# Expansivity and structurally stability of composition operators on Orlicz
spaces

Y. Estaremi Y. Estaremi<EMAIL_ADDRESS>Department of Mathematics, Faculty
of Sciences, Golestan University, Gorgan, Iran.

###### Abstract.

The notions of expansivity and positively expansivity for composition
operators on Orlicz spaces are investigated. We provide some necessary and
sufficient conditions for expansivity, positively expansivity and uniformly
expansivity of such operators. Moreover, we provide some equivalent conditions
for these concepts in the case that the system is dissipative. Our results are
essentially based on the results of [16].

###### Key words and phrases:

Orlicz spaces, Composition operators, Expansivity, structural stability.

###### 2020 Mathematics Subject Classification:

47B33, 37B65

##  1\.  Introduction and Preliminaries

The dynamics of continuous linear operators on Banach spaces have been
extensively studied during the last three decades. Many connections between
dynamical systems and other areas of mathematics, such as ergodic theory,
number theory and geometry of Banach spaces have been established. Interested
readers can refer to [4, 8, 1, 2, 6, 14]. In [11] Eisenberg and Hedlund
investigated relationships between expansivity and spectrum of operators on
Banach spaces. Also, in [17] the authors established relationships between
notions of expansivity and hypercyclicity, supercyclicity, Li–Yorke chaos and
shadowing. In the case that the Banach space is $c_{0}$ or $l_{p}$ $(1\leq
p<\infty)$, they have given complete characterizations of weighted shifts
which satisfy various notions of expansivity. In [16], the authors
investigated the two notions of expansivity and strong structural stability
for composition operators on $L^{p}(\mu)$-spaces, $1\leq p<\infty$. Necessary
and sufficient conditions for composition operators to be expansive have
provided, both in the general and the dissipative case. They also have shown
that, in the dissipative setting, the shadowing property implies the strong
structural stability and they have proved that these two notions are
equivalent under the extra hypothesis of positive expansivity. In this paper
we investigate the notions of expansivity and positively expansivity for
composition operators on Orlicz spaces. Indeed, first we extend the
definitions of some concepts like, dissipative system and bounded distortion
property to the setting of Orlicz spaces, in way that it coincides to the
definition of these concepts in $L^{p}(\mu)$-spaces, in special case. Then we
provide some necessary and sufficient conditions for expansivity, positively
expansivity and uniformly expansivity of such operators. Moreover, we provide
some equivalent conditions for these concepts in the case that the system is
dissipative.

In the sequel, for the convenience of the reader, we recall some essential
facts on Orlicz spaces for later use. For more details on Orlicz spaces, see
[15, 18].

A function $\Phi:\mathbb{R}\rightarrow[0,\infty]$ is called a Young’s function
if $\Phi$ is convex, $\Phi(-x)=\Phi(x)$, $\Phi(0)=0$ and
$\lim_{x\rightarrow\infty}\Phi(x)=+\infty$. With each Young’s function $\Phi$,
one can associate another convex function
$\Psi:\mathbb{R}\rightarrow[0,\infty]$ having similar properties defined by

$\Psi(y)=\sup\\{x\mid y\mid-\Phi(x):x\geq 0\\},\ \ \ \ \ y\in\mathbb{R}.$

The function $\Psi$ is called the complementary function to $\Phi$. It follows
from the definition that $\Psi(0)=0$, $\Psi(-y)=\Psi(y)$ and $\Psi(.)$ is a
convex increasing function satisfying
$\lim_{y\rightarrow\infty}\Psi(y)=+\infty$.

Let $\Phi$ be a Young’s function. Then we say $\Phi$ satisfies the
$\Delta_{2}$-condition or $\Phi$ is $\Delta_{2}$-regular, if $\Phi(2x)\leq
K\Phi(x)\;(x\geq x_{0})$ for some constants $K>0$ and $x_{0}>0$. Also, $\Phi$
satisfies the $\Delta_{2}$-condition globally if $\Phi(2x)\leq
K\Phi(x)\;(x\geq 0)$ for some $K>0$. If $x_{0}=0$, these conditions are said
to hold globally.

For a given complete $\sigma$-finite measure space$(X,\mathcal{F},\mu)$, let
$L^{0}(\mathcal{F})$ be the linear space of equivalence classes of
$\mathcal{F}$-measurable real-valued functions on $X$, that is, we identify
functions equal $\mu$-almost everywhere on $X$. The support $S(f)$ of a
measurable function $f$ is defined by $S(f):=\\{x\in X:f(x)\neq 0\\}$.

For a Young’s function $\Phi$, let
$\rho_{\Phi}:L^{\Phi}(\mu)\rightarrow\mathbb{R}^{+}$ such that
$\rho_{\Phi}(f)=\int_{X}\Phi(f)d\mu$ for all $f\in L^{\Phi}(\mu)$. Then the
space

$L^{\Phi}(\mu)=\left\\{f\in L^{0}(\mathcal{F}):\exists
k>0,\rho_{\Phi}(kf)<\infty\right\\}$

is called an Orlicz space. Define the functional

$N_{\Phi}(f)=\inf\\{k>0:\rho_{\Phi}(\frac{f}{k})\leq 1\\}.$

The functional $N_{\Phi}(.)$ is a norm on $L^{\Phi}(\mu)$ and is called guage
norm(or Luxemburge norm). Also, $(L^{\Phi}(\mu),N_{\Phi}(.))$ is a normed
linear space. If almost every where equal functions are identified, then
$(L^{\Phi}(\mu),N_{\Phi}(.))$ is a Banach space, the basic measure space
$(X,\mathcal{F},\mu)$ is unrestricted. Hence every element of $L^{\Phi}(\mu)$
is a class of measurable functions that are almost every where equal. Also,
there is another norm on $L^{\Phi}(\mu)$, defined as follow:

(1.1) $\|f\|_{\Phi}=\sup\\{\int_{X}\mid fg\mid d\mu:g\in
B_{\Psi}\\}=\sup\\{\mid\int_{X}fgd\mu\mid:g\in B_{\Psi}\\},$

in which $B_{\Psi}=\\{g\in L^{\Psi}(\mu):\int_{X}\Psi(\mid g\mid)d\mu\leq
1\\}$. The norm $\|.\|_{\Phi}$ is called Orlicz norm. For any $f\in
L^{\Phi}(\mu)$, $\Phi$ being a Young function, we have

(1.2) $N_{\Phi}(f)\leq\|f\|_{\Phi}\leq 2N_{\Phi}(f).$

For every $F\in\mathcal{F}$ with $0<\mu(F)<\infty$, we have
$N_{\Phi}(\chi_{F})=\frac{1}{\Phi^{-1}(\frac{1}{\mu(F)})}$. Moreover, if
$\Phi\in\Delta_{2}$, then $(L^{\Phi}(\mu))^{*}=L^{\Psi}(\mu)$, in which $\Psi$
is the complementary Young’s function to $\Phi$. Here we recall the next
theorem about convergence of sequences in Orlicz spaces.

######  Theorem 1.1.

[18] Let $\\{f_{n}\\}_{n\geq 1}$ be a sequence from $L^{\Phi}(\mu)$ and $f\in
L^{\Phi}(\mu)$. Then the following assertions hold:

(a) If $\|f_{n}-f\|_{\Phi}\rightarrow 0$ (or equivalently
$N_{\Phi}(f_{n}-f)\rightarrow 0$), then
$\rho_{\Phi}(f_{n})\rightarrow\rho_{\Phi}(f)$. The converse holds if $\Phi$ is
$\triangle_{2}$-regular.

(b) If $\Phi$ is $\triangle_{2}$-regular Young function, or if $\Phi$ is
continuous and concave $\Phi(0)=0$, $\Phi\nearrow$ as well,
$\rho_{\Phi}(f_{n})\rightarrow\rho_{\Phi}(f)$ as $n\rightarrow\infty$ and
$f_{n}\rightarrow f$ a.e., or in $\mu$-measure, then $f_{n}\rightarrow f$ in
norm.

Throughout this paper, $(X,\mathcal{F},\mu)$ will be a measure space, that is,
$X$ is a set, $\mathcal{F}$ is a sigma algebra on $X$ and $\mu$ is a positive
measure on $\mathcal{F}$. Also, we assume that $\varphi:X\rightarrow X$ is a
non-singular measurable transformation, that is,
$\varphi^{-1}(F)\in\mathcal{F}$, for every $F\in\mathcal{F}$ and
$\mu(\varphi^{-1}(F))=0$, if $\mu(F)=0$. Moreover, for the non-singular
measurable transformation $\varphi:X\rightarrow X$, if there exists a positive
constant $c$ for which

$\mu(\varphi^{-1}(F))\leq c\mu(F),\ \ \ \ \text{for every}F\in\mathcal{F},$

then the linear operator

$C_{\varphi}:L^{\Phi}(\mu)\rightarrow L^{\Phi}(\mu),\ \ \ \ \ \
C_{\varphi}(f)=f\circ\varphi,$

is well-defined and continuous on the Orlicz space $L^{\Phi}(\mu)$ and is
called composition operator. For more details on composition operators on
Orlicz spaces one can refer to [9].

##  2\. Expansivity for composition operators

Here we recall some results of [7].

######  Proposition 2.1.

Let $T$ be an operator on a Banach space $X$. Then:
(a) $T$ is positively expansive if and only if
$\sup_{n\in\mathbb{N}}\|T^{n}x\|=\infty$, for every nonzero $x\in X$.
(b) $T$ is uniformly positively expansive if and only if
$\lim_{n\rightarrow\infty}\|T^{n}x\|=\infty$, uniformly on $S_{X}$.

If, in addition, $T$ is invertible, then:
(c) $T$ is expansive if and only if $\sup_{n\in\mathbb{Z}}\|T^{n}x\|=\infty$,
for every nonzero $x\in X$.
(d) $T$ is uniformly expansive if and only if $S_{X}=A\cup B$, where
$\lim_{n\rightarrow\infty}\|T^{n}x\|=\infty$, uniformly on $A$ and
$\lim_{n\rightarrow\infty}\|T^{-n}x\|=\infty$, uniformly on $B$.

Let $C_{\varphi}:L^{\Phi}(\mu)\rightarrow L^{\Phi}(\mu)$ be composition
operator and $\mathcal{F}^{+}=\\{F\in\mathcal{F}:0<\mu(F)<\infty\\}$. Here we
rewrite the definition of composition dynamical system, that is used in [10]
and similar papers for composition operators on $L^{p}$-spaces, for
composition operators on Orlicz spaces $L^{\Phi}(\mu)$, in which
$\Phi\in\Delta_{2}$.

######  Definition 2.2.

A composition dynamical system is a quintuple
$(X,\mathcal{F},\mu,\varphi,C_{\varphi})$ where

  1. (1)

$(X,\mathcal{F},\mu)$ is a $\sigma$-finite measure space,

  2. (2)

$\varphi:X\rightarrow X$ is an injective bimeasurable transformation, i.e.,
$\varphi(F)\in\mathcal{F}$ and $\varphi^{-1}(F)\in\mathcal{F}$ for every
$F\in\mathcal{F}$,

  3. (3)

there is $c>0$ such that

(2.1) $\mu(\varphi^{-1}(A)\leq c\mu(A)\ \ \ \ \ \ \text{for every}\ \
A\in\mathcal{F},$

  4. (4)

$C_{\varphi}:L^{\Phi}(\mu)\rightarrow L^{\Phi}(\mu)$ is the composition
operator induced by $\varphi$, that is

$C_{\varphi}f=f\circ\varphi.$

It is known that the condition 2.1 is a necessary and sufficient condition for
boundeness of $C_{\varphi}$ on Orlicz space $L^{\Phi}(\mu)$. In addition, if
$\varphi$ is bijective and $\varphi^{-1}$ satisfies in condition 2.1 then
$C_{\varphi^{-1}}$ is a bounded linear operators and
$C^{-1}_{\varphi}=C_{\varphi^{-1}}$. For more details on composition operators
one can refer to [19].

