_{T}$, and $\kappa_{T}$ being the
isothermal compressibility in the thermodynamic limit. We write the integrand
in Eq. (7) as

$\begin{split}&g(\mathbf{r};N_{0})\ln g(\mathbf{r};N_{0})\approx
g(\mathbf{r})\ln g(\mathbf{r})\\\
&\quad\quad\quad-\frac{\chi_{T}^{\infty}}{N_{0}}(1+\ln g(\mathbf{r}))\\\
&g(\mathbf{r};N_{0})-1=g(\mathbf{r})-1-\frac{\chi_{T}^{\infty}}{N_{0}}\,,\end{split}$
(9)

where in the first line in the previous expression, we have neglected terms of
the order $O\left(\frac{1}{N_{0}^{2}}\right)$. The two contributions
$\frac{\chi_{T}^{\infty}}{N_{0}}$ cancel out exactly. The contribution
$\frac{\chi_{T}^{\infty}}{N_{0}}\ln g(\mathbf{r})$ can be neglected by
assuming a large number of particles (there is no $V/V_{0}$ contribution, only
$1/V_{0}$, hence, we can neglect it). This indicates that the two-body excess
entropy is ensemble invariant, consistent with the result reported Ref.
Wallace1987 ; Baranyai1989 . We thus rewrite Eq. (7) as

$\begin{split}s_{2}(V)=-\frac{\rho}{2V}\int_{V}\int_{V}d\mathbf{r}_{1}\,\
d\mathbf{r}_{2}\,&\left[g(\mathbf{r})\ln g(\mathbf{r})\right.\\\
&\left.-(g(\mathbf{r})-1)\right]\,.\end{split}$ (10)

The volume $V$ is finite and embedded into the volume $V_{0}$. The integration
domains can be rearranged as
$\int_{V}\int_{V}(\cdots)=\int_{V}\int_{V_{0}}(\cdots)-\int_{V}\int_{V_{0}-V}(\cdots)$.
Using a similar argument as the one used to calculate the finite-size
compressibility Binder-etal-EPL6-585-1988  and Kirkwood-Buff integrals
schnell-etal-JPhysChemLett4-235-2013 , the term $\int_{V}\int_{V_{0}}(\cdots)$
gives $s_{2}^{\infty}$ and the term $\int_{V}\int_{V_{0}-V}(\cdots)$ scales as
$1/L$ with $L=V^{1/3}$ the linear size of the cubic simulation box. Thus,

$s_{2}(L)=s_{2}^{\infty}+\frac{\sigma}{L}\,,$ (11)

with $\sigma$ a constant that depends on intensive thermodynamic quantities
only. In the following section, we introduce a method to compute $s_{2}(L)$
and verify its scaling behaviour with the linear size of the simulation box.

Figure 2:  Plot of the two contributions, _Kirkwood-Buff_ ($g(r;N_{0})-1$) and
_Information_ ($g(r;N_{0})\ln g(r;N_{0})$), to the truncated integral
$s_{2}^{R}$ for a system of linear size $L/\sigma_{\rm LJ}=35$ at $k_{\rm
B}T=2.0\epsilon$. It is apparent that the two terms oscillate out-of-phase for
small values of $R$, and their sum converges to $s_{2}^{\infty}$ when
$R\to\infty$.

To finish this section, we compare our results with the usual truncation of
Eq. (6) up to a cutoff radius $R$, namely

$\begin{split}s_{2}^{R}=-2\pi\rho\int_{0}^{R}dr\,r^{2}&\left[g(r;N_{0})\ln
g(r;N_{0})\right.\\\ &\left.-(g(r;N_{0})-1)\right]\,.\end{split}$ (12)

