# A characterization of Calderón–Zygmund operators on RBMO

Evgueni Doubtsov  St. Petersburg Department of Steklov Mathematical Institute,
Fontanka 27, St. Petersburg 191023, Russia<EMAIL_ADDRESS>and  Andrei V.
Vasin  St. Petersburg Department of Steklov Mathematical Institute, Fontanka
27, St. Petersburg 191023, Russia and Admiral Makarov State University of
Maritime and Inland Shipping, Dvinskaya st. 5/7, St. Petersburg 198035, Russia
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathrm{RBMO}(\mu)=\mathrm{RBMO}(\mathbb{R}^{m},\mu)$ denote the regular
BMO space introduced by X. Tolsa for an $n$-dimensional finite positive
measure on $\mathbb{R}^{m}$, $0<n\leq m$. We characterize the bounded
Calderón–Zygmund operators $T:\mathrm{RBMO}(\mu)\to\mathrm{RBMO}(\mu)$ in
terms of the function $T1$.

###### Key words and phrases:

T1 theorem, non-doubling measure, regular BMO space

This research was supported by the Russian Science Foundation (grant No.
23-11-00171), https://rscf.ru/project/23-11-00171/.

##  1\. Introduction

The main results of classical harmonic analysis are related to homogeneous
spaces $(\mathbb{R}^{m},\mu)$; see, for example, monograph [10]. By
definition, this means that $\mu$ has the standard doubling property
$\mu(2Q)\leq C\mu(Q)$ for an arbitrary cube $Q\subset\mathbb{R}^{m}$. The
principal motivation of the present work is to consider the Calderón–Zygmund
operators in a non-classical setting, that is, on non-homogeneous spaces.
Indeed, the main object in this work is the functional space
$\mathrm{RBMO}(\mathbb{R}^{m},\mu)$ introduced by X. Tolsa [11] for an
$n$-dimensional measure $\mu$. Thus, the measure $\mu$ may not have the
standard doubling property. Other generalizations of the space $\mathrm{BMO}$
were also considered for such measures; see, for example, [7, 8]. However,
$\mathrm{RBMO}(\mathbb{R}^{m},\mu)$ has various genuine properties of the
classical space $\mathrm{BMO}$, for example, the John–Nirenberg inequality
holds for $\mathrm{RBMO}(\mathbb{R}^{m},\mu)$.

Let $T$ be a Calderón–Zygmund operator such that its kernel satisfies the
appropriate cancellation property (see (1.4)). In [2], the authors obtained a
condition on the function $T1$, which guarantees that the operator $T$ is
bounded on $\mathrm{RBMO}(\mu)=\mathrm{RBMO}(\mathbb{R}^{m},\mu)$. In this
work, we prove that the condition from [2], after a slight modification, is
not only sufficient, but also necessary for the boundedness of $T$. In other
words, in the present paper, we obtain a so-called $T1$ theorem for the
operator $T$ on $\mathrm{RBMO}(\mu)$, that is, we give a characterization of
the bounded Calderón–Zygmund operators
$T:\mathrm{RBMO}(\mu)\to\mathrm{RBMO}(\mu)$ in terms of the function $T1$.

We introduce certain definitions and facts from [11] and [2].

###  1.1. Cubes and $n$-dimensional measures

Unless otherwise stated, everywhere below $\mu$ is a finite positive measure
defined on $\mathbb{R}^{m}$. By definition, cube is a closed cube in
$\mathbb{R}^{m}$, its sides are parallel to the coordinate axes and its center
belongs to the support of $\mu$. The side length of $Q$ is denoted by
$\ell=\ell(Q)$.

As in [11], we always assume that $\mu$ is an $n$-dimensional measure on
$\mathbb{R}^{m}$, where $0<n\leq m$. By definition, it means that

(1.1) $\mu(Q)\leq C\ell^{n}(Q)\ \textrm{for every cube\
}Q\subset\mathbb{R}^{m},\ \ell(Q)>0,$

with a constant $C>0$.

###  1.2. Calderón–Zygmund operators.

A Calderón–Zygmund kernel associated with an $n$-dimensional measure $\mu$ on
$\mathbb{R}^{m}$ is a measurable function $\mathcal{K}(x,y)$ defined on
$\mathbb{R}^{m}\times\mathbb{R}^{m}\setminus\\{(x,x):x\in\mathbb{R}^{m}\\}$
and satisfying the following conditions:

(1.2) $|\mathcal{K}(x,y)|\leq Cd^{-n}(x,y),$ (1.3)
$\displaystyle|\mathcal{K}(x_{1},y)-\mathcal{K}(x_{2},y)|$
$\displaystyle+|\mathcal{K}(y,x_{1})-\mathcal{K}(y,x_{2})|$ $\displaystyle\leq
C\frac{d^{\delta}(x_{1},x_{2})}{d^{n+\delta}(x_{1},y)},\quad\textrm{whenever}\
\ 2d(x_{1},x_{2})\leq d(x_{1},y),$

where $C>0$ is a universal constant and $\delta$, $0<\delta\leq 1$, is a
regularity constant. Restrictions (1.2) and (1.3) are standard by now. Also,
remark that property (1.3) is used in the celebrated $T1$ theorem of David and
Journé [1].

Also, in the main theorem of the present paper, we use the following more
specific cancellation property:

(1.4) $\left|\int_{Q(x,R)\setminus Q(x,r)}\mathcal{K}(x,y)\,d\mu(y)\right|\leq
C,\quad 0<r<R,$

where $C>0$ is a universal constant. This type of restriction is sufficiently
common (see, for example, [10, Ch. 7, Exercise. 5.17] in the classical
setting, see also [3] in the case of nonhomogeneous spaces).

The Calderón–Zygmund operator associated to the kernel $\mathcal{K}(x,y)$ and
the measure $\mu$ is defined as

$Tf(x)=\int_{\mathbb{R}^{m}}\mathcal{K}(x,y)f(y)\,d\mu(y)$

for $x\notin\textrm{supp}(f\mu)$. In the general situation, one introduces the
truncated operators $T_{\varepsilon}$, $\varepsilon>0$:

$T_{\varepsilon}f(x)=\int_{\mathbb{R}^{m}\setminus
Q(x,\varepsilon)}\mathcal{K}(x,y)f(y)\,d\mu(y).$

We say that $T$ is bounded on $L^{p}(\mu)$ provided that the operators
$T_{\varepsilon}$ are bounded on $L^{p}(\mu)$ uniformly in $\varepsilon>0$.

###  1.3. Regular BMO space

To define $\mathrm{RBMO}(\mu)$, we need coefficients $K(Q,R)$ and doubling
cubes.

####  1.3.1. Coefficients $K(Q,R)$

For embedded cubes $Q\subset R$ in $\mathbb{R}^{m}$, put

$K(Q,R)=1+\sum_{j=1}^{N_{Q,R}}\frac{\mu(2^{j}Q)}{\ell^{n}(2^{j}Q)},$

where $N_{Q,R}$ is the minimal integer $k$ such that
$\ell(2^{k}Q)\geq\ell(R)$. Clearly, $K(Q,R)\geq 1$. Also, remark that $K(Q,R)$
is estimated by $C\log(\ell(R)/\ell(Q))$, since $\mu$ is $n$-dimensional.

####  1.3.2. Doubling cubes

Let $\alpha>1$, $\beta>\alpha^{n}$. A cube $Q$ is called
$(\alpha,\beta)$-doubling if

$\mu(\alpha Q)<\beta\mu(Q).$

Let $\mu$ be a Radon measure defined on $\mathbb{R}^{m}$ and $\alpha>1$. As
indicated in [11], it is known that, for a sufficiently large parameter
$\beta=\beta(\alpha,n)$, for $\mu$-almost all $x\in\mathbb{R}^{m}$ there
exists a sequence of $(\alpha,\beta)$-doubling cubes
$\\{Q_{k}\\}_{k=1}^{\infty}$ centered at $x$ and such that $\ell(Q_{k})\to 0$
as $k\to\infty$. For definiteness, let $\beta_{n}$ denote the double infimum
of the corresponding constants $\beta(10,n)$.

######  Definition 1.

A cube $Q\subset\mathbb{R}^{m}$ is called doubling (we write
$Q\in\mathcal{D}$) if $Q$ is $(10,\beta_{n})$-doubling.

######  Remark 1.

The analogous definition in [2] is based on $(4,\beta_{n})$-doubling cubes in
the place of $(10,\beta_{n})$-doubling ones. Also, remark that Tolsa [11] uses
$(2,\beta)$-doubling cubes in his original definition. However, further work
showed that in Tolsa theory in $\mathbb{R}^{m}$ one can use
$(\alpha,\beta)$-doubling cubes with $\alpha>1$ (see, for example, [5, 6]).

####  1.3.3. Definition of the space $\mathrm{RBMO}(\mu)$

The definition of the space $\mathrm{RBMO}(\mu)$ given below is used as the
main one in [2]. Remark that Tolsa [11] used a different equivalent definition
as his principal one.

######  Definition 2.

The space $\mathrm{RBMO}(\mu)$ consists of $f\in L^{1}_{loc}(\mu)$ such that
there exists a constant $C_{\mathfrak{E}}>0$ and a collection of constants
$\\{f_{Q}\\}$ (one constant for every doubling cube $Q\subset\mathbb{R}^{m}$)
such that

(1.5) $\displaystyle\frac{1}{\mu(Q)}\int_{Q}|f-f_{Q}|\,d\mu$
$\displaystyle\leq C_{\mathfrak{E}},$ (1.6) $\displaystyle|f_{Q}-f_{R}|$
$\displaystyle\leq C_{\mathfrak{E}}K(Q,R)$

for all cubes $Q,R\in\mathcal{D}$, $Q\subset R$. Let $\|f\|_{\mathfrak{E}}$
denote the infimum of the corresponding constants $C_{\mathfrak{E}}>0$.

It is not difficult to verify that $\|f\|_{\mathfrak{E}}$ is a norm on the
space $\mathrm{RBMO}(\mu)$ modulo constants. Next, $\mu$ is a finite
$n$-dimensional measure, hence, $\mathrm{RBMO}(\mu)\subset L^{1}(\mu)$ by
standard arguments. Direct inspection shows that $\mathrm{RBMO}(\mu)$ is a
Banach space with respect to the following norm:

$\|f\|_{*}:=\|f\|_{\mathfrak{E}}+\|f\|_{L^{1}(\mu)}.$

###  1.4. Main theorem

Recall that $\mu(\mathbb{R}^{m})<\infty$ by assumption. For a cube
$Q\subset\mathbb{R}^{m}$, put

$K(Q)=K(Q,2^{k}Q),$

where $k$ is the minimal positive integer such that
$\mu(2^{k}Q)>\frac{1}{2}\mu(\mathbb{R}^{m})$.

######  Theorem 1.

Let $\mu$ be a finite positive $n$-dimensional measure defined on
$\mathbb{R}^{m}$, $0<n\leq m$. Let $T$ be a Calderón–Zygmund operator such
that the cancellation property (1.4) holds. Then the following properties are
equivalent:

  * (i)

The operator $T$ is bounded on the Banach space $\mathrm{RBMO}(\mu)$.

  * (ii)

For every cube $Q\in\mathcal{D}$, there exists a constant $b_{Q}$ such that

(1.7) $\displaystyle\frac{1}{\mu(Q)}\int_{Q}|T1-b_{Q}|\,d\mu$
$\displaystyle\leq\frac{C}{K(Q)}\quad\textrm{for all cubes}\ Q\in\mathcal{D},$
(1.8) $\displaystyle|b_{Q}-b_{R}|$ $\displaystyle\leq
C\frac{K(Q,R)}{K(Q)}\quad\textrm{for all cubes}\ Q,R\in\mathcal{D},\ Q\subset
R,$

where the constant $C>0$ does not depend on $Q$ and $R$.

######  Remark 2.

By definition, $T$ is bounded on $RBMO(\mu)$ provided that the truncated
operators $T_{\varepsilon}$, $\varepsilon>0$, are uniformly bounded on the
space $RBMO(\mu)$. Analogously, by definition, inequalities (1.7) and (1.8)
mean that these estimates hold for the operators $T_{\varepsilon}$ uniformly
in $\varepsilon>0$.

######  Remark 3.

Firstly, under hypothesis of Theorem 1, one implicitly assumes that $T1\in
L^{\infty}(\mu)$. Indeed, this property follows from (1.4) by [2, Lemma 2.6].
Secondly, by Lemma 2.8 from [2] condition (1.4) implies that $T$ is a bounded
operator on $L^{2}(\mu)$.

###  1.5. Organization of the paper

Certain auxiliary results are collected in Section 2. In Section 3, we
construct test functions. These functions are used in the final Section 4 to
prove the implication (i)$\Rightarrow$(ii) in the main theorem.