Also, by the fact that for every $F\in\mathcal{F}$ with $0<\mu(F)<\infty$, we
have
$\|\chi_{F}\|_{p}=(\int_{X}|\chi_{F}|^{p}d\mu)^{\frac{1}{p}}=\mu(F)^{\frac{1}{p}}$
and $N_{\Phi}(\chi_{F})=\frac{1}{\Phi^{-1}(\frac{1}{\mu(F)})}$, we can rewrite
the definition of wandering set, dissipative system and dissipative system of
bounded distortion in Orlicz spaces, that were defined for $L_{p}$ spaces
before in some contexts such as [10].

######  Definition 2.3.

Let $(X,\mathcal{F},\mu)$ be a measure space and $\varphi:X\rightarrow X$ be
an invertible non-singular transformation. A measurable set $W\subseteq X$ is
called a wandering set for $\varphi$ if the sets
$\\{\varphi^{-n}(W)\\}_{n\in\mathbb{Z}}$ are pair-wise disjoint.

######  Definition 2.4.

Let $(X,\mathcal{F},\mu)$ be a measure space and $\varphi:X\rightarrow X$ be
an invertible non-singular transformation. The quadruple
$(X,\mathcal{F},\mu,\varphi)$ is called

  * •

a dissipative system generated by $W$, if
$X=\dot{\cup}_{k\in\mathbb{Z}}\varphi^{k}(W)$ for some $W\in\mathcal{F}$ with
$0<\mu(W)<\infty$ (the symbol $\dot{\cup}$ denotes pairwise disjoint union);

  * •

a dissipative system, of bounded distortion, generated by $W$, if there exists
$K>0$, such that

(2.2)
$\frac{1}{K}\frac{N_{\Phi}(C^{k}_{\varphi}(\chi_{W}))}{N_{\Phi}(\chi_{W})}\leq\frac{N_{\Phi}(C^{k}_{\varphi}(\chi_{F}))}{N_{\Phi}(\chi_{F})}\leq
K\frac{N_{\Phi}(C^{k}_{\varphi}(\chi_{W}))}{N_{\Phi}(\chi_{W})},$

for all $k\in\mathbb{Z}$ and $F\in\mathcal{F}_{W}=\\{F\cap
W:F\in\mathcal{F}\\}$. If we replace $N_{\Phi}$ by the norm of $L_{p}$, then
we will have the bounded distortion property in $L_{p}$-spaces.(Definition
2.6.4, [10])

######  Definition 2.5.

A composition dynamical system $(X,\mathcal{F},\mu,\varphi,C_{\varphi})$ is
called

  * •

dissipative composition dynamical system, generated by $W$, if
$(X,\mathcal{F},\mu,\varphi)$ is a dissipative system generated by $W$;

  * •

dissipative composition dynamical system, of bounded distortion, generated by
$W$, if $(X,\mathcal{F},\mu,\varphi)$ is a dissipative system of bounded
distortion, generated by $W$.

In the following we recall Proposition 2.4 of [12].

######  Proposition 2.6.

Let $(X,\mathcal{F},\mu,\varphi)$ be a dissipative system of bounded
distortion, generated by $W$. Then there exists $H>0$ such that

(2.3)
$\frac{1}{H}\frac{N_{\Phi}(C^{t+s}_{\varphi}(\chi_{W}))}{N_{\Phi}(C^{s}_{\varphi}(\chi_{W}))}\leq\frac{N_{\Phi}(C^{t+s}_{\varphi}(\chi_{F}))}{N_{\Phi}(C^{s}_{\varphi}(\chi_{F}))}\leq
H\frac{N_{\Phi}(C^{t+s}_{\varphi}(\chi_{W}))}{N_{\Phi}(C^{s}_{\varphi(\chi_{W})})},$

for all $F\in\mathcal{F}_{W}$ with $\mu(F)>0$ and all $s,t\in\mathbb{Z}$.

Now in the next Theorem we obtain equivalent conditions for composition
operators to be expansive, positively expansive and uniformly positively
expansive.

######  Theorem 2.7.

Let $(X,\mathcal{F},\mu,\varphi,C_{\varphi})$ be a composition dynamical
system. The followings hold.

  1. (1)

The composition operator $C_{\varphi}$ is positively expansive if and only if
for every $A\in\mathcal{F}$, with $0<\mu(A)<\infty$, we have

$\inf_{n\in\mathbb{N}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(A))})=0.$

  2. (2)

The composition operator $C_{\varphi}$ is expansive if and only if for every
$A\in\mathcal{F}$, with $0<\mu(A)<\infty$, we have

$\inf_{n\in\mathcal{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(A))})=0.$

  3. (3)

If $\Phi$ is $\Delta_{2}$-regular, then the composition operator $C_{\varphi}$
is uniformly positively expansive if and only if

$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(A)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(A))})}=\infty,$

for all $A\in\mathcal{F}$, with $0<\mu(A)<\infty$.

  4. (4)

If $\Phi$ is $\Delta_{2}$-regular, then the composition operator is uniformly
expansive if and only if $\mathcal{F}^{+}$ can be splitted as
$\mathcal{F}^{+}=\mathcal{F}^{+}_{A}\cup\mathcal{F}^{+}_{B}$ such that

$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{n}(F))})}=\infty,\
\ \ \text{uniformly on }\ \mathcal{F}^{+}_{A},$
$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}=\infty,\
\ \ \text{uniformly on }\ \mathcal{F}^{+}_{B}.$

###### Proof.

First we prove (2). Let $C_{\varphi}$ be expansive. Then by part (c) of the
Proposition 2.1 we have

$\sup_{n\in\mathbb{Z}}N_{\Phi}(C^{n}_{\varphi}f)=\sup_{n\in\mathbb{Z}}N_{\Phi}(C_{\varphi^{n}}f)=\infty,\
\ \ \ \ \text{for each}\ \ \ f\in L^{\Phi}(\mu)\setminus\\{0\\}.$

Now if $F\in\mathcal{F}$ with $0<\mu(F)<\infty$, then by setting $f=\chi_{F}$
and for every $n\in\mathcal{Z}$ we have

$N_{\Phi}(C^{n}_{\varphi}f)=N_{\Phi}(\chi_{\varphi^{-n}(F)})=\frac{1}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}.$

So we have

$\sup_{n\in\mathbb{Z}}N_{\Phi}(C^{n}_{\varphi}f)=\sup_{n\in\mathbb{Z}}\frac{1}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}=\infty$

and therefore

$\inf_{n\in\mathbb{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})=0.$

Now we prove the converse. Suppose that for each $F\in\mathcal{F}$, with
$0<\mu(F)<\infty$, we have
$\inf_{n\in\mathbb{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})=0$. If $f\in
L^{\Phi}(\mu)\setminus\\{0\\}$, then there exists a positive constant $\delta$
such that the set $F_{\delta}=\\{x\in X:|f(x)|>\delta\\}$ has positive
measure, i.e, $\mu(F_{\delta})>0$. Since $(X,\mathcal{F},\mu)$ is
$\sigma$-finite, then without loss of generality we can assume that
$\mu(F_{\delta})<\infty$. So for every $n\in\mathbb{Z}$:

$\displaystyle\int_{X}\Phi(\frac{\delta.\chi_{F_{\delta}}\circ\varphi^{n}}{N_{\Phi}(C^{n}_{\varphi})})d\mu$
$\displaystyle\leq\int_{\varphi^{-n}(F_{\delta})}\Phi(\frac{f\circ\varphi^{n}}{N_{\Phi}(C^{n}_{\varphi})})d\mu$
$\displaystyle\leq\int_{X}\Phi(\frac{f\circ\varphi^{n}}{N_{\Phi}(C^{n}_{\varphi})})d\mu$
$\displaystyle\leq 1.$

Hence $N_{\Phi}(\delta.\chi_{F_{\delta}})\leq N_{\Phi}(C^{n}_{\varphi}f)$ and
so

$\delta.\frac{1}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F_{\delta}))})}\leq
N_{\Phi}(C^{n}_{\varphi}f).$

This inequality implies that if

$\inf_{n\in\mathbb{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F_{\delta}))})=0,$

then $\sup_{n\in\mathbb{Z}}N_{\Phi}(C^{n}_{\varphi}f)=\infty$. Since $f$ is
chosen arbitrarily, then by part (c) of the Proposition 2.1 we get that
$C_{\varphi}$ is expansive on $L^{\Phi}(\mu)$.
For the proof of part (1) it suffices just to replace $\mathbb{Z}$ by
$\mathbb{N}$.
(3). Let $C_{\varphi}$ be uniformly positively expansive. Then by Proposition
2.1 we have

$\lim_{n\rightarrow\infty}N_{\Phi}(C^{n}_{\varphi}f)=\infty,\ \ \ \
\text{uniformly on}\ \ \ \ \ S_{L^{\Phi}(\mu)},$

in which $S_{L^{\Phi}(\mu)}=\\{f\in L^{\Phi}(\mu):N_{\Phi}(f)=1\\}$. Let
$f=\Phi^{-1}(\frac{1}{\mu(F)})\chi_{F}$, for each $F\in\mathcal{F}^{+}$. Then
we have $N_{\Phi}(f)=1$ and also

$N_{\Phi}(C^{n}_{\varphi}f)=\Phi^{-1}(\frac{1}{\mu(F)})N_{\Phi}(\chi_{\varphi^{-n}(F)})=\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}.$

So by our assumptions we get that

$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}=\infty,\
\ \ \ \ \text{uniformly on the sets}\ \ \ F\in\mathcal{F}^{+}.$

Conversely, since $\Phi\in\Delta_{2}$, then the simple functions are dens in
$L^{\Phi}(\mu)$. So it suffices to prove that for any simple function $f\in
S_{L^{\Phi}(\mu)}$, the equality

$\lim_{n\rightarrow\infty}N_{\Phi}(C^{n}_{\varphi}F)=\infty$

holds. By our assumptions, for each $M>0$ there exists $N\in\mathbb{N}$ such
that for every $F\in\mathcal{F}^{+}$, we have

$\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}>M,\
\ \ \ \ \ \text{for each}\ \ \ n\geq N.$

By equation 1.1, it is easy to see that for $f,g\in L^{\Phi}(\mu)$, with
$S(f)\cap S(g)\neq\emptyset$, there exists $c>0$ such that

$\|f+g\|_{\Phi}\geq c(\|f\|_{\Phi}+\|g\|_{\Phi}).$

Now for every simple function $f\in S_{L^{\Phi}(\mu)}$, with
$f=\Sigma^{m}_{i=1}\alpha_{i}\chi_{F_{i}}$,
$\alpha_{i}\in\mathbb{C}\setminus\\{0\\}$ and $F_{i}\in\mathcal{F}$, without
loss of generality we can assume that measurable sets $\\{F_{i}\\}^{m}_{i=1}$
are pairwise disjoint. Hence by inequalities 1.2, for any $n\geq N$, we have