We use this truncated integral to verify numerically that ensemble finite-size
contributions cancel out almost exactly. widom2019first  For a system of size
$L/\sigma_{\mathrm{LJ}}=35$ at $k_{\rm B}T=2.0\epsilon$, we separate the
$g(r;N_{0})-1$, _Kirkwood-Buff_ , and the $g(r;N_{0})\ln g(r;N_{0})$,
_Information_ , contributions and plot them as a function of the truncation
radius $R$ (See Figure 2). Both integrals diverge for large values of $R$,
Kirkwood-Buff to infinity and Information to minus infinity, which signals a
clear ensemble finite-size effect. However, these two finite-size
contributions balance each other, and the sum of the two integrals converges
to $s_{2}^{\infty}$ for $R>>1$. Due to this error cancellation, the truncation
Eq. (12) gives $s_{2}^{\infty}$ even for relatively small simulation boxes,
and its finite-size dependence has been commonly overlooked in the literature.

##  IV Finite-volume excess entropy

Based on previous work on finite-size isothermal compressibility Roman1999
and Kirkwood-Buff integrals Sevilla2022 , we define a finite-volume two-body
excess entropy as follows.

$s_{2}(V)=-\frac{\rho}{2V}\int\int
d\mathbf{r}_{1}\,d\mathbf{r}_{2}\,R(\mathbf{r}_{1})\,R(\mathbf{r}_{2})\,h(\mathbf{r})\,,$
(13)

with $R(\mathbf{r})$ a step function that defines the finite integration
subdomain, being equal to one inside and to zero outside the volume $V$
Roman1999 . The function $h(\mathbf{r})$ is defined as

$h(\mathbf{r})=g(\mathbf{r})\ln g(\mathbf{r})-(g(\mathbf{r})-1)\,.$ (14)

We write the double integral of $s_{2}(V)$ in Fourier space and include the
periodicity of the simulation of the box in $h(\mathbf{r})$ explicitly. Thus

$s_{2}(V)=-\frac{\rho}{2(2\pi)^{3}V}\int
d\mathbf{k}\,\tilde{R}(\mathbf{k})\,\tilde{R}(-\mathbf{k})\,\tilde{h}^{\rm
PBC}(\mathbf{k})\,,$ (15)

where Roman1999

$\tilde{h}^{\rm
PBC}(\mathbf{k})=\sum_{n_{x},n_{y},n_{z}}e^{-\mathbf{k}\cdot\mathbf{s}_{n_{x},n_{y},n_{z}}}\tilde{h}(\mathbf{k})\,,$
(16)

with $\tilde{h}(\mathbf{k})$ the Fourier transform of $h(\mathbf{r})$ and
$\mathbf{s}_{n_{x},n_{y},n_{z}}=(n_{x}\,L_{x},n_{y}\,L_{y},n_{z}\,L_{x})$ a
vector specifying the system’s periodic images such that $n_{x,y,z}$ takes
integer values. In the following, we consider a cubic simulation box with
$L_{x}=L_{y}=L_{z}=L$. As before Sevilla2022 , we choose $|n_{x}|\leq 1$,
$|n_{y}|\leq 1$ and $|n_{z}|\leq 1$ to compute Eq. (16). Finally, we assume a
homogeneous and isotropic fluid such that $\tilde{h}(\mathbf{k})=\tilde{h}(k)$
with $k=\sqrt{\mathbf{k}\cdot\mathbf{k}}$.

Figure 3: Running $s_{2}$ as a function of the ratio $R/L$ for the case
$L/\sigma_{\mathrm{LJ}}=5$ at $k_{\rm B}T=2.0\epsilon$. The black line
corresponds to the truncation Eq. 12, and the red and blue curves are the
result of Eq. (15) including ($|n_{x}|\leq 1$, $|n_{y}|\leq 1$ and
$|n_{z}|\leq 1$) and not including ($|n_{x}|=|n_{y}|=|n_{z}|=0$) PBC,
respectively. By including PBC, the integral Eq. (15) converges to the
thermodynamic limit.