###  1.6. Notation

The symbol $C$ is used to denote a positive constant whose value may vary from
line to line. If $A,B>0$ and $A\leq CB$ for a constant $C>0$, then we write
$A\lesssim B$. Notation $A\approx B$ means that $A\lesssim B$ and $B\lesssim
A$.

##  2\. Auxiliary results

###  2.1. An equivalent norm on $\mathrm{RBMO}(\mu)$

######  Definition 3.

Let $f\in L^{1}_{loc}(\mu)$. Fix a constant $\rho>1$. Let
$\|f\|_{\mathfrak{A},\rho}$ denote the infimum of the constants
$C_{\mathfrak{A}}=C_{\mathfrak{A},\rho}>0$ with the following properties: for
every cube $Q$, there exists a constant $f_{Q}\in\mathbb{R}$ such that

(2.1) $\displaystyle\sup_{Q}\frac{1}{\mu(\rho
Q)}\int_{Q}|f(x)-f_{Q}|\,d\mu(x)$ $\displaystyle\leq C_{\mathfrak{A}},$ (2.2)
$\displaystyle|f_{Q}-f_{R}|$ $\displaystyle\leq
C_{\mathfrak{A}}K(Q,R)\quad\text{for all cubes\ }Q\subset R.$

Direct inspection shows that $\|\cdot\|_{\mathfrak{A}}$ is a seminorm on
$\mathrm{RBMO}(\mu)$.

######  Remark 4.

The original seminorm introduced by Tolsa [11] on $\mathrm{RBMO}(\mu)$ differs
from the seminorms given in Definitions 2 and 3. However, it is known that all
these seminorms are equivalent (see [11] and [2, Section 2]).

###  2.2. Decomposition of a function from $\mathrm{RBMO}(\mu)$ into three
terms

Let $T$ denote a Carderón–Zygmund operator bounded on $L^{2}(\mu)$. Assume
that $f\in\mathrm{RBMO}(\mu)$ and $\rho=2$. Applying Definition 3, for every
cube $2Q\subset\mathbb{R}^{m}$, we select a constant $f_{2Q}$ such that
properties (2.1) and (2.2) hold for the constant
$C_{\mathfrak{A},\rho}=2\|f\|_{\mathfrak{A},\rho}$ and for $2Q$ in the place
of $Q$, that is,

(2.3) $\displaystyle\sup_{2Q}\frac{1}{\mu(2\rho Q)}\int_{2Q}$
$\displaystyle|f(x)-f_{2Q}|\,d\mu(x)\leq 2\|f\|_{\mathfrak{A},\rho},$ (2.4)
$\displaystyle|f_{2Q}-f_{2R}|$ $\displaystyle\leq
2\|f\|_{\mathfrak{A},\rho}K(2Q,2R)\quad\text{for all cubes\ }Q\subset R.$

Introduce the following functions:

$\displaystyle f_{1}=f_{1,Q}$ $\displaystyle=f_{2Q},$ $\displaystyle
f_{2}=f_{2,Q}$ $\displaystyle=(f-f_{2Q})\chi_{2Q},$ $\displaystyle
f_{3}=f_{3,Q}$ $\displaystyle=(f-f_{2Q})\chi_{\mathbb{R}^{m}\setminus 2Q}.$

Observe that $f=f_{1}+f_{2}+f_{3}.$ Similar decompositions are well known
(see, for example, [4]).

Put $b_{2,Q}=0$ and

$b_{3,Q}=\frac{1}{\mu(Q)}\int_{Q}Tf_{3,Q}(y)\,d\mu(y).$

We will need two lemmas from [2] about properties of the above functions and
constants.

######  Lemma 1.

There exists a constant $C>0$ such that

$\frac{1}{\mu(Q)}\int_{Q}|Tf_{k}-b_{k,Q}|\,d\mu\leq C\|f\|,\quad k=2,3,$

for any cube $Q\in\mathcal{D}$.

######  Lemma 2.

There exists a constant $C>0$ such that

$|b_{k,Q}-b_{k,R}|\leq C\|f\|K(Q,R),\quad k=2,3.$

for any cubes $Q\subset R$.

##  3\. Test functions

In this section, we construct test functions used to prove the implication
(i)$\Rightarrow$(ii). We introduce the following notation:

$|x|=\max\\{|x_{j}|:j=1,2,\dots,m\\},\quad x\in\mathbb{R}^{m}.$

Put

(3.1) $\varphi(y)=\varphi(|y|)=1+\int_{|y|<|t|}\frac{d\mu(t)}{|t|^{n}},\quad
y\in\mathbb{R}^{m}\setminus\\{0\\}.$

The measure $\mu$ is $n$-dimensional and $\mu(\mathbb{R}^{m})<\infty$, thus,
the above integral converges. The function $\varphi$ is continuous and
$\ell^{\infty}$-radial on its domain.

Let $Q(r)$, $r>0$, denote the cube centered at the origin and with side length
$2r$.

######  Remark 5.

Using the dyadic decomposition for the integral defining $\varphi(y)$, we
conclude that

$\varphi(y)\approx K(Q(|y|))$

for $y\in\mathbb{R}^{d}\setminus\\{0\\}$. See [11].

The following result is motivated by Lemma 1.1 from [9], where $\mu$ is
Lebesgue measure.

######  Lemma 3.

Let $\varphi$ be defined by (3.1), $\alpha\geq 5$ and $\beta>\alpha^{n}$. Then
$\varphi\in\mathrm{RBMO}(\mu)$. There exist constants $C$ and
$C(\alpha,\beta)>0$ such that

(3.2) $|\langle\varphi_{S}\rangle|\geq CK(S)-C(\alpha,\beta)$

for every $(\alpha,\beta)$-doubling cube $S$ centered at the origin.

######  Remark 6.

We apply Lemma 3 with a sufficiently large $K(S)$, therefore,
$CK(S)-C(\alpha,\beta)$ in estimate (3.2) becomes positive in this case.

###### Proof of Lemma 3.

The argument is based on the following plan: on the first two steps, for an
arbitrary cube $Q$, we verify conditions (2.1) and (2.2) from Definition 3
with $\rho=5$. The final third step is devoted to proving estimate (3.2) for
an $(\alpha,\beta)$-doubling cube $S$ centered at the origin.

Step I. On this step, we verify the oscillation condition (2.1) from
Definition 3 with $\rho=5$. To do this, fix a cube $Q$ with center $x_{Q}$ and
with side length $\ell_{Q}$. Let $\widetilde{Q}$ denote the smallest cube with
center at the origin and containing the cube $Q$. Choose $y_{Q}\in\partial Q$
such that $|y_{Q}|=\ell_{Q}/2+|x_{Q}|$. Put $\varphi_{Q}=\varphi(y_{Q})$. We
have

$\begin{split}I&=\int_{Q}|\varphi(z)-\varphi(y_{Q})|d\mu(z)=\int_{Q}\left|\int_{|t|>|z|}\frac{d\mu(t)}{|t|^{n}}-\int_{|t|>|y_{Q}|}\frac{d\mu(t)}{|t|^{n}}\right|d\mu(z)\\\
&=\int_{Q}d\mu(z)\int_{|z|<|t|<|y_{Q}|}\frac{d\mu(t)}{|t|^{n}}\\\
&\leq\int_{\widetilde{Q}}d\mu(z)\int_{|z|<|t|<|y_{Q}|}\frac{d\mu(t)}{|t|^{n}}.\end{split}$

Now, consider two cases depending on the fulfillment of the inequality

$\ell_{Q}>|x_{Q}|.$

If $\ell_{Q}>|x_{Q}|$, then the inequalities
$|y_{Q}|=|x_{Q}|+\ell_{Q}/2<3\ell_{Q}/2$ and $|y_{Q}|+|x_{Q}|<5\ell_{Q}/2$
imply that $\widetilde{Q}=Q(|y_{Q}|)$. Therefore,
$Q\subset\widetilde{Q}\subset 5Q$. Hence,

$I\leq\int_{\widetilde{Q}}d\mu(z)\int_{|z|<|t|<|y_{Q}|}\frac{d\mu(t)}{|t|^{n}}=\int_{|z|<|y_{Q}|}d\mu(z)\int_{|z|<|t|<|y_{Q}|}\frac{d\mu(t)}{|t|^{n}}.$

Changing the order of integration, we continue as

$I\leq\int_{|t|<|y_{Q}|}\frac{d\mu(t)}{|t|^{n}}\int_{|z|<|t|}d\mu(z)\leq\int_{|t|<|y_{Q}|}\frac{\mu(Q(|t|))}{|t|^{n}}d\mu(t).$

Since $\mu$ is $n$-dimensional, we have $\mu(Q(|t|))\leq C|t|^{n}$, therefore,

$I\lesssim\int_{|t|>|y_{Q}|}d\mu(t)=\mu(\widetilde{Q})\leq\mu(5Q),$

as required. The analysis of the first case is completed.

If $\ell_{Q}\leq|x_{Q}|$, then $|z|\geq|x_{Q}|/2$ for $z\in Q$. Also, we have
$|y_{Q}|\leq 3|x_{Q}|/2$. Hence, for the inner integral, we obtain

$\begin{split}\int_{|z|<|t|<|y_{Q}|}\frac{d\mu(t)}{|t|^{n}}&\leq\int_{|x_{Q}|/2<|t|<3|x_{Q}|/2}\frac{d\mu(t)}{|t|^{n}}\\\
&\leq\frac{2^{n}}{|x_{Q}|^{n}}\int_{|t|<3|x_{Q}|/2}d\mu(t)\\\
&=\frac{2^{n}}{|x_{Q}|^{n}}\mu(Q(3|x_{Q}|/2)\leq\frac{3^{n}}{|\ell_{Q}|^{n}}|x_{Q}|^{n}\leq
C,\end{split}$

since $\mu$ is $n$-dimensional.

Thus, for the entire integral $I$, we have

$I\lesssim\int_{Q}d\mu(z)=\mu(Q),$

which completes the analysis of the second case. Therefore, the oscillation
condition from Definition 3 is verified with parameter $\rho=5$.

Step II. On this step, we verify $K$-condition from Definition 3 with
$\rho=5$. Recall that $\varphi_{Q}=\varphi(y_{Q})$ for $y_{Q}\in\partial Q$
such that $|y_{Q}|=|x_{Q}|+\ell_{Q}/2$.

Consider cubes $Q\subset R$. Let $k$ be the minimal integer such that
$Q_{0}:=2^{k}Q\supset R$. Let $\widetilde{Q}$, $\widetilde{Q}_{0}$ and
$\widetilde{R}$ denote the smallest cubes centered at zero and containing $Q$,
$Q_{0}$ and $R$, respectively.

We have

$|\varphi_{Q}-\varphi_{R}|\leq|\varphi_{Q}-\varphi_{Q_{0}}|+|\varphi_{Q_{0}}-\varphi_{R}|=A+B.$

The cubes $Q_{0}$ and $Q$ are concentric, thus,
$|y_{Q_{0}}|=|x_{Q}|+2^{k-1}\ell_{Q}$. Hence,

$A=\left|\int_{|t|>|y_{Q}|}\frac{d\mu(t)}{|t|}-\int_{|t|>|y_{Q_{0}}|}\frac{d\mu(t)}{|t|^{n}}\right|=\int_{|y_{Q}|<|t|<|y_{Q_{0}}|}\frac{d\mu(t)}{|t|^{n}}\approx
K(\widetilde{Q},\widetilde{Q}_{0})$

by the dyadic decomposition of the corresponding integral (cf. [11]).

Now, consider two cases depending on the fulfillment of the condition

$2|y_{Q}|\geq|y_{Q_{0}}|.$

If $2|y_{Q}|\geq|y_{Q_{0}}|$, then the cubes $\widetilde{Q}$ and
$\widetilde{Q}_{0}$ are comparable. Therefore,

$K(\widetilde{Q},\widetilde{Q}_{0})\leq C.$

If $2|y_{Q}|<|y_{Q_{0}}|$, then $|x_{Q}|<\ell_{Q_{0}}/2$. Hence, $0\in Q_{0}$.
Therefore, $Q_{0}\subset\widetilde{Q}_{0}\subset 3Q_{0}$ and we have

$\begin{split}K(\widetilde{Q},\widetilde{Q}_{0})&\leq
K(\widetilde{Q},3Q_{0})\\\ &\leq K(Q,3Q_{0})=K(Q,Q_{0})+K(Q_{0},3Q_{0})\leq
K(Q,Q_{0})+C\approx K(Q,R),\end{split}$

since the cubes $Q_{0}$ and $R$ are comparable.