$\displaystyle N_{\Phi}(C^{n}_{\varphi}f)$
$\displaystyle=N_{\Phi}(\Sigma^{m}_{i=1}\alpha_{i}\chi_{F_{i}}\circ\varphi^{n})$
$\displaystyle\geq\frac{c}{2}\Sigma^{m}_{i=1}|\alpha_{i}|N_{\Phi}\chi_{\varphi^{-n}(F_{i})}$
$\displaystyle=\frac{c}{2}\Sigma^{m}_{i=1}|\alpha_{i}|\frac{1}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F_{i}))})}$
$\displaystyle\geq\frac{cM}{2}\Sigma^{m}_{i=1}|\alpha_{i}|\frac{1}{\Phi^{-1}(\frac{1}{\mu(F_{i})})}$
$\displaystyle=\frac{cM}{2}\Sigma^{m}_{i=1}|\alpha_{i}|N_{\Phi}(\chi_{F_{i}})$
$\displaystyle\geq\frac{cM}{2}N_{\Phi}(f)$ $\displaystyle=\frac{cM}{2}.$

This implies that
$\lim_{n\rightarrow\infty}N_{\Phi}(C^{n}_{\varphi}f)=\infty$. As is known, for
every $f\in S_{L^{\Phi}(\mu)}$, we have a decomposition as $f=f^{+}-f^{-}$, in
which $f^{+},f^{-}$ are positive measurable functions. We can find a sequence
of simple functions $\\{f_{k}\\}$ with $|f_{k}|\leq 2|f|$, such that converges
point-wisely to $f$ and so the sequence $\\{\Phi(f_{k})\\}$ is also convergent
point-wisely to $\Phi(f)$. Then by using Lebesgue Dominated Convergence
Theorem we have

$\lim_{k\rightarrow\infty}\rho_{\Phi}(f_{k})=\rho_{\Phi}(f).$

Hence by Theorem 1.1 we get that

$\lim_{k\rightarrow\infty}N_{\Phi}(f_{k})=N_{\Phi}(f)=1.$

Therefore by combining the above observations we get that
$\lim_{n\rightarrow\infty}N_{\Phi}(C^{n}_{\varphi}f)=\infty$.
(4) Let $C_{\varphi}$ be uniformly expansive. Then by part (d) of the
Proposition 2.1 we have a decomposition as $S_{L^{\Phi}(\mu)}=A\cup B$, where

$\lim_{n\rightarrow\infty}N_{\Phi}(C^{n}_{\varphi}f)=\infty,\ \ \ \
\text{uniformly on}\ \ \ \ A$

and

$\lim_{n\rightarrow\infty}N_{\Phi}(C^{-n}_{\varphi}f)=\infty,\ \ \ \
\text{uniformly on}\ \ \ \ B.$

If we put $\mathcal{F}^{+}=\\{F\in\mathcal{F}:0<\mu(F)<\infty\\}$, then some
basic properties of Orlicz spaces $L^{\Phi}(\mu)$ and the fact that simple
functions are dense in them give us:
$\mathcal{F}^{+}=\mathcal{F}^{+}_{A}\cup\mathcal{F}^{+}_{B}$ in which

$\mathcal{F}^{+}_{A}=\\{F\in\mathcal{F}^{+}:\Phi^{-1}(\frac{1}{\mu(F)})\chi_{F}\in
A\\}\ \ \ \text{and}\ \ \ \
\mathcal{F}^{+}_{B}=\\{F\in\mathcal{F}^{+}:\Phi^{-1}(\frac{1}{\mu(F)})\in
B\\}.$

Since

$\Phi^{-1}(\frac{1}{\mu(F)})\chi_{F}\in S_{L^{\Phi}(\mu)}\ \ \text{and}\ \ \
C^{n}_{\varphi}(\Phi^{-1}(\frac{1}{\mu(F)})\chi_{F})=\Phi^{-1}(\frac{1}{\mu(F)})\chi_{\varphi^{-n}(F)},$

then by our assumptions we have

$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(F)}}{\Phi^{-1}(\frac{1}{\mu(\varphi^{n}(F))})}=\infty,\
\ \ \ \ \text{on}\ \ \ \mathcal{F}^{+}_{A}$

and

$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(F)}}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}=\infty,\
\ \ \ \ \text{on}\ \ \ \mathcal{F}^{+}_{B}.$

So we get the result. Now to prove the converse, we use again part (d) of the
Proposition 2.1. Hence it suffices to show that $S_{L^{\Phi}(\mu)}=A\cup B$,
such that

$\lim_{n\rightarrow\infty}N_{\Phi}(C^{n}_{\varphi}f)=\infty,\ \ \ \
\text{uniformly on}\ \ \ \ A$

and

$\lim_{n\rightarrow\infty}N_{\Phi}(C^{-n}_{\varphi}f)=\infty,\ \ \ \
\text{uniformly on}\ \ \ \ B.$

By the assumptions, for every $M>0$, there exists $N\in\mathbb{N}$ such that
for every $n\geq N$,

$\frac{\Phi^{-1}(\frac{1}{\mu(F)}}{\Phi^{-1}(\frac{1}{\mu(\varphi^{n}(F))})}>M\
\ \text{for each}\ F\in\mathcal{F}^{+}_{A}$

and

$\frac{\Phi^{-1}(\frac{1}{\mu(F)}}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}>M\
\ \text{for each}\ F\in\mathcal{F}^{+}_{B}.$

Now, let $f\in S_{L^{\Phi}(\mu)}$ be a simple function with
$f=\sum_{i=1}^{m}\alpha_{i}\chi_{F_{i}}$, in which $F_{i}$’s are pairwise
disjoint measurable sets with positive measure. Since
$\mathcal{F}^{+}=\mathcal{F}^{+}_{\mathcal{A}}\cup\mathcal{F}^{+}_{\mathcal{B}}$,
then we can define

$f_{\mathcal{F}^{+}_{A}}=\sum_{\\{1\leq i\leq
m:F_{i}\in\mathcal{F}^{+}_{A}\\}}\alpha_{i}\chi_{F_{i}}$

and

$f_{\mathcal{F}^{+}_{B}}=\sum_{\\{1\leq i\leq
m:F_{i}\in\mathcal{F}^{+}_{B}\\}}\alpha_{i}\chi_{F_{i}}.$

It is clear that $f=f_{\mathcal{F}^{+}_{A}}+f_{\mathcal{F}^{+}_{B}}$. So by
equation 1.1 there exists $c>0$ such that

$\|f\|_{\Phi}\geq
c(\|f_{\mathcal{F}^{+}_{A}}\|_{\Phi}+\|f_{\mathcal{F}^{+}_{B}}\|_{\Phi})\geq
c\|f\|_{\Phi}$

and since $N_{\Phi}(f)=1$, then by inequalities 1.2 we have

$2\geq
c(\|f_{\mathcal{F}^{+}_{A}}\|_{\Phi}+\|f_{\mathcal{F}^{+}_{B}}\|_{\Phi})\geq
2c.$

This means that

$\|f_{\mathcal{F}^{+}_{A}}\|_{\Phi}\geq 1\ \ \ \ \ \text{or}\ \ \ \ \
\|f_{\mathcal{F}^{+}_{B}}\|_{\Phi}\geq 1$

and so by inequalities 1.2 we have

$N_{\Phi}(f_{\mathcal{F}^{+}_{A}})\geq\frac{1}{2}\ \ \ \ \ \text{or}\ \ \ \ \
N_{\Phi}(f_{\mathcal{F}^{+}_{B}})\geq\frac{1}{2}.$

Hence if $N_{\Phi}(f_{\mathcal{F}^{+}_{A}})\geq\frac{1}{2}$, then for every
$n\geq N$

$\displaystyle N_{\Phi}(C^{-n}_{\varphi}f)$
$\displaystyle\geq\frac{1}{2}\|C^{-n}_{\varphi}f\|_{\Phi}$
$\displaystyle=\frac{1}{2}\|C^{-n}_{\varphi}(f_{\mathcal{F}^{+}_{A}}+f_{\mathcal{F}^{+}_{B}})\|_{\Phi}$
$\displaystyle\geq\frac{c}{2}(\|C^{-n}_{\varphi}(f_{\mathcal{F}^{+}_{A}}\|_{\Phi}+\|f_{\mathcal{F}^{+}_{B}})\|_{\Phi})$
$\displaystyle\geq\frac{c}{2}\|C^{-n}_{\varphi}f_{\mathcal{F}^{+}_{A}}\|_{\Phi}$
$\displaystyle\geq\frac{c^{2}}{2}\sum_{\\{1\leq i\leq
m:F_{i}\in\mathcal{F}^{+}_{A}\\}}|\alpha_{i}|\frac{1}{\Phi^{-1}(\frac{1}{\mu(\varphi^{n}(F_{i}))})}$
$\displaystyle\geq\frac{c^{2}M}{2}\sum_{\\{1\leq i\leq
m:F_{i}\in\mathcal{F}^{+}_{A}\\}}|\alpha_{i}|\frac{1}{\Phi^{-1}(\frac{1}{\mu(F_{i})})}$
$\displaystyle\geq\frac{c^{2}M}{2}N_{\Phi}(f_{\mathcal{F}^{+}_{A}})$
$\displaystyle\geq\frac{c^{2}M}{4}.$

By a similar way we get that if
$N_{\Phi}(f_{\mathcal{F}^{+}_{B}})\geq\frac{1}{2}$, then we have
$N_{\Phi}(C^{n}_{\varphi}f)\geq\frac{c^{2}M}{4}$. If we set

$S^{0}_{L^{\Phi}(\mu)}=\\{f\in S_{L^{\Phi}(\mu)}:f\ \ \text{is a simple
function}\\},$

then from the above observations we get that $S^{0}_{L^{\Phi}(\mu)}=A_{0}\cup
B_{0}$, in which

$A_{0}=\\{f\in
S^{0}_{L^{\Phi}(\mu)}:N_{\Phi}(f_{\mathcal{F}^{+}_{A}})\geq\frac{1}{2}\\}\ \ \
\ \text{and}\ \ \ \ B_{0}=\\{f\in
S^{0}_{L^{\Phi}(\mu)}:N_{\Phi}(f_{\mathcal{F}^{+}_{B}})\geq\frac{1}{2}\\}.$

Let $f\in S_{L^{\Phi}(\mu)}$ be an arbitrary element. Then we can write
$f=f^{+}-f^{-}$ in which $f^{+}$ and $f^{-}$ are positive and negative parts
of $f$, respectively. Hence we can find two non-decreasing sequences of simple
functions $\\{f^{+}_{k}\\}_{k\in\mathbb{N}}$ and
$\\{f^{+}-_{k}\\}_{k\in\mathbb{N}}$ such that converge pointwisely to $f^{+}$
and $f^{-}$, respectively. If we define $f_{k}=f^{+}_{k}-f^{-}_{k}$, then we
get that the sequence $\\{f_{k}\\}_{k\in\mathbb{N}}$ converges pointwisely to
$f$. Since $|f_{k}|=|f^{+}_{k}-f^{-}_{k}|\leq 2|f|$ and so
$\Phi(|f_{k}|)\leq\Phi(2|f|)$, then by Lebesgue Dominated Convergence Theorem
we have