To validate our approach, we verify that Eqs (15) and (12) converge to the
same value in the thermodynamic limit. To this aim, we consider a system with
linear size $L/\sigma_{\mathrm{LJ}}=50$ at $k_{\rm B}T=2.0\epsilon$, compute
the RDF and evaluate the truncated integral Eq. (12). According to Eq. (11),
implicit finite-size effects are the most relevant in this case. Hence, by
considering a sufficiently large simulation box, the large $R$ limit of Eq.
(12) converges to the TL value. We present this result in Fig. 3 (black solid
curve). To evaluate Eq. (15), we take the RDF from the simulation box with
linear size $L/\sigma_{\mathrm{LJ}}=20$ and perform the Fourier transform
procedure described above to obtain $\tilde{h}(\mathbf{k})$. It is apparent,
as expected, that with explicit PBC, the finite-size $s_{2}$ gives the TL
value (red dashed curve). Instead, by removing PBC, there is a significant
deviation from the TL value that we attribute to the $1/L$ dependence in Eq.
(11) (blue solid curved).

Figure 4: $s_{2}$ as a function of the inverse of the simulation box size $L$
for systems at $k_{\rm B}T=2.0\epsilon$. The black triangles are calculated
with the truncated integral (Eq. (12)), the red triangles and blue squares
were calculated with the double integral (Eq. (15)) including and excluding
PBC, respectively.

We verify this $1/L$ dependence in the finite-size $s_{2}$. In Figure 4, we
plot the result of $s_{2}^{R}$, Eq. (12), as a function of $1/L$ (black
inverted triangles). There, it is apparent that the integral converges when
the linear size of the system is $L/\sigma_{\mathrm{LJ}}>10$. The result of
using $s_{2}(V)$, Eq. (15), with explicit PBC, always converges to the TL
value (red triangles), regardless of the linear size of the system. More
interestingly, by removing PBC from Eq. (15), we observe a clear linear
dependence with $1/L$ (blue squares). Furthermore, by extrapolating this
behaviour (blue dashed line) to the axis $1/L=0$, we obtain a linear
extrapolation to $s_{2}^{\infty}$. This result completes the validation of
both, Eqs (11) and (15).

##  V Finite-size excess-entropy scaling

Figure 5: -$s_{2}$ as a function of $1/L$ for a LJ system at
$\rho\sigma_{\mathrm{LJ}}^{3}=0.864$ and different temperatures. All data
points were obtained with the RDF for the system of linear size
$L/\sigma_{\mathrm{LJ}}=20$ and using Eq. (15) without PBC.

In this section, we investigate the finite-size effects of the self-
diffusivity entropy scaling, Eq. (2). To this aim, we verify that the scaling
of $s_{2}$ with $1/L$ is valid in a wide temperature range. We present these
results for a LJ system with density $\rho\sigma_{\mathrm{LJ}}^{3}=0.864$ in
the range of temperatures $k_{\rm B}T=[0.7\epsilon,7\epsilon]$. The results in
Figure 5 indicate that the $1/L$ scaling is apparent for all temperatures
considered here.
We now collect all our data to investigate the scaling of Eq. (2) with the
simulation box size. The result is presented in Figure 6 where the diffusion
constant $D^{*}$ is plotted against $-s_{2}$. A clear trend with system size
emerges, indicating that Eq. (2) remains valid even for the smallest
simulation boxes considered and showing that the parameters $A$ and $\alpha$
are also size dependent. By extrapolating $D^{*}$ and $-s_{2}$ to the limit
$1/L\to 0$, we obtain the TL values given by the black empty triangles that
well agree with the reference scaling provided by Eq. (2) (black dashed line).
Indeed, we report $A^{\infty}=0.048\pm 0.001$ and $\alpha^{\infty}=1.000\pm
0.013$ in the TL, in good agreement with the value originally estimated in
Ref. Dzugutov1996 .

Figure 6: Reduced self-diffusion coefficient $D^{*}$ as a function of $-s_{2}$
for different system sizes and temperatures. The empty black triangles show
the thermodynamic limit values for $-s_{2}$ and $D^{*}$.