Next, we have

$B=|\varphi_{Q_{0}}-\varphi_{R}|=\left|\int_{|t|>|y_{Q_{0}}|}\frac{d\mu(t)}{|t|^{n}}-\int_{|t|>|y_{R}|}\frac{d\mu(t)}{|t|^{n}}\right|\approx
K(\widetilde{R},\widetilde{Q}_{0}).$

Consider two cases depending on the fulfillment of the condition

$2|y_{R}|\geq|y_{Q_{0}}|.$

If $2|y_{R}|\geq|y_{Q_{0}}|$, then the cubes $\widetilde{R}$ and
$\widetilde{Q}_{0}$ are comparable. Hence,

$K(\widetilde{R},\widetilde{Q}_{0})\leq C.$

Suppose that $2|y_{R}|<|y_{Q_{0}}|$. Since $R\supset Q$ and the cubes $Q$ and
$Q_{0}$ are concentric, we have

$|x_{Q}|<|y_{R}|<|y_{Q_{0}}|/2=1/2(|x_{Q}|+\ell_{Q_{0}}/2).$

Therefore, $|x_{Q}|<\ell_{Q_{0}}/2$ and $0\in Q_{0}$. Thus,
$Q_{0}\subset\widetilde{Q}_{0}\subset 3Q_{0}$ and we obtain

$B\leq K(\widetilde{R},\widetilde{Q}_{0})\leq
K(R,3Q_{0})=K(R,Q_{0})+K(Q_{0},3Q_{0})\leq C,$

since the cubes $Q_{0}$ and $R$ are comparable.

Step III. On this step, we assume that $S$ is a centered at the origin
$(\alpha,\beta)$-doubling cube, $\alpha\geq 5$, that is,
$\mu(5S)\leq\beta\mu(S)$. We have

$|\langle\varphi\rangle_{S}|-|\varphi_{S}|\leq\frac{1}{\mu(S)}\int_{S}|\varphi-\varphi_{S}|\,d\mu\leq\frac{\mu(5S)}{\mu(5S)\mu(S)}\int_{S}|\varphi-\varphi_{S}|\,d\mu\leq\beta
C$

by the oscillation property proven on Step I.

Recall that $\varphi_{S}=\varphi(y_{S})$. The cube $S$ is centered at the
origin, thus, $y_{S}=\ell_{S}/2$. Hence, by Remark 5, we obtain

$\varphi(y_{S})=\int_{|y_{S}|<|t|}\frac{d\mu(t)}{|t|^{n}}\approx K(S).$

Therefore,

$|\varphi_{S}|\geq|\langle\varphi\rangle_{S}|-C\beta\geq CK(S)-C\beta,$

which completes the proof of the lemma. ∎

######  Lemma 4.

For a sufficiently small constant $c>0$, let $\varphi_{x}=c\varphi(\cdot-x)$,
$x\in\mathbb{R}^{m}$, $\alpha\geq 5$ and $\beta>\alpha^{n}$. Then
$\|\varphi_{x}\|_{*}\leq 1$. There exist constants $C$ and $C(\alpha,\beta)>0$
such that

$|\langle\varphi_{x}\rangle_{Q}|\geq CK(Q)-C(\alpha,\beta)$

uniformly with respect to $x\in\mathbb{R}^{m}$ for every
$(\alpha,\beta)$-doubling cube $Q$ centered at $x$.

###### Proof.

Repeating the arguments from the proof of Lemma 3 for an arbitrary point
$x\in\mathbb{R}^{m}$, we see that the constants at all steps can be chosen
independently of $x$. This observation provides the required estimate for
$|\langle\varphi_{x}\rangle_{Q}|$ as well as a bound for the oscillation and
$K$-condition. To estimate $\|\varphi_{x}\|_{*}$, it remains to obtain a
uniform bound for $\|\varphi_{x}\|_{L^{1}(\mu)}$.

Applying Fubini’s theorem, we obtain

(3.3)
$\begin{split}\left\|\varphi_{x}(y)-1\right\|_{L^{1}(\mu)}=\int_{\mathbb{R}^{m}}\int_{|y-x|<|t|}\frac{d\mu(t)}{|t|^{n}}d\mu(y)&=\int_{\mathbb{R}^{m}}\int_{|y-x|<|t|}\frac{d\mu(y)}{|t|^{n}}d\mu(t)\\\
&\leq\int_{\mathbb{R}^{m}}Cd\mu(t)\leq C,\end{split}$

since $\mu$ is an $n$-dimensional measure and $\mu(\mathbb{R}^{m})=1$. Observe
that the above argument does not depend on $x\in\mathbb{R}^{m}$. Thus,
multiplying $\varphi$ by a sufficiently small constant, we guarantee that
$\|\varphi_{x}\|_{*}\leq 1$. The proof of the lemma is complete. ∎

######  Lemma 5.

There exists a family $\\{\varphi_{x}\\}_{x\in\mathbb{R}^{m}}$ of test
functions such that $\|\varphi_{x}\|_{*}\leq 1$ and

$|\langle\varphi_{x}\rangle_{2Q}|\geq CK(2Q)-C$

for every cube $Q\in\mathcal{D}$ centered at $x$.

###### Proof.

By definition, property $Q\in\mathcal{D}$ means that $Q$ is a
$(10,\beta_{n})$-doubling cube. Therefore, applying Lemma 4 to $2Q$, a
$(5,\beta_{n})$-doubling cube, completes the proof. Indeed, the constant
$\beta_{n}$ depends on $\mu$ only. ∎

##  4\. Proof of Theorem 1

(ii)$\Rightarrow$(i) This implication is proven in [2] up to the following
details: the arguments in [2] are given for other doubling cubes, namely, by
definition, these are $(4,\beta_{n})$-doubling cubes. In fact, the proof
remains the same for $(\alpha,\beta_{n})$-doubling cubes, $\alpha\geq 4$.
Thus, to prove the required implication, it is enough to repeat the argument
from [2] for $\alpha=10$, using the definition of the doubling cubes from the
present work.

(i)$\Rightarrow$(ii) For two doubling cubes $Q\subset R\subset\mathbb{R}^{m}$,
we have to verify that properties (1.7) and (1.8) hold with appropriate
constants $b_{Q},b_{R}$. Without loss of generality, we assume that $K(Q)$ is
sufficiently large.

Let $x$ denote the center of $Q$ and let $\varphi=\varphi_{x}$ denote the test
function provided by Lemma 5. By property (i), we have $\|T\varphi\|_{*}\leq
C\|\varphi\|_{*}$, in particular,

$\|T\varphi\|_{\mathfrak{}}\leq C\|\varphi\|_{*}\leq C$

Therefore, there exist constants $\beta_{Q}$ and $\beta_{R}$ such that

(4.1) $\displaystyle\frac{1}{\mu(Q)}\int_{Q}|T\varphi-\beta_{Q}|\,d\mu$
$\displaystyle\leq C,$ (4.2) $\displaystyle|\beta_{Q}-\beta_{R}|$
$\displaystyle\leq CK(Q,R),$

where the constant $C>0$ does not depend on $Q$ and $R$.

Applying the construction from Section 2.2, we obtain a decomposition

(4.3) $\varphi=\varphi_{2Q}+\varphi_{2}+\varphi_{3}$

and constants $b_{2,Q}$ and $b_{3,Q}$. Observe that, generally speaking,
$\varphi_{2Q}$ differs from the analogous constant constructed for the cube
$2Q$ in the proof of Lemma 3. By (2.3) with $\varphi$ in the place of $f$, we
have

$\frac{1}{\mu(4Q)}\int_{2Q}|\varphi-\varphi_{2Q}|\,d\mu\leq
2\|\varphi\|_{\mathfrak{A},2}.$

The cube $Q$ is doubling, thus,
$\mu(4Q)\leq\mu(10Q)\leq\beta_{n}\mu(Q)\leq\beta_{n}\mu(2Q)$, hence,

$|\langle\varphi\rangle_{2Q}|-|\varphi_{2Q}|\leq\frac{\mu(4Q)}{\mu(2Q)\mu(4Q)}\int_{2Q}|\varphi-\varphi_{2Q}|\,d\mu\leq
2\beta_{n}\|\varphi\|_{\mathfrak{A},2}.$

Therefore, by Lemma 5, we have

(4.4) $|\varphi_{2Q}|\geq|\langle\varphi\rangle_{2Q}|-C\|\varphi\|\geq
CK(2Q)-C-C\geq CK(2Q),$

since $K(2Q)$ is assumed to be sufficiently large.

Finally, we put

$\gamma_{Q}=\beta_{Q}-b_{2,Q}-b_{3,Q}$

and we directly check the conditions (1.7) and (1.8).

#### Oscillation condition (1.7)

Put $b_{Q}={\gamma_{Q}}/{\varphi_{2Q}}.$ Then we have

$\begin{split}\frac{\varphi_{2Q}}{\mu(Q)}\int_{Q}\left|T1-b_{Q}\right|\,d\mu&=\frac{1}{\mu(Q)}\int_{Q}|T\varphi-\beta_{Q}+b_{2,Q}-T\varphi_{2}+b_{3,Q}-T\varphi_{3}|\,d\mu\\\
&\leq C\end{split}$

by decomposition (4.3), property (4.1) and Lemma 1. Thus,

(4.5)
$\frac{1}{\mu(Q)}\int_{Q}\left|T1-b_{Q}\right|\,d\mu\leq\frac{C}{|\varphi_{2Q}|}\leq\frac{C}{K(Q)}$

by (4.4). Therefore, property (1.7) holds, as required.

####  $K$-condition (1.8)

Applying the definitions of $b_{Q}$ and $b_{R}$, we have

$\displaystyle|b_{Q}-b_{R}|\leq\left|\frac{\gamma_{Q}}{\varphi_{2Q}}-\frac{\gamma_{R}}{\varphi_{2Q}}\right|+\left|\frac{\gamma_{R}}{\varphi_{2Q}}-\frac{\gamma_{R}}{\varphi_{2R}}\right|$
$\displaystyle=\frac{|\gamma_{Q}-\gamma_{R}|}{|\varphi_{2Q}|}+|b_{R}|\frac{|\varphi_{2R}-\varphi_{2Q}|}{|\varphi_{2Q}|}$
$\displaystyle:=A+B.$

Combining Lemmas 2 and 3 and properties (4.2) and (4.4), we obtain

$A\leq C\frac{K(Q,R)}{K(2Q)}.$

Since $T1\in L^{\infty}(\mu)$, property (4.5) implies the estimate
$|b_{R}|\leq C$. Hence, applying (2.4) and the estimate
$\|\varphi\|_{\mathfrak{A},2}\leq C$, we obtain

$B\leq C\frac{K(2Q,2R)}{K(2Q)}.$

In sum, we have

$|b_{Q}-b_{R}|\leq A+B\leq C\frac{K(Q,R)}{K(Q)},$

that is, condition (1.8) holds. The proof of the theorem is finished.

## References

  * [1] G. David and J.-L. Journé, _A boundedness criterion for generalized Calderón-Zygmund operators_ , Ann. of Math. (2) 120 (1984), no. 2, 371–397. MR 763911
  * [2] E. Doubtsov and A. V. Vasin, _Calderón-Zygmund operators on RBMO_ , Proc. Amer. Math. Soc. 151 (2023), no. 2, 595–610. MR 4520012
  * [3] A. E. Gatto, _Boundedness on inhomogeneous Lipschitz spaces of fractional integrals singular integrals and hypersingular integrals associated to non-doubling measures_ , Collect. Math. 60 (2009), no. 1, 101–114. MR 2490753
  * [4] T. Hansson, _On Hardy spaces in complex ellipsoids_ , Ann. Inst. Fourier (Grenoble) 49 (1999), no. 5, 1477–1501. MR 1723824
  * [5] G. Hu, Ya. Meng, and Da. Yang, _A new characterization of regularized BMO spaces on non-homogeneous spaces and its applications_ , Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 1, 3–27. MR 3076796
  * [6] T. Hytönen, Da. Yang, and Do. Yang, _The Hardy space $H^{1}$ on non-homogeneous metric spaces_, Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 1, 9–31. MR 2943664
  * [7] J. Mateu, P. Mattila, A. Nicolau, and J. Orobitg, _BMO for nondoubling measures_ , Duke Math. J. 102 (2000), no. 3, 533–565. MR 1756109
  * [8] F. Nazarov, S. Treil, and A. Volberg, _The $Tb$-theorem on non-homogeneous spaces_, Acta Math. 190 (2003), no. 2, 151–239. MR 1998349
  * [9] T. Sjödin, _On properties of functions with conditions on their mean oscillation over cubes_ , Ark. Mat. 20 (1982), no. 2, 275–291. MR 686176
  * [10] E. M. Stein, _Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals_ , Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. MR 1232192
  * [11] X. Tolsa, _BMO, $H^{1}$, and Calderón-Zygmund operators for non doubling measures_, Math. Ann. 319 (2001), no. 1, 89–149. MR 1812821


# 3D Lip Event Detection via Interframe Motion Divergence at Multiple Temporal
Resolutions

Jie Zhang
School of Artificial Intelligence, Beijing Technology and Business University,
China
<EMAIL_ADDRESS>Robert B. Fisher
School of Informatics, The University of Edinburgh,UK
<EMAIL_ADDRESS>
###### Abstract

The lip is a dominant dynamic facial unit when a person is speaking. Detecting
lip events is beneficial to speech analysis and support for the hearing
impaired. This paper proposes a 3D lip event detection pipeline that
automatically determines the lip events from a 3D speaking lip sequence. We
define a motion divergence measure using 3D lip landmarks to quantify the
interframe dynamics of a 3D speaking lip. Then, we cast the interframe motion
detection in a multi-temporal-resolution framework that allows the detection
to be applicable to different speaking speeds. The experiments on the S3DFM
Dataset investigate the overall 3D lip dynamics based on the proposed motion
divergence. The proposed 3D pipeline is able to detect opening and closing lip
events across 100 sequences, achieving a state-of-the-art performance.