$\lim_{k\rightarrow\infty}\rho_{\Phi}(f_{k})=\rho_{\Phi}(f).$

Hence by Theorem 1.1 we get that

$\lim_{k\rightarrow\infty}N_{\Phi}(f_{k})=N_{\Phi}(f)=1.$

So there exists $k_{0}$ such that for every $k\geq k_{0}$,
$N_{\Phi}(f_{k})>\frac{1}{2}$. By the above observations we get that

$N_{\Phi}(C^{n}_{\varphi}(\frac{f_{k}}{N_{\Phi}(f_{k})})<4N_{\Phi}(C^{n}_{\varphi}(f)$

and also at least one of the following sets must be infinite:

$I_{1}(f)=\\{k\in\mathbb{N}:\frac{f_{k}}{N_{\Phi}(f_{k})}\in A_{0}\\};\ \ \ \
I_{1}(f)=\\{k\in\mathbb{N}:\frac{f_{k}}{N_{\Phi}(f_{k})}\in B_{0}\\}.$

Therefore, if $I_{1}(f)$ is infinite, then there is a subsequence

$\\{\frac{f_{k_{j}}}{N_{\Phi}(f_{k_{j}})}\\}_{j\in\mathbb{N}}\subseteq A_{0}$

such that for every $M>0$, there exists $N\in\mathbb{N}$ such that for each
$n\geq N$,

$N_{\Phi}(C^{-n}_{\varphi}(\frac{f_{k_{j}}}{N_{\Phi}(f_{k_{j}})})>\frac{M}{2}.$

Thus we have

$N_{\Phi}(C^{-n}_{\varphi}(f))>\frac{M}{8},\ \ \ \ \ \text{for every}\ \ \
n\geq N.$

Similarly, if $I_{2}(f)=\infty$, then for each $M>0$ we can find
$N\in\mathbb{N}$, such that for every $n\geq N$,
$N_{\Phi}(C^{n}_{\varphi}(f))>\frac{M}{8}$. Finally if we set

$A=\\{f\in S_{L^{\Phi}(\mu)}:I_{2}(f)=\infty\\}\ \ \ \text{and}\ \ \ B=\\{f\in
S_{L^{\Phi}(\mu)}:I_{1}(f)=\infty\\},$

then we have $S_{L^{\Phi}(\mu)}=A\cup B$, such that

$\lim_{n\rightarrow\infty}N_{\Phi}(C^{n}_{\varphi}f)=\infty,\ \ \ \
\text{uniformly on}\ \ \ \ A$

and

$\lim_{n\rightarrow\infty}N_{\Phi}(C^{-n}_{\varphi}f)=\infty,\ \ \ \
\text{uniformly on}\ \ \ \ B.$

This completes the proof. ∎

######  Theorem 2.8.

Let $(X,\mathcal{F},\mu,\varphi,C_{\varphi})$ be a dissipative composition
dynamical system of bounded distortion, generated by $W$. Then the following
statements hold.

  1. (1)

$C_{\varphi}$ is positively expansive if and only if
$\inf_{n\in\mathbb{N}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})=0$.

  2. (2)

$C_{\varphi}$ is expansive if and only if
$\inf_{n\in\mathbb{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{n}(W))})=0$.

  3. (3)

$C_{\varphi}$ is uniformly positively expansive if and only if

$\lim_{n\rightarrow\infty}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})=0.$

  4. (4)

$C_{\varphi}$ is uniformly expansive if and only if one of the following
conditions hold:

(2.4)
$\lim_{n\rightarrow\infty}\inf_{k\in\mathbb{Z}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}\right)=\infty,$
(2.5)
$\lim_{n\rightarrow\infty}\inf_{k\in\mathbb{Z}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k-n}(W))})}\right)=\infty,$
(2.6)
$\lim_{n\rightarrow\infty}\inf_{k\in\mathbb{Z}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}\right)=\infty\
\ \ \text{and}\ \ \
\lim_{n\rightarrow\infty}\inf_{k\in-\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k-n}(W))})}\right)=\infty.$

###### Proof.

First we prove part (2), then for the proof of part (1) it suffices just to
replace $\mathbb{Z}$ by $\mathbb{N}$.
In Theorem 2.7 we can replace $\mathcal{F}$ by $\mathcal{F}^{+}$, i.e., we can
write ”the composition operator $C_{\varphi}$ is expansive if and only if for
every $A\in\mathcal{F}^{+}$,

$\inf_{n\in\mathcal{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(A))})=0.$

Hence for the proof of (2), it suffices to prove

$\inf_{n\in\mathcal{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(A))})=0,\ \ \
\forall A\in\mathcal{F}^{+}\ \ \Leftrightarrow\ \
\inf_{n\in\mathbb{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{n}(W))})=0.$

The implication

$\inf_{n\in\mathcal{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(A))})=0,\ \ \
\forall A\in\mathcal{F}^{+}\ \ \Rightarrow\ \
\inf_{n\in\mathbb{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{n}(W))})=0$

is clear. So we prove the converse. Let $F\in\mathcal{F}^{+}$. Since
$X=\cup_{n\in\mathbb{Z}}\varphi^{n}(W)$, in which $\varphi^{n}(W)$’s are pair-
wise disjoint and $\mu(F)>0$, then there exists $n_{0}\in\mathbb{Z}$ such that
$\mu(F\cap\varphi^{n_{0}}(W))>0$. On the other hand
$\varphi^{-n_{0}}(F\cap\varphi^{n_{0}}(W))=\varphi^{n_{0}}(F)\cap W$. Hence if
we take $A=\varphi^{-n_{0}}(F\cap\varphi^{n_{0}}(W)$, then
$A\in\mathcal{F}_{W}$ and $\mu(A)\leq\mu(W)$. Now by using bounded distortion
property for we have

$\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(A))})\leq\frac{KN_{\Phi}(\chi_{W})}{N_{\Phi}(\chi_{A})}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))}),\
\ \forall n\in\mathbb{Z}$

and therefore

$\inf_{n\in\mathbb{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(A))})\leq\frac{KN_{\Phi}(\chi_{W})}{N_{\Phi}(\chi_{A})}\inf_{n\in\mathbb{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})=0.$

Consequently we have

$\inf_{n\in\mathbb{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})\leq\inf_{n\in\mathbb{Z}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(A))})=0.$

(3). By Theorem 2.7 it suffices to prove that

$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}=\infty,\
\ \forall F\in\mathcal{F}^{+}\ \ \Leftrightarrow\ \
\lim_{n\rightarrow\infty}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})=0.$

It is obvious that if

$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}=\infty,\
\ \forall F\in\mathcal{F}^{+},$

then
$\lim_{n\rightarrow\infty}\frac{1}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})}=\infty$
and so $\lim_{n\rightarrow\infty}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})=0$.
Now we prove the converse, suppose that
$\lim_{n\rightarrow\infty}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})=0$ and
$F\in\mathcal{F}^{+}$. Similar to pat (2) we can find $n_{0}\in\mathbb{Z}$
such that $0<\mu(F\cap\varphi^{n_{0}}(W))<\infty$ and

$A=\varphi^{-n_{0}}(F\cap\varphi^{n_{0}}(W))=\varphi^{-n_{0}}(F)\cap W.$

Hence $0<\mu(A)<\infty$ and $A\in\mathcal{F}_{W}$. Now by using bounded
distortion property we have

$\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(A))})\leq\frac{KN_{\Phi}(\chi_{W})}{N_{\Phi}(\chi_{A})}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))}),\
\ \ \forall n\in\mathbb{Z}$

and so

$\Phi^{-1}(\frac{1}{\mu(\varphi^{-n-n_{0}}(F))})\leq\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(A))})\leq\frac{KN_{\Phi}(\chi_{W})}{N_{\Phi}(\chi_{A})}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})$

Thus by taking $\lim$ we have

$\lim_{n\rightarrow\infty}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n-n_{0}}(F))})\leq\frac{KN_{\Phi}(\chi_{W})}{N_{\Phi}(\chi_{A})}\lim_{n\rightarrow\infty}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})=0$

hence
$\lim_{n\rightarrow\infty}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n-n_{0}}(F))})=0$
and consequently

$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}=\infty.$

By Theorem 2.7, part (3) we get the proof.
(4) Suppose that $C_{\varphi}$ is uniformly expansive. By Theorem 2.7 we have
that $\mathcal{F}^{+}=\mathcal{F}^{+}_{A}\cup\mathcal{F}^{+}_{B}$ such that

$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{n}(F))})}=\infty,\
\ \ \text{uniformly on }\ \mathcal{F}^{+}_{A},$
$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}=\infty,\
\ \ \text{uniformly on }\ \mathcal{F}^{+}_{B}.$

If we set

$I=\\{k\in\mathbb{Z}:\varphi^{k}(W)\in\mathcal{F}_{A}\\}\ \ \ \text{and}\ \ \
J=\\{k\in\mathbb{Z}:\varphi^{k}(W)\in\mathcal{F}_{B}\\},$

then by our assumptions we get that

$\lim_{n\rightarrow\infty}\inf_{k\in
I}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}\right)=\infty$

$\lim_{n\rightarrow\infty}\inf_{k\in
J}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k-n}(W))})}\right)=\infty.$

By these observations, if $J=\emptyset$, then the condition 2.4 holds, if
$I=\emptyset$, then the condition 2.5 holds and if $I$ and $J$ are both non-
empty, then there exist $i,j\in\mathbb{Z}$ such that
$[i,\infty)\cap\mathbb{Z}\subseteq I$ and $(-\infty,j]\cap\mathbb{Z}\subseteq
J$, this implies that the condition 2.6 holds.
For the converse, for every $F\in\mathcal{F}^{+}$, by our assumptions we have
$F=\cup_{k\in\mathbb{Z}}F_{k}$, where $F_{k}=F\cap\varphi^{k}(W)$ and
$F_{k}$’s are pair-wise disjoint and also by Proposition 2.6, for every
$n,k\in\mathbb{Z}$ there exists $H>0$ such that

$\frac{1}{H}\frac{N_{\Phi}(C^{k+n}_{\varphi}(\chi_{W}))}{N_{\Phi}(C^{k}_{\varphi}(\chi_{W}))}N_{\Phi}(\chi_{F_{k}})\leq
N_{\Phi}(C^{n}_{\varphi}(\chi_{F_{k}}))\leq
H\frac{N_{\Phi}(C^{k+n}_{\varphi}(\chi_{W}))}{N_{\Phi}(C^{k}_{\varphi}(\chi_{W}))}N_{\Phi}(\chi_{F_{k}}).$

Also, we have $0<\mu(F)<\infty$. Hence there exists $k\in\mathbb{Z}$ such that
$\mu(F_{k})>0$. Therefore , for each $n\in\mathbb{Z}$,