Finally, we investigate the relation between the coefficients $\delta$ and
$\sigma$ of the finite-size scaling of $D^{*}$ and $s^{2}$, respectively. In
Figure 7, we plot $\sigma$ as a function of $\delta$ and observe a power law
relation of the form $\sigma=a\delta^{b}$ with $a=1.256\pm 0.118$ and
$b=-0.513\pm 0.020$.

Figure 7: Coefficients $\sigma(T)$ as a function of $\delta(T)$ for all the
temperatures considered here. We used a power law $\sigma=a\delta^{b}$ to fit
the data.

##  VI Summary and outlook

We define a finite-size two-body excess entropy $s_{2}(L)$ integral equation
with $L$ the linear size of the simulation box. Using analytical arguments and
simulations of a prototypical Lennard-Jones liquid at different densities and
temperatures, we show that $s_{2}(L)=s_{2}^{\infty}+\sigma/L$ with $\sigma$ a
constant that depends on intensive thermodynamic quantities. Given the well-
know finite-size scaling of the self-diffusivity,
$D^{*}(L)=D^{*\infty}-\delta/L$, we show that the universal scaling relation
between entropy and diffusion $D^{*}=A\exp{(\alpha s_{2})}$ also exhibits a
finite-size dependence and, by extrapolating to the TL, report $A=0.048\pm
0.001$ and $\alpha=1.000\pm 0.013$, in good agreement with values reported in
the literature. Finally, and perhaps more interestingly, we show that the
scaling coefficients $\sigma$ and $\delta$ of $s_{2}$ and $D^{*}$,
respectively, are related by a somewhat simple power law $\sigma=a\delta^{b}$
with $a=1.256\pm 0.118$ and $b=-0.513\pm 0.020$.
The finite-size scaling of $s_{2}$ can be rationalised in terms of the
thermodynamics of small systems hill ; Puglisi_etal_Entropy2018 . In
particular, the statistical mechanics of a few model small systems in
confinement has been derived recently Braten_etal_JCP2021 . The authors have
shown that given the high surface area–to–volume ratio of small systems,
thermodynamic properties include surface contributions. In the case of
entropy, these contributions include $1/L$ terms with $L$, the linear size of
the system. In this context, we feel that the finite-size entropy scaling
investigated here might play a role in understanding the non-equilibrium
thermodynamics of confined, small systems nanothermobook .
The power law relation between the scaling coefficients of self-diffusion and
two-body excess entropy is somewhat intriguing. On the one hand, the size
scaling in the self-diffusion appears as a consequence of the conservation of
linear momentum Hummer2004 . On the other hand, the finite-size scaling in the
two-body entropy results from a surface contribution due to the confinement of
the system Braten_etal_JCP2021 . Admittedly, we do not have a satisfactory
explanation for this connection.
Nevertheless, we point out that the ratio $\delta^{b}/\sigma=1/a$ might be
related to a constant viscosity/entropy ratio. Indeed, $\delta$ is inversely
proportional to the system’s viscosity, and a simple dimensional analysis
tells us that $\sigma$ has units of entropy times length. Interestingly,
string theory methods have been used to conjecture that, for fluids in
equilibrium, the viscosity to entropy density ratio has a lower bound at
$\hbar/4\pi k_{\rm B}$ Kovtun_etal_PRL2005  with $\hbar$ the reduced Planck
constant. This relation, tested for various fluid systems
Angiella_etal_PhysLettA2009 ; Faussurier_etal_HEDP2014 ; Hohm_ChemPhys2014 ,
has been originally derived by considering that the entropy density of a black
hole is proportional to the surface to volume ratio of its event horizon, i.e.
a $1/L$ contribution. We find this connection fascinating, and, in our
opinion, it deserves further investigation.

###### Acknowledgements.

We are grateful to Kurt Kremer for his insightful discussions. We also thank
Denis Andrienko for his critical reading of the manuscript. R.C.-H. gratefully
acknowledges funding from SFB-TRR146 of the German Research Foundation (DFG).
Simulations have been performed on the THINC cluster at the Max Planck
Institute for Polymer Research and the COBRA cluster at the Max Planck
Computing and Data Facility.

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