##  1 Introduction

Speaking is a spontaneous behavior involving multiple biological modalities,
including voice, visual speech [10], dynamic face motions [20], etc. There are
many applications related with speaking, e.g. speech recognition, lip-reading
[8], or identity recognition [10, 20]. The lip is a dominant dynamic facial
unit when speaking, and the visual lip acts as an important counterpart to the
audio information. For visual lip related applications, detecting the lip
events of opening and closing is a significant prerequisite for analyzing lip
behavior.

Lip event detection aims at localizing and tracking the lip region across a
video sequence and then determining the starting and ending times of the
speaking behavior along the temporal domain. The core of lip event analysis
focuses on the spatio-temporal representation of interframe motion, instead of
the whole lip dynamics. The challenge always lies at fine-grained temporal
detection (i.e. frame-level motion decision).

Existing lip event analysis algorithms are mainly based on 2D image sequences.
A group of lip event detection methods are based on dense motion field
analysis. For example, Karlsson and Bigun [13] improved 2D optical flow
estimation and constructed a low-level lip dynamics feature for lip event
detection. Liu et al. [15] constructed oriented histograms of regional optical
flow (OH-ROF) over 2D lip sequences to represent frame-level lip motion, and
then proposed a low-rank affinity pursuit approach to determine the starting
and ending of a lip event. This method is efficient as it is free of prior
learning.

There are two drawbacks to using 2D motion fields: 1) the inner mouth is
cluttered and its motion field affects the lip event detection; 2) the motion
field of sequential 2D intensity images is sensitive to facial pose
variations. Another family of methods focuses on lip shape deformation [18, 5]
and motion features [19, 16]. Taeyup et al. [19] proposed a chaos-like lip
motion measure - fractal trajectories observed in phase space, which is
especially robust against illumination changes. Patrona et al. [16] utilized
multiple intensity image features and dense trajectories of keypoints to
represent both the local shape and motion of a speaking face. The
representations are then integrated into a bag of words model for later
classification. Recently, an end-to-end network - HiCA [17] was designed to
extract the local and global temporal features and achieves visual voice
activity detection. There are also some dual-modality approaches [9, 4, 3]
that combine audio and intensity video information for speaking event
detection. The two modalities complement each other. Overall, lip event
detection via 2D sequences always suffers from the common drawbacks of
intensity images. It is sensitive to facial pose variations, nonuniform
illumination, scale changes, etc.

Lip event detection via 3D sequence data is a promising alternative approach
for the task. To the best of our knowledge, 3D lip event analysis is a less
investigated approach. 3D lip event detection is challenging when dealing with
3D noisy data, cluttered backgrounds, and frame-level non-rigid deformation.
Since the 3D mouth cavity is usually reconstructed with lower quality due to
the darkness and occlusion of oral components, the 3D motion field based
approach is not a preferred solution. Besides, speakers with low speaking
speeds generate weak deformations between consecutive frames, which are tough
to be detected and compared.

To tackle the above problems, we focus on 3D lip event detection based on the
interframe motion of 3D lip landmarks. The landmark-based motion
representation is more immune to the cluttered and dark background of the
mouth cavity. The main contributions of this paper are:

  * •

We define a new interframe motion representation for the 3D speaking lip - 3D
motion divergence. The motion signature quantifies the overall deformation of
3D lip landmarks via a reference sphere over the 3D lip. (Sec. 3.2)

  * •

We propose a new 3D lip event detection pipeline that determines the lip
opening and closing frames from the 3D video in multiple temporal resolutions.
The coarse-to-fine temporal strategy is beneficial when dealing with various
motion speeds. (Sec. 3.3)

The proposed algorithms were verified on a 3D speaking face dataset (S3DFM
[20, 22])111http://groups.inf.ed.ac.uk/trimbot2020/DYNAMICFACES/ and has good
detection performance over 100 sequences with 200 events.

##  2 Overview of the proposed pipeline

We define two lip events as 1) the moment lips first start to open and 2) the
moment lips have finished closing.

The proposed pipeline for 3D lip event detection is shown in Fig.1. Given a 3D
lip video sequence, we first perform noise reduction to improve the overall
quality of the 3D data. Then, the 3D landmarks of the dynamic lips are
extracted using a non-rigid 3D registration algorithm with a 3D deformable
model [2, 11]. The registration also facilitates the rigid pose correction of
the lip. The motions of lip landmarks are fed into the proposed interframe
motion representation, following a framework of dynamic temporal resolutions.
The event region proposal is gradually refined to an event frame. More details
are given in the next section.

Figure 1: The proposed pipeline of 3D lip event detection. Given a 3D lip
sequence, the lip event detection begins with a coarse temporal resolution
$ts_{0}$ and an event reference frame $f_{1}$, and it gradually refines the
temporal region proposal by increasing temporal resolution $ts_{i}$ as well as
updating the reference frame. At each temporal resolution, we sequentially
compute the interframe divergence and classify the frames.

##  3 Interframe 3D lip motion representation

###  3.1 Preprocessing 3D lip sequence

The preprocessing incorporates two main steps: denoising and rigid pose
correction. The raw 3D point cloud sequences usually suffer from some spatial
noise and temporal fluctuations, due to the sensor technology and data capture
procedure. To improve the overall quality of the 3D data, we firstly denoise
the 3D point cloud sequence using a multi-frame fusion algorithm [21], but do
not reduce the frame rate. On the other hand, facial pose is likely to
slightly change while a person is speaking. The rigid lip pose variation mixed
with non-rigid lip deformation will affect the interframe lip motion analysis.
The lip landmarks extraction involves rigid registration that is used for
correcting the lip pose.

###  3.2 Interframe 3D lip motion signature

It is interesting to note that the lip events of opening and closing are not
the same for every person and every phrase that the speaker is about to
pronounce. According to the research on lip motion-based behaviometrics [7,
14, 20], the lip event is a person-specific trait. Besides, the motion vector
of an individual lip landmark cannot represent the opening or closing state of
a lip. E.g. an opening lip can contain both diverging and converging lip
landmarks. Based on the above observations, the interframe motion signature of
a 3D speaking lip should capture common and global properties of the frame-
level 3D lip dynamics, and cope with both person-specific and syllable-
specific differences. Another common property of the 3D lip dynamics while
speaking is the symmetry, which means that speaking is a regular and text-
constrained motion.

For raw 3D lip motion data, we only focus on the landmarks of the 3D lip,
instead of a dense point-wise motion field. This is to avoid the impact of
unstable motion from the inner mouth. Thus, we represent a 3D lip as a set of
3D landmarks $\\{{\bf{L}}^{t}_{i}\in R^{3}\\}(i=1...n)$ at time $t$. Each lip
landmark generates a motion vector ${\bf{m}}^{t}_{i}\in R^{3}$ during speaking
events. Motivated by divergence in flow analysis, we define an interframe
motion signature over all the 3D lip landmarks. The signature measures the
energy and overall type of a lip motion between two frames. It is a global
metric summarizing the interframe lip dynamics.

Figure 2: Reference sphere (blue ball) for calculating the interframe motion
divergence of a 3D lip at one time. ${\bf{L}}_{i}$ is a 3D lip landmark.
${\bf{m}}_{i}$ and ${\bf{v}}_{i}$ is the 3D motion vector and reference vector
of the $i_{th}$ lip landmark, respectively. The motion divergence can be
calculated according to Eqn.2.

Lip motion signature energy. For a lip event, we assume that the motion of
every lip landmark brings a unit of “energy” into or out of the mouth region.
To measure the motion energy of all the lip landmarks, we establish a
reference sphere $S$ within the lip region in a 3D coordinate system $o-xyz$,
as shown in Fig.2. The center $O_{S}$ of the sphere $S$ is the 3D center of
mass of all the lip landmarks at the reference frame. Since every frame has
been registered with the reference frame before interframe motion analysis,
the center is static across the sequence. The surface area of the sphere is a
constant $V_{S}$. We focus on the “energy” change at the center $O_{S}$, which
is defined precisely below.

For each point on the surface of the sphere, we define the radial direction as
a unit normal motion direction ${\bf{v}}_{i}$. The motion vector at time $t$
and lip landmark $i$ is
${\bf{m}}^{t}_{i}={\bf{L}}^{t}_{i}-{\bf{L}}^{[t-1]}_{i}$. A set of motion
vectors ${\bf{M}}^{t}\\{{\bf{m}}^{t}_{i}\in R^{3}\\}$ (as defined by the
motion of lip landmark ${\bf{L}}^{t}_{i}$) can only generate the motion energy
in the normal motion direction. Thus, the overall lip motion energy is defined
as Eqn.1.

$Div_{lip}^{t}=\mathop{\lim}\limits_{S\to{O_{S}}}\frac{1}{{{V_{S}}}}\mathop{{\int\\!\\!\\!\\!\\!\int}\mkern-21.0mu\bigcirc}\limits_{\Omega}{{{\bf{m}}^{t}_{i}}\cdot{{\bf{v}}_{i}}dS}$
(1)

where $\Omega$ is the closed surface of the sphere, with $dS$ is a unit of
area. $V_{S}$ is the sphere’s surface area. When all the energy of the motion
converges at the center, i.e. $S\to{O_{S}}$, the divergence at the center
measures the energy and type of the motion.

As the number $n$ of the lip landmarks is finite, we discretize the divergence
of interframe lip motion at time $t$ as

$Div_{lip}^{t}=\frac{1}{{{V_{S}}}}\sum\limits_{i=1}^{n}{{{\bf{m}}^{t}_{i}}\cdot\amalg({{\bf{L}}^{t}_{i}}-{{\bf{O}}^{t}_{S}})\Delta
S}$ (2)

with

$V_{S}=Area(\Omega)=\sum\limits_{i=1}^{n}{\Delta S}$ (3)

where $\Delta S$ is a discrete fraction of the sphere’s surface area, which is
a constant. $\amalg(\cdot)$ is the vector normalization.

Lip motion signature category. Every interframe motion can be represented by
the signature $Div_{lip}^{t}$. The event state of one lip landmark is defined
by the angle between the 3D landmark motion vector ${\bf{m}}^{t}_{i}$ and its
reference vector ${\bf{v}}_{i}=\amalg({{\bf{L}}^{t}_{i}}-{{\bf{O}}_{S}})$.
When the angle is over 90 degrees, the landmark is closing, and vice versa.
That is,

${\mathop{\rm sgn}}(Div_{lm}^{t})={\mathop{\rm
sgn}}\langle{{\bf{m}}^{t}_{i}},{({\bf{L}}^{t}_{i}}-{\bf{O}}_{S})\rangle$ (4)

where $sgn(\cdot)$ is a signum. $Div_{lm}^{t}$ is the motion divergence of a
lip landmark. When the mouth is opening, the overall interframe motion
divergence of all lip landmarks is a positive value and vice versa. We define
the lip as static if the motion signature satisfies a threshold
${\varepsilon_{silence}}$, as Eqn.5. When the units of 3D lip points are in
millimeters, we set ${\varepsilon_{silence}}=1$ in practice.

$\left|{Div^{t}_{lip}}\right|<{\varepsilon_{silence}}$ (5)

Otherwise

$\begin{array}[]{l}LipState_{t}=\left\\{{\begin{array}[]{*{20}{c}}{opening}&{{\mathop{\rm
sgn}}(Di{v_{lip}^{t}})=1}\\\ {closing}&{{\mathop{\rm
sgn}}(Di{v_{lip}^{t}})=-1}\end{array}}\right.\\\
\begin{array}[]{*{20}{c}}{s.t.}\end{array}\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}{\left|{Div_{left}^{t}-Div_{right}^{t}}\right|<\varepsilon_{symmetry}}\end{array}}\end{array}\end{array}$
(6)

where $sgn(\cdot)$ is a signum. $Div_{left}^{t}$ and $Div_{right}^{t}$ are the
divergences of the left side and right side of a lip at time $t$,
respectively, and both satisfy Eqn.2. We incorporate the symmetry as another
constraint in the lip event detection. That is, if the motion divergences of
left lip $Div_{left}^{t}$ and right lip $Div_{right}^{t}$ are almost the same
within a tolerance $\varepsilon_{symmetry}$, the interframe motion satisfies
symmetry. Otherwise, the interframe motion will be rejected as a state of
interest. In our implementation, the tolerance $\varepsilon_{symmetry}$ is set
as 0.4 empirically.