$\displaystyle N_{\Phi}(C^{n}_{\varphi}(\chi_{F}))$
$\displaystyle\geq\frac{1}{2}\|C^{n}_{\varphi}(\chi_{F})\|_{\Phi}$
$\displaystyle=\frac{1}{2}\|\chi_{\varphi^{-n}(F)}\|_{\Phi}$
$\displaystyle\geq\frac{c}{2}\sum_{k\in\mathbb{Z}}\|\chi_{\varphi^{-n}(F_{k})}\|_{\Phi}$
$\displaystyle\geq\frac{c}{2}\sum_{k\in\mathbb{Z}}N_{\Phi}(\chi_{\varphi^{-n}(F_{k})})$
$\displaystyle=\frac{c}{2}\sum_{k\in\mathbb{Z}}N_{\Phi}(C^{n}_{\varphi}(\chi_{F_{k}}))$
$\displaystyle\geq\frac{c}{2H}\sum_{k\in\mathbb{Z}}\frac{N_{\Phi}(C^{k+n}_{\varphi}(\chi_{W}))}{N_{\Phi}(C^{k}_{\varphi}(\chi_{W}))}N_{\Phi}(\chi_{F_{k}})$
$\displaystyle\geq\frac{c}{2H}\inf_{k\in\mathbb{Z}}\frac{N_{\Phi}(C^{k+n}_{\varphi}(\chi_{W}))}{N_{\Phi}(C^{k}_{\varphi}(\chi_{W}))}\sum_{k\in\mathbb{Z}}N_{\Phi}(\chi_{F_{k}})$
$\displaystyle\geq\frac{c}{2H}\inf_{k\in\mathbb{Z}}\frac{N_{\Phi}(C^{k+n}_{\varphi}(\chi_{W}))}{N_{\Phi}(C^{k}_{\varphi}(\chi_{W}))}N_{\Phi}(\chi_{F}).$

By these observations we get that

$\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{n}(F))})}\geq\frac{c}{2H}\inf_{k\in\mathbb{Z}}\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}.$

This inequality implies that if we set

$\mathcal{F}^{+}_{A}=\\{F\cap(\cup_{k\geq
0}\varphi^{k}(W)):F\in\mathcal{F}^{+}\\}$

and

$\mathcal{F}^{+}_{B}=\\{F\cap(\cup_{k<0}\varphi^{k}(W)):F\in\mathcal{F}^{+}\\},$

then $\mathcal{F}^{+}=\mathcal{F}^{+}_{A}\cup\mathcal{F}^{+}_{B}$ and also if
one of conditions 2.4, 2.5 or 2.6 holds, then we get that

$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{n}(F))})}=\infty,\
\ \ \text{uniformly on }\ \mathcal{F}^{+}_{A},$

$\lim_{n\rightarrow\infty}\frac{\Phi^{-1}(\frac{1}{\mu(F)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(F))})}=\infty,\
\ \ \text{uniformly on }\ \mathcal{F}^{+}_{B}.$

Thus by Theorem 2.7 the composition operator $C_{\varphi}$ is uniformly
expansive. ∎

In the next Proposition we obtain some sufficient conditions for composition
operator $C_{\varphi}$ on Orlicz spaces to decline structural stability and
even strongly structurally stability.

######  Proposition 2.9.

Let $(X,\mathcal{F},\mu,\varphi,C_{\varphi})$ be a dissipative composition
dynamical system of bounded distortion, generated by $W$ and
$\Phi\in\Delta^{\prime}$. If

$\overline{\lim}_{n\rightarrow\infty}\sup_{k\in\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}\right)^{\frac{1}{n}}<1\
\ \ \text{and}\ \ \
\underline{\lim}_{n\rightarrow\infty}\inf_{k\in-\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k-n}(W))})}\right)^{\frac{1}{n}}>1,$

then the composition operator $C_{\varphi}$ can not be structurally stable and
so can not be even strongly structurally stable.

###### Proof.

Since

$\underline{\lim}_{n\rightarrow\infty}\inf_{k\in-\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k-n}(W))})}\right)^{\frac{1}{n}}>1,$

then there exists $n_{0}\in\mathbb{N}_{0}$ and $t>1$ such that for every
$n\geq n_{0}$,

$\inf_{k\in-\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k-n}(W))})}\right)^{\frac{1}{n}}\geq
t>1.$

Hence

$\frac{\Phi^{-1}(\frac{1}{\mu(W)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})}\geq\inf_{k\in-\mathbb{N}_{0}}\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k-n}(W))})}\geq
t^{n}.$

This implies that
$\sup_{n\in\mathbb{N}}\frac{\Phi^{-1}(\frac{1}{\mu(W)})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})}=\infty$
and therefore

$\inf_{n\in\mathbb{N}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})=0.$

Thus by theorem 2.7, we get that $C_{\varphi}$ is positively expansive.
Moreover, for every $n\geq n_{0}$,

$\sup_{k\in-\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k-n}(W))})}\right)^{\frac{1}{n}}\geq\inf_{k\in-\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k-n}(W))})}\right)^{\frac{1}{n}}\geq
t>1.$

Hence we get that

$\overline{\lim}_{n\rightarrow\infty}\sup_{k\in\mathbb{Z}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k-n}(W))})}\right)^{\frac{1}{n}}\geq\overline{\lim}_{n\rightarrow\infty}\sup_{k\in-\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k-n}(W))})}\right)^{\frac{1}{n}}>1.$

In addition, since

$\overline{\lim}_{n\rightarrow\infty}\sup_{k\in\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}\right)^{\frac{1}{n}}<1,$

then there exists $m_{0}\in\mathbb{N}_{0}$ and $0<t<1$ such that for every
$n\geq m_{0}$,

$\sup_{k\in\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}\right)^{\frac{1}{n}}\leq
t<1.$

This implies that

$\inf_{k\in\mathbb{Z}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}\right)^{\frac{1}{n}}\leq\inf_{k\in\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}\right)^{\frac{1}{n}}\leq
t<1$

and consequently

$\underline{\lim}_{n\rightarrow\infty}\inf_{k\in\mathbb{Z}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}\right)^{\frac{1}{n}}\leq\underline{\lim}_{n\rightarrow\infty}\inf_{k\in\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}\right)^{\frac{1}{n}}<1.$

By the above observations we get that none of conditions 2.3, 2.4 and 2.5 of
the Corollary 2.20 of [12] holds. This means that $C_{\varphi}$ doesn’t have
the shadowing property and so it is not hyperbolic. Therefore $C_{\varphi}$ is
positively expansive but not hyperbolic, hence by Theorem 6 of [5] we get that
$C_{\varphi}$ is not structurally stable and therefore, not even strongly
structurally stable. ∎

######  Theorem 2.10.

Let $(X,\mathcal{F},\mu,\varphi,C_{\varphi})$ be a dissipative composition
dynamical system of bounded distortion, generated by $W$ and
$\Phi\in\Delta^{\prime}$. If at least one of the following inequalities holds:

(2.7)
$\overline{\lim}_{n\rightarrow\infty}\sup_{k\in\mathbb{Z}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}\right)^{\frac{1}{n}}<1$
(2.8)
$\underline{\lim}_{n\rightarrow\infty}\inf_{k\in\mathbb{Z}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}\right)^{\frac{1}{n}}>1$
(2.9)
$\overline{\lim}_{n\rightarrow\infty}\sup_{k\in-\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k-n}(W))})}\right)^{\frac{1}{n}}<1\
\ \text{and}\ \
\underline{\lim}_{n\rightarrow\infty}\inf_{k\in\mathbb{N}_{0}}\left(\frac{\Phi^{-1}(\frac{1}{\mu(\varphi^{k}(W))})}{\Phi^{-1}(\frac{1}{\mu(\varphi^{k+n}(W))})}\right)^{\frac{1}{n}}>1,$

then the composition operator $C_{\varphi}$ is strongly structurally stable.

###### Proof.

If one of the conditions 2.7, 2.8 and 2.9 holds, then Theorem 2.16 of [12] and
Theorem 1 of [3] we get that $C_{\varphi}$ is strongly structurally stable. ∎

Now by using Theorem 2.10 and also Corollary CSC of [12] we have the next
corollary.

######  Corollary 2.11.

Let $(X,\mathcal{F},\mu,\varphi,C_{\varphi})$ be a dissipative composition
dynamical system of bounded distortion, generated by $W$ and
$\Phi\in\Delta^{\prime}$. Then the composition operator $C_{\varphi}$ is
strongly structurally stable when it has shadowing property or equivalently is
generalized hyperbolic.

######  Theorem 2.12.

Let $(X,\mathcal{F},\mu,\varphi,C_{\varphi})$ be a dissipative composition
dynamical system of bounded distortion, generated by $W$,
$\inf_{n\in\mathbb{N}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})=0$ and
$\Phi\in\Delta^{\prime}$. Then the composition operator $C_{\varphi}$ is
strongly structurally stable if and only if it has the shadowing property.

###### Proof.

If $C_{\varphi}$ has the shadowing property, then by Corollary 2.11 we get
that $C_{\varphi}$ is strongly structurally stable. Conversely, if
$C_{\varphi}$ is strongly structurally stable, then by the assumption
$\inf_{n\in\mathbb{N}}\Phi^{-1}(\frac{1}{\mu(\varphi^{-n}(W))})=0$ and by
Theorem 2.8 we have $C_{\varphi}$ is positively expansive. Therefore by
Theorem 6 of [5] we get the proof. ∎

Declarations
Conflict of interest. The authors have not disclosed any competing interests.
Acknowledgement. My manuscript has no associate data.

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also at ]Jawaharlal Nehru Centre For Advanced Scientific Research, Jakkur,
Bangalore, India.

# The spreading of viruses by airborne aerosols: lessons from a first-passage-
time problem for tracers in turbulent flows

Akhilesh Kumar Verma<EMAIL_ADDRESS>Centre for Condensed Matter Theory,
Department of Physics, Indian Institute of Science, Bangalore 560012, India.
Akshay Bhatnagar<EMAIL_ADDRESS>NORDITA, KTH Royal Institute of
Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm,
Sweden.  Dhrubaditya Mitra<EMAIL_ADDRESS>NORDITA, KTH Royal
Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691
Stockholm, Sweden.  Rahul Pandit<EMAIL_ADDRESS>[  Centre for Condensed
Matter Theory, Department of Physics, Indian Institute of Science, Bangalore
560012, India<EMAIL_ADDRESS>Centre for Condensed Matter Theory,
Department of Physics, Indian Institute of Science, Bangalore 560012, India.
<EMAIL_ADDRESS>NORDITA, KTH Royal Institute of Technology and Stockholm
University, Roslagstullsbacken 23, 10691 Stockholm, Sweden.
<EMAIL_ADDRESS>NORDITA, KTH Royal Institute of Technology and
Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden.
<EMAIL_ADDRESS>[  Centre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560012, India.