The lip event is sequential and is defined as the first frame whose state is
opening or the last frame whose state is closing.

###  3.3 Event detection at multi-temporal-resolutions

We cast the interframe motion detection into a framework with multiple
temporal resolutions, as shown in Fig. 3. As a temporal region with a coarse
temporal resolution covers more motion energy, the pipeline begins with the
detection at a coarse time scale (bottom axis in Fig. 3) and gradually
increases the temporal detection resolution for fine detection (top axis in
Fig. 3).

Figure 3: Lip event detection at multiple temporal resolutions: the detection
begins at a coarse temporal resolution (averagely downsampling the original
frames in the time domain), determine an interframe interval as a region
proposal, increase the temporal resolution of the region by downsampling the
original frames with smaller sampling interval, repeat the detection at each
level of temporal resolution.

Given a 3D point cloud sequence, we define a coarse temporal resolution by
averagely downsampling the original frames in the time domain. Assume that the
initial temporal detection resolution is $ts_{0}\times$ frames. That is, the
initial frame rate is $1/ts_{0}$ of the original frame rate. We set the first
frame of the sequence as a reference frame. The interframe motion detection is
sequentially performed along the time axis. The detection algorithm generates
a temporal region proposal which includes the potential opening frame (or
closing frame) of a lip event. Then, we update the reference frame as the
first frame of the region and increase the temporal resolution by downsampling
the original frames in the region proposal again (but with the downsampling
interval smaller than that of the last round). The detection is conducted in
each round. The region proposal is finally refined to one frame as the
temporal detection resolution increases.

The merits of this hierarchical strategy are two-fold: 1) the coarse-to-fine
detection hierarchically reduces the searching space for the target frame,
which is beneficial to reduce the false detection rate; 2) coarse temporal
resolution allows motion event detection for people with lower speaking
speeds. When the speaking speed is slow, the interframe with the finest
temporal resolution may not generate enough motion divergence for detection.
The event response can be given at a coarser temporal resolution.

Finally, the proposed pipeline outputs the opening and closing frames of a 3D
lip event. The overall algorithm is shown in Table 1.

Algorithm 1 Proposed pipeline of 3D lip event detection

1:3D speaking face sequence $\\{f_{t}\\}(t=1...m)$;

2:Lip event frames $\\{f_{open},f_{close}\\}$ (Note: we only present the
procedure of lip opening detection as an example below, as the closing event
detection is similar);

3:Initialize reference frame as $f_{1}$;

4:Construct a reference sphere at the 3D lip center;

5:procedure Interframe Motion Representation

6: for each temporal resolution
$ts=\\{30\times,15\times,7\times,3\times,1\times\\}$ original interframe
interval do

7: for each 3D frame $f_{ts\times j}(j=1\cdots m/ts)$ do

8: Denoise current 3D frame;

9: Register $f_{ts\times j}$ with reference frame;

10: Extract 3D lip landmarks $\\{{\bf{L}}^{ts\times j}_{n}\\}$;

11: Compute motion vectors of lip landmarks;

12: Compute 3D interframe motion divergence;

13: Determine event state of current lip;

14: if lip opening event happens then

15: Update reference frame with $f_{ts\times(j-1)}$;

16: Update event frame with $f_{open}=f_{ts\times j}$;

17: Break;

18: end if

19: end for

20: end for

21:end procedure

22:return $\\{f_{open},f_{close}\\}$;

##  4 Experiments and discussion

Figure 4: 3D lip dynamics analysis: (a) an example 3D point cloud frame; (b)
interframe motion divergence across a 3D speaking face sequence. The first 250
interframe motions are measured for opening detection, with the first frame as
the reference frame. The last 250 interframe motions are used for closing
detection, with the last frame as the reference frame.

This section reports the experiments on a 3D speaking face dataset. We analyze
the full lip dynamics of speaking and investigate the comparative performance
of the proposed pipeline and some representative counterparts.

###  4.1 Dataset

The proposed pipeline was verified on a publicly available dynamic face
dataset - Speech-driven 3D Facial Motion Dataset (S3DFM) [1, 22]. The dataset
has multi-modality data from 77 subjects covering more than 20 nationalities.
The facial dynamics is generated from the subject speaking a one-second short
phrase “ni’hao”. Each sample set contains a 2D intensity sequence $+$ a 3D
point cloud sequence $+$ a synchronized audio sequence. The audio and video
modalities of a lip event were collected with a light flash as a
synchronization trigger. We only use the 3D speaking face modality in the
algorithm presented here, while the audio and intensity modalities help
determine the ground truth labeling. We set the frame rate of each sequence as
250 frames per second (fps) for better labeling. Each frame is a 3D point
cloud with the resolution of 600 points $\times$ 600 points. An example 3D
point cloud frame is shown in Fig.4a.

We manually labelled the lip opening and closing frames using both the pixel-
wise registered intensity sequences and the synchronized audio sequences. The
audio clip firstly gave a coarse time localization and then we compared the
consecutive frames around the coarse time to finally decide a fine ground-
truth event frame.

###  4.2 Qualitative analysis

3D lip landmarks dynamics. The 3D motion divergence of a whole sequence is
shown in Fig.4b, where we separate a whole sequence with 250 frames into the
opening event region proposal of the first half frames and the closing event
region proposal of the last half frames. We can see that the 3D motion
divergence is increasing during the start of speaking (blue bars), and vice
versa (red bars). Ax example of the dynamics of 3D lip landmarks across the
whole sequence is shown in Fig.5. The 3D motion vectors are diverging when the
opening event happens, and they are converging in the closing event.

Figure 5: 3D speaking lip motion in a 3D sequence of 500 frames: (a)-(c)
opening frames at 3 moments; (d)-(f) closing frames at 3 moments.

3D lip landmarks divergence. We set the first frame $f_{1}$ and the last frame
$f_{250}$ as initial reference frames for detecting the opening event and
closing event, respectively. Since the video clip is synchronized with the
audio clip, the first frame is earlier than the starting of the speech and the
last frame covers the end of the speech. It is noted that the lip motion is
sequential, so the motion state along the time axis is invariant to the
reference time. The temporal detection resolutions are set as
$\\{30\times,15\times,7\times,3\times 1\times\\}$ original interframe interval
(More analysis on temporal resolutions are presented in Sec. 4.5 below). At
each temporal resolution, the proposed pipeline calculated the 3D interframe
motion divergence along the time axis. Fig.6 shows an example from a single
frame of motion divergence of 3D lip landmarks at the temporal detection
resolution of $30\times$. We can see that the closing motion allows an angle
of over 90 degrees with its reference motion vector (the blue vector), thus
generating a negative motion divergence measure (Fig.6a), and vice versa
(Fig.6b).

Figure 6: Motion divergence of 3D lip landmarks: (a) Lip opening event with
red 3D motion vectors of lip landmarks, whose motion divergences are positive.
The blue vectors are 3D reference vectors to the lip landmarks from the center
of the reference sphere. (b) lip closing event with green 3D motion vectors of
lip landmarks, whose motion divergences are negative. (Note: for a better
view, we show the 3D vectors in the XY view and scale the length of the
vectors.)

###  4.3 Quantitative performance

####  4.3.1 Metrics

We define three hierarchical metrics to evaluate both frame-level and event-
level performance of the algorithm as below.

$\bullet$ Framewise Accuracy (F-Acc): our lip event detection issue is a
sequence-to-sequence classification task. Each frame refers to one of the
three activity states: static, opening, and closing. The F-Acc measures the
classification rate of all the frame states.

$\bullet$ Event Frame Deviation (F-Dev): F-Dev evaluates the mean deviation
between the detected events $T_{i}$ and corresponding ground truth events
$gt_{i}$, as an event deviation $\delta=\left|T_{i}-gt_{i}\right|$. The
deviation is caused by the event frames being detected earlier or later than
the ground-truth frame.

$\bullet$ Event Recall Rate (E-RR): For a detected event, if its event frame
deviation is within a F-Dev tolerance, the event is regarded as a true
response. E-RR is the ratio of the correctly detected events to all of the
events. The false events contain missing ones and incorrectly classified
events, where the former is related to the magnitude of the motion energy, and
the latter is caused by the wrong sign of 3D motion divergence. Thus, the E-RR
increases with a larger F-Dev tolerance.

Table 1: Multi-indictor comparison of different methods or configurations on the dataset Methods | F-Acc (%) | F-Dev (opening/closing) | E-RR (%) | T-Dev (ms)
---|---|---|---|---
Lip feature [22] | 85.14 | 12.28/24.87 | 80.5 | 74.3
TCN (2-fold CV) [6] | 85.36 | 11.82/24.77 | 80.5 | 73.2
LSTM (2-fold CV) [12] | 82.79 | 14.84/28.18 | 84.0 | 86.0
Single-scale (noisy features) | 82.20 | 14.01/30.5 | 81.0 | 89.0
Single-scale (smoothed features) | 88.16 |  9.81/19.78 | 88.0 | 59.2
Ours (noisy features) | 87.58 | 14.08/16.98 | 90.0 | 62.1
Ours (smoothed features) | 89.46 | 12.63/13.72 | 91.5 | 52.7
Note: F-Acc is Framewise Accuracy; F-Dev is Event Frame Deviation; E-RR is
Event Recall Rate with the F-Dev tolerance of 40 frames; T-Dev is average Time
Deviation of the events in frames converted to msec (frame per second is 250).
All the indictors are mean values across the samples. CV is cross-validation.

####  4.3.2 Ablation study on temporal resolutions

The temporal detection resolution is set to be 2 configurations: multi-
temporal-resolutions and single frame resolution. We calculate multiple
performance metrics for our algorithm across all the 100 test sequences (200
events including opening and closing). We test both configurations on noisy
features and smoothed features (Sec.3.1) to investigate the robustness. Fig.7
presents 10 example results of true event detection by our algorithm on
smoothed features. It shows that the opening and closing event frames detected
are closely consistent with the ground truth. More comparative results are
listed in the last four rows of Table 1 (the rest results are mentioned in the
next section). We set the frame deviation tolerance as 40 frames (16 msec) and
calculate an event recall rate (E-RR) for each method or configuration.

Figure 7: 10 example results of lip event detection: (a) 3D lip opening; (b)
3D lip closing. The deviation between the detected moment and ground truth
moment is measured by Event Frame Deviation (F-Dev).

From the comparative results, we can see that our pipeline with multi-
temporal-resolution on smoothed data achieves higher E-RR of $91.5\%$ and
lower T-Dev of 52.7 msec. More detailed F-Dev results are shown in Fig.8. For
the pipeline with multi-temporal-resolutions, the mean F-Dev. of opening and
closing events are 12.6 frames and 13.7 frames, respectively. The multi-
temporal resolution detection is more robust to noisy data, as the event
proposal generated when using larger scale motion energy focuses the detection
range.

Failure cases analysis. For the missed event samples, the motion energy was
too small to be detected. The false detection samples are mainly caused by
irregular lip motions and incorrect interframe motion. For the irregular lip
motion, the lips of a few speakers went through some frames of deforming
before opening and were somewhat widening while closing, which allows the lip
landmarks to generate irregular motion divergence and thus influence the event
decision. Besides, 3D lip landmark deviations or registration error will also
influence the divergence feature, which could be improved by using more
advanced lip landmark tracking or detection algorithms.

Figure 8: Event Frame deviation (F-Dev) of different methods or
configurations: A is lip distance-based features [20]; B is LSTM [12]; C is
TCN [6]; D is single-resolution on noisy features; E is single-resolution on
smoothed features; F is multi-resolution on noisy features; G is multi-
resolution on smoothed features. Each compared group consists of performance
on opening events and closing events.

###  4.4 Comparison on different methods

We compare the performance of the proposed 3D lip detection algorithm with
existing dynamic lip features and temporal detection methods on the same 3D
point cloud streams. The comparison pipelines are 3D dynamic lip features [20]
plus change time decision, divergence features plus temporal modeling method
LSTM [12], and divergence features plus temporal modeling TCN [6]. Both
temporal models are trained under a 2-fold cross validation mode. The overall
multi-indicator results are presented in Table 1, and Fig.8 shows the frame
deviation distribution of all the samples. We also plot the E-RR vs. F-Dev
tolerance curves in Fig.9 to indicate the correlation of the two metrics and
the comprehensive performance of the compared methods.

We can see from the multi-aspect indicators that our pipeline with multi-
temporal resolution outperforms others in terms of mean indicators. However,
there are some outliers with large frame deviations (shown in Fig.8), which
thus degrades the event recall rate when the F-Dev tolerance is larger than 60
frames (in Fig.9).

Figure 9: Event Recall Rate (E-RR) vs. varying framewise event deviation
tolerance (F-Dev tolerance)

###  4.5 Discussion on multi-temporal-resolutions

In our framework, $ts_{t}$ is an updated temporal resolution at which an event
frame proposal is generated. We set the dynamic temporal resolution updated
according to Eqn.7.

$ts_{t+1}=\lceil{\frac{ts_{t}}{k}}\rceil\;(ts_{t+1}\geq 1)$ (7)

where $k$ is an integral number that was set as $k=2$ in the experiments. The
number $n$ of the temporal resolutions is related to the parameter $k$,
satisfying $mod({\frac{ts_{0}}{k^{(}n-1)}})=0$.

At each temporal resolution $ts_{t}$, $gt_{t}$ is a relative ground truth
frame updated with the reference frame according to Eqn.8.

$gt_{t+1}=w(gt_{t}-\lfloor{\frac{gt_{t}}{ts_{t}}}\rfloor ts_{t})+(1-w)ts_{t}$
(8)

where $w\in[0,1]$ is a binary parameter that satisfies

$w=sgn[mod(gt_{t},ts_{t})]$ (9)

Detection mode. The event detection at each temporal resolution can be in a
sequential or a parallel mode. In the sequential mode, the numbers of
interframe event detections is related to the relative ground truth $gt_{t}$
and the resolution $ts_{t}$, as Eqn.10. The first item of Eqn.10 indicates the
number of interframe detections when the ground truth frame $gt_{t}$ is not
integral multiples of the current temporal resolution $ts_{t}$, and the second
item counts the number when $gt_{t}$ is integral multiples of $ts_{t}$.

$DetNum=\sum_{t=0}^{n}w(\lfloor{\frac{gt_{t}}{ts_{t}}}\rfloor+1)+(1-w){\frac{gt_{t}}{ts_{t}}}$
(10)

Based on the sequential mode analysis, we generated a set of synthetic ground
truth event times $gt_{0}$ and initial temporal resolutions $ts_{0}$ to
investigate how the number of interframe detections changes with both
parameters. The results are shown in Fig.10. The lip event are a progressive
motion, so the number of interframe detections changes with the event
happening moment. A large initial temporal detection resolution saves the
computational cost in the sequential interframe detection mode.

Figure 10: Number of interframe event detections for varying starting times of
a lip event at different initial temporal resolutions $ts_{0}$ ($ts_{0}$ is
measured by temporal interval of original resolution).

###  4.6 Discussion on applicability

Various speaking speeds. The test sequences exhibit different speaking speeds
due to individual behaviors. Speaking slowly typically generates lower
interframe motion energy. The interframe motion signature of lower motion
energy is more likely to be influenced by data noise and inexact landmark
localization. The multi-resolution strategy enlarges the interframe motion
energy via an initially low detection resolution and reduces the search space
gradually. For a slow speaker, a frame rate higher than that of the event is
unnecessary. Thus, if an event is detected at a coarse resolution but not
responded at the next finer resolution, our algorithm outputs the result at
the coarse resolution as the event.

Algorithm efficiency. The time complexity of the pipeline is $O(n_{ts}f)$,
which is related to the number of temporal resolutions $n_{ts}$ and the number
of frames $f$ at one temporal resolution. If the interframe detection of a
sequence at one temporal resolution is performed in a parallel mode, the
computational complexity becomes $O(n_{ts})$. Besides, since the 3D data
quality would influence the landmark extraction and pose correction, if
higher-quality data acquisition allows the pipeline to be free from data
denoising, the efficiency would be improved greatly. The pipeline is promising
to be applied in real time.

##  5 Conclusion

This paper proposes a 3D lip event detection method at multiple temporal
resolutions, with an interframe motion representation called 3D motion
divergence. The method avoids the drawbacks of 2D intensity data and is a
promising alternative for lip event detection. The experimental results
demonstrate that the proposed pipeline achieves a state-of-the-art
performance. The strategy of the multi-temporal resolution improves the
robustness to various speaking speeds. The motion representation based on 3D
lip landmarks avoids the complexity of the inner mouth. The 3D lip event
pipeline helps automatically segment the 3D dynamic clips of interest and is
beneficial for later global lip dynamics analysis.

In the future, we would like to focus on temporal modeling of 3D dynamic lips
that represents both the interframe motion and the global dynamics, and seek
to achieve 3D lip event detection and behavior analysis in an end-to-end
fashion.

## Acknowledgment

The work is supported by Natural Science Foundation of China (NSFC) under
Grant No.61906004.

## References

  * [1] Speech-driven 3d facial motion dataset (s3dfm).  http://groups.inf.ed.ac.uk/trimbot2020/DYNAMICFACES/.  Accessed July 30, 2021.
  * [2] Brian Amberg, Sami Romdhani, and Thomas Vetter.  Optimal step nonrigid ICP algorithms for surface registration.  In 2007 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2007), 18-23 June 2007, Minneapolis, Minnesota, USA, 2007.
  * [3] I. Ariav and I. Cohen.  An end-to-end multimodal voice activity detection using wavenet encoder and residual networks.  IEEE Journal of Selected Topics in Signal Processing, 13(2):265–274, 2019.
  * [4] Ido Ariav, David Dov, and Israel Cohen.  A deep architecture for audio-visual voice activity detection in the presence of transients.  Signal Processing, 142:69–74, 2018.
  * [5] A. Aubrey, B. Rivet, Y. Hicks, L. Girin, J. Chambers, and C. Jutten.  Two novel visual voice activity detectors based on appearance models and retinal filtering.  In 2007 15th European Signal Processing Conference, pages 2409–2413, 2007.
  * [6] Shaojie Bai, J. Zico Kolter, and Vladlen Koltun.  An empirical evaluation of generic convolutional and recurrent networks for sequence modeling.  CoRR, abs/1803.01271, 2018.
  * [7] Lanthao Benedikt, Darren Cosker, Paul L. Rosin, and A. David Marshall.  Assessing the uniqueness and permanence of facial actions for use in biometric applications.  IEEE Trans. Systems, Man, and Cybernetics, Part A, 40(3):449–460, 2010.
  * [8] Joon Son Chung, Andrew Senior, Oriol Vinyals, and Andrew Zisserman.  Lip reading sentences in the wild.  In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017.
  * [9] David Dov, Ronen Talmon, and Israel Cohen.  Audio-visual voice activity detection using diffusion maps.  IEEE/ACM Transactions on Audio, Speech, and Language Processing, 23(4):732–745, 2015.
  * [10] Cheng Feng, Wang Shi-Lin, and Liew Alan Wee-Chung.  Visual speaker authentication with random prompt texts by a dual-task cnn framework.  Pattern Recognition, 83:340–352, 2018.
  * [11] Thomas Gerig, Andreas Morel-Forster, Clemens Blumer, Bernhard Egger, Marcel Lüthi, Sandro Schönborn, and Thomas Vetter.  Morphable face models - an open framework.  In 13th IEEE International Conference on Automatic Face & Gesture Recognition, FG 2018, Xi’an, China, May 15-19, 2018, pages 75–82, 2018\.
  * [12] Sepp Hochreiter and Jürgen Schmidhuber.  Long short-term memory.  Neural Comput., 9(8):1735–1780, 1997.
  * [13] Stefan M. Karlsson and Josef Bigün.  Lip-motion events analysis and lip segmentation using optical flow.  In 2012 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, Providence, RI, USA, June 16-21, 2012, pages 138–145, 2012.
  * [14] Xin Liu and Yiu-ming Cheung.  Learning multi-boosted hmms for lip-password based speaker verification.  IEEE Trans. Information Forensics and Security, 9(2):233–246, 2014.
  * [15] Xin Liu, Yiu-ming Cheung, and Yuan Yan Tang.  Lip event detection using oriented histograms of regional optical flow and low rank affinity pursuit.  Comput. Vis. Image Underst., 148:153–163, 2016.
  * [16] F. Patrona, A. Iosifidis, A. Tefas, N. Nikolaidis, and I. Pitas.  Visual voice activity detection in the wild.  IEEE Transactions on Multimedia, 18(6):967–977, 2016.
  * [17] R. Sharma, K. Somandepalli, and S. Narayanan.  Toward visual voice activity detection for unconstrained videos.  In 2019 IEEE International Conference on Image Processing (ICIP), pages 2991–2995, 2019.
  * [18] D. Sodoyer, B. Rivet, L. Girin, J. . Schwartz, and C. Jutten.  An analysis of visual speech information applied to voice activity detection.  In 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings, 2006.
  * [19] Taeyup Song, Kyungsun Lee, and Hanseok Ko.  Visual voice activity detection via chaos based lip motion measure robust under illumination changes.  IEEE Transactions on Consumer Electronics, 60(2):251–257, 2014\.
  * [20] Jie Zhang and Robert B. Fisher.  3d visual passcode: Speech-driven 3d facial dynamics for behaviometrics.  Signal Process., 160:164–177, 2019.
  * [21] Jie Zhang, Christos Maniatis, Luis Horna, and Robert B. Fisher.  Dynamic 3d reconstruction improvement via intensity video guided 4d fusion.  J. Vis. Commun. Image Represent., 55:540–547, 2018.
  * [22] Jie Zhang, Korin Richmond, and Robert B. Fisher.  Dual-modality talking-metrics: 3d visual-audio integrated behaviometric cues from speakers.  In 24th International Conference on Pattern Recognition, ICPR 2018, Beijing, China, August 20-24, 2018, pages 3144–3149, 2018.


# Mpemba Effect and Superuniversality across Orders of Magnetic Phase
Transition

Sohini Chatterjee  Theoretical Sciences Unit and School of Advanced Materials,
Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O.,
Bangalore 560064, India  Soumik Ghosh  Theoretical Sciences Unit and School of
Advanced Materials, Jawaharlal Nehru Centre for Advanced Scientific Research,
Jakkur P.O., Bangalore 560064, India  Nalina Vadakkayil  Theoretical Sciences
Unit and School of Advanced Materials, Jawaharlal Nehru Centre for Advanced
Scientific Research, Jakkur P.O., Bangalore 560064, India  Complex Systems and
Statistical Mechanics, Department of Physics and Materials Science, University
of Luxembourg, Luxembourg, L-1511, Luxembourg  Tanay Paul  Theoretical
Sciences Unit and School of Advanced Materials, Jawaharlal Nehru Centre for
Advanced Scientific Research, Jakkur P.O., Bangalore 560064, India  Sanat K.
Singha  Theoretical Sciences Unit and School of Advanced Materials, Jawaharlal
Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560064,
India  Assam Energy Institute, A Centre of Rajiv Gandhi Institute of Petroleum
Technology, Sivasagar 785697, India  Subir K. Das<EMAIL_ADDRESS>Theoretical Sciences Unit and School of Advanced Materials, Jawaharlal Nehru
Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560064, India

The quicker freezing of hotter water, than a colder sample, when quenched to a
common lower temperature, is referred to as the Mpemba effect (ME). While this
counter-intuitive fact remains a surprize since long, efforts have begun to
identify similar effect in other systems. Here we investigate the ME in a
rather general context concerning magnetic phase transitions. From Monte Carlo
simulations of model systems, viz., the $q$-state Potts model and the Ising
model, with varying range of interaction and space dimension, we assert that
hotter paramagnets undergo ferromagnetic ordering faster than the colder ones.
The above conclusion we have arrived at following the analyses of the
simulation results on decay of energy and growth in ordering following
quenches from different starting temperatures, to fixed final temperatures
below the Curie points. We have obtained a unique scaling picture, on the
strength of the effect, with respect to the variation in spatial correlation
in the initial states. These results are valid irrespective of the order of
transition and relevant to the understanding of ME in other systems, including
water.

##  I Introduction

If two bodies of liquid water, differing in temperature, are placed in contact
with a thermal reservoir, operating at a subzero temperature ($<0\degree$C),
the most common prediction will be that the colder between the two will freeze
faster. The report in Ref. [1], by Mpemba and Osborne, however, contradicts
such an expectation. There has been a surge [1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29] in
interest in further investigating this forgotten counter-intuitive fact, which
found mention even in the works of Aristotle [30], now referred to as the
Mpemba effect (ME). In recent times, questions relevant to the ME are posed in
more general ways [2], going much beyond the domain of water [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11]: When two samples of the same material, from two different
temperatures, are quenched to a common lower temperature, which one will reach
the new equilibrium quicker? If there exists a point of phase transition,
between the initial and the final temperatures, for which starting temperature
the transformation will occur earlier? There is a rapid growth in interest in
studying systems of different types. Experimental studies on systems such as
colloidal systems [10], clathrate hydrates [11], carbon nanotube resonators
[12] and magnetic alloys [13] show the presence of ME. In the theoretical and
computational literature, the varieties that are studied include granular
matter [14, 15, 16, 17], spin glass [18] and few other systems of magnetic
origin [19, 20, 21, 22]. Nevertheless, the underlying reason remains a puzzle,
inviting the need for stronger theoretical intervention. The pertaining new
questions are fundamental from statistical mechanical and other theoretical
points of view. In addition, the effect can be exploited to much practical
advantages [28].