###### Abstract

We study the spreading of viruses, such as SARS-CoV-2, by airborne aerosols,
via a new first-passage-time problem for Lagrangian tracers that are advected
by a turbulent flow: By direct numerical simulations of the three-dimensional
(3D) incompressible, Navier-Stokes equation, we obtain the time $t_{R}$ at
which a tracer, initially at the origin of a sphere of radius $R$, crosses the
surface of the sphere for the first time. We obtain the probability
distribution function $\mathcal{P}(R,t_{R})$ and show that it displays two
qualitatively different behaviors: (a) for $R\ll L_{\rm I}$,
$\mathcal{P}(R,t_{R})$ has a power-law tail $\sim t_{R}^{-\alpha}$, with the
exponent $\alpha=4$ and $L_{\rm I}$ the integral scale of the turbulent flow;
(b) for $L_{\rm I}\lesssim R$, the tail of $\mathcal{P}(R,t_{R})$ decays
exponentially. We develop models that allow us to obtain these asymptotic
behaviors analytically. We show how to use $\mathcal{P}(R,t_{R})$ to develop
social-distancing guidelines for the mitigation of the spreading of airborne
aerosols with viruses such as SARS-CoV-2.

airborne aerosols; first-passage time; turbulence

##  I Introduction

By 1 June 2020 (14:31 GMT) the COVID-19 coronavirus pandemic had affected
$213$ countries and territories and $2$ international conveyances; the numbers
of cases and deaths were, respectively, $6,300,444$ and $374,527$ worldometer
. Social distancing has played an important role in mitigation strategies that
have been used in several countries to arrest the spread of COVID-19
socialdistancing . To optimise social-distancing guidelines we must ask: How
far, and how fast, do small respiratory droplets or virus-bearing aerosols
spread in turbulent flows? Given the ongoing COVID-19 pandemic, it is
important and extremely urgent to have at least a semi-quantitative answer to
this question. SARS-CoV-2, the virus that causes COVID-19, spreads,
principally, in two different ways: (1) First: Respiratory droplets, ejected
by the sneeze or cough of a patient, fall on nearby surfaces or persons; in
this case, approximate estimates of the distance, over which droplets are
likely to travel, are available NYTarticle ; bourouiba2014 ; bourouiba2020 .
(2) Second: Transmission of this virus can occur because of airborne aerosols,
such as, (a) a cloud of fine droplets, with diameters smaller than 5
micrometers, emitted by an infected person while speaking loudly laserlink  or
(b) the SARS-CoV-2 RNA on fine, suspended particulate matter Setti2020 . These
aerosols may remain suspended in the air for a long time. Indeed, they have
been reported in two hospitals in Wuhan Liu2020 ; and there is growing
evidence that the SARS-CoV-2 virus could also spread via airborne aerosols
laserlink ; Setti2020 ; Liu2020 ; Prather2020 ; Somsen2020 , typically indoors
Indoor2020 . Other diseases can also spread because of airborne aerosols;
examples include measles Riley1978 , chickenpox Leclair1980 , tuberculosis
Escombe2007 , and avian flu Zhao2019 .

The typical sedimentation speed for such aerosols is comparable to their
thermal speed. Therefore, at the simplest level, it is natural to model these
aerosol particles as neutrally-buoyant Lagrangian tracers, which are advected
by the flow, but are passive, in the sense that they do not affect the flow
velocity. We can then study the spread of viruses, such as SARS-CoV-2, via the
airborne-aerosol route, by considering the advection of such tracers by
turbulent fluid flows. There have been extensive studies of such tracers in
the fluid-dynamics literature Falkovich2001 ; Toschi2009 ; and models for such
tracers have been used, inter alia, to model the dispersion of pheromones by
lepidoptera Celani2014 .

We would like to determine the time that an aerosol particle (one of the red
particles in the schematic diagram of Fig. 1) takes to travel a distance $R$
from its source (the man at the centre of Fig. 1). In a turbulent flow, this
time is random; furthermore, a tracer particle can go past the distance $R$,
turn back, and reach $R$ again. It is important, therefore, to calculate the
time it takes for an aerosol particle to reach the distance $R$ for the first
time and to calculate the probability distribution function (PDF) of the
first-passage time of a tracer in a turbulent flow. We carry out this
calculation below.

Specifically, we consider Lagrangian tracer particles that emanate from a
point source in a turbulent fluid. If $t_{R}$ is the time at which a tracer,
initially at the origin of a sphere of radius $R$, crosses the surface of the
sphere for the first time, what is the probability distribution function (PDF)
$\mathcal{P}(R,t_{R}$)? The answer to this question is of central importance
in both fundamental nonequilibrium statistical mechanics bray2013persistence ;
chandrasekhar1943stochastic ; redner2001guide ; balakrishnan2008elements ;
ralf2014first  and in understanding the dispersal of tracers by a turbulent
flow, a problem whose significance cannot be overemphasized, for it is of
relevance to the advection of (a) airborne viruses, as we have noted above,
and (b) pollutants in the atmosphere. First-passage-time problems have been
studied extensively chandrasekhar1943stochastic ; redner2001guide ;
balakrishnan2008elements ; ralf2014first  and they have found applications in
a variety of areas in physics and astronomy, chemistry weiss1967first ,
biology ricciardi1999outline , and finance chicheportiche2014some . In the
fluid-turbulence context, different groups have studied zero crossings of
velocity fluctuations kailasnath1993zero  or various statistical measures of
two-particle dispersion, including exit-time statistics for such dispersion in
two- and three-dimensional (2D and 3D) turbulent flows boffetta2002statistics
; vulpiani2001exit . In contrast to these earlier studies (e.g., Refs.
boffetta2002statistics ; vulpiani2001exit ; lalescu2018tracer ), the first-
passage-time problem we pose considers one tracer in a turbulent flow that is
statistically homogeneous and isotropic. To the best of our knowledge, this
first-passage problem has not been studied hitherto. For such a particle we
show, via extensive direct numerical simulations (DNSs), that
$\mathcal{P}(R,t_{R})$ displays a crossover between two qualitatively
different behaviors: (a) for $R\ll L_{\rm I}$, $\mathcal{P}(R,t_{R})\sim
t_{R}^{-\alpha}$, with $L_{\rm I}$ the integral scale of the turbulent flow
and the exponent $\alpha=4$; (b) for $L_{\rm I}\lesssim R$,
$\mathcal{P}(R,t_{R})$ has an exponentially decaying tail (Fig. 2). We develop
models that allow us to obtain these two asymptotic behaviors analytically.
Most important, we show how to use $\mathcal{P}(R,t_{R})$ to obtain estimates
of social-distancing guidelines for the mitigation of the spreading of
airborne aerosols with viruses such as SARS-CoV-2.

Figure 1:  A schematic diagram illustrating how aerosols with viruses (red
points) may be advected by a turbulent flow, from the person at the centre
(the source) to other persons at different distances from the centre.

##  II Models, Methods, and Results

(a)(b)

Figure 2:  Plots of the complementary cumulative probability distribution
functions (CPDFs) $\mathcal{Q}$ versus the scaled first-passage time $t_{R}$
(see text): (a) Log-log plots of $\mathcal{Q}(\frac{t_{R}/T_{\rm
eddy}}{R/L_{\rm I}})$ for $R/L_{\rm I}=0.06$ (blue), $R/L_{\rm I}=0.10$
(purple), $R/L_{\rm I}=0.14$ (green), and $(\frac{t_{R}/T_{\rm eddy}}{R/L_{\rm
I}})^{-3}$ (black dashed line); the inset shows log-log plots of
$\mathcal{Q}(t_{R}/T_{\rm eddy})$ for the same values of $R/L_{\rm I}$. (b)
Semi-log plots of $\mathcal{Q}(t_{R}/T_{\rm eddy})$ for $R/L_{\rm I}=1.02$
(pink), $R/L_{\rm I}=1.43$ (blue), $R/L_{\rm I}=1.84$ (purple), and $R/L_{\rm
I}=2.04$ (green).

The 3D incompressible, Navier-Stokes equation is

$\partial_{t}{\bm{u}}+\left({\bm{u}}\cdot{\bm{\nabla}}\right){\bm{u}}=-{\bm{\nabla}}p+\nu\nabla^{2}{\bm{u}}+\bm{f}\/,$
(1a) and ${\bm{\nabla}}\cdot{\bm{u}}=0\/.$ (1b)

Here, ${\bm{u}}({\bm{x}},t)$ is the Eulerian velocity at position ${\bm{x}}$
at time $t$, $p({\bm{x}},t)$ is the pressure field, and $\nu$ is the kinematic
viscosity of the fluid; the constant density is chosen to be unity. Our direct
numerical simulation (DNS) uses the pseudo-spectral method canuto1988spectral
, with the $2/3$ rule for dealiasing, in a triply periodic cubical domain with
$N^{3}$ collocation points; we employ the second-order, exponential, Adams-
Bashforth scheme for time stepping bhatnagar2016long . We obtain a
nonequilibrium, statistically stationary turbulent state via a forcing term
$\bm{f}$, which imposes a constant rate of energy injection
lamorgese2005direct ; sahoo2011systematics , in wave-number shells $k=1$ and
$k=2$ in Fourier space; this turbulent state is statistically homogeneous and
isotropic.

To obtain the statistical properties of Lagrangian tracers, which are advected
by this turbulent flow, we seed the flow with $\mathcal{N}_{\rm p}$
independent, identical, tracer particles. If the Lagrangian displacement of a
tracer, which was at position ${\bf r}_{0}$ at time $t_{0}$, is
${\bm{r}}(t\mid{\bf r}_{0},t_{0})$, then its temporal evolution is given by

$\frac{d}{dt}{\bm{r}}={\bm{v}}(t\mid{\bf
r}_{0},t_{0})={\bm{u}}({\bm{r}},t)\/,$ (2)

where ${\bm{v}}$ is its Lagrangian velocity. In Eq. (2), we need the Eulerian
flow velocity at off-grid points; we obtain this by tri-linear interpolation;
and we use the first-order Euler method for time marching (see, e.g., Ref.
bhatnagar2016long ). We give important parameters for our DNS runs in Table 1.
These include the time step $dt$, the energy dissipation rate
$\epsilon=2\nu\sum_{k}k^{2}E(k)$, where
${E}(k)=\sum_{k-1/2<k^{\prime}<k+1/2}{\bm{u}}({\bf
k}^{\prime})\cdot{\bm{u}}({-\bf k}^{\prime})$ is the energy spectrum, the
Taylor-microscale $\lambda=\sqrt{\frac{2\nu E}{\epsilon}}$, where the total
energy $E=\sum_{k}E(k)$, the Taylor-microscale Reynolds number
$Re_{\lambda}\equiv\lambda u_{\rm rms}/\nu$, where $u_{\rm rms}=\sqrt{2E}$ is
the root-mean-square velocity of the flow; $L_{\rm
I}=\frac{\sum_{k}E(k)/k}{\sum_{k}E(k)}$ is the integral length scale and
${T_{\rm eddy}}=L_{\rm I}/u_{\rm rms}$ is the integral-scale eddy-turnover
time; $\eta=(\nu^{3}/\epsilon)^{1/4}$ and $\tau_{\eta}=(\nu/\epsilon)^{1/2}$
are, respectively, the Kolmogorov dissipation length and time scale; and
$k_{\rm{max}}$ is the maximum wave number that we use in our DNS.

$N$ | $\nu$ | $dt$ | $Re_{\lambda}$ | $\epsilon$ | $\eta$ | $k_{\rm{max}}\eta$ | $\lambda$ | $L_{\rm I}$ | $T_{\rm eddy}$ | $\tau_{\eta}$ | $\mathcal{N}_{\rm p}$
---|---|---|---|---|---|---|---|---|---|---|---
$512$ | $1.2\times 10^{-3}$ | $2\times 10^{-4}$ | $82$ | $0.67$ | $7.12\times 10^{-3}$ | $1.21$ | $0.08$ | $0.49$ | $0.43$ | $4.23\times 10^{-2}$ | $100000$
Table 1: Parameters (see text for definitions) for our DNS runs: $N^{3}$ is
the total number of collocation points; $\nu$ is the kinematic viscosity; $dt$
is the time step; $Re_{\lambda}$ is the Taylor-microscale Reynolds number;
$\epsilon$ is the energy-dissipation rate; $\eta$ is the Kolmogorov
dissipation length scale; $k_{\rm{max}}$ is the maximum wave number that we
use; $L_{\rm I}$ is the integral length scale; $T_{\rm eddy}$ is the integral-
scale eddy-turnover time; $\tau_{\eta}=(\nu/\epsilon)^{1/2}$ is the Kolmogorov
dissipation time scale; and $\mathcal{N}_{\rm p}$ is the number of tracer
particles.