In the case of water, for a transformation from liquid to solid, overcoming
metastability is a serious problem [31]. Nucleation there, as well as in
general, is strongly influenced by the choice of the final temperature
($T_{f}$), the type and the volume fraction of impurity, etc. However, how the
role of the starting temperature ($T_{s}$) enters the picture is an important
new fundamental issue. Is it that the above mentioned metastable aspect gets
affected by the nature of the initial thermodynamic state in an unexpected
order? With in-built frustration in the model Hamiltonian, some of the
theoretical works perhaps set the objective of exploring this angle. Such
studies are by primarily involving the magnetic systems with anti-
ferromagnetic interactions, including spin glasses. However, simpler systems
must also be considered. If the effect is found to be present in their
evolution dynamics, route to a proper understanding can be easier.

In fact, the standard ferromagnetic Ising model, without any impurity, is seen
to exhibit the effect [21]. The positive observation of the ME there was
attributed to the structural changes associated with the critical divergence
of the spatial correlation with the variation of $T_{s}$ in the paramagnetic
phase [2, 21]. To justify the validity of the attribution and estimate the
corresponding quantitative influence, it is important to study the model in
other situations like in different space dimensions ($d$) and with varying
range of interactions. Very importantly, it should be checked what are the
effects of the order of transitions [2, 29, 22]. This is by considering the
fact that in the case of water, the transition is of first order character,
while for the Ising model the problem was designed [21] in such a way that the
influence of second order transition is captured. Keeping these in mind [2,
29, 22], here we chose the $q$-state Potts model [32, 33] for which the order
of transition changes with the variation of its number of states $q$.
Interestingly, we observe the presence of the ME in all the above mentioned
cases. For the Potts model, it appears that with the increase of $q$ the
effect gets weaker, as far as the variation in $T_{s}$ is concerned. However,
interestingly, regarding the effects of spatial correlation in the initial
states there exists a unique scaling, not only for the simple variation of
$q$, but also with the change of the order of transition and space dimension.
We believe, in addition to being important for the class of systems
considered, this study also provides crucial angle with respect to the
interpretation of the effect in water.

##  II Models and Methods

As already mentioned, here we study a class of discrete spin systems with pure
ferromagnetic interactions. We investigate the Ising model [32] in different
dimensions, having short and long-range inter-site potentials [34].
Furthermore, generalization of this two-component system into multi-component
ones have also been considered. In general, the Hamiltonian can be written as
$H=-\sum_{i(\neq)j}J(r_{ij})\delta_{S_{i},S_{j}};\,\,S_{i},S_{j}=1,2,...,q;$
$r_{ij}$ being the separation between lattice sites. For $q=2$, this Potts
model Hamiltonian corresponds to the Ising model, differing only by a factor
of $2$. The latter, by correcting for the factor, we have studied in $d=2$ and
$3$, with nearest neighbor (NN) interactions, by setting the interaction
strength $J$ to unity, on square and simple cubic lattices, respectively. In
$d=2$, we have also presented results for the long-range (LR) version of the
Ising model with [34] $J(r_{ij})=1/r_{ij}^{2+\sigma}$, for $\sigma=0.8$, again
using the square lattice. Most extensive results are obtained for the (short-
range) Potts model, $q$ varying between $2$ and $10$. The critical temperature
for this model has the $q$-dependence [32] $T_{c}=J/[k_{B}\ln(1+\sqrt{q})]$,
$k_{B}$ being the Boltzmann constant, to be set to unity. Depending on the
value of $q$ the order of transition can alter. For $q>4$ the model loses its
“critical” character [33], the transition being of first order. For the Ising
case, the values of $T_{c}$ in $d=2$ and $3$ are $\simeq 2.269\,J/k_{B}$ and
$\simeq 4.51\,J/k_{B}$, respectively [32]. For the long-range case we have
used [35] $T_{c}=9.765\,J/k_{B}$.

The kinetics of transition from para to ferro states are studied via Monte
Carlo simulations [32], with the Glauber spin-turn mechanism [32]. The
preparation of the initial configurations near the critical point encounters
critical slowing down [32]. To avoid this, we have used cluster algorithms. In
the case of Ising or Potts model, this is done by implementing the Wolff
algorithm [36], and for the LR Ising model, we have used the Fukui-Todo
algorithm [37, 35]. The presence of the correlated spatial fluctuations in a
system and its variation with temperature can be quantified via the
calculation of the structure factor:
$S(k,t)=\langle\psi_{k}\,(t)\,\psi_{-k}(t)\rangle,$ $\psi_{k}\,(t)$ being the
Fourier transform of the order parameter [38] $\psi\,(\vec{r},t)$
$(=\exp(i\,\theta(\vec{r}));\>\>\theta=2\pi n/q,\>\>n=1,\dots,q$). In the
small $k$ regime [$\in(0,0.4)$ or shorter], for the short-range cases,
$S(k,t)$ is described well by the Ornstein-Zernike relation [39, 40],
$S(k)=k_{B}\>T_{s}\>\chi/(1+k^{2}\>\xi^{2}),$ $\chi$ being the susceptibility
and $\xi$ the correlation length. The average domain length, $\ell(t)$, of the
clusters, during an evolution towards the ferromagnetic state, has been
estimated via the first moment of the domain size distribution function [41],
$P(\ell_{d})$, i.e., $\ell(t)=\int P(\ell_{d},t)\ell_{d}d\ell_{d}$, where
$\ell_{d}$ is the distance between two consecutive interfaces along a given
direction.

##  III Results

Figure 1: Typical equilibrium configurations, for $q=2$ and $5$ state Potts
models, are shown in (a) and (b) from different starting temperatures $T_{s}$
that are located above the respective critical temperatures $T_{c}$. (c) The
plots of the correlation lengths, $\xi$, versus $\epsilon$
($=(T_{s}-T_{c})/T_{c}$), for the same Potts models. For $q=2$, $\xi$ diverges
as $\epsilon^{-\nu}$, with $\nu=1$ (see the solid line). These results are
obtained for $L=256$.

In Fig. 1 we depict how the choice of $T_{s}$, in the case of Potts model, can
influence the structural features in the initial configurations, for different
values of $q$. For both the considered cases, viz., $q=2$ and $5$, the
configurations at higher $T_{s}$ [see the snapshots at the bottom of the
columns in (a) and (b)] appear random or structureless. With the decrease of
$T_{s}$, spatial correlations emerge. This is more clearly identifiable in the
case of $q=2$ for which one expects the critical divergence [39, 40]
$\xi\sim\epsilon^{-\nu}$, with $\nu=1$. For a wide range of $\epsilon$
($=|T_{s}-T_{c}|/T_{c}$) such a behavior can be appreciated from Fig. 1(c).
For $q=5$, the phase transition is of first order [33], and we do not
associate any exponent with the data set. The enhancement in the value of
$\xi$ can be appreciated for this $q$ as well. In this case, the bending on
the log-log scale over the whole presented range can well be, in addition to
the finite-size effects, due to the first-order nature of the transition. Our
expectation is that for this $q$ the effect will be weaker. We proceed with
the objective of quantifying it and to investigate if there exists any scaling
rule [2] for arbitrary $q$ that can as well comply with the other considered
models.

Figure 2: Demonstration of relaxation in the 5-state Potts model, following
quenches to $T_{f}=0.5\,T_{c}$, from different $T_{s}$ values, with equal
fraction of different spin states in boxes having $L=256$. In (a) and (b) we
show evolution snapshots, taken at different times, in units of standard Monte
Carlo steps, for the systems initially at (a) $T_{s}=0.9$ and (b) $T_{s}=1.1$.
Different colours represent $q$ different Potts states. (c) Plots of energy
versus time, following quenches from several $T_{s}$ values. The upper half
corresponds to early time behavior and the lower one shows the late time
behavior. The dashed horizontal lines are drawn to extract the times,
$t_{c,E^{{\rm ref},i}},i=1,2$, corresponding to the crossings of certain
values of energy $E^{{\rm ref},i}$ by the energy curves of the systems with
different $T_{s}$. (d) Plots of $t_{c,E^{{\rm ref},1}}$ (upper panel) and
$t_{c,E^{{\rm ref},2}}$ (lower panel), versus $T_{s}$.

We quench the systems from different [42] $T_{s}$ to $T_{f}=0.5\,T_{c}$, for a
large set of $q$ values. In Fig. 2 we show results obtained during evolutions
following such quenches, for the $5$-state Potts model. In parts (a) and (b)
we show the snapshots at different stages of evolutions for $T_{s}=0.9$ and
$1.1$, respectively. It is evident that the system from the higher $T_{s}$
reaches the final equilibrium faster. This comparative picture is true not
only for the chosen set of initial configurations, the observation indeed
stands correct for a vast majority of the combinations of starting
configurations. In part (c) we look at the decay of the average energy ($E$)
of the systems during the relaxation processes. We have included results for
several $T_{s}$. Each of the data sets is presented after averaging over
$300000$ independent initial configurations. For clarity, we have enlarged the
early and late time behavior separately, in the upper and lower panels,
respectively. The orders of appearances of the plots, in terms of $T_{s}$, are
systematic and opposite in the two panels. This implies that there are
crossings amongst the energy curves, due to faster equilibrations of
configurations prepared at higher $T_{s}$. This is the essence of Mpemba
effect [1, 2, 18, 21]. For a demonstration of systematicity, we perform the
following exercise. The dashed horizontal lines in these panels correspond to
two reference energy values, $E^{{\rm ref},1}$ and $E^{{\rm ref},2}$. We
calculate the crossing time between a dashed line and the energy curve of the
systems starting from each of the $T_{s}$ values. Such a crossing time is
denoted by $t_{c,E^{{\rm ref},i}},i=1,2$. Part (d) of Fig. 2 shows
$t_{c,E^{{\rm ref},i}}$ as a function of $T_{s}$. The upper and lower panels
capture the early and late time behavior, respectively. The early time
quantity, i.e., $t_{c,E^{{\rm ref},1}}$, increases with the increase in
$T_{s}$ but, at late times we see a different behavior, i.e., $t_{c,E^{{\rm
ref},2}}$ decreases with the increase in $T_{s}$. This implies faster
relaxation of the systems with higher $T_{s}$, indicating the presence of ME.

Figure 3: (a) The plots of $\ell(t)$ versus $t$, for the 5-state Potts model,
for quenches from various $T_{s}$ to $T_{f}=0.5\,T_{c}$. The lower frame
captures the early time trend, while the upper frame depicts the late time
behavior. (b) Plots of time, $t_{c,\ell_{\infty}}$, corresponding to crossing
between growth curves for systems starting at $T_{s}=\infty$ and a finite
$T_{s}$, as a function of $T_{s}-T_{c}$, for different $q$ values. (c) Plot of
$t_{c,\ell_{\infty}}$ versus $q$ for systems prepared at $T_{s}=1.3\,T_{c}$.
All results correspond to $L=256$.

The faster relaxation of the higher $T_{s}$ systems can as well be quantified
by calculating the average domain length, $\ell(t)$, a key probe to
investigate coarsening dynamics [41, 42, 43]. In Fig. 3(a), we plot $\ell(t)$,
vs $t$, for different $T_{s}$ values, for $q=5$. The early time behavior for
different $T_{s}$ are presented in the lower part of the divided graph. The
late time comparisons are in the upper part. The systems starting at higher
$T_{s}$ tend to approach the new equilibrium earlier. This conveys a picture
same as that derived from the energy decay, further strongly suggesting the
presence of the Mpemba effect. We record the times at which the domain lengths
of the systems at different finite $T_{s}$ ($<\infty$) values are crossed or
overtaken by the corresponding plots for the systems starting from
$T_{s}=\infty$. We denote this by $t_{c,\ell_{\infty}}$. In Fig. 3(b) we have
plotted $t_{c,\ell_{\infty}}$ as a function of $T_{s}-T_{c}$, for a few values
of $q$, covering transitions of first as well as second order varieties. For
each $q$, the crossing time increases with the approach of $T_{s}$ to the
corresponding $T_{c}$. Given that depending upon the value of $q$ the nature
of critical fluctuation is different, presence of any unique scaling behavior
may not emerge from this figure. It appears, nevertheless, that for a given
distance of $T_{s}$ from $T_{c}$, the crossing time is longer for higher $q$.
A quantitative picture for this is shown in Fig. 3(c). This is a signature
that the ME gets weaker with the increase of $q$. Considering the influence of
both $q$ and $T_{s}$, the issue, however, is complex, that we address later.

Figure 4: (a) Plots of $\ell(t)$ versus $t$, corresponding to a few different
$T_{s}$ values, for quenches to $0.3\,T_{c}$, for the LR Ising model. The
value of $\sigma$ is $0.8$ and we have $L=1024$. (b) Same as (a) but here the
results are for the 3D nearest-neighbor (NN) Ising model with
$T_{f}=0.6\,T_{c}$, and $L=256$.

Next we check whether the same scenario is true for the case of the LR Ising
model. Due to the demanding computation, we analyze results for this case
after averaging over $100$ independent initial configurations. Note that the
LR systems encounter finite-size effects much faster than its short-range
counterpart, due to faster growth [34, 44, 45] with the decrease of $\sigma$.
To avoid this problem, we choose big systems and a large value of $\sigma$,
viz., $\sigma=0.8$, which, nevertheless, falls well within the long-range
interaction domain [34]. In Fig. 4(a) we plot $\ell(t)$, vs $t$, for quenches
to $T_{f}=0.3\,T_{c}$, from three $T_{s}$ values, with $L=1024$. From these
plots it is clear that the systems with the highest $T_{s}$ have the largest
$\ell(t)$, at late times. Thus, ME appears to be present in the LR Ising model
as well. Note that because of the above mentioned reasons we have used Ewald
summation [35, 46], and parallelized our codes, in this case, to speed up the
output.

So far we have dealt with $2\mbox{D}$ systems. Now we present results from the
$3\mbox{D}$ NN Ising model in Fig. 4(b), where also the faster relaxation of
the systems for the higher $T_{s}$ value is quite clear. Here we have quenched
the systems from different initial $T_{s}$ values to $T_{f}=0.6\,T_{c}$. These
results are presented after averaging over runs with 1440 independent initial
configurations, with $L=256$.

Returning to the Potts results in Fig. 3, we recall that an important
objective of our work is to obtain a scaling picture [2]. Note that for
different $q$ values, one expects differing fluctuations in the critical
vicinity. Thus, a unique behavior of the data sets in Fig. 3(b) should not be
expected. It is more instructive to replace the abscissa variable there by
$\xi$. Results from such an exercise is shown in Fig. 5(a). On a log-log scale
it appears that the data sets from different $q$ are reasonably parallel to
each other. In Fig. 5(b), thus, we introduce a prefactor $a$, for the
abscissa, constant for a particular value of $q$, to obtain an overlap of the
data sets in Fig. 5(a). A nice collapse of the data sets can be appreciated.
In fact, the results for the $2\mbox{D}$ and $3\mbox{D}$ Ising models also
comply with that. It is worth mentioning here that accurate estimations of the
crossing times require huge statistics.

Figure 5: (a) Plots of $t_{c,\ell_{\infty}}$ versus $\xi$, for the Potts
model, with a few different $q$ values, on a double-log scale. (b) Same as (a)
but here the abscissa of the data sets are scaled by constant factors to
obtain a “possible” overlap. In addition to the results from the Potts cases
($q\geq 3$), here we have included data for the NN Ising model, from different
space dimensions. We expect discrepancy in scaling very close to $T_{c}$ due
to strong finite-size effects from multiple sources. Dashed lines represent
power-laws.

##  IV Conclusion

We have investigated the presence of the Mpemba effect [1, 2, 3, 4, 5] during
para- to ferromagnetic transitions in several model systems with discrete spin
values. These include short-range Ising model in $d=2$ and $3$, as well as
long-range Ising model in $d=2$. Very extensive set of results are presented
for the $q$-state Potts model for a wide range of $q$ values. It is important
to note that in none of the considered models there exist in-built
frustration. Irrespective of the space dimension, range of interaction and
order of transition, we have observed the Mpemba effect. It has interesting
connection with the length of spatial correlations at the considered initial
temperatures. The relative delay in approach to the final equilibrium,
following quenches from para to ferro regions, with the lowering of starting
temperatures, has unique dependence upon $\xi$. For second order transitions
we have obtained a universal scaling for models with critical exponent $\nu$
varying nearly by a factor of $1.6$. More interestingly, the scaling is valid
for even first order transitions. This implies, for a given model, if two
initial temperatures possess nearly same spatial correlations, possible for
large $q$, configurations from these states will equilibrate almost
simultaneously at the final temperature, showing no detectable ME, even for
large differences in the $T_{s}$, in terms of times taken for reaching the
final destination. We believe that our results contain important message for
the understanding of the effect in water. Particularly the observation of it
in the cases of first order transition can shed light on the mystery with
respect to the latter. It will be interesting to investigate how the power-law
dependence upon $\xi$, with exponent $0.9$, may be connected to the scaling
picture described in Ref. [2].

Author contributions: SKD proposed the topic, designed the problem,
participated in the analyses, supervised the work and wrote the manuscript. NV
oversaw a few coding details and took part in progress on all the models at
the initial stages, alongside contributing to the writing. SC obtained all the
final results on the Potts model, analyzed these, and contributed to the
writing. SG and TP obtained and analyzed the results on the long-range and the
3D Ising models, respectively. Simulations of SKS provided the first hints of
the Mpemba effect in the 3D Ising model.

Acknowledgments: SKD acknowledges a discussion with R. Pandit at an early
stage and partial financial support from Science and Engineering Research
Board, India, via Grant No. MTR/2019/001585. The authors are thankful to the
supercomputing facility, PARAM Yukti, at JNCASR, under National Supercomputing
Mission.

## References

  * [1] E.B. Mpemba and D.G. Osborne, Cool?, Physics Education 4, 172 (1969).
  * [2] S.K. Das, Perspectives on a Few Puzzles in Phase Transformations: When Should the Farthest Reach the Earliest?, Langmuir (2023).
  * [3] J Bechhoefer, A Kumar, R Chétrite, A Fresh Understanding of the Mpemba Effect, Nat. Rev. Phys. 3, 534 (2021).
  * [4] D. Auerbach, Supercooling and the Mpemba Effect: When Hot Water Freezes Quicker Than Cold, Am. J. Phys. 63, 882 (1995).
  * [5] M. Jeng, The Mpemba Effect: When Can Hot Water Freeze Faster Than Cold?, Am. J. Phys. 74, 514 (2006).
  * [6] X. Zhang et al., Hydrogen-bond Memory and Water-skin Supersolidity Resolving the Mpemba Paradox, Phys. Chem. Chem. Phys. 16, 22995 (2014).
  * [7] J. Jin and W.A. Goddard III, Mechanisms Underlying the Mpemba Effect in Water from Molecular Dynamics Simulations, J. Phys. Chem. C 119, 2622 (2015).
  * [8] Y. Tao, W. Zou, J. Jia, W. Li, and D. Cremer, Different Ways of Hydrogen Bonding in Water - Why Does Warm Water Freeze Faster than Cold Water?, J. Chem. Theory and Computation 13, 55 (2017).
  * [9] Z. Tang, W. Huang, Y. Zhang, Y. Liu, and L. Zhao, Direct Observation of the Mpemba Effect with Water: Probe the Mysterious Heat Transfer, InfoMat 5, e12352 (2022).
  * [10] A. Kumar and J. Bechhoefer, Exponentially Faster Cooling in a Colloidal System, Nature 584, 64 (2020).
  * [11] Y.-H. Ahn, H. Kang, D.-Y. Koh, and H. Lee, Experimental Verifications of Mpemba-Like Behaviors of Clathrate Hydrates, Korean J. Chem. Eng. 33, 1903 (2016).
  * [12] P.A. Greaney, G. Lani, G. Cicero, and J.C. Grossman, Mpemba-Like Behavior in Carbon Nanotube Resonators, Metall. Mater. Trans. A 42, 3907 (2011).
  * [13] P. Chaddah, S. Dash, K. Kumar, and A. Banerjee, Overtaking While Approaching Equilibrium, arXiv.1011.3598 (2010).
  * [14] A. Lasanta, F.V. Reyes, A. Prados, and A. Santos, When the Hotter Cools More Quickly: Mpemba Effect in Granular Fluids, Phys. Rev. Lett. 119, 148001 (2017).
  * [15] A. Torrente, M.A. López-Castaño, A. Lasanta, F.V. Reyes, A. Prados, and A. Santos, Large Mpemba-Like Effect in a Gas of Inelastic Rough Hard Spheres, Phys. Rev. E 99, 060901(R) (2019).
  * [16] A. Biswas, V.V. Prasad, O. Raz, and R. Rajesh, Mpemba Effect in Driven Granular Maxwell Gases, Phys. Rev. E 102, 012906 (2020).
  * [17] R. Gómez González and V. Garzó, Time-dependent homogeneous states of binary granular suspensions, Physics of Fluids 33, 093315 (2021).
  * [18] M. Baity-Jesi et al., The Mpemba Effect in Spin Glasses is a Persistent Memory Effect, Proc. Natl. Acad. Sci. U. S. A. 116, 15350 (2019).
  * [19] Z. Lu and O. Raz, Nonequilibrium Thermodynamics of the Markovian Mpemba Effect and its Inverse, Proc. Natl. Acad. Sci. U. S. A. 114, 5083 (2017).
  * [20] A. Gal and O. Raz, Precooling Strategy Allows Exponentially Faster Heating, Phys. Rev. Lett. 124, 060602 (2020).
  * [21] N. Vadakkayil and S.K. Das, Should a Hotter Paramagnet Transform Quicker To a Ferromagnet? Monte Carlo Simulation Results for Ising Model, Phys. Chem. Chem. Phys. 23, 11186 (2021).
  * [22] R. Holtzman and O. Raz, Landau theory for the Mpemba effect through phase transitions, Commun Phys 5, 280 (2022).
  * [23] A. Biswas, R. Rajesh, and A. Pal, Mpemba effect in a Langevin system: population statistics, metastability, and other exact results, J. Chem. Phys. 159, 044120 (2023).
  * [24] M.R. Walker and M. Vucelja, Anomalous thermal relaxation of Langevin particles in a piecewise-constant potential, J. Stat. Mech.: Theory and Expt., 113105 (2021).
  * [25] M. Vynnycky and S. Kimura, Can Natural Convection Alone Explain the Mpemba Effect?, Int. J. Heat Mass Transf. 80, 243 (2015).
  * [26] Z.Y. Yang and J.-X. Hou, Mpemba effect of a mean-field system: The phase transition time, Phys. Rev. E 105, 014119 (2022).
  * [27] F.J. Schwarzendahl and H. Löwen, Anomalous Cooling and Overcooling of Active Colloids, Phys. Rev. Lett. 129, 138002 (2022).
  * [28] Z. Cao, R. Bao, J. Zheng, and Z. Hou, Fast Functionalization with High Performance in the Autonomous Information Engine, J. Phys. Chem. Lett. 14, 66 (2023).
  * [29] S. Zhang and J.-X. Hou, Theoretical model for the Mpemba effect through the canonical first-order phase transition, Phys. Rev E 106, 034131 (2022).
  * [30] Aristotle, Meteorologica, translated by H.D.P. Lee (Harvard University Press, 1962), Book I, Chap. XII, pp. 85-87.
  * [31] M. Matsumoto, S. Saito, and I. Ohmine, Molecular Dynamics Simulation of the Ice Nucleation and Growth Process Leading To Water Freezing, Nature 416, 409 (2002).
  * [32] D.P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, Cambridge, 2009).
  * [33] K. Binder, Static and dynamic critical phenomena of the two-dimensional $q$-state Potts model, J. Stat. Phys. 24, 69 (1981).
  * [34] A.J. Bray, Domain-growth Scaling in Systems with Long-range Interactions, Phys. Rev. E 47, 3191 (1993).
  * [35] T. Horita, H. Suwa, and S. Todo, Upper and Lower Critical Decay Exponents of Ising Ferromagnets with Long-range Interaction, Phys. Rev. E 95, 012143 (2017).
  * [36] U. Wolff, Collective Monte Carlo Updating for Spin Systems, Phys. Rev. Lett. 62, 361 (1989).
  * [37] K. Fukui and S. Todo, Order-N Cluster Monte Carlo Method for Spin Systems with Long-Range Interactions, J. Comp. Phys. 228, 2629 (2009).
  * [38] S. Puri, R. Ahluwalia, and A.J. Bray, Dynamical crossover in the clock model with a conserved order parameter, Phys. Rev. E 55, 2345 (1997).
  * [39] M.E. Fisher, The theory of equilibrium critical phenomena, Rep. Prog. Phys. 30, 615 (1967).
  * [40] H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Clarendon Press, Oxford, 1971).
  * [41] S. Majumder and S.K. Das, Diffusive domain coarsening: Early time dynamics and finite-size effects, Phys. Rev. E 84, 021110 (2011).
  * [42] S. Chakraborty, S.K. Das, Role of initial correlation in coarsening of a ferromagnet, Eur. Phys. J. B 88, 1-9 (2015).
  * [43] A.J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 51, 481 (2002).
  * [44] H. Christiansen, S. Majumder, and W. Janke, Phase ordering kinetics of the long-range Ising model, Phys. Rev. E 99, 011301(R) (2019).
  * [45] S. Ghosh, S.K. Das, Aging During Phase Separation in Long-Range Ising Model, arXiv:2304.04996 (2023).
  * [46] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids (Oxford University Press, 1991).