Clearly, $t_{R}$ is the first time at which $|{\bm{r}}|$ becomes equal to $R$.
Instead of computing the PDF (or histogram) of $t_{R}$ numerically, we
calculate the complementary cumulative probability distribution function
(CPDF) $\mathcal{Q}(t_{R})$, by using the rank-order method mit+bec+pan+fri05
, to circumvent binning errors. In Fig. 2, we present log-log and semi-log
plots of $\mathcal{Q}(t_{R})$ versus $t_{R}/T_{\rm eddy}$, for several values
of $R$. From Fig. 2 (a) we conclude that, for $R\ll L_{\rm I}$,
$\mathcal{Q}(t_{R}/T_{\rm eddy})\sim(t_{R}/T_{\rm eddy})^{-\alpha+1}$, for
large $t_{R}/T_{\rm eddy}$, with $\alpha\simeq 4$; note that, in this power-
law scaling regime, the complementary CPDFs for different values of $R/L_{\rm
I}$ collapse onto a universal scaling form, if we plot
$\mathcal{Q}(\frac{t_{R}/T_{\rm eddy}}{R/L_{\rm I}})$. In contrast, Fig. 2 (b)
shows that, for $L_{\rm I}\lesssim R$, the tail of $\mathcal{Q}(t_{R}/T_{\rm
eddy})$ decays exponentially. For the first-passage-time PDF, these results
imply that

$\mathcal{P}(R,t_{R}/T_{\rm eddy})\sim\begin{cases}(t_{R}/T_{\rm
eddy})^{-4}\quad{\rm for}\quad R\ll L_{\rm I};\\\ \exp(-(t_{R}/T_{\rm
eddy}))\quad{\rm for}\quad L_{\rm I}\lesssim R.\\\ \end{cases}$ (3)

We now develop models that allow us to understand these two asymptotic
behaviors analytically.

For the power-law behavior of $\mathcal{P}(R,t_{R})$, in the range $R\ll
L_{\rm I}$, we construct the following, natural, ballistic model: Tracer
particles emanate from the origin with (a) a velocity whose magnitude $v$ is a
random variable with a PDF $p(v)$; and (b) when it starts out from the origin,
the tracer’s velocity vector points in a random direction. Tracers move
ballistically, for short times. Therefore, for $R\ll L_{\rm I}$, the first-
passage time $t_{R}=R/v$; and the first-passage PDF is

$\mathcal{P}(R,t_{R})=\int\delta(t_{R}-R/v)p(v)dv.$ (4)

In statistically homogeneous and isotropic and incompressible-fluid
turbulence, each component of the Eulerian velocity has a PDF that is very
close to Gaussian pramanareview , so $p(v)$ has the Maxwellian
gotoh2002velocity  form

$p(v)=C_{d}v^{d-1}\exp(-v^{2}/\sigma^{2}),$ (5)

where $C_{d}$ depends on the spatial dimension $d$, and $C_{d}=4\pi$ for
$d=3$, and $\sigma=\sqrt{\langle v^{2}\rangle}=u_{\rm rms}$. We substitute Eq.
(5) in Eq. (4); then, by integrating over $v$, we obtain

$\mathcal{P}(R,t_{R})=C_{d}\frac{R^{3}}{{t_{R}}^{d+1}}\exp(-R^{2}/({t_{R}}^{2}\sigma^{2})).$
(6)

Therefore, in the limit of small $R$ and large $t_{R}$, the first-passage-time
probability is

$\mathcal{P}(R,t_{R})\sim R^{3}/{t_{R}}^{4},\hskip 14.22636pt{\rm for}\hskip
14.22636ptd=3;$ (7)

this power-law exponent is the same as the one we have obtained from our DNSs
above (Table 1 and Fig. 2).

We can obtain the tail $\mathcal{P}(R,t_{R}/T_{\rm
eddy})\sim\exp(-(t_{R}/T_{\rm eddy}))$ for $L_{\rm I}\lesssim R$ as follows.
At times that are larger than the typical auto-correlation time of velocities
in the Lagrangian description, we follow Taylor tay22  and assume that the
motion of a tracer particle is diffusive. Therefore, we consider a Brownian
particle in three dimensions (3D). To calculate the first-passage-time PDF, we
must first obtain the survival probability $S(t,R|0)$, i.e., the probability
that the particle has not reached the surface of the sphere of radius $R$ up
to time $t$, if it has started from the origin of this sphere. We start with
the forward Fokker-Planck equation balakrishnan2008elements ; Risken  for the
PDF of finding the particle at a distance $r$ from the origin at time $t$:

$\frac{\partial P(r,t)}{\partial t}=K\bigg{(}\frac{\partial^{2}}{\partial
r^{2}}+\frac{2}{r}\frac{\partial}{\partial r}\bigg{)}P(r,t)\/,$ (8)

where $K$ is the diffusion constant; this PDF satisfies the initial condition,
$P(r,0)=\delta(r)/(4\pi r^{2})$ and the absorbing boundary condition
$P(R,t)=0$, for all $t$ at $r=R$. We obtain the following solution:

$P(r,t)=\frac{1}{2R^{2}}\sum_{n=0}^{\infty}\frac{n}{r}\sin\bigg{(}\frac{n\pi
r}{R}\bigg{)}\exp\bigg{(}-Kn^{2}\pi^{2}t/R^{2}\bigg{)},$ (9)

whence we get

$\displaystyle S(R,t_{R})$ $\displaystyle=\int_{0}^{R}P(r,t)4\pi r^{2}dr\/$
$\displaystyle=2\sum_{n=0}^{\infty}(-1)^{n+1}\exp(-Kn^{2}\pi^{2}t/R^{2})\/,$
(10)

where, in the last step, we have used Eq. (9). The first-passage-time
probability is

$\displaystyle\mathcal{P}(R,t_{R})$ $\displaystyle=-\frac{\partial}{\partial
t_{R}}S(R,t_{R})$
$\displaystyle=\frac{2K\pi^{2}}{R^{2}}\sum_{n=0}^{\infty}(-1)^{n+1}n^{2}\exp(-Kn^{2}\pi^{2}t_{R}/R^{2}).$
(11)

At large times, the first term ($n=1$) is the dominant one; therefore,

$\mathcal{P}(R,t_{R})\sim(1/R^{2})\exp(-K\pi^{2}t_{R}/R^{2}),$ (12)

the exponential form that we have obtained from our DNS (Fig. 2 (b)); the
$1/R^{2}$ pre-factor cannot be extracted reliably from our DNS data, because
this requires much longer runs than are possible with our computational
resources.

We now show that both the small- and large-$R/L_{\rm I}$ behaviors of
$\mathcal{P}(R,t_{R})(R,t_{R})$ in Eq. (3) can be obtained from one stochastic
model for the motion of a particle. The simplest such model uses a particle
that obeys the following Ornstein-Uhlenbeck (OU) model:

$\displaystyle\frac{d{x}_{i}}{dt}$ $\displaystyle={v_{i}}\/,$ (13a)
$\displaystyle\frac{d{v}_{i}}{dt}$ $\displaystyle=-\gamma
v_{i}+\frac{\sqrt{\Gamma}}{m}\zeta_{i}\/.$ (13b)

Here, $\gamma$ and $\Gamma$ are positive constants; $x_{i}$ and $v_{i}$ are
the Cartesian components of the position and velocity of the particle; in
three dimensions, $i=1,2$, and $3$; $\zeta_{i}(t)$ is a zero-mean Gaussian
white noise with $\langle\zeta_{i}\rangle=0$ and
$\langle\zeta_{i}(t)\zeta_{j}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime})$;
this noise is such that the fluctuations-dissipation theorem (FDT) holds. Note
that there is no FDT for turbulence. However, for the one-particle statistics
that we consider, the simple OU model is adequate. We use $N_{p}=50,000$
particles; for each particle, the initial-position components $x_{i}(t=0)$ are
distributed randomly and uniformly on the interval $[0,2\pi]$; and the
velocity components $v_{i}(t=0)$ are chosen from a Gaussian distribution. For
each particle, we obtain, numerically, the time $t_{R}$ at which it reaches a
distance $R$ from the origin for the first time. We then obtain the first-
passage-time complementary CPDF $\mathcal{Q}(t_{R})$, which we plot in Fig. 3,
for $R\ll L$ and $L\lesssim R$, where $L=\sqrt{\frac{\Gamma}{\gamma^{3}}}$,
the natural length scale for Eq. (13), plays the role of $L_{\rm I}$ in our
DNSs above (Table 1 and Fig. 2). We find

$\displaystyle\mathcal{P}(R,t_{R})$ $\displaystyle\sim$
$\displaystyle\Bigl{[}\frac{\gamma t_{R}}{(R/L)}\Bigr{]}^{-4},\,{\rm
for}\,R\ll L;$ $\displaystyle\mathcal{P}(R,t_{R})$ $\displaystyle\sim$
$\displaystyle\exp\Bigl{(}-\frac{\gamma t_{R}}{(R/L)^{2}}\Bigr{)},\,{\rm
for}\,L\lesssim R;$ (14)

these are the OU-model analogs of our DNS results Eq. (3). We have carried out
two OU-model simulations: (a) we have designed the first, with $\gamma=0.01$,
to explore the form of $\mathcal{P}(R,t_{R})$ in the ballistic regime $R\ll
L$; (b) the second, with $\gamma=30$, allows us to uncover the form of
$\mathcal{P}(R,t_{R})$ in the diffusive regime $L\lesssim R$. (From a
numerical perspective, it is expensive to obtain the precise form of
$\mathcal{P}(R,t_{R})$ in both ballistic and diffusive regimes, with one value
of $\gamma$.) We now explore in detail the forms of $\mathcal{P}(R,t_{R})$ in
these two regimes. In Fig. 3(a), we present log-log plots of the complementary
CPDFs of the scaled first-passage time $t_{R}/R$, for $R\ll L$ and
$\gamma=0.01$. The complementary CPDFs of $t_{R}/R$, for $R/L=0.0002$,
$R/L=0.00035$, and $R/L=0.0005$, collapse onto one curve; i.e., in this
regime, $t_{R}$ scales as $R$, which is a clear manifestation of ballistic
motion. In Fig. 3(b), we present semi-log plots of the complementary CPDFs of
the scaled first-passage time $t_{R}/R^{2}$, for $L\lesssim R$ and
$\gamma=30$. The complementary CPDFs of $t_{R}/R^{2}$, for $R/L=10$, $R/L=14$,
$R/L=18$, and $R/L=20$, collapse onto one curve; from this we conclude that,
in this regime, $t_{R}$ scales as $R^{2}$, which is a clear signature of
diffusive motion.

(a)(b)

Figure 3: (a) Log-log plots of the complementary CPDFs
$\mathcal{Q}(\frac{\gamma t_{R}}{R/L})$ of the scaled first-passage time
$\frac{\gamma t_{R}}{R/L}$ , for $R\ll L$ and $\gamma=0.01$; the complementary
CPDFs, for $R/L=0.0002$ (green), $R/L=0.00035$ (blue), and $R/L=0.0005$
(orange), collapse onto one curve; (b) semi-log plots of the complementary
CPDFs of the scaled first-passage time $t_{R}/R^{2}$, for $L\lesssim R$ and
$\gamma=30$. The complementary CPDF of $t_{R}/R^{2}$, for $R/L=10$ (purple),
$R/L=14$ (green), $R/L=18$ (blue), and $R/L=20$ (orange), collapse onto one
curve. Plots of the complementary CPDFs $\mathcal{Q}(\gamma t_{R})$ versus
$\gamma t_{R}$ are shown in the insets.

##  III Conclusions and Discussion

We have defined and studied a new first-passage-time problem for Lagrangian
tracers that are advected by a 3D turbulent flow that is statistically steady,
homogeneous and isotropic. Our work shows that the first-passage-time PDF
$\mathcal{P}(R,t_{R})$ has tails that cross over from a power-law form to an
exponentially decaying form as we move from the regime $R\ll L_{\rm I}$ to
$L_{\rm I}\lesssim R$ (Eq. (3)). We develop ballistic-transport and diffusive
models, for which we can obtain these limiting asymptotic behaviors of
$\mathcal{P}(R,t_{R})$ analytically. We also demonstrate that an OU model,
with Gaussian white noise, which mimics the effects of turbulence, suffices to
obtain the crossover between these limiting forms. Of course, such a simple
stochastic model cannot be used for more complicated multifractal properties
of turbulent flows vulpiani2001exit ; pramanareview ; arneodo .

Earlier studies have concentrated on two-particle relative dispersion by using
doubling-time statistics, in 2D fluid turbulence; in particular, they have
shown that the PDF of this doubling time has an exponential tail
boffetta2002statistics . Studies of velocity zero crossings kailasnath1993zero
, in a turbulent boundary layer, have shown that PDFs of the zero-crossing
times have exponential tails.

The single-particle first-passage-time statistics that we study have not been
explored so far. Furthermore, $\mathcal{P}(R,t_{R})$ can be used to to develop
social-distancing guidelines for the mitigation of the spreading of airborne
aerosols with viruses such as SARS-CoV-2 as we show below.

Given a pseudospectral DNS, of the type we have carried out, we can obtain the
integral scale $L_{\rm I}$ and $u_{\rm rms}$ from the energy spectrum $E(k)$,
as we have noted above. A recent study of COVID-19 in $320$ municipalities in
China suggests that a very large fraction of COVID-19 infections occur because
of indoor transmission of the SARS-CoV-2 virus Indoor2020 . Therefore, it is
important to study such transmission in rooms and offices; a comprehensive DNS
study of the Navier-Stokes equation, with the correct boundary conditions
enforced at every wall and surface in the room and accurate forcing functions
(dictated, e.g., by fans and vents), is a considerable challenge. Furthermore,
it is not possible to carry out such a DNS for every room with a different
arrangement of the furniture in it. Hence, it is important to come up with
semi-quantitative criteria that help us to understand, and mitigate, the
indoor transmission of such virues. Turbulence models have been used to study
the flow of air in rooms and offices li2005multi ; zhang2007evaluation ; from
these models and related experiments we obtain the estimate $u_{\rm rms}\simeq
0.05$ m/s in a typical office. We must also estimate $L_{\rm I}$, for it is an
important crossover length scale in our analysis of $\mathcal{P}(R,t_{R})$. In
our DNS $L_{\rm I}$ is $\simeq 0.1L$, where $L$ is the linear size of our
simulation domain. In a typical office or a train, with fixed forcing, via
fans or vents, we use $L_{\rm I}$ to be approximately a few meters; of course,
$L_{\rm I}$ must depend on the degree of crowding on a train or the number of
cubicles in a large office room. Now consider one infected person who is at a
distance $R$ from another person. The probability of virus-laden aerosol
particles not reaching the second person, up until time $t$ is related to
$\mathcal{P}(R,t_{R})$ as follows:

$W(R,t)=1-\frac{\mathcal{P}(R,t_{R}=t)}{\Sigma_{t_{R}}\mathcal{P}(R,t_{R})},$
(15)

which we calculate, by using Eq. (11), and depict in Fig. 4(a) and Fig. 4(b),
in the diffusive regime; Fig. 4(a) is a surface plot of $W(R,t)$ versus $R$
and $t$, for the representative values $L_{\rm I}=2$ m and $u_{\rm rms}=0.05$
m/s; Fig. 4(b) gives a surface plot of $W$ versus the dimensionless parameters
$R/L_{\rm I}$ and $t/T$, where $T=L_{\rm I}/u_{\rm rms}$. (We give similar
plots for the ballistic regime in the Supplemental Material.) In Table 2 we
give the values of $W(R,t)$ for different values of $R$ and $t$. These figures
and Table 2 lead to three clear observations:

  1. 1.

If the separation $R\ll L_{\rm I}$, i.e., we have to consider the ballistic
regime (see the Supplementary Material), then $W(R,t)$ goes very rapidly to
$0$ (i.e., the aerosol particle reaches the second person), even if $t$ is
very small.

  2. 2.

The smaller the separation $R$, between two persons, the shorter the time $t$
in which $W(R,t)$ becomes very small, i.e., the aerosol particles reach from
one person to the other.

  3. 3.

Our calculation leads to quantitative predictions; e.g., if the separation
$L_{\rm I}\lesssim R$, i.e., we have to consider the diffusive regime, then
$W(R,t)$ goes to $0$ in tens of seconds, if $R=2$ m, and in hundreds of
seconds, $R=10$ m, for the representative parameters that we use to obtain
Table 2. A recent study vancovid  has suggested that the SARS-CoV-2 virus
remains viable in aerosols for nearly $3$ hours. Therefore, if the
concentration of virus-laden aerosols is high in a poorly ventilated room,
then we must employ more stringent social-distancing norms than are in place
now, even if people spend only tens of minutes together in such a room.

|  $t=10$ s |  $t=30$ s |  $t=60$ s |  $t=100$ s |  $t=120$ s |  $t=150$ s |  $t=180$ s |  $t=240$ s
---|---|---|---|---|---|---|---|---
$R=2$ m | $0.169$ | $0.001$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$
$R=3$ m | $0.642$ | $0.074$ | $0.003$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$
$R=4$ m | $0.917$ | $0.313$ | $0.049$ | $0.004$ | $0.001$ | $0.000$ | $0.000$ | $0.000$
$R=5$ m | $0.989$ | $0.594$ | $0.187$ | $0.039$ | $0.018$ | $0.005$ | $0.001$ | $0.000$
$R=6$ m | $0.999$ | $0.805$ | $0.383$ | $0.129$ | $0.074$ | $0.032$ | $0.014$ | $0.002$
$R=7$ m | $1.000$ | $0.923$ | $0.579$ | $0.263$ | $0.174$ | $0.093$ | $0.049$ | $0.011$
$R=8$ m | $1.000$ | $0.097$ | $0.739$ | $0.412$ | $0.299$ | $0.182$ | $0.107$ | $0.030$
$R=9$ m | $1.000$ | $0.099$ | $0.850$ | $0.553$ | $0.428$ | $0.283$ | $0.180$ | $0.059$
$R=10$ m | $1.000$ | $0.998$ | $0.920$ | $0.673$ | $0.547$ | $0.386$ | $0.260$ | $0.093$
Table 2: Table of values of $W(R,t)$, the probability of virus-laden aerosol
particles not reaching a person (at a distance $R$ from an infected person),
up until time $t$ (the first-passage time) in the diffusive region, for the
representative values $K=0.1{\rm m}^{2}/{\rm s}$, $u_{\rm rms}=0.05$ m/s, and
$L_{\rm I}=2$m.

(a)(b)

Figure 4:  Surface plots of $W(R,t)$, the probability of virus-laden aerosol
particles not reaching a person (at a distance $R$ from an infected person),
up until time $t$ (the first-passage time) in (a) versus $R$ and $t$, and (b)
versus $R/L_{I}$ and $t/T$, in the diffusive region, for the representative
values $K=0.1{\rm m}^{2}/{\rm s}$, $u_{\rm rms}=0.05$ m/s, and $L_{\rm I}=2$m.

The methods that we have developed can be applied, mutatis mutandis, (a) in
sophisticated models for virus particles or droplets, e.g., those that use
inertial particles Cencini2006 ; Salazar2009  or multi-phase flows
Balachandar2010 ; Pal2016  and (b) in turbulent flows that are not
statistically homogeneous and isotropic. We will examine these in future work.
At the moment, it is important to use our minimal model to obtain semi-
quantitative for social-distancing guidelines, as we have done above.

###### Acknowledgements.

D.M. and A.B. thank John Wettlaufer for useful discussions. A.K.V. and R.P.
thank Jaya Kumar Alageshan for discussions and for help with Fig. 1, and CSIR
(IN) and DST (IN) for financial support. A.B. and D.M. are supported by the
grant Bottlenecks for Particle Growth in Turbulent Aerosols from the Knut and
Alice Wallenberg Foundation (Dnr. KAW 2014.0048); the computations were
performed on resources provided by the Swedish National Infrastructure for
Computing (SNIC) at PDC and SERC (at IISc).

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Supplemental Materials:The spreading of viruses by airborne aerosols: lessons
from a first-passage-time problem for tracers in turbulent flows Akhilesh
Kumar Verma Akshay Bhatnagar Dhrubaditya Mitra Rahul Pandit

also at ]Jawaharlal Nehru Centre For Advanced Scientific Research, Jakkur,
Bangalore, India.

In this Supplemental Material, we provide the following:

(a) Consider one infected person, who is at a distance $R$ from another
person. We give surface plots of the probability $W(R,t)$ of a virus-laden
aerosol particle not reaching the second person, up until time $t$, in the
ballistic regime (Fig. S1). We also give surface plots of $(1-W(R,t))$, the
probability that a virus-laden aerosol particle reaches the second person, at
time $t$, for the first time for ballistic (Fig. S2) and diffusive (Fig. S3)
cases.

(b) The energy spectrum ${E}(k)=\sum_{k-1/2<k^{\prime}<k+1/2}{\bm{u}}({\bf
k}^{\prime})\cdot{\bm{u}}({-\bf k}^{\prime})$ from our direct numerical
simulation of the three-dimensional Navier-Stokes equation (see the main
paper). A log-log plot of this spectrum (red curve) is given in Fig. S4; for
comparison, we show the Kolmogorov scaling form $E(k)\sim k^{-5/3}$ (black
line).

(a)(b)

Figure S1:  Surface plots of $W(R,t)$ versus (a) $R$ and $t_{R}$ and (b) (b)
the scaled radius $R/L_{\rm I}$ and scaled time $t/T$ (see the main paper), in
the ballistic region, for the representative values $u_{\rm rms}=0.05$ m/s,
and $L_{\rm I}=2$m.

(a)(b)

Figure S2:  Surface plots of $(1-W(R,t))$ versus (a) $R$ and $t$ and (b) (b)
the scaled radius $R/L_{\rm I}$ and scaled time $t/T$ (see the main paper), in
the ballistic region, for the representative values $u_{\rm rms}=0.05$ m/s,
and $L_{\rm I}=2$m.

(a)(b)

Figure S3:  Surface plots of $(1-W(R,t))$ versus (a) $R$ and $t$ and (b) (b)
the scaled radius $R/L_{\rm I}$ and scaled time $t/T$ (see the main paper), in
the diffusive region, for the representative values $u_{\rm rms}=0.05$ m/s,
$K=0.1{\rm m}^{2}/{\rm s}$, and $L_{I}=2$ m.

Figure S4: Log-log plot of the energy spectrum ${E}(k)$ (red curve) versus the
wave number $k$ from our direct numerical simulation of the three-dimensional
Navier-Stokes equation (see the main paper); for comparison, we show the
Kolmogorov scaling form $E(k)\sim k^{-5/3}$ (black line).